· The Course Introduction The Brain. Course objectives Course outcomes Textbooks Course...

The CourseIntroduction

The Brain.

Computational Neuroscience. Session 1-1

Dr. Marco A Roque Sol

05/29/2018

Dr. Marco A Roque Sol Computational Neuroscience. Session 1-1


The Brain.

Course objectivesCourse outcomesTextbooks

Course objectives

Students will gain understanding of cell physiologyunderlying neuronal excitability.

Students will learn the Hodgkin-Huxley model of actionpotential generation and propagation.

Students will learn models of neuronal spiking and burstingof different levels of complexity.

Students will learn how to qualitatively analyze thebehavior of solutions of ordinary differential equationsusing phase-plane analysis.



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Course objectives

Students will understand the concept of bifurcation of adynamical system, and use it to analyze models ofneuronal excitability.

Students will learn how to use MATLAB to graphicallyanalyze and numerically solve ordinary differentialequations arising in neuronal modeling.

Students can describe physiological mechanismsunderlying an action potential in an excitable cell.



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Course objectives






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Course objectives






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Course objectives






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Course outcomes

Students are able to analyze the behavior of a non-linearordinary differential equation using phase-plane analysis.

Students are able to build and analyze models of spikingand bursting neurons.

Students are able to write a MATLAB program tonumerically solve ordinary differential equations arising inthe modeling of neural excitability.




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Course outcomes







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Textbooks

Ermentrout G Bard, Terman David H . MathematicalFoundations of Neuroscience. 2010 Springer Science.

Izhikevich Eugene M. Dynamical Systems inNeuroscience, The Geometry of Excitability and Bursting .2007 MIT Press.

Dayan Peter, Abbot L F . Theoretical Neuroscience,Computational and Mathematical Modeling of NeuralSystem. 2001 MIT Press.

Squirre Larry S, Berg Darwing, Bloom Floyd E, Du LacSascha, Gosh Arnivan . Fundamental Neuroscience. 4thEdition, 2012 Academic Press.



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Textbooks







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Textbooks







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Textbooks







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Textbooks







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Textbooks

Johnston Daniel, Miao-Sin Wu Samuel . Foundations ofCellular Neurophysiology, with illustrations and simulationsby Richard Gray. 1995 MIT Press.

Wallisch Pascal, Lusignan Michael E, Benayoun Marc D,Baker Tanya I, Dickey Adam Seth, Hatsopoulos NicholasG. MATLAB for Neuroscientists, An introduction toScientific Computing in MATLAB. 2th Edition, 2014Academic Press.

Ermentrout G Bard. Simulating, Analyzing, and AnimatingDynamical Systems: A Guide to XPPAUT for Researchersand Students. 2012 SIAM.



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Textbooks






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Textbooks






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Textbooks






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Table of ContentsIntroduction to Neuroscience.

Table of contents

1.- Introduction to Neuroscience.

2.- Basic introduction to the brain.

3- ODE Review .

4.-Intro to MATLAB.

5.- MATLAB and ODES’s

6.- The resting Potential.

7.- Nernst-Planck equation, Nernst equation, GHKequation. How to solve ODE.



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Table of contents



3- ODE Review .

4.-Intro to MATLAB.






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Table of contents



3- ODE Review .

4.-Intro to MATLAB.






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Table of contents



3- ODE Review .

4.-Intro to MATLAB.






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Table of contents



3- ODE Review .

4.-Intro to MATLAB.






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Table of contents



3- ODE Review .

4.-Intro to MATLAB.






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Table of contents



3- ODE Review .

4.-Intro to MATLAB.






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Table of contents



3- ODE Review .

4.-Intro to MATLAB.






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Table of contents

8.- Dynamics of Passive membrane.

9.- The Huxley-Hodgkin Model. Integrate-and-FireModels I.

10.-The Huxley-Hodgkin Model. Integrate-and-FireModels II.

11.- The Huxley-Hodgkin Model.The Cable equation I.

12.- The Huxley-Hodgkin Model. The Cable equation II



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Table of contents








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Table of contents








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Table of contents








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Table of contents








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Table of contents








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Table of contents

13.- Introduction to Dynamical Systems for Neural Network.Reduced one and two-dimensional networks I (***).

14.- Introduction to Dynamical Systems for NeuralNetwork. Reduced one and two-dimensional networks II.

15.- One-dimensional Neural Model.Phase-Space Analysis I

∗ ∗ ∗ The idea in this part, in a regular class duringFall/Spring Semester, is to get support from either theDepartment of Biology or the Institute of Neurosciences, tosee some in vitro experiments.



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Table of contents







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Table of contents

16.- Two-dimensional Neural Model.Phase-Space Analysis I

17.- Two-dimensional Neural Model.Phase-Space Analysis II

18.- Subthreshold Oscillations. Subthreshold andSuprathreshold Resonance



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Table of contents






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Table of contents






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Table of contents






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Introduction to Neuroscience.

The field of Neuroscience had a breakthrough in 1952

when A.F. Huxley and A. L. Hodgkin published its paper where theydescribed, through a set of nonlinear Partial DifferentialEquations, the fundamental role of the action potential in theaxon of a giant squid.

Since that moment a great number of mathematicians havecontributed to the understanding of the field.

At the same time the growth of computer sciences has allowedto have a different perspective of the problem.



The Brain.



The field of Neuroscience had a breakthrough in 1952 when A.F. Huxley and

A. L. Hodgkin published its paper where theydescribed, through a set of nonlinear Partial DifferentialEquations, the fundamental role of the action potential in theaxon of a giant squid.





The Brain.



The field of Neuroscience had a breakthrough in 1952 when A.F. Huxley and A. L. Hodgkin

published its paper where theydescribed, through a set of nonlinear Partial DifferentialEquations, the fundamental role of the action potential in theaxon of a giant squid.





The Brain.



The field of Neuroscience had a breakthrough in 1952 when A.F. Huxley and A. L. Hodgkin published its paper where theydescribed,

through a set of nonlinear Partial DifferentialEquations, the fundamental role of the action potential in theaxon of a giant squid.





The Brain.



The field of Neuroscience had a breakthrough in 1952 when A.F. Huxley and A. L. Hodgkin published its paper where theydescribed, through a set of nonlinear Partial DifferentialEquations,

the fundamental role of the action potential in theaxon of a giant squid.





The Brain.



The field of Neuroscience had a breakthrough in 1952 when A.F. Huxley and A. L. Hodgkin published its paper where theydescribed, through a set of nonlinear Partial DifferentialEquations, the fundamental role of the action potential

in theaxon of a giant squid.





The Brain.



The field of Neuroscience had a breakthrough in 1952 when A.F. Huxley and A. L. Hodgkin published its paper where theydescribed, through a set of nonlinear Partial DifferentialEquations, the fundamental role of the action potential in theaxon of a giant squid.





The Brain.




Since that moment

a great number of mathematicians havecontributed to the understanding of the field.




The Brain.




Since that moment a great number of mathematicians

havecontributed to the understanding of the field.




The Brain.




Since that moment a great number of mathematicians havecontributed

to the understanding of the field.




The Brain.








The Brain.





At the same time

the growth of computer sciences has allowedto have a different perspective of the problem.



The Brain.





At the same time the growth of computer sciences

has allowedto have a different perspective of the problem.



The Brain.





At the same time the growth of computer sciences has allowedto have

a different perspective of the problem.



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Thus,

using mathematical analysis and computer modeling, thearea of brain research has two powerful tools to study anddetermine how the nervous systems works.

In this course we will have the opportunity to realize howdifferent areas of the knowledge interact to give an explanationto physical phenomena.

In this way, we will make use of the concepts in Biology,Chemistry, Physics, and Physiology to generate Mathematicalmodels to understand the brain behavior.



The Brain.



Thus, using mathematical analysis and

computer modeling, thearea of brain research has two powerful tools to study anddetermine how the nervous systems works.





The Brain.



Thus, using mathematical analysis and computer modeling,

thearea of brain research has two powerful tools to study anddetermine how the nervous systems works.





The Brain.



Thus, using mathematical analysis and computer modeling, thearea of brain research

has two powerful tools to study anddetermine how the nervous systems works.





The Brain.



Thus, using mathematical analysis and computer modeling, thearea of brain research has two powerful tools

to study anddetermine how the nervous systems works.





The Brain.



Thus, using mathematical analysis and computer modeling, thearea of brain research has two powerful tools to study anddetermine

how the nervous systems works.





The Brain.



Thus, using mathematical analysis and computer modeling, thearea of brain research has two powerful tools to study anddetermine how the nervous systems works.





The Brain.




In this course

we will have the opportunity to realize howdifferent areas of the knowledge interact to give an explanationto physical phenomena.




The Brain.




In this course we will have the

opportunity to realize howdifferent areas of the knowledge interact to give an explanationto physical phenomena.




The Brain.




In this course we will have the opportunity to realize

howdifferent areas of the knowledge interact to give an explanationto physical phenomena.




The Brain.




In this course we will have the opportunity to realize howdifferent areas

of the knowledge interact to give an explanationto physical phenomena.




The Brain.




In this course we will have the opportunity to realize howdifferent areas of the knowledge interact

to give an explanationto physical phenomena.




The Brain.




In this course we will have the opportunity to realize howdifferent areas of the knowledge interact to give

an explanationto physical phenomena.




The Brain.








The Brain.





In this way,

we will make use of the concepts in Biology,Chemistry, Physics, and Physiology to generate Mathematicalmodels to understand the brain behavior.



The Brain.





In this way, we will make use

of the concepts in Biology,Chemistry, Physics, and Physiology to generate Mathematicalmodels to understand the brain behavior.



The Brain.





In this way, we will make use of the concepts in Biology,

Chemistry, Physics, and Physiology to generate Mathematicalmodels to understand the brain behavior.



The Brain.





In this way, we will make use of the concepts in Biology,Chemistry,

Physics, and Physiology to generate Mathematicalmodels to understand the brain behavior.



The Brain.





In this way, we will make use of the concepts in Biology,Chemistry, Physics, and

Physiology to generate Mathematicalmodels to understand the brain behavior.



The Brain.





In this way, we will make use of the concepts in Biology,Chemistry, Physics, and Physiology

to generate Mathematicalmodels to understand the brain behavior.



The Brain.





In this way, we will make use of the concepts in Biology,Chemistry, Physics, and Physiology to generate Mathematicalmodels

to understand the brain behavior.



The Brain.








The Brain.








The Brain.



At the beginning ...

The Greek physician Galen

thought that the brain was agland, from which nerves carry fluid to the extremities. Inthe mid-nineteenth century, it was discovered that electricalactivity in a nerve has predictable effects on neighboringneurons.

Camillo Golgi and Santiago Ramon y Cajal made the firstdetailed descriptions of nerve cells in the late nineteenthcentury, and one of Cajal’s drawings is shwon below:



The Brain.




The Greek physician Galen thought that the brain was agland,

from which nerves carry fluid to the extremities. Inthe mid-nineteenth century, it was discovered that electricalactivity in a nerve has predictable effects on neighboringneurons.




The Brain.




The Greek physician Galen thought that the brain was agland, from which nerves

carry fluid to the extremities. Inthe mid-nineteenth century, it was discovered that electricalactivity in a nerve has predictable effects on neighboringneurons.




The Brain.




The Greek physician Galen thought that the brain was agland, from which nerves carry fluid to the extremities.

Inthe mid-nineteenth century, it was discovered that electricalactivity in a nerve has predictable effects on neighboringneurons.




The Brain.




The Greek physician Galen thought that the brain was agland, from which nerves carry fluid to the extremities. Inthe mid-nineteenth century,

it was discovered that electricalactivity in a nerve has predictable effects on neighboringneurons.




The Brain.




The Greek physician Galen thought that the brain was agland, from which nerves carry fluid to the extremities. Inthe mid-nineteenth century, it was discovered

that electricalactivity in a nerve has predictable effects on neighboringneurons.




The Brain.




The Greek physician Galen thought that the brain was agland, from which nerves carry fluid to the extremities. Inthe mid-nineteenth century, it was discovered that electricalactivity in a nerve

has predictable effects on neighboringneurons.




The Brain.




The Greek physician Galen thought that the brain was agland, from which nerves carry fluid to the extremities. Inthe mid-nineteenth century, it was discovered that electricalactivity in a nerve has predictable effects

on neighboringneurons.




The Brain.




The Greek physician Galen thought that the brain was agland, from which nerves carry fluid to the extremities. Inthe mid-nineteenth century, it was discovered that electricalactivity in a nerve has predictable effects on neighboringneurons.




The Brain.





Camillo Golgi and Santiago Ramon y Cajal

made the firstdetailed descriptions of nerve cells in the late nineteenthcentury, and one of Cajal’s drawings is shwon below:



The Brain.





Camillo Golgi and Santiago Ramon y Cajal made the firstdetailed descriptions

of nerve cells in the late nineteenthcentury, and one of Cajal’s drawings is shwon below:



The Brain.





Camillo Golgi and Santiago Ramon y Cajal made the firstdetailed descriptions of nerve cells

in the late nineteenthcentury, and one of Cajal’s drawings is shwon below:



The Brain.





Camillo Golgi and Santiago Ramon y Cajal made the firstdetailed descriptions of nerve cells in the late nineteenthcentury, and

one of Cajal’s drawings is shwon below:



The Brain.





Camillo Golgi and Santiago Ramon y Cajal made the firstdetailed descriptions of nerve cells in the late nineteenthcentury, and one of Cajal’s drawings

is shwon below:



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The Brain.








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Ramon y Cajal drawing of a single neuron



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Ramon y Cajal drawing of a single neuron



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Ross Harrison discovered that the axon and dendritesgrow from the cell body in isolated culture [1]; also see [2].

Pharmacologists discovered that drugs affect cells bybinding to receptors, which opened the door to thediscovery of chemical communication between neurons atsynapses.



The Brain.



At the beginning ...Ross Harrison

discovered that the axon and dendritesgrow from the cell body in isolated culture [1]; also see [2].




The Brain.



At the beginning ...Ross Harrison discovered that

the axon and dendritesgrow from the cell body in isolated culture [1]; also see [2].




The Brain.



At the beginning ...Ross Harrison discovered that the axon and dendrites

grow from the cell body in isolated culture [1]; also see [2].




The Brain.



At the beginning ...Ross Harrison discovered that the axon and dendritesgrow from the cell body

in isolated culture [1]; also see [2].




The Brain.



At the beginning ...Ross Harrison discovered that the axon and dendritesgrow from the cell body in isolated culture

[1]; also see [2].




The Brain.



At the beginning ...Ross Harrison discovered that the axon and dendritesgrow from the cell body in isolated culture [1]; also see [2].




The Brain.







The Brain.







The Brain.







The Brain.




Pharmacologists discovered

that drugs affect cells bybinding to receptors, which opened the door to thediscovery of chemical communication between neurons atsynapses.



The Brain.




Pharmacologists discovered that drugs affect cells

bybinding to receptors, which opened the door to thediscovery of chemical communication between neurons atsynapses.



The Brain.




Pharmacologists discovered that drugs affect cells bybinding to receptors,

which opened the door to thediscovery of chemical communication between neurons atsynapses.



The Brain.




Pharmacologists discovered that drugs affect cells bybinding to receptors, which opened the door

to thediscovery of chemical communication between neurons atsynapses.



The Brain.




Pharmacologists discovered that drugs affect cells bybinding to receptors, which opened the door to thediscovery of

chemical communication between neurons atsynapses.



The Brain.




Pharmacologists discovered that drugs affect cells bybinding to receptors, which opened the door to thediscovery of chemical communication

between neurons atsynapses.



The Brain.




Pharmacologists discovered that drugs affect cells bybinding to receptors, which opened the door to thediscovery of chemical communication between neurons

atsynapses.



The Brain.







The Brain.







The Brain.







The Brain.




Over the past century neuroscience has now grown into abroad and diverse field. Molecular neuroscience studiesthe detailed structure and dynamics of neurons, synapsesand small networks;

systems neuroscience studies larger-scale networks thatperform tasks and interact with other such networks (orbrain areas) to form pathways for higher-level functions;

and cognitive neuroscience studies the relationshipbetween the underlying physiology (neural substrates) andbehavior, thought, and cognition.



The Brain.




Over the past century

neuroscience has now grown into abroad and diverse field. Molecular neuroscience studiesthe detailed structure and dynamics of neurons, synapsesand small networks;





The Brain.




Over the past century neuroscience has now grown

into abroad and diverse field. Molecular neuroscience studiesthe detailed structure and dynamics of neurons, synapsesand small networks;





The Brain.




Over the past century neuroscience has now grown into abroad and diverse field.

Molecular neuroscience studiesthe detailed structure and dynamics of neurons, synapsesand small networks;





The Brain.




Over the past century neuroscience has now grown into abroad and diverse field. Molecular neuroscience

studiesthe detailed structure and dynamics of neurons, synapsesand small networks;





The Brain.




Over the past century neuroscience has now grown into abroad and diverse field. Molecular neuroscience studiesthe detailed structure and

dynamics of neurons, synapsesand small networks;





The Brain.




Over the past century neuroscience has now grown into abroad and diverse field. Molecular neuroscience studiesthe detailed structure and dynamics of neurons,

synapsesand small networks;





The Brain.




Over the past century neuroscience has now grown into abroad and diverse field. Molecular neuroscience studiesthe detailed structure and dynamics of neurons, synapsesand

small networks;





The Brain.









The Brain.





systems neuroscience

studies larger-scale networks thatperform tasks and interact with other such networks (orbrain areas) to form pathways for higher-level functions;




The Brain.





systems neuroscience studies larger-scale networks

thatperform tasks and interact with other such networks (orbrain areas) to form pathways for higher-level functions;




The Brain.





systems neuroscience studies larger-scale networks thatperform tasks and

interact with other such networks (orbrain areas) to form pathways for higher-level functions;




The Brain.





systems neuroscience studies larger-scale networks thatperform tasks and interact with other

such networks (orbrain areas) to form pathways for higher-level functions;




The Brain.





systems neuroscience studies larger-scale networks thatperform tasks and interact with other such networks (orbrain areas)

to form pathways for higher-level functions;




The Brain.





systems neuroscience studies larger-scale networks thatperform tasks and interact with other such networks (orbrain areas) to form pathways

for higher-level functions;




The Brain.









The Brain.






and cognitive neuroscience

studies the relationshipbetween the underlying physiology (neural substrates) andbehavior, thought, and cognition.



The Brain.






and cognitive neuroscience studies the relationship

between the underlying physiology (neural substrates) andbehavior, thought, and cognition.



The Brain.






and cognitive neuroscience studies the relationshipbetween the underlying physiology (neural substrates) and

behavior, thought, and cognition.



The Brain.






and cognitive neuroscience studies the relationshipbetween the underlying physiology (neural substrates) andbehavior,

thought, and cognition.



The Brain.






and cognitive neuroscience studies the relationshipbetween the underlying physiology (neural substrates) andbehavior, thought, and

cognition.



The Brain.









The Brain.









The Brain.




Mathematical treatments of the nervous system began inthe mid 20th century. One of the first examples is the bookof Norbert Wiener, based on work done with the Mexicanphysiologist Arturo Rosenblueth, and originally publishedin 1948 [3].

Rosenbleuth and Wiener



The Brain.




Mathematical treatments

of the nervous system began inthe mid 20th century. One of the first examples is the bookof Norbert Wiener, based on work done with the Mexicanphysiologist Arturo Rosenblueth, and originally publishedin 1948 [3].




The Brain.




Mathematical treatments of the nervous system

began inthe mid 20th century. One of the first examples is the bookof Norbert Wiener, based on work done with the Mexicanphysiologist Arturo Rosenblueth, and originally publishedin 1948 [3].




The Brain.




Mathematical treatments of the nervous system began inthe mid 20th century.

One of the first examples is the bookof Norbert Wiener, based on work done with the Mexicanphysiologist Arturo Rosenblueth, and originally publishedin 1948 [3].




The Brain.




Mathematical treatments of the nervous system began inthe mid 20th century. One of the first examples

is the bookof Norbert Wiener, based on work done with the Mexicanphysiologist Arturo Rosenblueth, and originally publishedin 1948 [3].




The Brain.




Mathematical treatments of the nervous system began inthe mid 20th century. One of the first examples is the bookof Norbert Wiener,

based on work done with the Mexicanphysiologist Arturo Rosenblueth, and originally publishedin 1948 [3].




The Brain.




Mathematical treatments of the nervous system began inthe mid 20th century. One of the first examples is the bookof Norbert Wiener, based on work done

with the Mexicanphysiologist Arturo Rosenblueth, and originally publishedin 1948 [3].




The Brain.




Mathematical treatments of the nervous system began inthe mid 20th century. One of the first examples is the bookof Norbert Wiener, based on work done with the Mexicanphysiologist Arturo Rosenblueth, and

originally publishedin 1948 [3].




The Brain.








The Brain.





Rosenbleuth and WienerDr. Marco A Roque Sol Computational Neuroscience. Session 1-1


The Brain.





Rosenbleuth and WienerDr. Marco A Roque Sol Computational Neuroscience. Session 1-1


The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis, information theory and feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced

ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis, information theory and feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems,

symmetry groups, statistical mechanics, timeseries analysis, information theory and feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups,

statistical mechanics, timeseries analysis, information theory and feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics,

timeseries analysis, information theory and feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis,

information theory and feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis, information theory and

feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis, information theory and feedback control.

He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis, information theory and feedback control.He also discussed

the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis, information theory and feedback control.He also discussed the relationship between

digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis, information theory and feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and

neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis, information theory and feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits,

atheme that John von Neumann subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis, information theory and feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann

subsequently addressed ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis, information theory and feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed

ina book written in 1955-57 and published in the year afterhis death [4].



The Brain.




Weiner introduced ideas from dissipative dynamicalsystems, symmetry groups, statistical mechanics, timeseries analysis, information theory and feedback control.He also discussed the relationship between digitalcomputers (then in their infancy) and neural circuits, atheme that John von Neumann subsequently addressed ina book written in 1955-57 and

published in the year afterhis death [4].



The Brain.







The Brain.







The Brain.







The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact,

while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing

one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers

(JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),

von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate

some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations

of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4],

see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann).

It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that,

in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics,

Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on

von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works

in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis,

ergodic theory, computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory,

computation andgame theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation and

game theory, as well his own studies of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies

of Brownian motion(now known as Wiener processes ).



The Brain.




In fact, while developing one of the first programmabledigital computers (JONIAC, built at the Institute forAdvanced Study in Princeton after the second World War),von Neumann had “tried to imitate some of the knownoperations of the live brain” ( [4], see the Preface by Klaravon Neumann). It is also notable that, in developingcybernetics, Wiener drew heavily on von NeumannâAZsearlier works in analysis, ergodic theory, computation andgame theory, as well his own studies of Brownian motion

(now known as Wiener processes ).



The Brain.







The Brain.







The Brain.




These books [3],[4] were directed at the brain and nervoussystem in toto, although much of the former was based ondetailed experimental studies of heart and leg muscles inanimals.

The first cellular-level mathematical model of a singleneuron was developed in the early 1950’s by the Britishphysiologists Alan Hodgkin and Andrew Huxley [5].



The Brain.








The Brain.




These books [3],[4]

were directed at the brain and nervoussystem in toto, although much of the former was based ondetailed experimental studies of heart and leg muscles inanimals.




The Brain.




These books [3],[4] were directed at the brain and

nervoussystem in toto, although much of the former was based ondetailed experimental studies of heart and leg muscles inanimals.




The Brain.




These books [3],[4] were directed at the brain and nervoussystem in toto, although much of the former

was based ondetailed experimental studies of heart and leg muscles inanimals.




The Brain.




These books [3],[4] were directed at the brain and nervoussystem in toto, although much of the former was based ondetailed experimental studies

of heart and leg muscles inanimals.




The Brain.








The Brain.





The first cellular-level mathematical model

of a singleneuron was developed in the early 1950’s by the Britishphysiologists Alan Hodgkin and Andrew Huxley [5].



The Brain.





The first cellular-level mathematical model of a singleneuron

was developed in the early 1950’s by the Britishphysiologists Alan Hodgkin and Andrew Huxley [5].



The Brain.





The first cellular-level mathematical model of a singleneuron was developed

in the early 1950’s by the Britishphysiologists Alan Hodgkin and Andrew Huxley [5].



The Brain.





The first cellular-level mathematical model of a singleneuron was developed in the early 1950’s

by the Britishphysiologists Alan Hodgkin and Andrew Huxley [5].



The Brain.





The first cellular-level mathematical model of a singleneuron was developed in the early 1950’s by the Britishphysiologists

Alan Hodgkin and Andrew Huxley [5].



The Brain.








The Brain.








The Brain.




This work, which won them the Nobel Prize in Physiologyin 1963, grew out of a long series of experiments on thegiant axon of the squid Loligo by themselves and others (see Huxley’s obituary [6] )

Since their pioneering work, mathematical neurosciencehas grown i nto a subdiscipline, served worldwide bycourses short and long, a growing list of textbooks (e.g. [7],[8], [9], [10],[11], [12]) and review articles such as [13],[14], [15], [16].



The Brain.








The Brain.




This work,

which won them the Nobel Prize in Physiologyin 1963, grew out of a long series of experiments on thegiant axon of the squid Loligo by themselves and others (see Huxley’s obituary [6] )




The Brain.




This work, which won them the Nobel Prize in Physiologyin 1963,

grew out of a long series of experiments on thegiant axon of the squid Loligo by themselves and others (see Huxley’s obituary [6] )




The Brain.




This work, which won them the Nobel Prize in Physiologyin 1963, grew out of a long series

of experiments on thegiant axon of the squid Loligo by themselves and others (see Huxley’s obituary [6] )




The Brain.




This work, which won them the Nobel Prize in Physiologyin 1963, grew out of a long series of experiments

on thegiant axon of the squid Loligo by themselves and others (see Huxley’s obituary [6] )




The Brain.




This work, which won them the Nobel Prize in Physiologyin 1963, grew out of a long series of experiments on thegiant axon

of the squid Loligo by themselves and others (see Huxley’s obituary [6] )




The Brain.




This work, which won them the Nobel Prize in Physiologyin 1963, grew out of a long series of experiments on thegiant axon of the squid Loligo

by themselves and others (see Huxley’s obituary [6] )




The Brain.




This work, which won them the Nobel Prize in Physiologyin 1963, grew out of a long series of experiments on thegiant axon of the squid Loligo by themselves and others

(see Huxley’s obituary [6] )




The Brain.








The Brain.





Since their pioneering work,

mathematical neurosciencehas grown i nto a subdiscipline, served worldwide bycourses short and long, a growing list of textbooks (e.g. [7],[8], [9], [10],[11], [12]) and review articles such as [13],[14], [15], [16].



The Brain.





Since their pioneering work, mathematical neurosciencehas grown i

nto a subdiscipline, served worldwide bycourses short and long, a growing list of textbooks (e.g. [7],[8], [9], [10],[11], [12]) and review articles such as [13],[14], [15], [16].



The Brain.





Since their pioneering work, mathematical neurosciencehas grown i nto a subdiscipline,

served worldwide bycourses short and long, a growing list of textbooks (e.g. [7],[8], [9], [10],[11], [12]) and review articles such as [13],[14], [15], [16].



The Brain.





Since their pioneering work, mathematical neurosciencehas grown i nto a subdiscipline, served worldwide

bycourses short and long, a growing list of textbooks (e.g. [7],[8], [9], [10],[11], [12]) and review articles such as [13],[14], [15], [16].



The Brain.





Since their pioneering work, mathematical neurosciencehas grown i nto a subdiscipline, served worldwide bycourses short and long,

a growing list of textbooks (e.g. [7],[8], [9], [10],[11], [12]) and review articles such as [13],[14], [15], [16].



The Brain.





Since their pioneering work, mathematical neurosciencehas grown i nto a subdiscipline, served worldwide bycourses short and long, a growing list of textbooks

(e.g. [7],[8], [9], [10],[11], [12]) and review articles such as [13],[14], [15], [16].



The Brain.





Since their pioneering work, mathematical neurosciencehas grown i nto a subdiscipline, served worldwide bycourses short and long, a growing list of textbooks (e.g. [7],[8], [9], [10],[11], [12]) and

review articles such as [13],[14], [15], [16].



The Brain.





Since their pioneering work, mathematical neurosciencehas grown i nto a subdiscipline, served worldwide bycourses short and long, a growing list of textbooks (e.g. [7],[8], [9], [10],[11], [12]) and review articles

such as [13],[14], [15], [16].



The Brain.





Since their pioneering work, mathematical neurosciencehas grown i nto a subdiscipline, served worldwide bycourses short and long, a growing list of textbooks (e.g. [7],[8], [9], [10],[11], [12]) and review articles such as

[13],[14], [15], [16].



The Brain.








The Brain.








The Brain.



Mathematical Neuroscience

Mathematical Neuroscience here means an area ofneuroscience where mathematics is the primary tool forelucidating the fundamental mechanisms responsible forexperimentally observed behaviour.

Drawing together, the field provides the possibility of acritical discussion of the relevant experimental facts and ofvarious mathematical methods and techniques that havebeen successfully applied to date.



The Brain.








The Brain.





here means an area ofneuroscience where mathematics is the primary tool forelucidating the fundamental mechanisms responsible forexperimentally observed behaviour.




The Brain.




Mathematical Neuroscience here means

an area ofneuroscience where mathematics is the primary tool forelucidating the fundamental mechanisms responsible forexperimentally observed behaviour.




The Brain.




Mathematical Neuroscience here means an area ofneuroscience

where mathematics is the primary tool forelucidating the fundamental mechanisms responsible forexperimentally observed behaviour.




The Brain.




Mathematical Neuroscience here means an area ofneuroscience where mathematics is the primary tool

forelucidating the fundamental mechanisms responsible forexperimentally observed behaviour.




The Brain.




Mathematical Neuroscience here means an area ofneuroscience where mathematics is the primary tool forelucidating

the fundamental mechanisms responsible forexperimentally observed behaviour.




The Brain.




Mathematical Neuroscience here means an area ofneuroscience where mathematics is the primary tool forelucidating the fundamental mechanisms

responsible forexperimentally observed behaviour.




The Brain.




Mathematical Neuroscience here means an area ofneuroscience where mathematics is the primary tool forelucidating the fundamental mechanisms responsible for

experimentally observed behaviour.




The Brain.








The Brain.





Drawing together,

the field provides the possibility of acritical discussion of the relevant experimental facts and ofvarious mathematical methods and techniques that havebeen successfully applied to date.



The Brain.





Drawing together, the field provides

the possibility of acritical discussion of the relevant experimental facts and ofvarious mathematical methods and techniques that havebeen successfully applied to date.



The Brain.





Drawing together, the field provides the possibility of

acritical discussion of the relevant experimental facts and ofvarious mathematical methods and techniques that havebeen successfully applied to date.



The Brain.





Drawing together, the field provides the possibility of acritical discussion of

the relevant experimental facts and ofvarious mathematical methods and techniques that havebeen successfully applied to date.



The Brain.





Drawing together, the field provides the possibility of acritical discussion of the relevant experimental facts and

ofvarious mathematical methods and techniques that havebeen successfully applied to date.



The Brain.





Drawing together, the field provides the possibility of acritical discussion of the relevant experimental facts and ofvarious mathematical methods and techniques

that havebeen successfully applied to date.



The Brain.








The Brain.








The Brain.




More importantly, it can draw attention to, and develop,those pieces of mathematical theory which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point it is worth telling the story of Wilfrid Rall, who inthe 1960s developed the cable model of the dendritic tree(see [18] for a survey of his work)

Cable theory uses coupled PDEs (Partial Differential;Equations) to describe how membrane potential spreadsalong the dendritic branches in response to a localconductance change (synaptic input).



The Brain.




More importantly,

it can draw attention to, and develop,those pieces of mathematical theory which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point it is worth telling the story of Wilfrid Rall, who inthe 1960s developed the cable model of the dendritic tree(see [18] for a survey of his work)




The Brain.




More importantly, it can draw attention to,

and develop,those pieces of mathematical theory which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point it is worth telling the story of Wilfrid Rall, who inthe 1960s developed the cable model of the dendritic tree(see [18] for a survey of his work)




The Brain.




More importantly, it can draw attention to, and develop,

those pieces of mathematical theory which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point it is worth telling the story of Wilfrid Rall, who inthe 1960s developed the cable model of the dendritic tree(see [18] for a survey of his work)




The Brain.




More importantly, it can draw attention to, and develop,those pieces of

mathematical theory which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point it is worth telling the story of Wilfrid Rall, who inthe 1960s developed the cable model of the dendritic tree(see [18] for a survey of his work)




The Brain.




More importantly, it can draw attention to, and develop,those pieces of mathematical theory

which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point it is worth telling the story of Wilfrid Rall, who inthe 1960s developed the cable model of the dendritic tree(see [18] for a survey of his work)




The Brain.




More importantly, it can draw attention to, and develop,those pieces of mathematical theory which are likely to berelevant

to future studies of the brain [17]. In illustration ofthis point it is worth telling the story of Wilfrid Rall, who inthe 1960s developed the cable model of the dendritic tree(see [18] for a survey of his work)




The Brain.




More importantly, it can draw attention to, and develop,those pieces of mathematical theory which are likely to berelevant to future studies of the brain [17].

In illustration ofthis point it is worth telling the story of Wilfrid Rall, who inthe 1960s developed the cable model of the dendritic tree(see [18] for a survey of his work)




The Brain.




More importantly, it can draw attention to, and develop,those pieces of mathematical theory which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point

it is worth telling the story of Wilfrid Rall, who inthe 1960s developed the cable model of the dendritic tree(see [18] for a survey of his work)




The Brain.




More importantly, it can draw attention to, and develop,those pieces of mathematical theory which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point it is worth telling the story

of Wilfrid Rall, who inthe 1960s developed the cable model of the dendritic tree(see [18] for a survey of his work)




The Brain.




More importantly, it can draw attention to, and develop,those pieces of mathematical theory which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point it is worth telling the story of Wilfrid Rall,

who inthe 1960s developed the cable model of the dendritic tree(see [18] for a survey of his work)




The Brain.




More importantly, it can draw attention to, and develop,those pieces of mathematical theory which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point it is worth telling the story of Wilfrid Rall, who inthe 1960s

developed the cable model of the dendritic tree(see [18] for a survey of his work)




The Brain.




More importantly, it can draw attention to, and develop,those pieces of mathematical theory which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point it is worth telling the story of Wilfrid Rall, who inthe 1960s developed the cable model

of the dendritic tree(see [18] for a survey of his work)




The Brain.




More importantly, it can draw attention to, and develop,those pieces of mathematical theory which are likely to berelevant to future studies of the brain [17]. In illustration ofthis point it is worth telling the story of Wilfrid Rall, who inthe 1960s developed the cable model of the dendritic tree

(see [18] for a survey of his work)




The Brain.








The Brain.








The Brain.





Cable theory

uses coupled PDEs (Partial Differential;Equations) to describe how membrane potential spreadsalong the dendritic branches in response to a localconductance change (synaptic input).



The Brain.





Cable theory uses coupled PDEs (Partial Differential;Equations)

to describe how membrane potential spreadsalong the dendritic branches in response to a localconductance change (synaptic input).



The Brain.





Cable theory uses coupled PDEs (Partial Differential;Equations) to describe

how membrane potential spreadsalong the dendritic branches in response to a localconductance change (synaptic input).



The Brain.





Cable theory uses coupled PDEs (Partial Differential;Equations) to describe how membrane potential

spreadsalong the dendritic branches in response to a localconductance change (synaptic input).



The Brain.





Cable theory uses coupled PDEs (Partial Differential;Equations) to describe how membrane potential spreadsalong the dendritic branches

in response to a localconductance change (synaptic input).



The Brain.





Cable theory uses coupled PDEs (Partial Differential;Equations) to describe how membrane potential spreadsalong the dendritic branches in response to

a localconductance change (synaptic input).



The Brain.








The Brain.








The Brain.




Using his mathematical formalism, Rall showed that thereis a subclass of trees that is electrically equivalent to asingle cylinder whose diameter is that of the stem dendrite.

To a first approximation, many neurons (e.g.α-motoneuron) belong to this subclass (though cortical andhippocampal pyramidal cells do not). Importantly Rall’s“equivalent cylinder” model allows for a simple analyticalsolution and this has provided the main insights regardingthe spread of electrical signals in passive dendritic trees.



The Brain.




Using his mathematical formalism,

Rall showed that thereis a subclass of trees that is electrically equivalent to asingle cylinder whose diameter is that of the stem dendrite.




The Brain.




Using his mathematical formalism, Rall showed that

thereis a subclass of trees that is electrically equivalent to asingle cylinder whose diameter is that of the stem dendrite.




The Brain.




Using his mathematical formalism, Rall showed that thereis a subclass of trees

that is electrically equivalent to asingle cylinder whose diameter is that of the stem dendrite.




The Brain.




Using his mathematical formalism, Rall showed that thereis a subclass of trees that is electrically equivalent

to asingle cylinder whose diameter is that of the stem dendrite.




The Brain.




Using his mathematical formalism, Rall showed that thereis a subclass of trees that is electrically equivalent to asingle cylinder

whose diameter is that of the stem dendrite.




The Brain.




Using his mathematical formalism, Rall showed that thereis a subclass of trees that is electrically equivalent to asingle cylinder whose diameter is

that of the stem dendrite.




The Brain.








The Brain.








The Brain.





To a first approximation,

many neurons (e.g.α-motoneuron) belong to this subclass (though cortical andhippocampal pyramidal cells do not). Importantly Rall’s“equivalent cylinder” model allows for a simple analyticalsolution and this has provided the main insights regardingthe spread of electrical signals in passive dendritic trees.



The Brain.





To a first approximation, many neurons (e.g.α-motoneuron)

belong to this subclass (though cortical andhippocampal pyramidal cells do not). Importantly Rall’s“equivalent cylinder” model allows for a simple analyticalsolution and this has provided the main insights regardingthe spread of electrical signals in passive dendritic trees.



The Brain.





To a first approximation, many neurons (e.g.α-motoneuron) belong to this subclass

(though cortical andhippocampal pyramidal cells do not). Importantly Rall’s“equivalent cylinder” model allows for a simple analyticalsolution and this has provided the main insights regardingthe spread of electrical signals in passive dendritic trees.



The Brain.





To a first approximation, many neurons (e.g.α-motoneuron) belong to this subclass (though cortical andhippocampal pyramidal cells do not).

Importantly Rall’s“equivalent cylinder” model allows for a simple analyticalsolution and this has provided the main insights regardingthe spread of electrical signals in passive dendritic trees.



The Brain.





To a first approximation, many neurons (e.g.α-motoneuron) belong to this subclass (though cortical andhippocampal pyramidal cells do not). Importantly Rall’s“equivalent cylinder”

model allows for a simple analyticalsolution and this has provided the main insights regardingthe spread of electrical signals in passive dendritic trees.



The Brain.





To a first approximation, many neurons (e.g.α-motoneuron) belong to this subclass (though cortical andhippocampal pyramidal cells do not). Importantly Rall’s“equivalent cylinder” model allows for a simple analyticalsolution and

this has provided the main insights regardingthe spread of electrical signals in passive dendritic trees.



The Brain.





To a first approximation, many neurons (e.g.α-motoneuron) belong to this subclass (though cortical andhippocampal pyramidal cells do not). Importantly Rall’s“equivalent cylinder” model allows for a simple analyticalsolution and this has provided the main insights

regardingthe spread of electrical signals in passive dendritic trees.



The Brain.





To a first approximation, many neurons (e.g.α-motoneuron) belong to this subclass (though cortical andhippocampal pyramidal cells do not). Importantly Rall’s“equivalent cylinder” model allows for a simple analyticalsolution and this has provided the main insights regardingthe spread of electrical signals

in passive dendritic trees.



The Brain.








The Brain.








The Brain.




As another example we turn to work on neural fieldequations in the 1970’s, by people such as Hugh Wilson,Jack Cowan, Bard Ermentrout, Shun-ichi Amari, PaulNunez and Hermann Haken (for a recent overview see[19])

These are tissue level models that describe thespatio-temporal evolution of coarse-grained variables suchas synaptic or firing rate activity in populations of neurons,and often take the form of integro-differential equations.



The Brain.




As another example

we turn to work on neural fieldequations in the 1970’s, by people such as Hugh Wilson,Jack Cowan, Bard Ermentrout, Shun-ichi Amari, PaulNunez and Hermann Haken (for a recent overview see[19])




The Brain.




As another example we turn to work

on neural fieldequations in the 1970’s, by people such as Hugh Wilson,Jack Cowan, Bard Ermentrout, Shun-ichi Amari, PaulNunez and Hermann Haken (for a recent overview see[19])




The Brain.




As another example we turn to work on neural fieldequations in the 1970’s,

by people such as Hugh Wilson,Jack Cowan, Bard Ermentrout, Shun-ichi Amari, PaulNunez and Hermann Haken (for a recent overview see[19])




The Brain.




As another example we turn to work on neural fieldequations in the 1970’s, by people such as

Hugh Wilson,Jack Cowan, Bard Ermentrout, Shun-ichi Amari, PaulNunez and Hermann Haken (for a recent overview see[19])




The Brain.




As another example we turn to work on neural fieldequations in the 1970’s, by people such as Hugh Wilson,

Jack Cowan, Bard Ermentrout, Shun-ichi Amari, PaulNunez and Hermann Haken (for a recent overview see[19])




The Brain.




As another example we turn to work on neural fieldequations in the 1970’s, by people such as Hugh Wilson,Jack Cowan,

Bard Ermentrout, Shun-ichi Amari, PaulNunez and Hermann Haken (for a recent overview see[19])




The Brain.




As another example we turn to work on neural fieldequations in the 1970’s, by people such as Hugh Wilson,Jack Cowan, Bard Ermentrout,

Shun-ichi Amari, PaulNunez and Hermann Haken (for a recent overview see[19])




The Brain.




As another example we turn to work on neural fieldequations in the 1970’s, by people such as Hugh Wilson,Jack Cowan, Bard Ermentrout, Shun-ichi Amari,

PaulNunez and Hermann Haken (for a recent overview see[19])




The Brain.




As another example we turn to work on neural fieldequations in the 1970’s, by people such as Hugh Wilson,Jack Cowan, Bard Ermentrout, Shun-ichi Amari, PaulNunez and

Hermann Haken (for a recent overview see[19])




The Brain.








The Brain.





These are tissue level models

that describe thespatio-temporal evolution of coarse-grained variables suchas synaptic or firing rate activity in populations of neurons,and often take the form of integro-differential equations.



The Brain.





These are tissue level models that describe thespatio-temporal

evolution of coarse-grained variables suchas synaptic or firing rate activity in populations of neurons,and often take the form of integro-differential equations.



The Brain.





These are tissue level models that describe thespatio-temporal evolution of coarse-grained variables

suchas synaptic or firing rate activity in populations of neurons,and often take the form of integro-differential equations.



The Brain.





These are tissue level models that describe thespatio-temporal evolution of coarse-grained variables suchas synaptic or firing rate activity

in populations of neurons,and often take the form of integro-differential equations.



The Brain.





These are tissue level models that describe thespatio-temporal evolution of coarse-grained variables suchas synaptic or firing rate activity in populations of neurons,and

often take the form of integro-differential equations.



The Brain.





These are tissue level models that describe thespatio-temporal evolution of coarse-grained variables suchas synaptic or firing rate activity in populations of neurons,and often take the form

of integro-differential equations.



The Brain.








The Brain.








The Brain.




The sorts of dynamic behaviour

that are typically observedin neural field models include spatially and temporallyperiodic patterns (beyond a Turing instability[morphogenesis]), localised regions of activity andtravelling waves.

The mathematical study of such equations and theirsolutions has proven relevant to understanding EEGrhythms( Electroencephalography), mechanisms forshort-term memory, motion perception and drug-inducedvisual hallucinations.



The Brain.




The sorts of dynamic behaviour that are typically observed

in neural field models include spatially and temporallyperiodic patterns (beyond a Turing instability[morphogenesis]), localised regions of activity andtravelling waves.




The Brain.




The sorts of dynamic behaviour that are typically observedin neural field models

include spatially and temporallyperiodic patterns (beyond a Turing instability[morphogenesis]), localised regions of activity andtravelling waves.




The Brain.




The sorts of dynamic behaviour that are typically observedin neural field models include spatially and temporallyperiodic patterns

(beyond a Turing instability[morphogenesis]), localised regions of activity andtravelling waves.




The Brain.




The sorts of dynamic behaviour that are typically observedin neural field models include spatially and temporallyperiodic patterns (beyond a Turing instability[morphogenesis]),

localised regions of activity andtravelling waves.




The Brain.




The sorts of dynamic behaviour that are typically observedin neural field models include spatially and temporallyperiodic patterns (beyond a Turing instability[morphogenesis]), localised regions of activity and

travelling waves.




The Brain.




The sorts of dynamic behaviour that are typically observedin neural field models include spatially and temporallyperiodic patterns (beyond a Turing instability[morphogenesis]), localised regions of activity andtravelling waves.




The Brain.





The mathematical study

of such equations and theirsolutions has proven relevant to understanding EEGrhythms( Electroencephalography), mechanisms forshort-term memory, motion perception and drug-inducedvisual hallucinations.



The Brain.





The mathematical study of such equations and

theirsolutions has proven relevant to understanding EEGrhythms( Electroencephalography), mechanisms forshort-term memory, motion perception and drug-inducedvisual hallucinations.



The Brain.





The mathematical study of such equations and theirsolutions

has proven relevant to understanding EEGrhythms( Electroencephalography), mechanisms forshort-term memory, motion perception and drug-inducedvisual hallucinations.



The Brain.





The mathematical study of such equations and theirsolutions has proven relevant

to understanding EEGrhythms( Electroencephalography), mechanisms forshort-term memory, motion perception and drug-inducedvisual hallucinations.



The Brain.





The mathematical study of such equations and theirsolutions has proven relevant to understanding EEGrhythms( Electroencephalography),

mechanisms forshort-term memory, motion perception and drug-inducedvisual hallucinations.



The Brain.





The mathematical study of such equations and theirsolutions has proven relevant to understanding EEGrhythms( Electroencephalography), mechanisms forshort-term memory,

motion perception and drug-inducedvisual hallucinations.



The Brain.





The mathematical study of such equations and theirsolutions has proven relevant to understanding EEGrhythms( Electroencephalography), mechanisms forshort-term memory, motion perception and

drug-inducedvisual hallucinations.



The Brain.








The Brain.








The Brain.




In this latter context the use of symmetric bifurcation theoryhas shown that neural activity patterns underlying commonvisual hallucinations can be accounted for in terms ofcertain symmetry properties of the anisotropicsynapticconnections in visual cortex (requiring the use of anovel representation of the planar Euclidean group) [20]

As well as the above examples of the practice ofmathematical neuroscience, it is as well to mention someof the tools in the arsenal of the mathematicalneuroscientist world.



The Brain.








The Brain.




In this latter context

the use of symmetric bifurcation theoryhas shown that neural activity patterns underlying commonvisual hallucinations can be accounted for in terms ofcertain symmetry properties of the anisotropicsynapticconnections in visual cortex (requiring the use of anovel representation of the planar Euclidean group) [20]




The Brain.




In this latter context the use of symmetric bifurcation theory

has shown that neural activity patterns underlying commonvisual hallucinations can be accounted for in terms ofcertain symmetry properties of the anisotropicsynapticconnections in visual cortex (requiring the use of anovel representation of the planar Euclidean group) [20]




The Brain.




In this latter context the use of symmetric bifurcation theoryhas shown that

neural activity patterns underlying commonvisual hallucinations can be accounted for in terms ofcertain symmetry properties of the anisotropicsynapticconnections in visual cortex (requiring the use of anovel representation of the planar Euclidean group) [20]




The Brain.




In this latter context the use of symmetric bifurcation theoryhas shown that neural activity patterns

underlying commonvisual hallucinations can be accounted for in terms ofcertain symmetry properties of the anisotropicsynapticconnections in visual cortex (requiring the use of anovel representation of the planar Euclidean group) [20]




The Brain.




In this latter context the use of symmetric bifurcation theoryhas shown that neural activity patterns underlying commonvisual hallucinations

can be accounted for in terms ofcertain symmetry properties of the anisotropicsynapticconnections in visual cortex (requiring the use of anovel representation of the planar Euclidean group) [20]




The Brain.




In this latter context the use of symmetric bifurcation theoryhas shown that neural activity patterns underlying commonvisual hallucinations can be accounted for

in terms ofcertain symmetry properties of the anisotropicsynapticconnections in visual cortex (requiring the use of anovel representation of the planar Euclidean group) [20]




The Brain.




In this latter context the use of symmetric bifurcation theoryhas shown that neural activity patterns underlying commonvisual hallucinations can be accounted for in terms ofcertain symmetry properties

of the anisotropicsynapticconnections in visual cortex (requiring the use of anovel representation of the planar Euclidean group) [20]




The Brain.




In this latter context the use of symmetric bifurcation theoryhas shown that neural activity patterns underlying commonvisual hallucinations can be accounted for in terms ofcertain symmetry properties of the

anisotropicsynapticconnections in visual cortex (requiring the use of anovel representation of the planar Euclidean group) [20]




The Brain.




In this latter context the use of symmetric bifurcation theoryhas shown that neural activity patterns underlying commonvisual hallucinations can be accounted for in terms ofcertain symmetry properties of the anisotropicsynapticconnections

in visual cortex (requiring the use of anovel representation of the planar Euclidean group) [20]




The Brain.




In this latter context the use of symmetric bifurcation theoryhas shown that neural activity patterns underlying commonvisual hallucinations can be accounted for in terms ofcertain symmetry properties of the anisotropicsynapticconnections in visual cortex

(requiring the use of anovel representation of the planar Euclidean group) [20]




The Brain.




In this latter context the use of symmetric bifurcation theoryhas shown that neural activity patterns underlying commonvisual hallucinations can be accounted for in terms ofcertain symmetry properties of the anisotropicsynapticconnections in visual cortex (requiring the use

of anovel representation of the planar Euclidean group) [20]




The Brain.




In this latter context the use of symmetric bifurcation theoryhas shown that neural activity patterns underlying commonvisual hallucinations can be accounted for in terms ofcertain symmetry properties of the anisotropicsynapticconnections in visual cortex (requiring the use of anovel representation

of the planar Euclidean group) [20]




The Brain.








The Brain.





As well as the above examples

of the practice ofmathematical neuroscience, it is as well to mention someof the tools in the arsenal of the mathematicalneuroscientist world.



The Brain.





As well as the above examples of the practice ofmathematical neuroscience,

it is as well to mention someof the tools in the arsenal of the mathematicalneuroscientist world.



The Brain.





As well as the above examples of the practice ofmathematical neuroscience, it is as well to mention

someof the tools in the arsenal of the mathematicalneuroscientist world.



The Brain.





As well as the above examples of the practice ofmathematical neuroscience, it is as well to mention someof the tools

in the arsenal of the mathematicalneuroscientist world.



The Brain.





As well as the above examples of the practice ofmathematical neuroscience, it is as well to mention someof the tools in the arsenal

of the mathematicalneuroscientist world.



The Brain.








The Brain.








The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that

techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and

mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics

have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date.

Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed,

seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding

nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials,

dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and

the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG,

mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved

on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass

increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools

of modern applied mathematics (Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics

(Topology, Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology,

Algrebraic Geometry, Category Theory ).



The Brain.




It is clear that techniques from nonlinear dynamicalsystems theory and mathematical physics have provenuseful to date. Indeed, seeded by successes inunderstanding nerve action potentials, dendriticprocessing, and the neural basis of EEG, mathematicalneuroscience has moved on to encompass increasinglysophisticated tools of modern applied mathematics (Topology, Algrebraic Geometry,

Category Theory ).



The Brain.







The Brain.







The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are

Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques

forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and

bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue

levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and

EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21],

heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling

in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22],

the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory

in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23],

using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations

to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],

spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network

evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25],

the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis

of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26],

the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and

the use of information geometry in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry

in developingnovel brain-style computations [28].



The Brain.




Included among these are Evans function techniques forstudying wave stability and bifurcation in tissue levelmodels of synaptic and EEG activity [21], heterocliniccycling in theories of olfactory coding [22], the use ofgeometric singular perturbation theory in understandingrhythmogenesis [23], using stochastic differentialequations to treat inherent neuronal noise [24],spike-density approaches for modelling network evolution[25], the weakly nonlinear analysis of pattern formation[26], the role of canards in organising neural dynamics[27], and the use of information geometry in developingnovel

brain-style computations [28].



The Brain.







The Brain.







The Brain.




The field is now in the healthy state where not only ismathematics having an impact on neuroscience, the latteris simultaneously motivating important research inmathematics. In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now

in the healthy state where not only ismathematics having an impact on neuroscience, the latteris simultaneously motivating important research inmathematics. In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now in the healthy state

where not only ismathematics having an impact on neuroscience, the latteris simultaneously motivating important research inmathematics. In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now in the healthy state where not only ismathematics

having an impact on neuroscience, the latteris simultaneously motivating important research inmathematics. In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now in the healthy state where not only ismathematics having an impact on neuroscience,

the latteris simultaneously motivating important research inmathematics. In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now in the healthy state where not only ismathematics having an impact on neuroscience, the latter

is simultaneously motivating important research inmathematics. In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now in the healthy state where not only ismathematics having an impact on neuroscience, the latteris simultaneously motivating important research

inmathematics. In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now in the healthy state where not only ismathematics having an impact on neuroscience, the latteris simultaneously motivating important research inmathematics.

In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now in the healthy state where not only ismathematics having an impact on neuroscience, the latteris simultaneously motivating important research inmathematics. In recent years

a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now in the healthy state where not only ismathematics having an impact on neuroscience, the latteris simultaneously motivating important research inmathematics. In recent years a number of high profilemathematical institutes,

including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now in the healthy state where not only ismathematics having an impact on neuroscience, the latteris simultaneously motivating important research inmathematics. In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004),

theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now in the healthy state where not only ismathematics having an impact on neuroscience, the latteris simultaneously motivating important research inmathematics. In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and

the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now in the healthy state where not only ismathematics having an impact on neuroscience, the latteris simultaneously motivating important research inmathematics. In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops

with the title“Mathematical Neuroscience“ (of at least three daysduration)



The Brain.




The field is now in the healthy state where not only ismathematics having an impact on neuroscience, the latteris simultaneously motivating important research inmathematics. In recent years a number of high profilemathematical institutes, including the MathematicalSciences Research Institute (Berkeley; 2004), theInternational Centre for Mathematical Sciences(Edinburgh; 2005), and the Centre de Recerca Matematica(Andorra; 2006), have held workshops with the title“Mathematical Neuroscience“

(of at least three daysduration)



The Brain.







The Brain.







The Brain.







The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience

has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted

thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics

is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease

associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony.

In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular,

there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attempt

by the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community

to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation

(a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatment

involving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation

of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses

to specific parts of the brain) affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain)

affectsneuronal dynamics in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics

in a curative manner for Parkinson’sdisease [29]



The Brain.




One area in which neuroscience has already prompted thedevelopment of novel mathematics is that of neurologicaldisease associated with abnormalities in neural networksynchrony. In particular, there is now a concerted attemptby the mathematical neuroscience community to uncoverjust how deep-brain-stimulation (a surgical treatmentinvolving the implantation of a device which sendselectrical impulses to specific parts of the brain) affectsneuronal dynamics in a curative manner

for Parkinson’sdisease [29]



The Brain.







The Brain.







The Brain.




Mathematical Neuroscience workshop

to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held

at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques,

Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal

in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007

(co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin).

The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony”

is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example

of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience,

where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps

more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known

for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion

in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and

brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Mathematical Neuroscience workshop to be held at theCentre de Recherches Mathematiques, Universite deMontreal in September 2007 (co-organised by S Coombes,with A Longtin and J Rubin). The issue of “synchrony” is agood example of the relevance of mathematics inneuroscience, where it is perhaps more well known for itsdiscussion in relation to the binding problem [30] and brainrhythms in general [31].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed,

there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neuroscience

that have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified

further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement.

For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example,

one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that

a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network

can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto

is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that

cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies

(without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and

that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect

correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits

in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32].

Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another,

is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells,

which fire strongly when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly

when an animalis in locations that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations

that tessellate the environment in ahexagonal pattern [33].



The Brain.




Indeed, there are many current advances in neurosciencethat have identified further need for mathematiciansinvolvement. For example, one area that a MathematicalNeuroscience Network can make a significant contributionto is the recent discovery that cannabinoids candesynchronise neuronal assemblies (without affectingaverage firing rates), and that this effect correlates withmemory deficits in individuals [32]. Another, is thediscovery of grid cells, which fire strongly when an animalis in locations that tessellate the environment

in ahexagonal pattern [33].



The Brain.







The Brain.







The Brain.

Basic introduction to the brain.References

Basic introduction to the brain

Motivation.

This lectures are intended to give students ageneral intuition for basic mathematical language used todescribe and model neurons. These principles will serveas the foundation for future sections.



The Brain.



Motivation. This lectures are

intended to give students ageneral intuition for basic mathematical language used todescribe and model neurons. These principles will serveas the foundation for future sections.



The Brain.



Motivation. This lectures are intended to give students

ageneral intuition for basic mathematical language used todescribe and model neurons. These principles will serveas the foundation for future sections.



The Brain.



Motivation. This lectures are intended to give students ageneral intuition

for basic mathematical language used todescribe and model neurons. These principles will serveas the foundation for future sections.



The Brain.



Motivation. This lectures are intended to give students ageneral intuition for basic mathematical language

used todescribe and model neurons. These principles will serveas the foundation for future sections.



The Brain.



Motivation. This lectures are intended to give students ageneral intuition for basic mathematical language used todescribe and model neurons.

These principles will serveas the foundation for future sections.



The Brain.



Motivation. This lectures are intended to give students ageneral intuition for basic mathematical language used todescribe and model neurons. These principles

will serveas the foundation for future sections.



The Brain.



Motivation. This lectures are intended to give students ageneral intuition for basic mathematical language used todescribe and model neurons. These principles will serveas

the foundation for future sections.



The Brain.



Motivation. This lectures are intended to give students ageneral intuition for basic mathematical language used todescribe and model neurons. These principles will serveas the foundation for future sections.



The Brain.






The Brain.






The Brain.


Basic introduction to the brain.

Basic organization of the brain.

The brain is typicallydivided into 4 lobes. The temporal lobe contains neuralmachinery for processing speech and sounds, spatialinformation, and for encoding episodic (autobiographical)memories. The parietal lobe is involved with sensoryperception, sensory integration, and memory.



The Brain.



Basic organization of the brain. The brain

is typicallydivided into 4 lobes. The temporal lobe contains neuralmachinery for processing speech and sounds, spatialinformation, and for encoding episodic (autobiographical)memories. The parietal lobe is involved with sensoryperception, sensory integration, and memory.



The Brain.



Basic organization of the brain. The brain is typicallydivided into 4 lobes.

The temporal lobe contains neuralmachinery for processing speech and sounds, spatialinformation, and for encoding episodic (autobiographical)memories. The parietal lobe is involved with sensoryperception, sensory integration, and memory.



The Brain.



Basic organization of the brain. The brain is typicallydivided into 4 lobes. The temporal lobe

contains neuralmachinery for processing speech and sounds, spatialinformation, and for encoding episodic (autobiographical)memories. The parietal lobe is involved with sensoryperception, sensory integration, and memory.



The Brain.



Basic organization of the brain. The brain is typicallydivided into 4 lobes. The temporal lobe contains neuralmachinery

for processing speech and sounds, spatialinformation, and for encoding episodic (autobiographical)memories. The parietal lobe is involved with sensoryperception, sensory integration, and memory.



The Brain.



Basic organization of the brain. The brain is typicallydivided into 4 lobes. The temporal lobe contains neuralmachinery for processing speech and sounds,

spatialinformation, and for encoding episodic (autobiographical)memories. The parietal lobe is involved with sensoryperception, sensory integration, and memory.



The Brain.



Basic organization of the brain. The brain is typicallydivided into 4 lobes. The temporal lobe contains neuralmachinery for processing speech and sounds, spatialinformation, and

for encoding episodic (autobiographical)memories. The parietal lobe is involved with sensoryperception, sensory integration, and memory.



The Brain.



Basic organization of the brain. The brain is typicallydivided into 4 lobes. The temporal lobe contains neuralmachinery for processing speech and sounds, spatialinformation, and for encoding episodic (autobiographical)memories.

The parietal lobe is involved with sensoryperception, sensory integration, and memory.



The Brain.



Basic organization of the brain. The brain is typicallydivided into 4 lobes. The temporal lobe contains neuralmachinery for processing speech and sounds, spatialinformation, and for encoding episodic (autobiographical)memories. The parietal lobe is involved

with sensoryperception, sensory integration, and memory.



The Brain.



Basic organization of the brain. The brain is typicallydivided into 4 lobes. The temporal lobe contains neuralmachinery for processing speech and sounds, spatialinformation, and for encoding episodic (autobiographical)memories. The parietal lobe is involved with sensoryperception,

sensory integration, and memory.



The Brain.



Basic organization of the brain. The brain is typicallydivided into 4 lobes. The temporal lobe contains neuralmachinery for processing speech and sounds, spatialinformation, and for encoding episodic (autobiographical)memories. The parietal lobe is involved with sensoryperception, sensory integration, and memory.



The Brain.






The Brain.






The Brain.



Basic organization of the brain.The frontal lobe is associated

with personality, reasoning,planning, problem solving, working memory, andmovement. The occipital lobe is primiarily involved withvision and visual processing. The central sulcus separatesthe primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



The Brain.



Basic organization of the brain.The frontal lobe is associated with personality,

reasoning,planning, problem solving, working memory, andmovement. The occipital lobe is primiarily involved withvision and visual processing. The central sulcus separatesthe primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



The Brain.



Basic organization of the brain.The frontal lobe is associated with personality, reasoning,

planning, problem solving, working memory, andmovement. The occipital lobe is primiarily involved withvision and visual processing. The central sulcus separatesthe primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



The Brain.



Basic organization of the brain.The frontal lobe is associated with personality, reasoning,planning,

problem solving, working memory, andmovement. The occipital lobe is primiarily involved withvision and visual processing. The central sulcus separatesthe primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



The Brain.



Basic organization of the brain.The frontal lobe is associated with personality, reasoning,planning, problem solving,

working memory, andmovement. The occipital lobe is primiarily involved withvision and visual processing. The central sulcus separatesthe primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



The Brain.



Basic organization of the brain.The frontal lobe is associated with personality, reasoning,planning, problem solving, working memory, and

movement. The occipital lobe is primiarily involved withvision and visual processing. The central sulcus separatesthe primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



The Brain.



Basic organization of the brain.The frontal lobe is associated with personality, reasoning,planning, problem solving, working memory, andmovement.

The occipital lobe is primiarily involved withvision and visual processing. The central sulcus separatesthe primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



The Brain.



Basic organization of the brain.The frontal lobe is associated with personality, reasoning,planning, problem solving, working memory, andmovement. The occipital lobe

is primiarily involved withvision and visual processing. The central sulcus separatesthe primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



The Brain.



Basic organization of the brain.The frontal lobe is associated with personality, reasoning,planning, problem solving, working memory, andmovement. The occipital lobe is primiarily involved

withvision and visual processing. The central sulcus separatesthe primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



The Brain.



Basic organization of the brain.The frontal lobe is associated with personality, reasoning,planning, problem solving, working memory, andmovement. The occipital lobe is primiarily involved withvision and visual processing.

The central sulcus separatesthe primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



The Brain.



Basic organization of the brain.The frontal lobe is associated with personality, reasoning,planning, problem solving, working memory, andmovement. The occipital lobe is primiarily involved withvision and visual processing. The central sulcus separates

the primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



The Brain.



Basic organization of the brain.The frontal lobe is associated with personality, reasoning,planning, problem solving, working memory, andmovement. The occipital lobe is primiarily involved withvision and visual processing. The central sulcus separatesthe primary motor cortex (frontal lobe)

from the primarysomatosensory cortex (parietal lobe).



The Brain.



Basic organization of the brain.The frontal lobe is associated with personality, reasoning,planning, problem solving, working memory, andmovement. The occipital lobe is primiarily involved withvision and visual processing. The central sulcus separatesthe primary motor cortex (frontal lobe) from the primarysomatosensory cortex (parietal lobe).



The Brain.






The Brain.






The Brain.



Basic organization of the brain.

The medial longitudinalfissure’ separates the right and left hemispheres of thebrain. The spinal cord sends signals from the primarymotor cortex to the body’s skeletal muscles.



The Brain.



Basic organization of the brain. The medial longitudinalfissure’

separates the right and left hemispheres of thebrain. The spinal cord sends signals from the primarymotor cortex to the body’s skeletal muscles.



The Brain.



Basic organization of the brain. The medial longitudinalfissure’ separates the right and left hemispheres of thebrain.

The spinal cord sends signals from the primarymotor cortex to the body’s skeletal muscles.



The Brain.



Basic organization of the brain. The medial longitudinalfissure’ separates the right and left hemispheres of thebrain. The spinal cord

sends signals from the primarymotor cortex to the body’s skeletal muscles.



The Brain.



Basic organization of the brain. The medial longitudinalfissure’ separates the right and left hemispheres of thebrain. The spinal cord sends signals

from the primarymotor cortex to the body’s skeletal muscles.



The Brain.



Basic organization of the brain. The medial longitudinalfissure’ separates the right and left hemispheres of thebrain. The spinal cord sends signals from the primarymotor cortex

to the body’s skeletal muscles.



The Brain.



Basic organization of the brain. The medial longitudinalfissure’ separates the right and left hemispheres of thebrain. The spinal cord sends signals from the primarymotor cortex to the body’s skeletal muscles.



The Brain.






The Brain.






The Brain.



Neuron anatomy.

Neurons are electrically excitable cellsin the brain. Neurons “listen“ to other cells via branch-likedendrites. Signals from the dendrites travel down to thecell body, or soma, which contains the nucleus of the cell.Neurons communicate with other cells by sendingelectrical impulses down their axons, which most oftensynapse onto the dendrites of other neurons.



The Brain.



Neuron anatomy. Neurons are electrically excitable cells

in the brain. Neurons “listen“ to other cells via branch-likedendrites. Signals from the dendrites travel down to thecell body, or soma, which contains the nucleus of the cell.Neurons communicate with other cells by sendingelectrical impulses down their axons, which most oftensynapse onto the dendrites of other neurons.



The Brain.



Neuron anatomy. Neurons are electrically excitable cellsin the brain.

Neurons “listen“ to other cells via branch-likedendrites. Signals from the dendrites travel down to thecell body, or soma, which contains the nucleus of the cell.Neurons communicate with other cells by sendingelectrical impulses down their axons, which most oftensynapse onto the dendrites of other neurons.



The Brain.



Neuron anatomy. Neurons are electrically excitable cellsin the brain. Neurons “listen“ to other cells

via branch-likedendrites. Signals from the dendrites travel down to thecell body, or soma, which contains the nucleus of the cell.Neurons communicate with other cells by sendingelectrical impulses down their axons, which most oftensynapse onto the dendrites of other neurons.



The Brain.



Neuron anatomy. Neurons are electrically excitable cellsin the brain. Neurons “listen“ to other cells via branch-likedendrites.

Signals from the dendrites travel down to thecell body, or soma, which contains the nucleus of the cell.Neurons communicate with other cells by sendingelectrical impulses down their axons, which most oftensynapse onto the dendrites of other neurons.



The Brain.



Neuron anatomy. Neurons are electrically excitable cellsin the brain. Neurons “listen“ to other cells via branch-likedendrites. Signals from the dendrites

travel down to thecell body, or soma, which contains the nucleus of the cell.Neurons communicate with other cells by sendingelectrical impulses down their axons, which most oftensynapse onto the dendrites of other neurons.



The Brain.



Neuron anatomy. Neurons are electrically excitable cellsin the brain. Neurons “listen“ to other cells via branch-likedendrites. Signals from the dendrites travel down to thecell body, or soma,

which contains the nucleus of the cell.Neurons communicate with other cells by sendingelectrical impulses down their axons, which most oftensynapse onto the dendrites of other neurons.



The Brain.



Neuron anatomy. Neurons are electrically excitable cellsin the brain. Neurons “listen“ to other cells via branch-likedendrites. Signals from the dendrites travel down to thecell body, or soma, which contains the nucleus of the cell.

Neurons communicate with other cells by sendingelectrical impulses down their axons, which most oftensynapse onto the dendrites of other neurons.



The Brain.



Neuron anatomy. Neurons are electrically excitable cellsin the brain. Neurons “listen“ to other cells via branch-likedendrites. Signals from the dendrites travel down to thecell body, or soma, which contains the nucleus of the cell.Neurons communicate with other cells

by sendingelectrical impulses down their axons, which most oftensynapse onto the dendrites of other neurons.



The Brain.



Neuron anatomy. Neurons are electrically excitable cellsin the brain. Neurons “listen“ to other cells via branch-likedendrites. Signals from the dendrites travel down to thecell body, or soma, which contains the nucleus of the cell.Neurons communicate with other cells by sendingelectrical impulses down their axons,

which most oftensynapse onto the dendrites of other neurons.



The Brain.



Neuron anatomy. Neurons are electrically excitable cellsin the brain. Neurons “listen“ to other cells via branch-likedendrites. Signals from the dendrites travel down to thecell body, or soma, which contains the nucleus of the cell.Neurons communicate with other cells by sendingelectrical impulses down their axons, which most oftensynapse onto

the dendrites of other neurons.



The Brain.



Neuron anatomy. Neurons are electrically excitable cellsin the brain. Neurons “listen“ to other cells via branch-likedendrites. Signals from the dendrites travel down to thecell body, or soma, which contains the nucleus of the cell.Neurons communicate with other cells by sendingelectrical impulses down their axons, which most oftensynapse onto the dendrites of other neurons.



The Brain.






The Brain.






The Brain.






The Brain.



Neuron anatomy.

There are only around 1011 neurons inthe human brain, much fewer than the number ofnon-neural cells such as glia.

Yet neurons are unique in the sense that only they cantransmit electrical signals over long distances.

From neuronal level we can go down to cell biophysics, tomolecular biology of gene regulation, etc. From neuronallevel we can go up to neuronal circuits, to corticalstructures, to the whole brain, and finally to the behavior ofthe organism



The Brain.



Neuron anatomy. There are only

around 1011 neurons inthe human brain, much fewer than the number ofnon-neural cells such as glia.





The Brain.



Neuron anatomy. There are only around 1011 neurons inthe human brain,

much fewer than the number ofnon-neural cells such as glia.





The Brain.



Neuron anatomy. There are only around 1011 neurons inthe human brain, much fewer

than the number ofnon-neural cells such as glia.





The Brain.



Neuron anatomy. There are only around 1011 neurons inthe human brain, much fewer than the number

ofnon-neural cells such as glia.





The Brain.



Neuron anatomy. There are only around 1011 neurons inthe human brain, much fewer than the number ofnon-neural cells such as glia.





The Brain.




Yet neurons are unique

in the sense that only they cantransmit electrical signals over long distances.




The Brain.




Yet neurons are unique in the sense that only they

cantransmit electrical signals over long distances.




The Brain.




Yet neurons are unique in the sense that only they cantransmit electrical signals

over long distances.




The Brain.








The Brain.





From neuronal level

we can go down to cell biophysics, tomolecular biology of gene regulation, etc. From neuronallevel we can go up to neuronal circuits, to corticalstructures, to the whole brain, and finally to the behavior ofthe organism



The Brain.





From neuronal level we can go down

to cell biophysics, tomolecular biology of gene regulation, etc. From neuronallevel we can go up to neuronal circuits, to corticalstructures, to the whole brain, and finally to the behavior ofthe organism



The Brain.





From neuronal level we can go down to cell biophysics,

tomolecular biology of gene regulation, etc. From neuronallevel we can go up to neuronal circuits, to corticalstructures, to the whole brain, and finally to the behavior ofthe organism



The Brain.





From neuronal level we can go down to cell biophysics, tomolecular biology of gene regulation, etc.

From neuronallevel we can go up to neuronal circuits, to corticalstructures, to the whole brain, and finally to the behavior ofthe organism



The Brain.





From neuronal level we can go down to cell biophysics, tomolecular biology of gene regulation, etc. From neuronallevel

we can go up to neuronal circuits, to corticalstructures, to the whole brain, and finally to the behavior ofthe organism



The Brain.





From neuronal level we can go down to cell biophysics, tomolecular biology of gene regulation, etc. From neuronallevel we can go up to neuronal circuits,

to corticalstructures, to the whole brain, and finally to the behavior ofthe organism



The Brain.





From neuronal level we can go down to cell biophysics, tomolecular biology of gene regulation, etc. From neuronallevel we can go up to neuronal circuits, to corticalstructures,

to the whole brain, and finally to the behavior ofthe organism



The Brain.





From neuronal level we can go down to cell biophysics, tomolecular biology of gene regulation, etc. From neuronallevel we can go up to neuronal circuits, to corticalstructures, to the whole brain, and

finally to the behavior ofthe organism



The Brain.








The Brain.








The Brain.



Action potentials. Neurons communicate w

ith each otherby changes in their membrane voltage (we’ll get to whatthis means in a bit). Small changes in membrane voltageare picked up by neurons up to approximately 1mm away.Thus, for nervous systems on the scale of 1mm, such asthe fruit fly nervous system, no other special mode ofcommunication is needed.



The Brain.



Action potentials. Neurons communicate w ith each otherby changes

in their membrane voltage (we’ll get to whatthis means in a bit). Small changes in membrane voltageare picked up by neurons up to approximately 1mm away.Thus, for nervous systems on the scale of 1mm, such asthe fruit fly nervous system, no other special mode ofcommunication is needed.



The Brain.



Action potentials. Neurons communicate w ith each otherby changes in their membrane voltage

(we’ll get to whatthis means in a bit). Small changes in membrane voltageare picked up by neurons up to approximately 1mm away.Thus, for nervous systems on the scale of 1mm, such asthe fruit fly nervous system, no other special mode ofcommunication is needed.



The Brain.



Action potentials. Neurons communicate w ith each otherby changes in their membrane voltage (we’ll get to whatthis means in a bit).

Small changes in membrane voltageare picked up by neurons up to approximately 1mm away.Thus, for nervous systems on the scale of 1mm, such asthe fruit fly nervous system, no other special mode ofcommunication is needed.



The Brain.



Action potentials. Neurons communicate w ith each otherby changes in their membrane voltage (we’ll get to whatthis means in a bit). Small changes in membrane voltage

are picked up by neurons up to approximately 1mm away.Thus, for nervous systems on the scale of 1mm, such asthe fruit fly nervous system, no other special mode ofcommunication is needed.



The Brain.



Action potentials. Neurons communicate w ith each otherby changes in their membrane voltage (we’ll get to whatthis means in a bit). Small changes in membrane voltageare picked up by neurons up

to approximately 1mm away.Thus, for nervous systems on the scale of 1mm, such asthe fruit fly nervous system, no other special mode ofcommunication is needed.



The Brain.



Action potentials. Neurons communicate w ith each otherby changes in their membrane voltage (we’ll get to whatthis means in a bit). Small changes in membrane voltageare picked up by neurons up to approximately 1mm away.

Thus, for nervous systems on the scale of 1mm, such asthe fruit fly nervous system, no other special mode ofcommunication is needed.



The Brain.



Action potentials. Neurons communicate w ith each otherby changes in their membrane voltage (we’ll get to whatthis means in a bit). Small changes in membrane voltageare picked up by neurons up to approximately 1mm away.Thus,

for nervous systems on the scale of 1mm, such asthe fruit fly nervous system, no other special mode ofcommunication is needed.



The Brain.



Action potentials. Neurons communicate w ith each otherby changes in their membrane voltage (we’ll get to whatthis means in a bit). Small changes in membrane voltageare picked up by neurons up to approximately 1mm away.Thus, for nervous systems on the scale of 1mm,

such asthe fruit fly nervous system, no other special mode ofcommunication is needed.



The Brain.



Action potentials. Neurons communicate w ith each otherby changes in their membrane voltage (we’ll get to whatthis means in a bit). Small changes in membrane voltageare picked up by neurons up to approximately 1mm away.Thus, for nervous systems on the scale of 1mm, such asthe fruit fly nervous system,

no other special mode ofcommunication is needed.



The Brain.



Action potentials. Neurons communicate w ith each otherby changes in their membrane voltage (we’ll get to whatthis means in a bit). Small changes in membrane voltageare picked up by neurons up to approximately 1mm away.Thus, for nervous systems on the scale of 1mm, such asthe fruit fly nervous system, no other special mode ofcommunication is needed.



The Brain.






The Brain.






The Brain.






The Brain.



Action potentials.

However, in larger nervous systems(e.g. ours), neurons fire action potentials - suddenchanges in voltage. These form the basic mode of neuralcommunication in the brain. Over the next few lectureswe’ll be trying to understand how action potentials comeabout by modeling neurons in increasing levels of detail.



The Brain.



Action potentials. However,

in larger nervous systems(e.g. ours), neurons fire action potentials - suddenchanges in voltage. These form the basic mode of neuralcommunication in the brain. Over the next few lectureswe’ll be trying to understand how action potentials comeabout by modeling neurons in increasing levels of detail.



The Brain.



Action potentials. However, in larger nervous systems

(e.g. ours), neurons fire action potentials - suddenchanges in voltage. These form the basic mode of neuralcommunication in the brain. Over the next few lectureswe’ll be trying to understand how action potentials comeabout by modeling neurons in increasing levels of detail.



The Brain.



Action potentials. However, in larger nervous systems(e.g. ours),

neurons fire action potentials - suddenchanges in voltage. These form the basic mode of neuralcommunication in the brain. Over the next few lectureswe’ll be trying to understand how action potentials comeabout by modeling neurons in increasing levels of detail.



The Brain.



Action potentials. However, in larger nervous systems(e.g. ours), neurons fire action potentials -

suddenchanges in voltage. These form the basic mode of neuralcommunication in the brain. Over the next few lectureswe’ll be trying to understand how action potentials comeabout by modeling neurons in increasing levels of detail.



The Brain.



Action potentials. However, in larger nervous systems(e.g. ours), neurons fire action potentials - suddenchanges in voltage.

These form the basic mode of neuralcommunication in the brain. Over the next few lectureswe’ll be trying to understand how action potentials comeabout by modeling neurons in increasing levels of detail.



The Brain.



Action potentials. However, in larger nervous systems(e.g. ours), neurons fire action potentials - suddenchanges in voltage. These form the basic mode

of neuralcommunication in the brain. Over the next few lectureswe’ll be trying to understand how action potentials comeabout by modeling neurons in increasing levels of detail.



The Brain.



Action potentials. However, in larger nervous systems(e.g. ours), neurons fire action potentials - suddenchanges in voltage. These form the basic mode of neuralcommunication

in the brain. Over the next few lectureswe’ll be trying to understand how action potentials comeabout by modeling neurons in increasing levels of detail.



The Brain.



Action potentials. However, in larger nervous systems(e.g. ours), neurons fire action potentials - suddenchanges in voltage. These form the basic mode of neuralcommunication in the brain.

Over the next few lectureswe’ll be trying to understand how action potentials comeabout by modeling neurons in increasing levels of detail.



The Brain.



Action potentials. However, in larger nervous systems(e.g. ours), neurons fire action potentials - suddenchanges in voltage. These form the basic mode of neuralcommunication in the brain. Over the next few lectures

we’ll be trying to understand how action potentials comeabout by modeling neurons in increasing levels of detail.



The Brain.



Action potentials. However, in larger nervous systems(e.g. ours), neurons fire action potentials - suddenchanges in voltage. These form the basic mode of neuralcommunication in the brain. Over the next few lectureswe’ll be trying to understand

how action potentials comeabout by modeling neurons in increasing levels of detail.



The Brain.



Action potentials. However, in larger nervous systems(e.g. ours), neurons fire action potentials - suddenchanges in voltage. These form the basic mode of neuralcommunication in the brain. Over the next few lectureswe’ll be trying to understand how action potentials

comeabout by modeling neurons in increasing levels of detail.



The Brain.



Action potentials. However, in larger nervous systems(e.g. ours), neurons fire action potentials - suddenchanges in voltage. These form the basic mode of neuralcommunication in the brain. Over the next few lectureswe’ll be trying to understand how action potentials comeabout by modeling neurons

in increasing levels of detail.



The Brain.



Action potentials. However, in larger nervous systems(e.g. ours), neurons fire action potentials - suddenchanges in voltage. These form the basic mode of neuralcommunication in the brain. Over the next few lectureswe’ll be trying to understand how action potentials comeabout by modeling neurons in increasing levels of detail.



The Brain.






The Brain.






The Brain.






The Brain.






The Brain.



What is a spike?.

A typical neuron receives inputs from more than 10,000other neurons through the contacts on its dendritic treecalled synapses:



The Brain.



What is a spike?.

A typical neuron receives

inputs from more than 10,000other neurons through the contacts on its dendritic treecalled synapses:



The Brain.



What is a spike?.

A typical neuron receives inputs from more than 10,000other neurons

through the contacts on its dendritic treecalled synapses:



The Brain.



What is a spike?.

A typical neuron receives inputs from more than 10,000other neurons through the contacts on its dendritic tree

called synapses:



The Brain.



What is a spike?.




The Brain.



What is a spike?.




The Brain.



What is a spike?.




The Brain.



What is a spike?.

The inputs produce electrical transmembrane currents thatchange the membrane potential of the neuron. Smallsynaptic currents produce small changes, calledpost-synaptic potentials (PSPs).

Larger currents produce significant PSPs that could beamplified by the voltage- sensitive channels embedded inneuronal membrane and lead to the generation of anaction potential or spike - an abrupt and transientchange of membrane voltage that propagates to otherneurons via a long protrusion called an axon.



The Brain.



What is a spike?.

The inputs produce electrical transmembrane currents

thatchange the membrane potential of the neuron. Smallsynaptic currents produce small changes, calledpost-synaptic potentials (PSPs).




The Brain.



What is a spike?.

The inputs produce electrical transmembrane currents thatchange the membrane potential of the neuron.

Smallsynaptic currents produce small changes, calledpost-synaptic potentials (PSPs).




The Brain.



What is a spike?.

The inputs produce electrical transmembrane currents thatchange the membrane potential of the neuron. Smallsynaptic currents produce small changes,

calledpost-synaptic potentials (PSPs).




The Brain.



What is a spike?.





The Brain.



What is a spike?.





The Brain.



What is a spike?.


Larger currents

produce significant PSPs that could beamplified by the voltage- sensitive channels embedded inneuronal membrane and lead to the generation of anaction potential or spike - an abrupt and transientchange of membrane voltage that propagates to otherneurons via a long protrusion called an axon.



The Brain.



What is a spike?.


Larger currents produce significant PSPs

that could beamplified by the voltage- sensitive channels embedded inneuronal membrane and lead to the generation of anaction potential or spike - an abrupt and transientchange of membrane voltage that propagates to otherneurons via a long protrusion called an axon.



The Brain.



What is a spike?.


Larger currents produce significant PSPs that could beamplified by the voltage- sensitive channels

embedded inneuronal membrane and lead to the generation of anaction potential or spike - an abrupt and transientchange of membrane voltage that propagates to otherneurons via a long protrusion called an axon.



The Brain.



What is a spike?.


Larger currents produce significant PSPs that could beamplified by the voltage- sensitive channels embedded inneuronal membrane and

lead to the generation of anaction potential or spike - an abrupt and transientchange of membrane voltage that propagates to otherneurons via a long protrusion called an axon.



The Brain.



What is a spike?.


Larger currents produce significant PSPs that could beamplified by the voltage- sensitive channels embedded inneuronal membrane and lead to the generation of anaction potential or spike -

an abrupt and transientchange of membrane voltage that propagates to otherneurons via a long protrusion called an axon.



The Brain.



What is a spike?.


Larger currents produce significant PSPs that could beamplified by the voltage- sensitive channels embedded inneuronal membrane and lead to the generation of anaction potential or spike - an abrupt and transientchange

of membrane voltage that propagates to otherneurons via a long protrusion called an axon.



The Brain.



What is a spike?.


Larger currents produce significant PSPs that could beamplified by the voltage- sensitive channels embedded inneuronal membrane and lead to the generation of anaction potential or spike - an abrupt and transientchange of membrane voltage that propagates to otherneurons

via a long protrusion called an axon.



The Brain.



What is a spike?.





The Brain.



What is a spike?.





The Brain.



What is a spike?.

Such spikes are the main means of communicationbetween neurons. In general, neurons do not fire on theirown, they get fired by the incoming spikes from otherneurons. One of the most fundamental question ofneuroscience is: what exactly makes neurons fire?

What is it in the incoming pulses that elicits a response inone neuron but not in another one? Why could twoneurons have different responses to exactly the same inputand identical responses to completely different inputs?



The Brain.



What is a spike?.

Such spikes

are the main means of communicationbetween neurons. In general, neurons do not fire on theirown, they get fired by the incoming spikes from otherneurons. One of the most fundamental question ofneuroscience is: what exactly makes neurons fire?




The Brain.



What is a spike?.

Such spikes are the main means of communicationbetween neurons.

In general, neurons do not fire on theirown, they get fired by the incoming spikes from otherneurons. One of the most fundamental question ofneuroscience is: what exactly makes neurons fire?




The Brain.



What is a spike?.

Such spikes are the main means of communicationbetween neurons. In general, neurons do not fire on theirown,

they get fired by the incoming spikes from otherneurons. One of the most fundamental question ofneuroscience is: what exactly makes neurons fire?




The Brain.



What is a spike?.

Such spikes are the main means of communicationbetween neurons. In general, neurons do not fire on theirown, they get fired by the incoming spikes from otherneurons.

One of the most fundamental question ofneuroscience is: what exactly makes neurons fire?




The Brain.



What is a spike?.

Such spikes are the main means of communicationbetween neurons. In general, neurons do not fire on theirown, they get fired by the incoming spikes from otherneurons. One of the most fundamental question ofneuroscience is:

what exactly makes neurons fire?




The Brain.



What is a spike?.





The Brain.



What is a spike?.


What is it

in the incoming pulses that elicits a response inone neuron but not in another one? Why could twoneurons have different responses to exactly the same inputand identical responses to completely different inputs?



The Brain.



What is a spike?.


What is it in the incoming pulses that

elicits a response inone neuron but not in another one? Why could twoneurons have different responses to exactly the same inputand identical responses to completely different inputs?



The Brain.



What is a spike?.


What is it in the incoming pulses that elicits a response inone neuron

but not in another one? Why could twoneurons have different responses to exactly the same inputand identical responses to completely different inputs?



The Brain.



What is a spike?.


What is it in the incoming pulses that elicits a response inone neuron but not in another one?

Why could twoneurons have different responses to exactly the same inputand identical responses to completely different inputs?



The Brain.



What is a spike?.


What is it in the incoming pulses that elicits a response inone neuron but not in another one? Why could twoneurons have different responses

to exactly the same inputand identical responses to completely different inputs?



The Brain.



What is a spike?.


What is it in the incoming pulses that elicits a response inone neuron but not in another one? Why could twoneurons have different responses to exactly the same inputand

identical responses to completely different inputs?



The Brain.



What is a spike?.


What is it in the incoming pulses that elicits a response inone neuron but not in another one? Why could twoneurons have different responses to exactly the same inputand identical responses

to completely different inputs?



The Brain.



What is a spike?.





The Brain.



What is a spike?.





The Brain.



What is a spike?.

To answer these questions, we need to understand thedynamics of spike-generation mechanisms of neurons.

Most introductory neuroscience books describe neuronsas integrators with a threshold: Neurons sum up incomingPSPs and ”compare” the integrated PSP with a certainvoltage value, called firing threshold.



The Brain.



What is a spike?.

To answer these questions,

we need to understand thedynamics of spike-generation mechanisms of neurons.




The Brain.



What is a spike?.

To answer these questions, we need to understand thedynamics

of spike-generation mechanisms of neurons.




The Brain.



What is a spike?.





The Brain.



What is a spike?.


Most introductory neuroscience books

describe neuronsas integrators with a threshold: Neurons sum up incomingPSPs and ”compare” the integrated PSP with a certainvoltage value, called firing threshold.



The Brain.



What is a spike?.


Most introductory neuroscience books describe neuronsas integrators

with a threshold: Neurons sum up incomingPSPs and ”compare” the integrated PSP with a certainvoltage value, called firing threshold.



The Brain.



What is a spike?.


Most introductory neuroscience books describe neuronsas integrators with a threshold:

Neurons sum up incomingPSPs and ”compare” the integrated PSP with a certainvoltage value, called firing threshold.



The Brain.



What is a spike?.


Most introductory neuroscience books describe neuronsas integrators with a threshold: Neurons sum up incomingPSPs and

”compare” the integrated PSP with a certainvoltage value, called firing threshold.



The Brain.



What is a spike?.


Most introductory neuroscience books describe neuronsas integrators with a threshold: Neurons sum up incomingPSPs and ”compare” the integrated PSP with a certainvoltage value,

called firing threshold.



The Brain.



What is a spike?.





The Brain.



What is a spike?.





The Brain.



What is a spike?.

If it is below the threshold, the neuron remains quiescent;when it is above the threshold, the neuron fires anall-or-none spike and resets its membrane potential.

To add theoretical plausibility to this argument, we have torefer to the Hodgkin-Huxley model of spike-generation insquid giant axons.



The Brain.



What is a spike?.

If it is below the threshold,

the neuron remains quiescent;when it is above the threshold, the neuron fires anall-or-none spike and resets its membrane potential.




The Brain.



What is a spike?.

If it is below the threshold, the neuron remains quiescent;when it is above the threshold, the neuron fires anall-or-none spike and

resets its membrane potential.




The Brain.



What is a spike?.





The Brain.



What is a spike?.


To add theoretical plausibility to this argument,

we have torefer to the Hodgkin-Huxley model of spike-generation insquid giant axons.



The Brain.



What is a spike?.


To add theoretical plausibility to this argument, we have torefer to the

Hodgkin-Huxley model of spike-generation insquid giant axons.



The Brain.



What is a spike?.


To add theoretical plausibility to this argument, we have torefer to the Hodgkin-Huxley model of spike-generation

insquid giant axons.



The Brain.



What is a spike?.





The Brain.



What is a spike?.





The Brain.



The neuron as a fluid-filled ball.

Each 3µm of cytoplasmcontains on the order of 1010 water molecules, 108 ions(e.g. sodium, potassium, calcium, chloride), 107 smallmolecules (e.g. amino acids, nucleicacids), and 105

proteins. Relative to the extracellular space, the inside ofthe cell is negatively charged (the difference is carried byabout 1 out of every 100,000 ions). This results in avoltage (V ) across themembrane of approximately−70mV .



The Brain.



The neuron as a fluid-filled ball. Each 3µm of cytoplasmcontains

on the order of 1010 water molecules, 108 ions(e.g. sodium, potassium, calcium, chloride), 107 smallmolecules (e.g. amino acids, nucleicacids), and 105




The Brain.



The neuron as a fluid-filled ball. Each 3µm of cytoplasmcontains on the order of 1010 water molecules,

108 ions(e.g. sodium, potassium, calcium, chloride), 107 smallmolecules (e.g. amino acids, nucleicacids), and 105




The Brain.



The neuron as a fluid-filled ball. Each 3µm of cytoplasmcontains on the order of 1010 water molecules, 108 ions(e.g. sodium, potassium, calcium, chloride),

107 smallmolecules (e.g. amino acids, nucleicacids), and 105




The Brain.



The neuron as a fluid-filled ball. Each 3µm of cytoplasmcontains on the order of 1010 water molecules, 108 ions(e.g. sodium, potassium, calcium, chloride), 107 smallmolecules (e.g. amino acids, nucleicacids), and

105




The Brain.



The neuron as a fluid-filled ball. Each 3µm of cytoplasmcontains on the order of 1010 water molecules, 108 ions(e.g. sodium, potassium, calcium, chloride), 107 smallmolecules (e.g. amino acids, nucleicacids), and 105

proteins.

Relative to the extracellular space, the inside ofthe cell is negatively charged (the difference is carried byabout 1 out of every 100,000 ions). This results in avoltage (V ) across themembrane of approximately−70mV .



The Brain.




proteins. Relative to the extracellular space,

the inside ofthe cell is negatively charged (the difference is carried byabout 1 out of every 100,000 ions). This results in avoltage (V ) across themembrane of approximately−70mV .



The Brain.




proteins. Relative to the extracellular space, the inside ofthe cell

is negatively charged (the difference is carried byabout 1 out of every 100,000 ions). This results in avoltage (V ) across themembrane of approximately−70mV .



The Brain.




proteins. Relative to the extracellular space, the inside ofthe cell is negatively charged

(the difference is carried byabout 1 out of every 100,000 ions). This results in avoltage (V ) across themembrane of approximately−70mV .



The Brain.




proteins. Relative to the extracellular space, the inside ofthe cell is negatively charged (the difference is carried byabout 1 out of every 100,000 ions). This results in avoltage (V )

across themembrane of approximately−70mV .



The Brain.







The Brain.







The Brain.







The Brain.







The Brain.



The neuron as a capacitor.

Excess negative charges inthe cell oppose each other and line up around inside ofmembrane. This attracts an equal number of extracellularpositive ions, which line up outside the cell. In this way, themembrane builds up charge - it’s acting as a capacitor!The amount of charge (Q) stored by the membrane isgiven by the following equation: CmV = Q



The Brain.



The neuron as a capacitor. Excess negative charges inthe cell

oppose each other and line up around inside ofmembrane. This attracts an equal number of extracellularpositive ions, which line up outside the cell. In this way, themembrane builds up charge - it’s acting as a capacitor!The amount of charge (Q) stored by the membrane isgiven by the following equation: CmV = Q



The Brain.



The neuron as a capacitor. Excess negative charges inthe cell oppose each other and

line up around inside ofmembrane. This attracts an equal number of extracellularpositive ions, which line up outside the cell. In this way, themembrane builds up charge - it’s acting as a capacitor!The amount of charge (Q) stored by the membrane isgiven by the following equation: CmV = Q



The Brain.



The neuron as a capacitor. Excess negative charges inthe cell oppose each other and line up around inside ofmembrane.

This attracts an equal number of extracellularpositive ions, which line up outside the cell. In this way, themembrane builds up charge - it’s acting as a capacitor!The amount of charge (Q) stored by the membrane isgiven by the following equation: CmV = Q



The Brain.



The neuron as a capacitor. Excess negative charges inthe cell oppose each other and line up around inside ofmembrane. This attracts an equal number of extracellularpositive ions,

which line up outside the cell. In this way, themembrane builds up charge - it’s acting as a capacitor!The amount of charge (Q) stored by the membrane isgiven by the following equation: CmV = Q



The Brain.



The neuron as a capacitor. Excess negative charges inthe cell oppose each other and line up around inside ofmembrane. This attracts an equal number of extracellularpositive ions, which line up outside the cell.

In this way, themembrane builds up charge - it’s acting as a capacitor!The amount of charge (Q) stored by the membrane isgiven by the following equation: CmV = Q



The Brain.



The neuron as a capacitor. Excess negative charges inthe cell oppose each other and line up around inside ofmembrane. This attracts an equal number of extracellularpositive ions, which line up outside the cell. In this way,

themembrane builds up charge - it’s acting as a capacitor!The amount of charge (Q) stored by the membrane isgiven by the following equation: CmV = Q



The Brain.



The neuron as a capacitor. Excess negative charges inthe cell oppose each other and line up around inside ofmembrane. This attracts an equal number of extracellularpositive ions, which line up outside the cell. In this way, themembrane builds up charge -

it’s acting as a capacitor!The amount of charge (Q) stored by the membrane isgiven by the following equation: CmV = Q



The Brain.



The neuron as a capacitor. Excess negative charges inthe cell oppose each other and line up around inside ofmembrane. This attracts an equal number of extracellularpositive ions, which line up outside the cell. In this way, themembrane builds up charge - it’s acting as a capacitor!

The amount of charge (Q) stored by the membrane isgiven by the following equation: CmV = Q



The Brain.



The neuron as a capacitor. Excess negative charges inthe cell oppose each other and line up around inside ofmembrane. This attracts an equal number of extracellularpositive ions, which line up outside the cell. In this way, themembrane builds up charge - it’s acting as a capacitor!The amount of charge (Q) stored by the membrane

isgiven by the following equation: CmV = Q



The Brain.



The neuron as a capacitor. Excess negative charges inthe cell oppose each other and line up around inside ofmembrane. This attracts an equal number of extracellularpositive ions, which line up outside the cell. In this way, themembrane builds up charge - it’s acting as a capacitor!The amount of charge (Q) stored by the membrane isgiven by the following equation:

CmV = Q



The Brain.



The neuron as a capacitor. Excess negative charges inthe cell oppose each other and line up around inside ofmembrane. This attracts an equal number of extracellularpositive ions, which line up outside the cell. In this way, themembrane builds up charge - it’s acting as a capacitor!The amount of charge (Q) stored by the membrane isgiven by the following equation: CmV = Q



The Brain.






The Brain.






The Brain.



That is,

the amount of charge stored by the membrane isequal to the ability of the membrane to store charge (i.e.,its capacitance) multiplied by the voltage difference acrossthe membrane. The total membrane capacitance (Cm ) isproportional to the surface area of the cell (A):

Cm = cmA



The Brain.



That is, the amount of charge stored by the membrane

isequal to the ability of the membrane to store charge (i.e.,its capacitance) multiplied by the voltage difference acrossthe membrane. The total membrane capacitance (Cm ) isproportional to the surface area of the cell (A):

Cm = cmA



The Brain.



That is, the amount of charge stored by the membrane isequal to the ability of the membrane

to store charge (i.e.,its capacitance) multiplied by the voltage difference acrossthe membrane. The total membrane capacitance (Cm ) isproportional to the surface area of the cell (A):

Cm = cmA



The Brain.



That is, the amount of charge stored by the membrane isequal to the ability of the membrane to store charge

(i.e.,its capacitance) multiplied by the voltage difference acrossthe membrane. The total membrane capacitance (Cm ) isproportional to the surface area of the cell (A):

Cm = cmA



The Brain.



That is, the amount of charge stored by the membrane isequal to the ability of the membrane to store charge (i.e.,its capacitance)

multiplied by the voltage difference acrossthe membrane. The total membrane capacitance (Cm ) isproportional to the surface area of the cell (A):

Cm = cmA



The Brain.



That is, the amount of charge stored by the membrane isequal to the ability of the membrane to store charge (i.e.,its capacitance) multiplied by the voltage difference

acrossthe membrane. The total membrane capacitance (Cm ) isproportional to the surface area of the cell (A):

Cm = cmA



The Brain.



That is, the amount of charge stored by the membrane isequal to the ability of the membrane to store charge (i.e.,its capacitance) multiplied by the voltage difference acrossthe membrane.

The total membrane capacitance (Cm ) isproportional to the surface area of the cell (A):

Cm = cmA



The Brain.



That is, the amount of charge stored by the membrane isequal to the ability of the membrane to store charge (i.e.,its capacitance) multiplied by the voltage difference acrossthe membrane. The total membrane capacitance (Cm )

isproportional to the surface area of the cell (A):

Cm = cmA



The Brain.



That is, the amount of charge stored by the membrane isequal to the ability of the membrane to store charge (i.e.,its capacitance) multiplied by the voltage difference acrossthe membrane. The total membrane capacitance (Cm ) isproportional to the surface area of the cell (A):

Cm = cmA



The Brain.



That is, the amount of charge stored by the membrane isequal to the ability of the membrane to store charge (i.e.,its capacitance) multiplied by the voltage difference acrossthe membrane. The total membrane capacitance (Cm ) isproportional to the surface area of the cell (A):

Cm = cmA



The Brain.



Specific capacitance (cm)

depends on conductance andthickness of membrane, which is about the same for allneurons - about 10nF /mm2. Neurons typically have asurface area of 0.01− 0.1mm2 , so Cm ranges from around0.1− 1nF . We can now compute the number of chargesstored by a given neuron (we’ll assume 1nF totalcapacitance and −70mV membrane potential):

1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.

Note: A Columb is 1 Farrad × volt.



The Brain.



Specific capacitance (cm) depends on conductance andthickness of membrane,

which is about the same for allneurons - about 10nF /mm2. Neurons typically have asurface area of 0.01− 0.1mm2 , so Cm ranges from around0.1− 1nF . We can now compute the number of chargesstored by a given neuron (we’ll assume 1nF totalcapacitance and −70mV membrane potential):

1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.




The Brain.



Specific capacitance (cm) depends on conductance andthickness of membrane, which is about the same for allneurons

- about 10nF /mm2. Neurons typically have asurface area of 0.01− 0.1mm2 , so Cm ranges from around0.1− 1nF . We can now compute the number of chargesstored by a given neuron (we’ll assume 1nF totalcapacitance and −70mV membrane potential):

1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.




The Brain.



Specific capacitance (cm) depends on conductance andthickness of membrane, which is about the same for allneurons - about 10nF /mm2.

Neurons typically have asurface area of 0.01− 0.1mm2 , so Cm ranges from around0.1− 1nF . We can now compute the number of chargesstored by a given neuron (we’ll assume 1nF totalcapacitance and −70mV membrane potential):

1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.




The Brain.



Specific capacitance (cm) depends on conductance andthickness of membrane, which is about the same for allneurons - about 10nF /mm2. Neurons typically have asurface area

of 0.01− 0.1mm2 , so Cm ranges from around0.1− 1nF . We can now compute the number of chargesstored by a given neuron (we’ll assume 1nF totalcapacitance and −70mV membrane potential):

1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.




The Brain.



Specific capacitance (cm) depends on conductance andthickness of membrane, which is about the same for allneurons - about 10nF /mm2. Neurons typically have asurface area of 0.01− 0.1mm2 ,

so Cm ranges from around0.1− 1nF . We can now compute the number of chargesstored by a given neuron (we’ll assume 1nF totalcapacitance and −70mV membrane potential):

1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.




The Brain.



Specific capacitance (cm) depends on conductance andthickness of membrane, which is about the same for allneurons - about 10nF /mm2. Neurons typically have asurface area of 0.01− 0.1mm2 , so Cm ranges from around0.1− 1nF .

We can now compute the number of chargesstored by a given neuron (we’ll assume 1nF totalcapacitance and −70mV membrane potential):

1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.




The Brain.



Specific capacitance (cm) depends on conductance andthickness of membrane, which is about the same for allneurons - about 10nF /mm2. Neurons typically have asurface area of 0.01− 0.1mm2 , so Cm ranges from around0.1− 1nF . We can now compute

the number of chargesstored by a given neuron (we’ll assume 1nF totalcapacitance and −70mV membrane potential):

1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.




The Brain.



Specific capacitance (cm) depends on conductance andthickness of membrane, which is about the same for allneurons - about 10nF /mm2. Neurons typically have asurface area of 0.01− 0.1mm2 , so Cm ranges from around0.1− 1nF . We can now compute the number of chargesstored by a given neuron

(we’ll assume 1nF totalcapacitance and −70mV membrane potential):

1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.




The Brain.



Specific capacitance (cm) depends on conductance andthickness of membrane, which is about the same for allneurons - about 10nF /mm2. Neurons typically have asurface area of 0.01− 0.1mm2 , so Cm ranges from around0.1− 1nF . We can now compute the number of chargesstored by a given neuron (we’ll assume 1nF totalcapacitance and

−70mV membrane potential):

1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.




The Brain.



Specific capacitance (cm) depends on conductance andthickness of membrane, which is about the same for allneurons - about 10nF /mm2. Neurons typically have asurface area of 0.01− 0.1mm2 , so Cm ranges from around0.1− 1nF . We can now compute the number of chargesstored by a given neuron (we’ll assume 1nF totalcapacitance and −70mV membrane potential):

1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.




The Brain.




1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.




The Brain.




1nF ×−70mV = 10−9F ×70×10−3V = 70×10−12C = 109charges.




The Brain.



Changes in current.

Membrane current is a measure ofthe number of charges per second that travel across themembrane. Current is measured in amps - 1amp is 1Columb per second:

I =dQdt

In order to compute the membrane current, we can takethe time derivative of the equation for determining howmuch charge the membrane stores:

CmdVdt

=dQdt

= I

Example: suppose Cm = 1nF . Then injecting I = 1nA ofcurrent causes the membrane voltage to rise by 1 volt persecond (i.e., 1mV per millisecond ).



The Brain.



Changes in current. Membrane current

is a measure ofthe number of charges per second that travel across themembrane. Current is measured in amps - 1amp is 1Columb per second:

I =dQdt


CmdVdt

=dQdt

= I




The Brain.



Changes in current. Membrane current is a measure ofthe number of charges per second

that travel across themembrane. Current is measured in amps - 1amp is 1Columb per second:

I =dQdt


CmdVdt

=dQdt

= I




The Brain.



Changes in current. Membrane current is a measure ofthe number of charges per second that travel across themembrane.

Current is measured in amps - 1amp is 1Columb per second:

I =dQdt


CmdVdt

=dQdt

= I




The Brain.



Changes in current. Membrane current is a measure ofthe number of charges per second that travel across themembrane. Current is measured in amps

- 1amp is 1Columb per second:

I =dQdt


CmdVdt

=dQdt

= I




The Brain.



Changes in current. Membrane current is a measure ofthe number of charges per second that travel across themembrane. Current is measured in amps - 1amp is 1Columb per second:

I =dQdt


CmdVdt

=dQdt

= I




The Brain.




I =dQdt


CmdVdt

=dQdt

= I




The Brain.




I =dQdt

In order to compute the membrane current,

we can takethe time derivative of the equation for determining howmuch charge the membrane stores:

CmdVdt

=dQdt

= I




The Brain.




I =dQdt

In order to compute the membrane current, we can takethe time derivative of the equation

for determining howmuch charge the membrane stores:

CmdVdt

=dQdt

= I




The Brain.




I =dQdt

In order to compute the membrane current, we can takethe time derivative of the equation for determining howmuch charge

the membrane stores:

CmdVdt

=dQdt

= I




The Brain.




I =dQdt


CmdVdt

=dQdt

= I




The Brain.




I =dQdt


CmdVdt

=dQdt

= I




The Brain.




I =dQdt


CmdVdt

=dQdt

= I

Example:

suppose Cm = 1nF . Then injecting I = 1nA ofcurrent causes the membrane voltage to rise by 1 volt persecond (i.e., 1mV per millisecond ).



The Brain.




I =dQdt


CmdVdt

=dQdt

= I

Example: suppose Cm = 1nF .

Then injecting I = 1nA ofcurrent causes the membrane voltage to rise by 1 volt persecond (i.e., 1mV per millisecond ).



The Brain.




I =dQdt


CmdVdt

=dQdt

= I

Example: suppose Cm = 1nF . Then injecting I = 1nA ofcurrent

causes the membrane voltage to rise by 1 volt persecond (i.e., 1mV per millisecond ).



The Brain.




I =dQdt


CmdVdt

=dQdt

= I

Example: suppose Cm = 1nF . Then injecting I = 1nA ofcurrent causes the membrane voltage to rise

by 1 volt persecond (i.e., 1mV per millisecond ).



The Brain.




I =dQdt


CmdVdt

=dQdt

= I

Example: suppose Cm = 1nF . Then injecting I = 1nA ofcurrent causes the membrane voltage to rise by 1 volt persecond

(i.e., 1mV per millisecond ).



The Brain.




I =dQdt


CmdVdt

=dQdt

= I




The Brain.




I =dQdt


CmdVdt

=dQdt

= I




The Brain.



Membrane current.

There are two components of current(I). The first is membrane current. The membrane containsion channels - these let specific neurons through. Theycan open and close. One type of channel is the sodiumchannel. The inside of the cell contains fewer sodium ionsthan outside the cell.



The Brain.



Membrane current. There are two components of current(I).

The first is membrane current. The membrane containsion channels - these let specific neurons through. Theycan open and close. One type of channel is the sodiumchannel. The inside of the cell contains fewer sodium ionsthan outside the cell.



The Brain.



Membrane current. There are two components of current(I). The first is membrane current.

The membrane containsion channels - these let specific neurons through. Theycan open and close. One type of channel is the sodiumchannel. The inside of the cell contains fewer sodium ionsthan outside the cell.



The Brain.



Membrane current. There are two components of current(I). The first is membrane current. The membrane containsion channels

- these let specific neurons through. Theycan open and close. One type of channel is the sodiumchannel. The inside of the cell contains fewer sodium ionsthan outside the cell.



The Brain.



Membrane current. There are two components of current(I). The first is membrane current. The membrane containsion channels - these let specific neurons through.

Theycan open and close. One type of channel is the sodiumchannel. The inside of the cell contains fewer sodium ionsthan outside the cell.



The Brain.



Membrane current. There are two components of current(I). The first is membrane current. The membrane containsion channels - these let specific neurons through. Theycan open and close.

One type of channel is the sodiumchannel. The inside of the cell contains fewer sodium ionsthan outside the cell.



The Brain.



Membrane current. There are two components of current(I). The first is membrane current. The membrane containsion channels - these let specific neurons through. Theycan open and close. One type of channel is the sodiumchannel.

The inside of the cell contains fewer sodium ionsthan outside the cell.



The Brain.



Membrane current. There are two components of current(I). The first is membrane current. The membrane containsion channels - these let specific neurons through. Theycan open and close. One type of channel is the sodiumchannel. The inside of the cell contains

fewer sodium ionsthan outside the cell.



The Brain.



Membrane current. There are two components of current(I). The first is membrane current. The membrane containsion channels - these let specific neurons through. Theycan open and close. One type of channel is the sodiumchannel. The inside of the cell contains fewer sodium ionsthan outside the cell.



The Brain.






The Brain.






The Brain.



When the sodium channels open,

sodium (positivelycharged) flows into the cell and causes the membranevoltage to increase. Diffusion of sodium and other ions(e.g. potassium) is called the membrane current, Im .



The Brain.



When the sodium channels open, sodium (positivelycharged) flows

into the cell and causes the membranevoltage to increase. Diffusion of sodium and other ions(e.g. potassium) is called the membrane current, Im .



The Brain.



When the sodium channels open, sodium (positivelycharged) flows into the cell and causes

the membranevoltage to increase. Diffusion of sodium and other ions(e.g. potassium) is called the membrane current, Im .



The Brain.



When the sodium channels open, sodium (positivelycharged) flows into the cell and causes the membranevoltage to increase.

Diffusion of sodium and other ions(e.g. potassium) is called the membrane current, Im .



The Brain.



When the sodium channels open, sodium (positivelycharged) flows into the cell and causes the membranevoltage to increase. Diffusion of sodium and

other ions(e.g. potassium) is called the membrane current, Im .



The Brain.



When the sodium channels open, sodium (positivelycharged) flows into the cell and causes the membranevoltage to increase. Diffusion of sodium and other ions

(e.g. potassium) is called the membrane current, Im .



The Brain.



When the sodium channels open, sodium (positivelycharged) flows into the cell and causes the membranevoltage to increase. Diffusion of sodium and other ions(e.g. potassium)

is called the membrane current, Im .



The Brain.



When the sodium channels open, sodium (positivelycharged) flows into the cell and causes the membranevoltage to increase. Diffusion of sodium and other ions(e.g. potassium) is called

the membrane current, Im .



The Brain.



When the sodium channels open, sodium (positivelycharged) flows into the cell and causes the membranevoltage to increase. Diffusion of sodium and other ions(e.g. potassium) is called the membrane current, Im .



The Brain.






The Brain.






The Brain.






The Brain.



Driving force and the equilibrium potential.

In additionto sodium being driven to flow down its concentrationgradient, one can make it more or less difficult for sodiumto enter the cell by changing the membrane voltage.Because sodium is positively charged, decreasing, orhyperpolarizing, the membrane voltage (inside relative tooutside) will make sodium ions more likely to flow into thecell.



The Brain.



Driving force and the equilibrium potential. In additionto sodium being driven to flow down

its concentrationgradient, one can make it more or less difficult for sodiumto enter the cell by changing the membrane voltage.Because sodium is positively charged, decreasing, orhyperpolarizing, the membrane voltage (inside relative tooutside) will make sodium ions more likely to flow into thecell.



The Brain.



Driving force and the equilibrium potential. In additionto sodium being driven to flow down its concentrationgradient,

one can make it more or less difficult for sodiumto enter the cell by changing the membrane voltage.Because sodium is positively charged, decreasing, orhyperpolarizing, the membrane voltage (inside relative tooutside) will make sodium ions more likely to flow into thecell.



The Brain.



Driving force and the equilibrium potential. In additionto sodium being driven to flow down its concentrationgradient, one can make it more or less difficult

for sodiumto enter the cell by changing the membrane voltage.Because sodium is positively charged, decreasing, orhyperpolarizing, the membrane voltage (inside relative tooutside) will make sodium ions more likely to flow into thecell.



The Brain.



Driving force and the equilibrium potential. In additionto sodium being driven to flow down its concentrationgradient, one can make it more or less difficult for sodiumto enter the cell

by changing the membrane voltage.Because sodium is positively charged, decreasing, orhyperpolarizing, the membrane voltage (inside relative tooutside) will make sodium ions more likely to flow into thecell.



The Brain.



Driving force and the equilibrium potential. In additionto sodium being driven to flow down its concentrationgradient, one can make it more or less difficult for sodiumto enter the cell by changing the membrane voltage.

Because sodium is positively charged, decreasing, orhyperpolarizing, the membrane voltage (inside relative tooutside) will make sodium ions more likely to flow into thecell.



The Brain.



Driving force and the equilibrium potential. In additionto sodium being driven to flow down its concentrationgradient, one can make it more or less difficult for sodiumto enter the cell by changing the membrane voltage.Because sodium is positively charged,

decreasing, orhyperpolarizing, the membrane voltage (inside relative tooutside) will make sodium ions more likely to flow into thecell.



The Brain.



Driving force and the equilibrium potential. In additionto sodium being driven to flow down its concentrationgradient, one can make it more or less difficult for sodiumto enter the cell by changing the membrane voltage.Because sodium is positively charged, decreasing, orhyperpolarizing,

the membrane voltage (inside relative tooutside) will make sodium ions more likely to flow into thecell.



The Brain.



Driving force and the equilibrium potential. In additionto sodium being driven to flow down its concentrationgradient, one can make it more or less difficult for sodiumto enter the cell by changing the membrane voltage.Because sodium is positively charged, decreasing, orhyperpolarizing, the membrane voltage

(inside relative tooutside) will make sodium ions more likely to flow into thecell.



The Brain.



Driving force and the equilibrium potential. In additionto sodium being driven to flow down its concentrationgradient, one can make it more or less difficult for sodiumto enter the cell by changing the membrane voltage.Because sodium is positively charged, decreasing, orhyperpolarizing, the membrane voltage (inside relative tooutside)

will make sodium ions more likely to flow into thecell.



The Brain.



Driving force and the equilibrium potential. In additionto sodium being driven to flow down its concentrationgradient, one can make it more or less difficult for sodiumto enter the cell by changing the membrane voltage.Because sodium is positively charged, decreasing, orhyperpolarizing, the membrane voltage (inside relative tooutside) will make sodium ions more likely

to flow into thecell.



The Brain.



Driving force and the equilibrium potential. In additionto sodium being driven to flow down its concentrationgradient, one can make it more or less difficult for sodiumto enter the cell by changing the membrane voltage.Because sodium is positively charged, decreasing, orhyperpolarizing, the membrane voltage (inside relative tooutside) will make sodium ions more likely to flow into thecell.



The Brain.



Driving force and the equilibrium potential. In additionto sodium being driven to flow down its concentrationgradient, one can make it more or less difficult for sodiumto enter the cell by changing the membrane voltage.Because sodium is positively charged, decreasing, orhyperpolarizing, the membrane voltage (inside relative tooutside) will make sodium ions more likely to flow into thecell.



The Brain.



Conversly,

increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing,

the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage

will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely

to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell.

The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow

of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops

is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E .

When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,

positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell.

When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell.

This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that

V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE .

Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E)

as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane.

When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative,

positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.

When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive,

positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



Conversly, increasing, or depolarizing, the membranevoltage will make sodium ions less likely to flow into thecell. The membrane potential at which net flow of an ionstops is called the equilibrium potential, E . When V > E ,positive ions flow out of the cell. When V < E , positiveions flow into the cell. This means that V is driven towardsE . Thus, we sometimes refer to the quantity (V − E) as thedriving force across the cell membrane. When the drivingforce is negative, positive ions are driven out of the cell.When the driving force is positive, positive ions are pulledinto the cell.



The Brain.



External current.

The second component of current iscurrent that is injected into the neuron from externalsources (e.g. if we stick an electrode into the neuron andpump in current). The change in membrane voltage V dueto some change in current I follows Ohm’s Law: V = IR,where R is the membrane resistance, described next.

Membrane resistance. Ion channels are like little holes inthe membrane. They let ions pass through them - i.e., theyconduct ions. A given unit area of membrane has somenumber of open channels, and we can measure the easewith which ions pass through those channels - the specificconductance, gm. The total conducatance is proportionalto the neuron’s area: Gm = gmA



The Brain.



External current. The second component of current

iscurrent that is injected into the neuron from externalsources (e.g. if we stick an electrode into the neuron andpump in current). The change in membrane voltage V dueto some change in current I follows Ohm’s Law: V = IR,where R is the membrane resistance, described next.




The Brain.



External current. The second component of current iscurrent that is injected

into the neuron from externalsources (e.g. if we stick an electrode into the neuron andpump in current). The change in membrane voltage V dueto some change in current I follows Ohm’s Law: V = IR,where R is the membrane resistance, described next.




The Brain.



External current. The second component of current iscurrent that is injected into the neuron

from externalsources (e.g. if we stick an electrode into the neuron andpump in current). The change in membrane voltage V dueto some change in current I follows Ohm’s Law: V = IR,where R is the membrane resistance, described next.




The Brain.



External current. The second component of current iscurrent that is injected into the neuron from externalsources

(e.g. if we stick an electrode into the neuron andpump in current). The change in membrane voltage V dueto some change in current I follows Ohm’s Law: V = IR,where R is the membrane resistance, described next.




The Brain.



External current. The second component of current iscurrent that is injected into the neuron from externalsources (e.g. if we stick an electrode into the neuron andpump in current).

The change in membrane voltage V dueto some change in current I follows Ohm’s Law: V = IR,where R is the membrane resistance, described next.




The Brain.



External current. The second component of current iscurrent that is injected into the neuron from externalsources (e.g. if we stick an electrode into the neuron andpump in current). The change in membrane voltage

V dueto some change in current I follows Ohm’s Law: V = IR,where R is the membrane resistance, described next.




The Brain.



External current. The second component of current iscurrent that is injected into the neuron from externalsources (e.g. if we stick an electrode into the neuron andpump in current). The change in membrane voltage V dueto some change in current I

follows Ohm’s Law: V = IR,where R is the membrane resistance, described next.




The Brain.



External current. The second component of current iscurrent that is injected into the neuron from externalsources (e.g. if we stick an electrode into the neuron andpump in current). The change in membrane voltage V dueto some change in current I follows Ohm’s Law:

V = IR,where R is the membrane resistance, described next.




The Brain.



External current. The second component of current iscurrent that is injected into the neuron from externalsources (e.g. if we stick an electrode into the neuron andpump in current). The change in membrane voltage V dueto some change in current I follows Ohm’s Law: V = IR,

where R is the membrane resistance, described next.




The Brain.



External current. The second component of current iscurrent that is injected into the neuron from externalsources (e.g. if we stick an electrode into the neuron andpump in current). The change in membrane voltage V dueto some change in current I follows Ohm’s Law: V = IR,where R is the membrane resistance, described next.




The Brain.




Membrane resistance.

Ion channels are like little holes inthe membrane. They let ions pass through them - i.e., theyconduct ions. A given unit area of membrane has somenumber of open channels, and we can measure the easewith which ions pass through those channels - the specificconductance, gm. The total conducatance is proportionalto the neuron’s area: Gm = gmA



The Brain.




Membrane resistance. Ion channels are like little holes inthe membrane.

They let ions pass through them - i.e., theyconduct ions. A given unit area of membrane has somenumber of open channels, and we can measure the easewith which ions pass through those channels - the specificconductance, gm. The total conducatance is proportionalto the neuron’s area: Gm = gmA



The Brain.




Membrane resistance. Ion channels are like little holes inthe membrane. They let ions pass through them

- i.e., theyconduct ions. A given unit area of membrane has somenumber of open channels, and we can measure the easewith which ions pass through those channels - the specificconductance, gm. The total conducatance is proportionalto the neuron’s area: Gm = gmA



The Brain.




Membrane resistance. Ion channels are like little holes inthe membrane. They let ions pass through them - i.e., theyconduct ions.

A given unit area of membrane has somenumber of open channels, and we can measure the easewith which ions pass through those channels - the specificconductance, gm. The total conducatance is proportionalto the neuron’s area: Gm = gmA



The Brain.




Membrane resistance. Ion channels are like little holes inthe membrane. They let ions pass through them - i.e., theyconduct ions. A given unit area of membrane

has somenumber of open channels, and we can measure the easewith which ions pass through those channels - the specificconductance, gm. The total conducatance is proportionalto the neuron’s area: Gm = gmA



The Brain.




Membrane resistance. Ion channels are like little holes inthe membrane. They let ions pass through them - i.e., theyconduct ions. A given unit area of membrane has somenumber of open channels, and

we can measure the easewith which ions pass through those channels - the specificconductance, gm. The total conducatance is proportionalto the neuron’s area: Gm = gmA



The Brain.




Membrane resistance. Ion channels are like little holes inthe membrane. They let ions pass through them - i.e., theyconduct ions. A given unit area of membrane has somenumber of open channels, and we can measure the easewith which ions pass through those channels

- the specificconductance, gm. The total conducatance is proportionalto the neuron’s area: Gm = gmA



The Brain.




Membrane resistance. Ion channels are like little holes inthe membrane. They let ions pass through them - i.e., theyconduct ions. A given unit area of membrane has somenumber of open channels, and we can measure the easewith which ions pass through those channels - the specificconductance, gm.

The total conducatance is proportionalto the neuron’s area: Gm = gmA



The Brain.




Membrane resistance. Ion channels are like little holes inthe membrane. They let ions pass through them - i.e., theyconduct ions. A given unit area of membrane has somenumber of open channels, and we can measure the easewith which ions pass through those channels - the specificconductance, gm. The total conducatance

is proportionalto the neuron’s area: Gm = gmA



The Brain.




Membrane resistance. Ion channels are like little holes inthe membrane. They let ions pass through them - i.e., theyconduct ions. A given unit area of membrane has somenumber of open channels, and we can measure the easewith which ions pass through those channels - the specificconductance, gm. The total conducatance is proportionalto the neuron’s area:

Gm = gmA



The Brain.




Membrane resistance. Ion channels are like little holes inthe membrane. They let ions pass through them - i.e., theyconduct ions. A given unit area of membrane has somenumber of open channels, and we can measure the easewith which ions pass through those channels - the specificconductance, gm. The total conducatance is proportionalto the neuron’s area:

Gm = gmA



The Brain.







The Brain.







The Brain.



By convention,

we tend to talk about the inverse ofconductance, which is called resistance. Whereasconductance is proportional to the surface area of theneuron, resistance is proportional to the inverse of thesurface area of the neuron:

Rm =rm

A

. Note that the membrane resistance often changes as afunction of voltage, which makes things interesting.



The Brain.



By convention, we tend to talk about the inverse ofconductance,

which is called resistance. Whereasconductance is proportional to the surface area of theneuron, resistance is proportional to the inverse of thesurface area of the neuron:

Rm =rm

A




The Brain.



By convention, we tend to talk about the inverse ofconductance, which is called resistance.

Whereasconductance is proportional to the surface area of theneuron, resistance is proportional to the inverse of thesurface area of the neuron:

Rm =rm

A




The Brain.



By convention, we tend to talk about the inverse ofconductance, which is called resistance. Whereas

conductance is proportional to the surface area of theneuron, resistance is proportional to the inverse of thesurface area of the neuron:

Rm =rm

A




The Brain.



By convention, we tend to talk about the inverse ofconductance, which is called resistance. Whereasconductance is proportional to the surface area of theneuron,

resistance is proportional to the inverse of thesurface area of the neuron:

Rm =rm

A




The Brain.



By convention, we tend to talk about the inverse ofconductance, which is called resistance. Whereasconductance is proportional to the surface area of theneuron, resistance is proportional to the inverse

of thesurface area of the neuron:

Rm =rm

A




The Brain.



By convention, we tend to talk about the inverse ofconductance, which is called resistance. Whereasconductance is proportional to the surface area of theneuron, resistance is proportional to the inverse of thesurface area of the neuron:

Rm =rm

A




The Brain.




Rm =rm

A

.

Note that the membrane resistance often changes as afunction of voltage, which makes things interesting.



The Brain.




Rm =rm

A

. Note that the membrane resistance

often changes as afunction of voltage, which makes things interesting.



The Brain.




Rm =rm

A

. Note that the membrane resistance often changes as afunction of voltage,

which makes things interesting.



The Brain.




Rm =rm

A




The Brain.




Rm =rm

A




The Brain.



The Neuron Equation. Previously we had:

CmdVdt

= I

which we can update to reflect that I is comprised of bothmembrane and external currents:

CmdVdt

= Ie + Im



The Brain.




CmdVdt

= I

which we can update to reflect that

I is comprised of bothmembrane and external currents:

CmdVdt

= Ie + Im



The Brain.




CmdVdt

= I

which we can update to reflect that I is comprised of bothmembrane and

external currents:

CmdVdt

= Ie + Im



The Brain.




CmdVdt

= I


CmdVdt

= Ie + Im



The Brain.




CmdVdt

= I


CmdVdt

= Ie + Im



The Brain.




CmdVdt

= I


CmdVdt

= Ie + Im



The Brain.



Im

depends on the driving force V − E and also difficultywith which ions flow through the membrane - i.e., themembrane resistance, Rm. In particular

Im =1

Rm(E − V )

The Neuron Equation. Note that the order of the E and Vterms in the driving force have been swapped. This isbecause the internal and external currents need to go inopposite directions. We can multiply both sides of theequation by Rm for convenience:

CmRmdVdt

= RmIe + (E − V )



The Brain.



Im depends on the driving force V − E and

also difficultywith which ions flow through the membrane - i.e., themembrane resistance, Rm. In particular

Im =1

Rm(E − V )


CmRmdVdt

= RmIe + (E − V )



The Brain.



Im depends on the driving force V − E and also difficultywith which ions flow

through the membrane - i.e., themembrane resistance, Rm. In particular

Im =1

Rm(E − V )


CmRmdVdt

= RmIe + (E − V )



The Brain.



Im depends on the driving force V − E and also difficultywith which ions flow through the membrane

- i.e., themembrane resistance, Rm. In particular

Im =1

Rm(E − V )


CmRmdVdt

= RmIe + (E − V )



The Brain.



Im depends on the driving force V − E and also difficultywith which ions flow through the membrane - i.e., themembrane resistance, Rm.

In particular

Im =1

Rm(E − V )


CmRmdVdt

= RmIe + (E − V )



The Brain.



Im depends on the driving force V − E and also difficultywith which ions flow through the membrane - i.e., themembrane resistance, Rm. In particular

Im =1

Rm(E − V )


CmRmdVdt

= RmIe + (E − V )



The Brain.




Im =1

Rm(E − V )


CmRmdVdt

= RmIe + (E − V )



The Brain.




Im =1

Rm(E − V )

The Neuron Equation.

Note that the order of the E and Vterms in the driving force have been swapped. This isbecause the internal and external currents need to go inopposite directions. We can multiply both sides of theequation by Rm for convenience:

CmRmdVdt

= RmIe + (E − V )



The Brain.




Im =1

Rm(E − V )

The Neuron Equation. Note that the order of the E and Vterms

in the driving force have been swapped. This isbecause the internal and external currents need to go inopposite directions. We can multiply both sides of theequation by Rm for convenience:

CmRmdVdt

= RmIe + (E − V )



The Brain.




Im =1

Rm(E − V )

The Neuron Equation. Note that the order of the E and Vterms in the driving force

have been swapped. This isbecause the internal and external currents need to go inopposite directions. We can multiply both sides of theequation by Rm for convenience:

CmRmdVdt

= RmIe + (E − V )



The Brain.




Im =1

Rm(E − V )

The Neuron Equation. Note that the order of the E and Vterms in the driving force have been swapped. This isbecause the internal and external currents

need to go inopposite directions. We can multiply both sides of theequation by Rm for convenience:

CmRmdVdt

= RmIe + (E − V )



The Brain.




Im =1

Rm(E − V )

The Neuron Equation. Note that the order of the E and Vterms in the driving force have been swapped. This isbecause the internal and external currents need to go inopposite directions.

We can multiply both sides of theequation by Rm for convenience:

CmRmdVdt

= RmIe + (E − V )



The Brain.




Im =1

Rm(E − V )


CmRmdVdt

= RmIe + (E − V )



The Brain.




Im =1

Rm(E − V )


CmRmdVdt

= RmIe + (E − V )



The Brain.




Im =1

Rm(E − V )


CmRmdVdt

= RmIe + (E − V )



The Brain.



Since Cm = cmA and

Rm = rmA , the A′s cancel, and we get

cmrm, which is independent of the cell’s surface area.Since cmrm determines the rate at which the cell’smembrane potential changes, it is given a special variable,τm, or the membrane time constant. The equation for allneuron models we’ll see in this class is:

τmdVdt

= RmIe + E − V



The Brain.



Since Cm = cmA and Rm = rmA ,

the A′s cancel, and we getcmrm, which is independent of the cell’s surface area.Since cmrm determines the rate at which the cell’smembrane potential changes, it is given a special variable,τm, or the membrane time constant. The equation for allneuron models we’ll see in this class is:

τmdVdt

= RmIe + E − V



The Brain.



Since Cm = cmA and Rm = rmA , the A′s cancel, and

we getcmrm, which is independent of the cell’s surface area.Since cmrm determines the rate at which the cell’smembrane potential changes, it is given a special variable,τm, or the membrane time constant. The equation for allneuron models we’ll see in this class is:

τmdVdt

= RmIe + E − V



The Brain.



Since Cm = cmA and Rm = rmA , the A′s cancel, and we get

cmrm,

which is independent of the cell’s surface area.Since cmrm determines the rate at which the cell’smembrane potential changes, it is given a special variable,τm, or the membrane time constant. The equation for allneuron models we’ll see in this class is:

τmdVdt

= RmIe + E − V



The Brain.




cmrm, which is independent of the cell’s surface area.

Since cmrm determines the rate at which the cell’smembrane potential changes, it is given a special variable,τm, or the membrane time constant. The equation for allneuron models we’ll see in this class is:

τmdVdt

= RmIe + E − V



The Brain.




cmrm, which is independent of the cell’s surface area.Since cmrm

determines the rate at which the cell’smembrane potential changes, it is given a special variable,τm, or the membrane time constant. The equation for allneuron models we’ll see in this class is:

τmdVdt

= RmIe + E − V



The Brain.




cmrm, which is independent of the cell’s surface area.Since cmrm determines the rate at which the cell’smembrane potential changes,

it is given a special variable,τm, or the membrane time constant. The equation for allneuron models we’ll see in this class is:

τmdVdt

= RmIe + E − V



The Brain.




cmrm, which is independent of the cell’s surface area.Since cmrm determines the rate at which the cell’smembrane potential changes, it is given a special variable,τm, or

the membrane time constant. The equation for allneuron models we’ll see in this class is:

τmdVdt

= RmIe + E − V



The Brain.




cmrm, which is independent of the cell’s surface area.Since cmrm determines the rate at which the cell’smembrane potential changes, it is given a special variable,τm, or the membrane time constant. The equation for allneuron models

we’ll see in this class is:

τmdVdt

= RmIe + E − V



The Brain.





τmdVdt

= RmIe + E − V



The Brain.





τmdVdt

= RmIe + E − V



The Brain.





τmdVdt

= RmIe + E − V



The Brain.



V∞. From the above equation,

we see that the change inmembrane voltage is some fraction (proportional to τm ) ofthe difference between RmIe + E and the currentmembrane voltage, V . By this equation the membranevoltage approaches RmIe + E over time. For conveniencewe can define

V∞ = RmIe + E

where V∞ is the membrane voltage that will be reachedgiven an external current and membrane resistance, andan infinite amount of time.



The Brain.



V∞. From the above equation, we see that the change inmembrane voltage

is some fraction (proportional to τm ) ofthe difference between RmIe + E and the currentmembrane voltage, V . By this equation the membranevoltage approaches RmIe + E over time. For conveniencewe can define

V∞ = RmIe + E




The Brain.



V∞. From the above equation, we see that the change inmembrane voltage is some fraction

(proportional to τm ) ofthe difference between RmIe + E and the currentmembrane voltage, V . By this equation the membranevoltage approaches RmIe + E over time. For conveniencewe can define

V∞ = RmIe + E




The Brain.



V∞. From the above equation, we see that the change inmembrane voltage is some fraction (proportional to τm )

ofthe difference between RmIe + E and the currentmembrane voltage, V . By this equation the membranevoltage approaches RmIe + E over time. For conveniencewe can define

V∞ = RmIe + E




The Brain.



V∞. From the above equation, we see that the change inmembrane voltage is some fraction (proportional to τm ) ofthe difference between RmIe + E and

the currentmembrane voltage, V . By this equation the membranevoltage approaches RmIe + E over time. For conveniencewe can define

V∞ = RmIe + E




The Brain.



V∞. From the above equation, we see that the change inmembrane voltage is some fraction (proportional to τm ) ofthe difference between RmIe + E and the currentmembrane voltage, V .

By this equation the membranevoltage approaches RmIe + E over time. For conveniencewe can define

V∞ = RmIe + E




The Brain.



V∞. From the above equation, we see that the change inmembrane voltage is some fraction (proportional to τm ) ofthe difference between RmIe + E and the currentmembrane voltage, V . By this equation the membranevoltage approaches

RmIe + E over time. For conveniencewe can define

V∞ = RmIe + E




The Brain.



V∞. From the above equation, we see that the change inmembrane voltage is some fraction (proportional to τm ) ofthe difference between RmIe + E and the currentmembrane voltage, V . By this equation the membranevoltage approaches RmIe + E over time.

For conveniencewe can define

V∞ = RmIe + E




The Brain.



V∞. From the above equation, we see that the change inmembrane voltage is some fraction (proportional to τm ) ofthe difference between RmIe + E and the currentmembrane voltage, V . By this equation the membranevoltage approaches RmIe + E over time. For conveniencewe can define

V∞ = RmIe + E




The Brain.




V∞ = RmIe + E




The Brain.




V∞ = RmIe + E

where V∞

is the membrane voltage that will be reachedgiven an external current and membrane resistance, andan infinite amount of time.



The Brain.




V∞ = RmIe + E

where V∞ is the membrane voltage

that will be reachedgiven an external current and membrane resistance, andan infinite amount of time.



The Brain.




V∞ = RmIe + E

where V∞ is the membrane voltage that will be reachedgiven an external current and

membrane resistance, andan infinite amount of time.



The Brain.




V∞ = RmIe + E

where V∞ is the membrane voltage that will be reachedgiven an external current and membrane resistance, and

an infinite amount of time.



The Brain.




V∞ = RmIe + E




The Brain.




V∞ = RmIe + E




The Brain.



Resting potential.

If you shut off the external current (i.e.,set Ie = 0), then V∞ = E . For this reason, we call E theresting potential of the cell - the potential the cellapproaches if we remove external forces.



The Brain.



Resting potential. If you shut off the external current

(i.e.,set Ie = 0), then V∞ = E . For this reason, we call E theresting potential of the cell - the potential the cellapproaches if we remove external forces.



The Brain.



Resting potential. If you shut off the external current (i.e.,set Ie = 0), then

V∞ = E . For this reason, we call E theresting potential of the cell - the potential the cellapproaches if we remove external forces.



The Brain.



Resting potential. If you shut off the external current (i.e.,set Ie = 0), then V∞ = E .

For this reason, we call E theresting potential of the cell - the potential the cellapproaches if we remove external forces.



The Brain.



Resting potential. If you shut off the external current (i.e.,set Ie = 0), then V∞ = E . For this reason,

we call E theresting potential of the cell - the potential the cellapproaches if we remove external forces.



The Brain.



Resting potential. If you shut off the external current (i.e.,set Ie = 0), then V∞ = E . For this reason, we call E

theresting potential of the cell - the potential the cellapproaches if we remove external forces.



The Brain.



Resting potential. If you shut off the external current (i.e.,set Ie = 0), then V∞ = E . For this reason, we call E theresting potential of the cell

- the potential the cellapproaches if we remove external forces.



The Brain.



Resting potential. If you shut off the external current (i.e.,set Ie = 0), then V∞ = E . For this reason, we call E theresting potential of the cell - the potential the cellapproaches

if we remove external forces.



The Brain.



Resting potential. If you shut off the external current (i.e.,set Ie = 0), then V∞ = E . For this reason, we call E theresting potential of the cell - the potential the cellapproaches if we remove external forces.



The Brain.






The Brain.






The Brain.






The Brain.



Computing the voltage at a particular time, t.

We needto solve the following differential equation for V :

τmdVdt

= RmIe + (E − V )

We know that, given enough time, V tends towards V∞ , sowe can say that at time t :

V (t) = V∞ + f (t)

Now we need to find f (t). We use the equation above,substituting in V∞ + f for V



The Brain.



Computing the voltage at a particular time, t. We needto solve

the following differential equation for V :

τmdVdt

= RmIe + (E − V )


V (t) = V∞ + f (t)




The Brain.



Computing the voltage at a particular time, t. We needto solve the following differential

equation for V :

τmdVdt

= RmIe + (E − V )


V (t) = V∞ + f (t)




The Brain.



Computing the voltage at a particular time, t. We needto solve the following differential equation for V :

τmdVdt

= RmIe + (E − V )


V (t) = V∞ + f (t)




The Brain.




τmdVdt

= RmIe + (E − V )


V (t) = V∞ + f (t)




The Brain.




τmdVdt

= RmIe + (E − V )

We know that,

given enough time, V tends towards V∞ , sowe can say that at time t :

V (t) = V∞ + f (t)




The Brain.




τmdVdt

= RmIe + (E − V )

We know that, given enough time,

V tends towards V∞ , sowe can say that at time t :

V (t) = V∞ + f (t)




The Brain.




τmdVdt

= RmIe + (E − V )

We know that, given enough time, V tends towards V∞ ,

sowe can say that at time t :

V (t) = V∞ + f (t)




The Brain.




τmdVdt

= RmIe + (E − V )


V (t) = V∞ + f (t)




The Brain.




τmdVdt

= RmIe + (E − V )


V (t) = V∞ + f (t)




The Brain.




τmdVdt

= RmIe + (E − V )


V (t) = V∞ + f (t)

Now we need

to find f (t). We use the equation above,substituting in V∞ + f for V



The Brain.




τmdVdt

= RmIe + (E − V )


V (t) = V∞ + f (t)

Now we need to find f (t).

We use the equation above,substituting in V∞ + f for V



The Brain.




τmdVdt

= RmIe + (E − V )


V (t) = V∞ + f (t)

Now we need to find f (t). We use the equation above,

substituting in V∞ + f for V



The Brain.




τmdVdt

= RmIe + (E − V )


V (t) = V∞ + f (t)




The Brain.




τmdVdt

= RmIe + (E − V )


V (t) = V∞ + f (t)




The Brain.



τmdfdt

= RmIe + E − V∞ − f

Since V∞ = RmIe + E , those terms cancel and we have:

τmdVdt

= −f

Integrating this differential equation we have

τmdfdt

= −f

τmdf = −fdt)



The Brain.



τmdfdt

= RmIe + E − V∞ − f

Since V∞ = RmIe + E ,

those terms cancel and we have:

τmdVdt

= −f


τmdfdt

= −f

τmdf = −fdt)



The Brain.



τmdfdt

= RmIe + E − V∞ − f


τmdVdt

= −f


τmdfdt

= −f

τmdf = −fdt)



The Brain.



τmdfdt

= RmIe + E − V∞ − f


τmdVdt

= −f


τmdfdt

= −f

τmdf = −fdt)



The Brain.



τmdfdt

= RmIe + E − V∞ − f


τmdVdt

= −f


τmdfdt

= −f

τmdf = −fdt)



The Brain.



τmdfdt

= RmIe + E − V∞ − f


τmdVdt

= −f


τmdfdt

= −f

τmdf = −fdt)



The Brain.



τmdfdt

= RmIe + E − V∞ − f


τmdVdt

= −f


τmdfdt

= −f

τmdf = −fdt)



The Brain.



τmdfdt

= RmIe + E − V∞ − f


τmdVdt

= −f


τmdfdt

= −f

τmdf = −fdt)



The Brain.



∫ f (0)

f (t)τmdf = −

∫ t

0fdt

τm[lnf (t)− lnf (0)] = −t

τmln[f (t)f (0)

] = −t

ln[f (t)f (0)

] = − tτm



The Brain.



∫ f (0)

f (t)τmdf = −

∫ t

0fdt


τmln[f (t)f (0)

] = −t

ln[f (t)f (0)

] = − tτm



The Brain.



∫ f (0)

f (t)τmdf = −

∫ t

0fdt


τmln[f (t)f (0)

] = −t

ln[f (t)f (0)

] = − tτm



The Brain.



∫ f (0)

f (t)τmdf = −

∫ t

0fdt


τmln[f (t)f (0)

] = −t

ln[f (t)f (0)

] = − tτm



The Brain.



∫ f (0)

f (t)τmdf = −

∫ t

0fdt


τmln[f (t)f (0)

] = −t

ln[f (t)f (0)

] = − tτm



The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm

Previously, we said that the voltage changes as a functionof the distance between the voltage at the present timeand V∞. So f (0) = V (0)− V∞ , and f (t) = f (0)e−

tτm .

Plugging f (t) back into the original equation gives:

V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm

This gives us a way to compute how long it will take tocharge up the neuron to an arbitrary voltage V (t), bysolving for t .



The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm


tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm

Previously,

we said that the voltage changes as a functionof the distance between the voltage at the present timeand V∞. So f (0) = V (0)− V∞ , and f (t) = f (0)e−

tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm

Previously, we said that the voltage changes

as a functionof the distance between the voltage at the present timeand V∞. So f (0) = V (0)− V∞ , and f (t) = f (0)e−

tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm

Previously, we said that the voltage changes as a functionof the distance

between the voltage at the present timeand V∞. So f (0) = V (0)− V∞ , and f (t) = f (0)e−

tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm

Previously, we said that the voltage changes as a functionof the distance between the voltage at the present timeand

V∞. So f (0) = V (0)− V∞ , and f (t) = f (0)e−t

τm .Plugging f (t) back into the original equation gives:

V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm

Previously, we said that the voltage changes as a functionof the distance between the voltage at the present timeand V∞.

So f (0) = V (0)− V∞ , and f (t) = f (0)e−t


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm

Previously, we said that the voltage changes as a functionof the distance between the voltage at the present timeand V∞. So f (0) = V (0)− V∞ ,

and f (t) = f (0)e−t


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm


tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm


tτm .

Plugging f (t) back into the original equation

gives:

V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm


tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm


tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm


tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm

This gives us

a way to compute how long it will take tocharge up the neuron to an arbitrary voltage V (t), bysolving for t .



The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm


tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm

This gives us a way to compute

how long it will take tocharge up the neuron to an arbitrary voltage V (t), bysolving for t .



The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm


tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm

This gives us a way to compute how long it will take

tocharge up the neuron to an arbitrary voltage V (t), bysolving for t .



The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm


tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm

This gives us a way to compute how long it will take tocharge up

the neuron to an arbitrary voltage V (t), bysolving for t .



The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm


tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm

This gives us a way to compute how long it will take tocharge up the neuron to an arbitrary voltage V (t),

bysolving for t .



The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm


tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.



f (t)f (0)

= e−t

τm

f (t) = f (0)e−t

τm


tτm .


V (t) = V∞ = V∞ + f (t) = V∞ + (V (0)− V∞)e−t

τm




The Brain.


References

[1] R. Harrison. The outgrowth of the nerve fiber as a mode ofprotoplasmic movement. J. Exp. Zool. , 9:787-846, 1910.

[2] H. Keshishian. Ross Harrison’s ’The Outgrowth of the NerveFiber as a Mode of Protoplasmic Movement’. J. Exp. Zool. ,301A:201-203, 2004.

[3] N. Wiener. Cybernetics: or Control and Communication inthe Animal and the Machine. M.I.T. Press, Cambridge, MA,1948. 2nd Edition, 1961.

[4] J. von Neumann. The Computer and the Brain . YaleUniversity Press, New Haven, CT, 1958. 2nd Edition, with aforeward by P.M. and P.S. Churchland, 2000.



The Brain.


References

[5] A.L. Hodgkin and A.F. Huxley. A quantitative description ofmembrane current and its application to conduction andexcitation in nerve. J. Physiol., 117:500-544, 1952.

[6] M.C. Mackey and M. Santillan. Andrew Fielding Huxley(1917-1952). AMS Notices, 60 (5):576-584, 2013.

[7] H. Wilson. Spikes, Decisions and Actions: The DynamicalFoundations of Neuroscience. Oxford University Press, Oxford,U.K., 1999. Currently out of print, downloadable fromhttp://cvr.yorku.ca/webpages/wilson.htm# book.

[8] P. Dayan and L.F. Abbott. Theoretical Neuroscience:Computational and Mathematical Modeling of Neural Systems.MIT Press, Cambridge, MA, 2001.



The Brain.


References

[9] E.M. Izhikevich. Dynamical systems in neuroscience: Thegeometry of excitab ility and bursting. MIT Press, Cambridge,MA, 2007.

[10] J. Keener and J. Sneyd. Mathematical Physiology.Springer, New York, 2009. 2nd Edition, 2 Vols

[11] G.B. Ermentrout and D. Terman. MathematicalFoundations of Neuroscience . Springer, New York, 2010.

[12] F. Gabbiani and S. Cox. Mathematics for Neuroscientists.Academic Press, San Diego, CA, 2010.



The Brain.


References

[13] X.J. Wang. Neurophysiological and computationalprinciples of cortical rhythms in cognition. Physiol. Rev. ,90:1195-1268, 2010.

[14] N. Kopell. Toward a theory of modelling central patterngenerators. In A.H Cohen, S. Rossig- nol, and S. Grillner,editors, Neural Control of Rhythmic Movements in Vertebrates,pages 3369âAS413. Wiley, New York, 1988.

[15] M.M. McCarthy, S. Ching, M.A. Whittington, and N. Kopell.Dynamical changes in neurological disease andanesthesia.Curr. Opin. Neurobiol. , 22 (4):693-703, 2012.

[16] G. Deco, E.T. Rolls, L. Albantakis, and R. Romo. Brainmechanisms for perceptual and reward-relateddecision-making. Prog. in Neurobiol. , 103:194-213, 2013



The Brain.


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[17] J S Griffith. Mathematical Neurobiology: An introduction tothe mathematics of the nervous system. Academic Press, 1971.

[18] I Segev, J Rinzel, and G M Shepherd, editors. Thetheoretical foundations of dendritic function: selected papers ofWilfrid Rall with commentaries. MIT Press, 199.

[19] S Coombes. Waves, bumps, and patterns in neural fieldtheories. Biological Cybernetics, 93:91âAS108, 2005.

[20] C Bressloff, J D Cowan, M Golubitsky, P J Thomas, and MWiener. Geometric visual hallucinations, Euclidean symmetryand the functional architecture of striate cortex. PhilosophicalTransactions of the Royal Society London B, 40:299âAS330,2001.



The Brain.


References

[21] S Coombes and M R Owen. Evans functions for integralneural field equations with Heaviside firing rate function. SIAMJournal on Applied Dynamical Systems , 34:574âAS600, 2004.

[22] P. Ashwin and M Timme. When instability makes sense.Nature, 436:36-37, 2005.

[23] J Rubin and D Terman. Handbook of Dynamical SystemsII, chapter Geometric Singular Perturbation Analysis ofNeuronal Dynamics. Elsevier, 2002.

[24] A Longtin and P Swain, editors. Stochastic Dynamics ofNeural and Genetic Networks . Special Focus Issue of CHAOS,Vol.16, 2006.



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[25] D Cai, L Tao, A V Rangan, and D W McLaughlin. Kinetictheory for neuronal network dynamics. Communications inMathematical Sciences, 4:97-127, 2006.

[26] P C Bressloff. Spatially periodic modulation of corticalpatterns by long-range horizontal connections. Physica D,185:131âAS157, 2003.

[27] Moehlis. Canards for a reduction of the Hodgkin-Huxleyequations. Journal of Mathematical Biology , 52:141âAS153,2006.

[28] S Ikeda, T Tanaka, and S Amari. Stochastic reasoning, freeenergy, and information geometry. Neural Computation,16:1779-1810, 2004.



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[29] M Rosenblum and A Pikovsky. Delayed feedback control ofcollective synchrony: An approach to suppression ofpathological brain rhythms. Physica Review E , 70(041904),2004.

[30] W Singer. Synchronization of cortical activity and itsputative role in information processing and learning. AnnualReview of Physiology, 55:349âAS374, 1993.

[31] G Buzsaki. Rhythms of the Brain. Oxford University Press,2006.

[32] D Robbe, S M Montgomery, A Thome, P E Rueda-Orozco,B L McNaughton, and G Buzs ÌAaki. Cannabinoids revealimportance of spike timing coordination in hippocampalfunction. Nature Neuroscience, 9:1526âAS1533, 2006



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[33] F Sargolini, M Fyhn, T Hafting, B L McNaughton, M PWitter, M-B Moser, and E I Moser. Conjunctive representationof position, direction, and velocity in entorhinal cortex. Science,312:758âAS762, 2006.


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