Broca’sarea

Parsopercularis

Motor cortex Somatosensory cortex

Sensory associativecortex

PrimaryAuditory cortex

Wernicke’sarea

Visual associativecortex

Visualcortex

Neurons

Synapses

• There are about 1012 neurons in the human brain.

• Neurons generate electrical signals (action potentials).

• Neurons communicate with each other at synapses.

• There are about 1015 synaptic connections.

Neurons and synapses

What the brain does results from neuronal activity patterns.What the brain does results from neuronal activity patterns.

A single neuron may exhibit complex firing patterns.

VPeriodic spiking

Burstingoscillation

Network Activity

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Synchrony Uncorrelated activity

Propagating waves

Mathematical Challenges

• How should one model neuronal networks?

• What types of activity patterns emerge in a model?

• How does these patterns change wrt parameters?

• How can we mathematically analyze the solutions?

• How does the brain use this information?

How do we model neuronal systems?

1) Single neurons

2) Synaptic connections between neurons

3) Network architecture

The Neuron

Electrical Signal: Action potential that propagates along axon

The Hodgkin-Huxley Model

Alan Lloyd Hodgkin Andrew Huxley

Hodgkin-Huxley Equations

V = Membrane potential

h, m, n = Channel state variables

Model for action potential in the squid giant axon

CVt = DVxx - gNam3h(V-Ena) - gKn4(V-EK) - gL(V-EL)

mt = (m(V) - m) / m(V)

ht = (h(V) - h) / h(V)

nt = (n(V) - n) / n(V)

Some basic biology

Cells have resting potential: potential difference betweeninside and outside of cell

Resting potential maintained by concentration differences of ions inside and outside of cell

There are channels in membrane selective to different ions. Channels may be open or closed.

Membrane potential changes as ions flow into or out of cell.

K+K+

K+

Na+

Na+

Na+

The action potential

K+K+

K+

Na+

Na+

Na+

CVt = -gNam3h(V-Ena) - gKn4(V-EK) - gL(V-EL) mt = (m(V) - m) / m(V) ht = (h(V) - h) / h(V) nt = (n(V) - n) / n(V)

The Morris-Lecar equations

CVt = -gCa m(V) (V-ECa) - gKn(V-EK) - gL(V-EL) + Iapp

nt = (n(V) - n) / n(V)

m(V) = .5(1+tanh((v-v1)/v2)n(V) = .5(1+tanh((v-v3)/v4)n(V) = 1/cosh((v-v3)/2v4)

We will write this system as:

V’ = f(V,n) + Iapp

n’ = g(V,n)

Class I: (SNIC) Axons have sharp thresholds, can have long to firing, and can fire at arbitrarily low frequencies

Class II: (Hopf) Axons have variable thresholds, short latency and a positive frequency.

Networks

Synaptic connections

There may be different types of synapses:- excitatory or inhibitory

- activate and/or inactivate at different time rates

v1’ = f(v1,w1) – gsyns2(v1 – vsyn)

w1’ = g(v1,w1)

s1’ = (1-s1)H(v1-)-s1

v2’ = f(v2,w2) – gsyns1(v2-vsyn)

w2’ = g(v2,w2)

s2’ = (1-s2)H(v2-)-s2

v1’ = f(v1,w1) – gsyns2(v1 – vsyn)

w1’ = g(v1,w1)

s1’ = (1-s1)H(v1-)-s1

v2’ = f(v2,w2) – gsyns1(v2-vsyn)

w2’ = g(v2,w2)

s2’ = (1-s2)H(v2-)-s2

Synapses may be excitatory or inhibitory

They may turn on or turn off at different rates

Cell 2

Cell 1

Model for two mutually coupled cells

Network Architecture

Example: excitatory-inhibitory network

Note: There are many different types of connectivities:

-- Sparse, global, random, structures, …

Sleep

Oscillatory processes with many time-scales:

• Circadian: 24 hours

• Slower: homeostatic sleep dept

• Internal sleep structure: minutes – hours

• Neuronal activity: milliseconds

Slow-Wave Activity: -- Spindles: 7 - 15 Hz ; Wax and Wane -- Delta: 1 - 4 Hz -- Slow Osc. .5 - 1 Hz

Stages of sleep form cyclical pattern

Intracellular aspects of spindling in the thalamocortical system

Sleep involves many parts of the brain

Hobson, Nature Reviews Neuroscience 2002

These sleep rhythms arise from interactions between cortical neurons and two groups of cells within the thalamus: RE and TC cell.

Thalamocortical Network

Ctx

RE TC+

+

+

-

Cells behave differently during Spindling and Delta

TC RE

Spindle

Delta

ClustersDo not fireevery cycle

7-15 HzSynchrony

1 - 4 HzSynchrony

Slow Rhythm < 1 Hz

Questions:

• How do we model this system?

• What mechanisms underlie these rhythms?

• Transitions between sleep stages?

BASAL GANGLIA

BASAL GANGLIA

• Involved in the control of movement

• Dysfunction associated with Parkinson’s and Huntington’s disease

• Site of surgical procedures -- Deep Brain Stimulation (DBS)

BASAL GANGLIA

SNc

Striatum

GPiSTN

GPe

ExcitationInhibition

CTX

Thalamus

dopamine

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Motivation of Computational Study

• Explain changes in firing patterns within the basal ganglia

• During PD, neurons display:

– Increased synchrony

– Increased bursting activity

• Mechanism underlying DBS mysterious