Biological Modeling of Neural Networks

Biological Modeling of Neural Networks

Week 2 – Biophysical modeling:

The Hodgkin-Huxley model

Wulfram GerstnerEPFL, Lausanne, Switzerland

2.1 Biophysics of neurons - Overview2.2 Reversal potential

- Nernst equation2.3 Hodgin-Huxley Model2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold?2.5. Detailed biophysical models - the zoo of ion channels

Week 2 – part 1: Biophysics of neurons



Week 2 – part 1: Biophysics of neurons

Neuronal Dynamics – 2.1. Introduction

visual cortex

motor cortex

frontal cortex

to motoroutput

Neuronal Dynamics – 2.1. Introduction

motor cortex

frontal cortex

to motoroutput

10 000 neurons3 km wires

1mm

Neuronal Dynamics – 2.1 Introduction

10 000 neurons3 km wires1mm

Signal:action potential (spike)

action potential

Ramon y Cajal

How is a spike generated?

Review of week 1: Integrate-and-Fire models

u

Spike emission

-spikes are events-triggered at threshold-spike/reset/refractoriness

Postsynaptic potential

synapse t

Neuronal Dynamics – week 2: Biophysics of neurons

Signal:action potential (spike)

action potential

Ca2+

Na+

K+

-70mV

Ions/proteins

Cell surrounded by membraneMembrane contains - ion channels - ion pumps

Neuronal Dynamics – week 2: Biophysics of neurons

Ca2+

Na+

K+

-70mV

Ions/proteins


Resting potential -70mV how does it arise?

Ions flow through channel in which direction?

Neuron emits action potentials why?

Neuronal Dynamics – 2. 1. Biophysics of neurons




Hodgkin-Huxley model Hodgkin&Huxley (1952) Nobel Prize 1963

Neuronal Dynamics – 2. 1. Biophysics of neurons


u

...du

dt u

( )I t

Neuronal Dynamics – Quiz In a natural situation, the electrical potential inside a neuron is [ ] the same as outside [ ] is different by 50-100 microvolt [ ] is different by 50-100 millivolt

If a channel is open, ions can [ ] flow from the surround into the cell [ ] flow from inside the cell into the surrounding liquid

Neurons and cells [ ] Neurons are special cells because they are surrounded by a membrane [ ] Neurons are just like other cells surrounded by a membrane [ ] Neurons are not cells

Ion channels are [ ] located in the cell membrane [ ] special proteins [ ] can switch from open to closed

Multiple answers possible!






Week 2 – part 2: Reversal potential and Nernst equation


Neuronal Dynamics – 2.2. Resting potential

Ca2+

Na+

K+

-70mV

Ions/proteins





Neuronal Dynamics – 2. 2. Resting potential





100

mV

0

inside

outside

Ka

Na

Ion channels Ion pumpkTE

en

density

Mathetical derivation

Ion pump Concentration difference

Neuronal Dynamics – 2. 2. Reversal potential

E

K

100

mV

0

inside

outside

Ka

Na

Ion channels Ion pump

Neuronal Dynamics – 2. 2. Nernst equation

kTE

en K

100

mV

0

inside

outside

Ka

Na


Reversal potential

1

2

( )1 2 ( )ln n u

n u

kTu u u

q

Neuronal Dynamics – 2. 2. Nernst equation

Concentration difference voltage difference

K

inside

outside

Ka

Na


Reversal potential

Neuronal Dynamics – 2. 2. Reversal potential

Ion pump Concentration difference

Concentration difference voltage difference

Nernst equation

K

Reversal potential

Exercise 1.1– Reversal potential of ion channels

1

2

( )1 2 ( )ln n u

n u

kTu u u

q

Calculate the reversal potentialfor Sodium Postassium Calciumgiven the concentrations

What happens if you changethe temperature T from 37to 18.5 degree?

Next Lecture 9:45

Neuronal Dynamics:Computational Neuroscienceof Single NeuronsWeek 2 – Biophysical modeling:




- Nernst equation2.3 Hodgkin-Huxley Model2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold?2.5. Detailed biophysical models - the zoo of ion channels

Week 2 – part 3 : Hodgkin-Huxley Model

Neuronal Dynamics – 2. 3. Hodgkin-Huxley Model


...du

dt u

( )I t

giant axonof squid

inside

outside

Ka

Na


C RlRK RNa

I

Mathematicalderivation

Hodgkin and Huxley, 1952Neuronal Dynamics – 2.3. Hodgkin-Huxley Model

C RlRK RNa

I

Neuronal Dynamics – 2.3. Hodgkin-Huxley Model

100

mV

0

)()()()( 43 tIEugEungEuhmgdt

duC llKKNaNa

)(

)(0u

umm

dt

dm

m

)(

)(0u

unn

dt

dn

n

)(

)(0u

uhh

dt

dh

h

stimulusleakI

inside

outside

Ka

Na


C glgK gNa

I

u u

n0(u))(un

Hodgkin and Huxley, 1952Neuronal Dynamics – 2.3. Hodgkin-Huxley Model

NaI KI

1 2n nion ionI g r s

)(

)(0u

urr

dt

dr

r

u u

r0(u))(ur

Neuronal Dynamics – 2.3. Ion channel

, ( )ion kk

duC I I tdt

0 ( )

( )r

s s uds

dt u

)()(21 tIEusrgdt

duC Na

nnion

)(

)(0u

urr

dt

dr

r

u u

r0(u))(ur

Exercise 2 and 1.2 NOW!! - Ion channel

Next lecture at: 10H40






Week 2 – part 4: Threshold in the Hodgkin-Huxley Model


Neuronal Dynamics – 2.4. Threshold in HH model inside

outside

Ka

Na



duC llKKNaNa

leakI

)(

)(0u

umm

dt

dm

m

)(

)(0u

uhh

dt

dh

h

inside

outside

Ka

Na


u u

h0(u)

m0(u) )(uh

)(um

C glgK gNa

I

Where is the threshold for firing?

Neuronal Dynamics – 2.4. Threshold in HH model

NaI KI

Constant current inputI0

C glgK gNa

I

Threshold? for repetitive firing

(current threshold)


inside

outside

Ka

Na


pulse inputI(t)

Threshold?

- AP if amplitude 7.0 units- No AP if amplitude 6.9 units

(pulse with 1ms duration)

(and pulse with 0.5 ms duration?)



duC llKKNaNa

)(

)(0u

umm

dt

dm

m

)(

)(0u

uhh

dt

dh

h

Stim. NaI KI leakI

u u

h0(u)

m0(u) )(uh

)(um

pulse input

I(t)

Mathematical explanation


3 ( ) ( )Na Na K leak

duC g m h u E I I I tdt

)(

)(0u

umm

dt

dm

m

)(

)(0u

uhh

dt

dh

h

Stim. NaI

u u

h0(u)

m0(u) )(uh

)(um

pulse input

I(t)


Where is the threshold?Blackboard!


duC llKKNaNa

)(

)(0u

umm

dt

dm

m

)(

)(0u

uhh

dt

dh

h

Stim. NaI KI leakI

u u

h0(u)

m0(u) )(uh

)(um

pulse input

I(t)


0 ( )

( )n

n n udn

dt u

Why start the explanation with m and not h?

What about n?

Where is the threshold?


3

4

( )

( )

( )

( )

Na Na

K K

l l

duC g m h u Edt

g n u E

g u E

I t


There is no strict threshold:

Coupled differential equations

‘Effective’ threshold in simulations?

First conclusion:

0 20ms

Strongstimulus

Action potential100

mV

0

strong stimuli

refractoriness

Strongstimulus

100

mV

0

Where is the firing threshold?

Refractoriness! Harder to elicit a second spike

Neuronal Dynamics – 2.4. Refractoriness in HH model

I(t)

100

mV

0

Stimulation withtime-dependentinput current

Neuronal Dynamics – 2.4. Simulations of the HH model

I(t)

0

5

-5

mV

mV

0

100100

mV

0

Subthresholdresponse

Spike

Neuronal Dynamics – 2.4. Simulations of the HH model

Step current inputI2

I


step input I2I

Where is the firing threshold?

pulse inputI(t)

ramp input

There is no threshold - no current threshold - no voltage threshold

‘effective’ threshold - depends on typical input

...)(3 NaNa Euhmgdt

duC


Response at firing threshold?

ramp input/constant input

I0

Type I type II

I0 I0

fff-I curve f-I curve

Neuronal Dynamics – 2.4. Type I and Type II

Hodgkin-Huxley modelwith standard parameters(giant axon of squid)

Hodgkin-Huxley modelwith other parameters(e.g. for cortical pyramidalNeuron )

Neuronal Dynamics – 2.4. Hodgkin-Huxley model -4 differential equations

-no explicit threshold-effective threshold depends on stimulus-BUT: voltage threshold good approximation

Giant axon of the squid cortical neurons-Change of parameters-More ion channels-Same framework

Exercise 3.1-3.3 – Hodgkin-Huxley – ion channel dynamics

)()(4 tIEungdt

duC KK

stimulus

inside

outside

Ka

Na


C gK

I

u u

n0(u)

)(un

Determine ion channel dynamics

adapted fromHodgkin&Huxley 1952

apply voltage step

voltage step u2u

Next Lecture at:11.38





- Nernst equation2.3 Hodgkin-Huxley Model2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold?2.5. Detailed biophysical models - the zoo of ion channels

Week 2 – part 5: Detailed Biophysical Models


inside

outside

Ka

Na


Neuronal Dynamics – 2.5 Biophysical models

b

a

Na+ channel from rat heart (Patlak and Ortiz 1985)A traces from a patch containing several channels. Bottom: average gives current time course.B. Opening times of single channel events

Steps:Different numberof channels

Ca2+

Na+

K+

Ions/proteins

Neuronal Dynamics – 2.5 Ion channels

inside

outside

Ka

Na



b

a

There are about 200identified ion channels

http://channelpedia.epfl.ch/

How can we know which ones are present in a given neuron?

b

a

a

b

Kv1 Kv2 Kv3

Kv

Voltage Activated K+ Channels

+ +

Kv1.1 Kv1.2 Kv1.3 Kv1.4 Kv1.5 Kv1.6 Kv2.2Kv2.1 Kv3.2Kv3.1 Kv3.3 Kv3.4 KChIP1 KChIP2 KChIP3

KChIP

Kv1 Kv3Kv2 Kv4 Kv5 Kv6 Kv8 Kv9

Kv

Kv4.3Kv4.1 Kv4.2

Ion Channels investigated in the study of

CGRP

CB

PV CR NPY

VIP

SOM

CCK

pENK

Dyn SP CRH

schematic mRNAExpression profile

C glgKv1gNa

I

gKv3

Toledo-Rodriguez, …, Markram (2004)

C glgKv1gNa

I

gKv3

Model of a hypothetical neuron

)()()()()( 233

411

3 tIEugEungEungEuhmgdt

duC llKKvKvKKvKvNaNa

)(

)(0u

umm

dt

dm

m

1 0,11

,1

( )

( )n

n n udn

dt u

)(

)(0u

uhh

dt

dh

h

stimulusNaI KI leakI

inside

outside

Ka

Na


Erisir et al, 1999Hodgkin and Huxley, 1952

1 0,33

,3

( )

( )n

n n udn

dt u

How many parameters per channel?

C glgKv1gNa

I

gKv3

Model of a hypothetical neuron

I(t)

Spike

Subthreshold

Detailed model, based on ion channels

C glgKv1gNa

I

gKv3

Model of a hypothetical neuron (type I)

Biophysical model, based on ion channels

Current pulse

constantcurrent

- Delayed AP initiation- Smooth f-I curve type I neuron

Neuronal Dynamics – 2.5 Adaptation

Functional roles of channels?- Example: adaptation

I

u

Neuronal Dynamics – 2.5 Adaptation: IM -current

M current: - Potassium current - Kv7 subunits - slow time constant

IM current is one of many potential sources of adaptationYamada et al., 1989

( )M M KI g m u E

Neuronal Dynamics – 2.5 Adaptation – INaP current current: - persistent sodium current - fast activation time constant - slow inactivation ( ~ 1s)

INaP current - increases firing threshold- source of adaptationAracri et al., 2006

( )NaP NaP NaI g m h u E

inside

outside

Ka

Na



b

Hodgkin-Huxley modelprovides flexible framework

Hodgkin&Huxley (1952) Nobel Prize 1963

Exercise 4 – Hodgkin-Huxley model – gating dynamics

( ) (1 ) ( )m m

dmu m u m

dt

0 ( ) 0.5{1 tanh[ ( )]m u u

1( ) ( )

1 exp[ ( ) / ]m mu uu a b

A) Often the gating dynamics is formulated as

B) Assume a form

How are a and b related to and in the equations 0 ( )

( )m

m m udm

dt u

C ) What is the time constant ? ( )m u

0 ( )

( )m

m m udm

dt u

0 ( ) ( )mm u and uCalculate

Now Computer Exercises:

Play with Hodgkin-Huxley model

Neuronal Dynamics – References and Suggested Reading

Reading: W. Gerstner, W.M. Kistler, R. Naud and L. Paninski,Neuronal Dynamics: from single neurons to networks and models of cognition. Chapter 2: The Hodgkin-Huxley Model, Cambridge Univ. Press, 2014OR W. Gerstner and W. M. Kistler, Spiking Neuron Models, Chapter 2, Cambridge, 2002

- Hodgkin, A. L. and Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol, 117(4):500-544. -Ranjan, R.,et al. (2011). Channelpedia: an integrative and interactive database for ion channels. Front Neuroinform, 5:36.

-Toledo-Rodriguez, M., Blumenfeld, B., Wu, C., Luo, J., Attali, B., Goodman, P., and Markram, H. (2004). Correlation maps allow neuronal electrical properties to be predicted from single-cell gene expression profiles in rat neocortex. Cerebral Cortex, 14:1310-1327.

-Yamada, W. M., Koch, C., and Adams, P. R. (1989). Multiple channels and calcium dynamics. In Koch, C. and Segev, I., editors, Methods in neuronal modeling, MIT Press.

- Aracri, P., et al. (2006). Layer-specic properties of the persistent sodium current in sensorimotor cortex. Journal of Neurophysiol., 95(6):3460-3468.

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Biological Modeling of Neural Networks

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