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Biological Modeling

of Neural Networks:

Week 13 – Membrane potential

densities and Fokker-Planck

Wulfram Gerstner

EPFL, Lausanne, Switzerland

13.1 Review: Integrate-and-fire

- stochastic spike arrival

13.2 Density of membrane potential - Continuity equation

13.3 Flux - jump flux

- drift flux

13.4. Fokker –Planck Equation

- free solution

13.5. Threshold and reset - time dependent activity

- network states

Week 13 – Membrane Potential Densities and Fokker-Planck Equation

Spike reception

iu

i j

Spike emission

-spikes are events

-threshold

-spike/reset/refractoriness

Week 13-part 1: Review: integrate-and-fire-type models

)()( tRIuuudt

deq

)();( equuVtRIVVdt

d

iui

I

j

Week 13-part 1: Review: leaky integrate-and-fire model

resetuu If firing:

)()( tRIuuudt

deq LIF

resetuu If firing:

I=0 u

dt

d

u

I>0 u

dt

d

u

resting

t

u

repetitive

t

flow (drift)

towards

threshold

Week 13-part 1: Review: leaky integrate-and-fire model

I(t)

)(tAn

Week 13-part 1: Review: microscopic vs. macroscopic

I(t)

?

t

t

tN

tttntA

);()(population

activity

Homogeneous

network: -each neuron receives input

from k neurons in network

-each neuron receives the same

(mean) external input

Week 13-part 1: Review: homogeneous population

Stochastic spike arrival:

excitation, total rate Re

inhibition, total rate Ri

u

0u

EPSC IPSC

Synaptic current pulses

)()()( ttIRuuudt

d meaneq

Langevin equation,

Ornstein Uhlenbeck process

)()()(

','

''

,

fk

fki

fk

fkeeq ttqttqRuuu

dt

d

Week 13-part 1: Review: diffusive noise/stochastic spike arrival

Biological Modeling

of Neural Networks:



Wulfram Gerstner




13.2 Density of membrane potential -

13.3 Flux -

-


- -

13.5. Threshold and reset -

Week 13 part 2 – Membrane Potential Densities

EPSC IPSC

For any arbitrary neuron in the population

Blackboard:

density of potentials?

))()((

','

''

,

fk

fki

fk

fke ttqttqRuu

dt

d

,

( ) ( )f extek

k f

qd uu t t I t

dt C

external current

input excitatory input spikes

Week 13-part 2: membrane potential density

( , ) ( , )d d

p u t J u tdt du

Week 13-part 2: continuity equation

u

p(u)

Exercise 1: flux caused by stochastic spike arrival

u

Membrane potential density

spike arrival rate

Next lecture:

10h15

b)

c) k

k

spike arrival rate

a) Jump at time t

Reference level u0

What is the flux

across u0?

( ) ( )}f

eq e

f

du u u R q t t

dt

Biological Modeling

of Neural Networks:



Wulfram Gerstner




13.2 Density of membrane potential - continuity equation

-

13.3 Flux - jump flux

- drift flux


-



Week 13-part 2: membrane potential density: flux by jumps

Week 13-part 2: membrane potential density: flux by drift

u

p(u)

flux – two possibilities

u


spike arrival rate a) Jumps caused at

Reference level u0

What is the flux

across u0?

b)

flux caused by jumps due to

stochastic spike arrival

flux caused by

systematic drift

Blackboard:

Slope and

density of potentials

( )( ) ( )}ext t f

eq e

f

du u u R I q t t

dt

Biological Modeling

of Neural Networks:



Wulfram Gerstner





13.3 Flux - continuity equation


- source and sink

-



EPSC IPSC

For any arbitrary neuron in the population

'

'

, ', '

1( ) ( ) ( )f f exte i

k k

k f k f

q qd uu t t t t I t

dt C C C

external input

( , ) ( , )p u t J u tt u

Continuity equation:

Flux: - jump (spike arrival)

- drift (slope of trajectory)

Blackboard:

Derive Fokker-Planck

equation

Week 13-part 4: from continuity equation to Fokker-Planck

22

2( , ) [ ( ) ( , )] ( , )p u t u p u t p u t

t u u

Fokker-Planck

drift diffusion

k

kk wuu )( 2 21

2k k

k

w

spike arrival rate


u

p(u)

Week 13-part 4: Fokker-Planck equation

u

p(u)

Exercise 2: solution of free Fokker-Planck equation

u

Membrane potential density: Gaussian

22

2( , ) [ ( ) ( , )] ( , )p u t u p u t p u t

t u u

Fokker-Planck

drift diffusion ( ) ( )k k

k

u u w RI t 2 21

2k k

k

w

spike arrival rate

constant input rates

no threshold

Next lecture:

11h25

Biological Modeling

of Neural Networks:



Wulfram Gerstner





13.3 Flux - continuity equation


- -



u

p(u)

u


22

2( , ) [ ( ) ( , )] ( , ) ( ) ( )resetp u t u p u t p u t A t u u

t u u

Fokker-Planck

drift diffusion ( ) k k

k

u u w RI 2 21

2k k

k

w

spike arrival rate

blackboard

Week 13-part 5: Threshold and reset (sink and source terms)

u

p(u)

u


blackboard

Week 13-part 5: population firing rate A(t)

Population Firing rate A(t): flux at threshold

I

0I)(tI

)()( tIRuuudt

deq

noiseo IItI )(

effective noise current

frequency f

0I

with noise

EPSC IPSC

Synaptic current pulses

)()()( ttIRuuudt

d meaneq

)()()(

','

''

,

fk

fki

fk

fkeeq ttqttqRuuu

dt

d

Week 13-part 5: population firing rate A(t) = single neuron rate

( )g I

Week 13-part 5: membrane potential density

Week 13-part 5: population activity, time-dependent

Nykamp+Tranchina,

2000

Week 13-part 5: network states

frequency f

0I

with noise

( )g I

EPSC IPSC

( ) ( ) ( )mean

eq

du u u R I t t

dt

)()()(

','

''

,

fk

fki

fk

fkeeq ttqttqRuuu

dt

d

mean I(T) depends on state

Variance/noise depends on state

Week 13-part 5: network states

Brunel 2000

u

p(u)

Exercise 3: Diffusive noise + Threshold

u


sourcetupu

tupuu

tupt

),(

)],()([

),(

2

22

21

Fokker-Planck A= f

0I

with noise

- Calculate distribution p(u)

- Determine population firing rate A

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