Biological Modeling of Neural Networks: Week 13 – Membrane potential densities and Fokker-Planck...

Biological Modeling of Neural Networks:

Week 13 – Membrane potential densities and Fokker-Planck

Wulfram GerstnerEPFL, Lausanne, Switzerland

13.1 Review: Integrate-and-fire - stochastic spike arrival13.2 Density of membrane potential

- Continuity equation

13.3 Flux - jump flux - drift flux13.4. 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

iuij

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=0u

dt

d

u

I>0u

dt

d

u

resting

t

u

repetitive

t

flow (drift)towardsthreshold

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

);(

)(populationactivity

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

u0u

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





-13.3 Flux - - 13.4. Fokker –Planck Equation - - 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 currentinputexcitatory input spikes

Week 13-part 2: membrane potential density

( , ) ( , )d dp u t J u t

dt 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?

( ) ( )}feq ef

du u u R q t t

dt





- continuity equation -

13.3 Flux - jump flux - drift flux13.4. Fokker –Planck Equation - 13.5. Threshold and reset -


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 ratea) Jumps caused at

Reference level u0

What is the flux

across u0?

b)

flux caused by jumps due to stochastic spike arrival

flux caused bysystematic drift

Blackboard:Slope and density of potentials

( )( ) ( )}ext t feq e

f

du u u R I q t t

dt





-

13.3 Flux - continuity equation 13.4. Fokker –Planck Equation - source and sink - 13.5. Threshold and reset -


EPSC IPSC

For any arbitrary neuron in the population''

, ', '

1( ) ( ) ( )f f exte i

k kk 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 diffusionk

kkwuu )( 2 21

2 k kk

w

spike arrival rate

Membrane potential densityu

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

2 k kk

w

spike arrival rate

constant input ratesno threshold

Next lecture: 11h25





-

13.3 Flux - continuity equation 13.4. Fokker –Planck Equation - - 13.5. Threshold and reset -


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

2 k kk

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

I0I

)(tI

)()( tIRuuudt

deq

noiseo IItI )(

effective noise current

frequencyf

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

frequencyf

0I

with noise

( )g I

EPSC IPSC

( ) ( ) ( )meaneq

du u u R I t t

dt

)()()(','

''

, fk

fki

fk

fkeeq ttqttqRuuu

dt

d

mean I(T) depends on stateVariance/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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Biological Modeling of Neural Networks: Week 13 – Membrane potential densities and Fokker-Planck...

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