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

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

-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

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 - 13.5. Threshold and reset -

Week 13 – Membrane Potential Densities and Fokker-Planck Equation

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

Membrane potential density

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

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

-

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

Week 13 – Membrane Potential Densities and Fokker-Planck Equation

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

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

-

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

Week 13 – Membrane Potential Densities and Fokker-Planck Equation

u

p(u)

u

Membrane potential density

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

Membrane potential density

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

Membrane potential density

sourcetupu

tupuu

tupt

),(

)],()([

),(

2

22

21

Fokker-Planck A=f

0I

with noise

- Calculate distribution p(u)

- Determine population firing rate A

Miniproject: 12h00

THE END

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