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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:
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 -
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 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:
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
-
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 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:
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 -
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 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
Membrane potential density
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:
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 -
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
2k k
k
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
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
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
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