Biological Modeling of Neural Networks Week 8 – Noisy input models: Barrage of spike arrivals Wulfram Gerstner EPFL, Lausanne, Switzerland 8.1 Variation

Biological Modeling of Neural Networks

Week 8 – Noisy input models:

Barrage of spike arrivals

Wulfram GerstnerEPFL, Lausanne, Switzerland

8.1 Variation of membrane potential - white noise approximation8.2 Autocorrelation of Poisson8.3 Noisy integrate-and-fire

- superthreshold and subthreshold

8.4 Escape noise - stochastic intensity8.5 Renewal models8.6 Comparison of noise models

Week 8 – part 1 :Variation of membrane potential

Crochet et al., 2011

awake mouse, cortex, freely whisking,

Spontaneous activity in vivo

Neuronal Dynamics – 8.1 Review: Variability in vivo Variability

- of membrane potential? - of spike timing?

Neuronal Dynamics – 8.1 Review. Variability

Fluctuations-of membrane potential-of spike times fluctuations=noise?

model of fluctuations?

relevance for coding?

source of fluctuations?

In vivo data looks ‘noisy’

In vitro data fluctuations

- Intrinsic noise (ion channels)

Na+

K+

-Finite number of channels-Finite temperature

-Network noise (background activity)

-Spike arrival from other neurons-Beyond control of experimentalist

Check intrinisic noise by removing the network

Neuronal Dynamics – 8.1. Review Sources of Variability


Na+

K+

-Network noise

Neuronal Dynamics – 8.1. Review: Sources of Variability

small contribution!

big contribution!

In vivo data looks ‘noisy’

In vitro data small fluctuations nearly deterministic

Neuronal Dynamics – 8.1 Review: Calculating the mean

)()( k

fk

fk

syn ttwtRI

)'()'(')( 1 k

fk

fkR

syn ttttdtwtI

k

fk

fkR

syn ttttdtwtII )'()'(')( 10

mean: assume Poisson process

k

kkRttdtwI )'('1

0

t

( ) ' ( ') ( ' )fkf

x t dt f t t t t

( ) ' ( ') ( ' )fkf

x t dt f t t t t

rate of inhomogeneousPoisson process

use for exercises

( ) ' ( ') ( ')x t dt f t t t

Neuronal Dynamics – 8.1. Fluctuation of potential

for a passive membrane, predict -mean -varianceof membrane potential fluctuations

Passive membrane=Leaky integrate-and-fire without threshold

)()( tIRuuudt

d synrest

Passive membrane

0RI

)()( k

fk

fk

syn ttwtRI

EPSC

Synaptic current pulses of shape a

)(tIR

)()( tIRuuudt

d synrest

Passive membrane

)()( 0 tIItI fluctsyn

Neuronal Dynamics – 8.1. Fluctuation of current/potential

Blackboard,

Math detour:

White noise

0( ) ( ) ( )synRI t RI t t

2

( ) 0

( ) ( ') ( ')

t

t t a t t

I(t) Fluctuating input current

0I

Neuronal Dynamics – 8.1 Calculating autocorrelations

( ) ' ( ') ( ) '

( ) ( ) ( )

x t dt f t t I t

x t ds f s I t s

( ) ( ) ( )x t ds f s I t s 0( ) ( ) [ ( ) ( ) ]x t ds f s I t s t s

Autocorrelation

( ) ( ')x t x t

ˆ ˆ( ) ( ) ' " ( ') ( ") ( ') ( ")x t x t dt dt f t t f t t I t I t

Mean:Blackboard,

Math detour

I(t)

Fluctuating input current

0 ( )I t

0( ) ( ) ( )I t I t t

0( ) ( ) ( )x t ds f s I t s

White noise: Exercise 1.1-1.2 now

2)()()()()( tututututu

)(tu)(tu

Expected voltage at time t ( ) ?u t

Input starts here

Assumption:far away from theshold

Variance of voltage at time t

Next lecture: 9:56

Diffusive noise (stochastic spike arrival)


)(tu

0( ) ( )u t u t

)()()( ttIRuuudt

drest

Math argument

( ') ( ) ( ) ( ') ( ) ( ')u t u t u t u t u t u t

2 2[ ( )] [1 exp( 2 / )]uu t t

Neuronal Dynamics – 8.1 Calculating autocorrelations

t

( ) ' ( ') ( ' )

' ( ') ( ')

fk

f

x t dt f t t t t

dt f t t S t

( ) ' ( ') ( ')x t dt f t t S t

rate of inhomogeneousPoisson process

( ) ( ) ( )x t ds f s t s

Autocorrelation

( ) ( ')x t x t

ˆ ˆ( ) ( ) ' " ( ') ( ") ( ') ( ")x t x t dt dt f t t f t t S t S t

Mean:

Blackboard,

Math detour

Stochastic spike arrival: excitation, total rate

)()( tSRuuudt

drest

u0u

Synaptic current pulses

Exercise 2.1-2.3 now: Diffusive noise (stochastic spike arrival)

1. Assume that for t>0 spikes arrive stochastically with rate - Calculate mean voltage

2. Assume autocorrelation - Calculate

( ) ( )fef

S t q t t

2( ) ( ') ( ')S t S t t t

?)()( tutu

)(tS

Next lecture: 9h58

Assumption: stochastic spiking rate

Poisson spike arrival: Mean and autocorrelation of filtered signal

( ) ( )f

f

S t t t ( ) ( ) ( )x t F s S t s ds

mean( )t

( )F s

( ) ( ) ( )x t F s S t s ds ( ) ( ) ( )x t F s t s ds

( ) ( ') ( ) ( ) ( ') ( ' ') 'x t x t F s S t s ds F s S t s ds ( ) ( ') ( ) ( ') ( ) ( ' ') 'x t x t F s F s S t s S t s dsds

Autocorrelation of output

Autocorrelation of input

Filter


Week 8 – Variability and Noise:

Autocorrelation



- superthreshold and subthreshold8.4 Escape noise -renewal model

Week 8 – part 2 : Autocorrelation of Poisson Process

Stochastic spike arrival:

Justify autocorrelation of spike input:Poisson process in discrete time

In each small time step Prob. Of firing

Firing independent between one time step and the next

tp t

Blackboard

Stochastic spike arrival: excitation, total rate

Exercise 2 now: Poisson process in discrete time

Show that autocorrelation for

2)'()'()( tttStS

TTN )(

In each small time step Prob. Of firing

Firing independent between one time step and the next

Show that in an a long interval of duration T, the expected number of spikes is

tp t

0t

Next lecture: 10:46

0FP v t Probability of spike in time step:

Neuronal Dynamics – 8.2. Autocorrelation of Poisson

t

spike train

math detour now!

Probability of spike in step n AND step k

20 0( ) ( ') ( ') [ ]S t S t v t t v

Autocorrelation (continuous time)

Quiz – 8.1. Autocorrelation of Poisson

t

spike train

( ) ( ')S t S t

The Autocorrelation (continuous time)

Has units

[ ] probability (unit-free)[ ] probability squared (unit-free)[ ] rate (1 over time)[ ] (1 over time)-squred



Autocorrelation



- superthreshold and subthreshold8.4 Escape noise -renewal model

Week 8 – part 3 : Noisy Integrate-and-fire

Neuronal Dynamics – 8.3 Noisy Integrate-and-firefor a passive membrane, we can analytically predict the mean of membrane potential fluctuations

Passive membrane=Leaky integrate-and-fire without threshold

)()( tIRuuudt

d synrest

Passive membrane

ADD THRESHOLD Leaky Integrate-and-Fire

I0I

)(tI

)()( tIRuuudt

drest

noiseo IItI )(

effective noise current

u(t)

( ) ( ) rIF u t THEN u t u

noisy input/diffusive noise/stochastic spikearrival

LIF

Neuronal Dynamics – 8.3 Noisy Integrate-and-fire

I(t)

fluctuating input current

fluctuating potential

Random spike arrival


stochastic spike arrival in I&F – interspike intervals

I0I

ISI distribution


0( ) ( ) ( )synRI t RI t t

white noise

Superthreshold vs. Subthreshold regime


u(t)

Neuronal Dynamics – 8.3. Stochastic leaky integrate-and-fire

noisy input/ diffusive noise/stochastic spike arrival

subthreshold regime: - firing driven by fluctuations - broad ISI distribution - in vivo like

ISI distribution

Crochet et al., 2011

awake mouse, freely whisking,

Spontaneous activity in vivo

Neuronal Dynamics – review- Variability in vivo

Variability of membrane potential?

Subthreshold regime

fluctuating potential22( ) ( ) [ ( )] ( )u t u t u t u t

Passive membrane( ) ( ' )

' ( ') ( ')k

fk k

k f

kk

u t w t t

w dt t t S t

for a passive membrane, we can analytically predict the amplitude of membrane potential fluctuations

Leaky integrate-and-firein subthreshold regime In vivo like

Stochastic spike arrival:

Neuronal Dynamics – 8.3 Summary:Noisy Integrate-and-fire




Week 8 – Noisy input models: barrage of spike arrivals

THE END








Week 8 – part 4 : Escape noise


Na+

K+

-Finite number of channels-Finite temperature

-Network noise (background activity)

-Spike arrival from other neurons-Beyond control of experimentalist

Neuronal Dynamics – Review: Sources of Variability

small contribution!

big contribution!Noise models?

escape process,stochastic intensity

stochastic spike arrival (diffusive noise)

Noise models

u(t)

t

)(t

))(()( tuftescape rate

)(tRIudt

dui

i

noisy integration

Relation between the two models: later

Now:Escape noise!

t̂ t̂

escape process

u(t)

t

)(t


u

Neuronal Dynamics – 8.4 Escape noise

1 ( )( ) exp( )

u tt

escape rate

( ) ( )rest

du u u RI t

dt

f frif spike at t u t u

Example: leaky integrate-and-fire model

t̂

escape process

u(t)

t

)(t


u

Neuronal Dynamics – 8.4 stochastic intensity

1 ( )( ) exp( )

( )

u tt

t

examples

Escape rate = stochastic intensity of point process

( ) ( ( ))t f u t

t̂

u(t)

t

)(t


u

Neuronal Dynamics – 8.4 mean waiting time

t

I(t)

( ) ( )rest

du u u RI t

dt

mean waiting time, after switch

1ms

1ms

Blackboard,

Math detour

u(t)

t

)(t

Neuronal Dynamics – 8.4 escape noise/stochastic intensity

Escape rate = stochastic intensity of point process

( ) ( ( ))t f u t

t̂

Neuronal Dynamics – Quiz 8.4Escape rate/stochastic intensity in neuron models[ ] The escape rate of a neuron model has units one over time[ ] The stochastic intensity of a point process has units one over time[ ] For large voltages, the escape rate of a neuron model always saturates at some finite value[ ] After a step in the membrane potential, the mean waiting time until a spike is fired is proportional to the escape rate [ ] After a step in the membrane potential, the mean waiting time until a spike is fired is equal to the inverse of the escape rate [ ] The stochastic intensity of a leaky integrate-and-fire model with reset only depends on the external input current but not on the time of the last reset[ ] The stochastic intensity of a leaky integrate-and-fire model with reset depends on the external input current AND on the time of the last reset



Autocorrelation



- superthreshold and subthreshold8.4 Escape noise - stochastic intensity8.5 Renewal models

Week 8 – part 5 : Renewal model

Neuronal Dynamics – 8.5. Interspike Intervals

( ) ( )du F u RI t

dt

f frif spike at t u t u

Example: nonlinear integrate-and-fire model

deterministic part of input( ) ( )I t u t

noisy part of input/intrinsic noise escape rate

( ) ( ( )) exp( ( ) )t f u t u t

Example: exponential stochastic intensity

t

escape process

u(t)Survivor function

t̂ t

)(t))(()( tuft

escape rate

)ˆ()()ˆ( ttStttS IIdtd

Neuronal Dynamics – 8.5. Interspike Interval distribution

t

t

escape processA

u(t)

t

t

dtt^

)')'(exp( )ˆ( ttS I

Survivor function

t

)(t


t

t

dttt^

)')'(exp()( )ˆ( ttPI

Interval distribution

Survivor functionescape rate

)ˆ()()ˆ( ttStttS IIdtd

Examples now

u

Neuronal Dynamics – 8.5. Interspike Intervals

t̂

t̂

))ˆ(exp())ˆ(()ˆ( ttuttuftt

escape rate)(t

Example: I&F with reset, constant input

t̂

Survivor function1

0ˆ( )S t t )')ˆ'(exp()ˆ( ̂

t

t

dtttttS


t̂

0ˆ( )P t t

)ˆ(

)')ˆ'(exp()ˆ()ˆ(ˆ

ttS

dtttttttP

dtd

t

t

Neuronal Dynamics – 8.5. Renewal theory

t̂

))ˆ(exp())ˆ(()ˆ( ttuttuftt

escape rate)(t

Example: I&F with reset, time-dependent input,

t̂

Survivor function1 ˆ( )S t t )')ˆ'(exp()ˆ( ̂

t

t

dtttttS


t̂

ˆ( )P t t)ˆ(

)')ˆ'(exp()ˆ()ˆ(ˆ

ttS

dtttttttP

dtd

t

t

Neuronal Dynamics – 8.5. Time-dependent Renewal theory

t̂

escape rate

0( )u

t for u

neuron with relative refractoriness, constant input

t̂

t̂

Survivor function1

)ˆ(0 ttS 0ˆ( )S t t


t̂

)ˆ(0 ttP 0ˆ( )P t t

0

Neuronal Dynamics – Homework assignement

Neuronal Dynamics – 8.5. Firing probability in discrete time

1

1( ) exp[ ( ') ']k

k

t

k k

t

S t t t dt

1t 2t 3t0 T

Probability to survive 1 time step

1( | ) exp[ ( ) ] 1 Fk k k kS t t t P

Probability to fire in 1 time stepFkP

Neuronal Dynamics – 8.5. Escape noise - experiments

1 ( )( ) exp( )

u tt

escape rate

1 exp[ ( ) ]Fk kP t

Jolivet et al. ,J. Comput. Neurosc.2006

Neuronal Dynamics – 8.5. Renewal process, firing probability

Escape noise = stochastic intensity

-Renewal theory - hazard function - survivor function - interval distribution-time-dependent renewal theory-discrete-time firing probability-Link to experiments

basis for modern methods of neuron model fitting








Week 8 – part 6 : Comparison of noise models

escape process (fast noise)


u(t)

^t ^tt

)(t


)(tRIudt

dui

i

noisy integration

Neuronal Dynamics – 8.6. Comparison of Noise Models

Assumption: stochastic spiking rate

Poisson spike arrival: Mean and autocorrelation of filtered signal

( ) ( )f

f

S t t t ( ) ( ) ( )x t F s S t s ds

mean( )t

( )F s

( ) ( ) ( )x t F s S t s ds ( ) ( ) ( )x t F s t s ds

( ) ( ') ( ) ( ) ( ') ( ' ') 'x t x t F s S t s ds F s S t s ds ( ) ( ') ( ) ( ') ( ) ( ' ') 'x t x t F s F s S t s S t s dsds

Autocorrelation of output

Autocorrelation of input

Filter

Stochastic spike arrival: excitation, total rate Re

inhibition, total rate Ri

)()()(','

''

, fk

fk

i

fk

fk

erest tt

C

qtt

C

quuu

dt

d

u0u

EPSC IPSC

Synaptic current pulses


)()()( ttIRuuudt

drest

Langevin equation,Ornstein Uhlenbeck process

Blackboard



)(tu)(tu

)()()( ttIRuuudt

drest

( ') ( ) ( ) ( ') ( ) ( ')u t u t u t u t u t u t

Math argument: - no threshold - trajectory starts at known value



)(tu

0( ) ( )u t u t

)()()( ttIRuuudt

drest

Math argument

( ') ( ) ( ) ( ') ( ) ( ')u t u t u t u t u t u t

2 2[ ( )] [1 exp( 2 / )]uu t t

uu

Membrane potential density: Gaussian

p(u)

constant input ratesno threshold

Neuronal Dynamics – 8.6. Diffusive noise/stoch. arrivalA) No threshold, stationary input

)(tRIudt

dui

i

noisy integration

uu

Membrane potential density: Gaussian at time t

p(u(t))

Neuronal Dynamics – 8.6 Diffusive noise/stoch. arrivalB) No threshold, oscillatory input

)(tRIudt

dui

i

noisy integration

u

Membrane potential density

u

p(u)

Neuronal Dynamics – 6.4. Diffusive noise/stoch. arrivalC) With threshold, reset/ stationary input

Superthreshold vs. Subthreshold regime

up(u) p(u)

Nearly Gaussian

subthreshold distr.

Neuronal Dynamics – 8.6. Diffusive noise/stoch. arrival

Image:Gerstner et al. (2013)Cambridge Univ. Press;See: Konig et al. (1996)

escape process (fast noise)


A C

u(t)

noise

white(fast noise)

synapse(slow noise)

(Brunel et al., 2001)

t

t

dttt^

)')'(exp()( )¦( ^ttPI : first passage

time problem

)¦( ^ttPI Interval distribution

^t ^tt

Survivor functionescape rate

)(t


)(tRIudt

dui

i

noisy integration

Neuronal Dynamics – 8.6. Comparison of Noise Models

Stationary input:-Mean ISI

-Mean firing rateSiegert 1951

Neuronal Dynamics – 8.6 Comparison of Noise Models

Diffusive noise - distribution of potential - mean interspike interval FOR CONSTANT INPUT

- time dependent-case difficult

Escape noise - time-dependent interval distribution


Noise models: from diffusive noise to escape rates

))(()( 0 tuftescape rate

noisy integration

)(0 tu

t

)2

))((exp(

2

20

tu

)/))((( 0 tuerf

)]('1[ 0 tu

tu '0

))('),(()( 00 tutuft

Comparison: diffusive noise vs. escape rates

))(()( 0 tuft

escape rate

)2

))((exp(

2

20

tu )]('1[ 0 tu))('),(()( 00 tutuft

subthresholdpotential

Probability of first spike

diffusiveescape

Plesser and Gerstner (2000)

Neuronal Dynamics – 8.6 Comparison of Noise Models Diffusive noise

- represents stochastic spike arrival - easy to simulate - hard to calculate

Escape noise - represents internal noise - easy to simulate - easy to calculate - approximates diffusive noise - basis of modern model fitting methods

Neuronal Dynamics – Quiz 8.4.A. Consider a leaky integrate-and-fire model with diffusive noise:[ ] The membrane potential distribution is always Gaussian.[ ] The membrane potential distribution is Gaussian for any time-dependent input.[ ] The membrane potential distribution is approximately Gaussian for any time-dependent input, as long as the mean trajectory stays ‘far’ away from the firing threshold.[ ] The membrane potential distribution is Gaussian for stationary input in the absence of a threshold.[ ] The membrane potential distribution is always Gaussian for constant input and fixed noise level.

B. Consider a leaky integrate-and-fire model with diffusive noise for time-dependent input. The above figure (taken from an earlier slide) shows that[ ] The interspike interval distribution is maximal where the determinstic reference trajectory is closest to the threshold.[ ] The interspike interval vanishes for very long intervals if the determinstic reference trajectory has stayed close to the threshold before - even if for long intervals it is very close to the threshold[ ] If there are several peaks in the interspike interval distribution, peak n is always of smaller amplitude than peak n-1.[ ] I would have ticked the same boxes (in the list of three options above) for a leaky integrate-and-fire model with escape noise.

Documents

Biological Modeling of Neural Networks Week 8 – Noisy input models: Barrage of spike arrivals Wulfram Gerstner EPFL, Lausanne, Switzerland 8.1 Variation