biological modeling of neural networks
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Week 8 – part 1 :Variation of membrane potential. 8 .1 Variation of membrane potential - white noise approximation 8.2 Autocorrelation of Poisson 8 .3 Noisy integrate -and- fire - superthreshold and subthreshold 8 .4 Escape noise - stochastic intensity - PowerPoint PPT PresentationTRANSCRIPT
Biological Modeling of Neural Networks
Week 8 – Noisy input models:Barrage of spike arrivalsWulfram 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. VariabilityFluctuations-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
- Intrinsic noise (ion channels)
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
kkR ttdtwI )'('10
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
)()( tIRuuudtd syn
rest
Passive membrane
0RI
)()( k
fk
fk
syn ttwtRI
EPSC
Synaptic current pulses of shape
)(tIR
)()( tIRuuudtd syn
rest
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)
2)()()()()( tututututu
)(tu
0( ) ( )u t u t
)()()( ttIRuuudtd
rest
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
)()( tSRuuudtd
rest
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
( ) ( )ff
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
Biological Modeling of Neural Networks
Week 8 – Variability and Noise:AutocorrelationWulfram GerstnerEPFL, Lausanne, Switzerland
8.1 Variation of membrane potential - white noise approximation8.2 Autocorrelation of Poisson8.3 Noisy integrate-and-fire
- 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
Biological Modeling of Neural Networks
Week 8 – Variability and Noise:AutocorrelationWulfram GerstnerEPFL, Lausanne, Switzerland
8.1 Variation of membrane potential - white noise approximation8.2 Autocorrelation of Poisson8.3 Noisy integrate-and-fire
- 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
)()( tIRuuudtd syn
rest
Passive membrane
ADD THRESHOLD Leaky Integrate-and-Fire
I0I
)(tI
)()( tIRuuudtd
rest
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
Neuronal Dynamics – 8.3 Noisy Integrate-and-fire
stochastic spike arrival in I&F – interspike intervals
I0I
ISI distribution
Neuronal Dynamics – 8.3 Noisy Integrate-and-fire
0( ) ( ) ( )synRI t RI t t
white noise
Superthreshold vs. Subthreshold regime
Neuronal Dynamics – 8.3 Noisy Integrate-and-fire
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
Biological Modeling of Neural Networks
8.1 Variation of membrane potential - white noise approximation8.2 Autocorrelation of Poisson8.3 Noisy integrate-and-fire
- superthreshold and subthreshold
Week 8 – Noisy input models: barrage of spike arrivals
THE END
Biological Modeling of Neural Networks
Week 8 – Noisy input models:Barrage of spike arrivalsWulfram 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 4 : Escape noise
- 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
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
)(tRIudtdu
ii
noisy integration
Relation between the two models: later
Now:Escape noise!
t̂ t̂
escape process
u(t)
t
)(t
))(()( tuftescape rate
u
Neuronal Dynamics – 8.4 Escape noise
1 ( )( ) exp( )u tt
escape rate
( ) ( )restd u u u RI tdt
f frif spike at t u t u
Example: leaky integrate-and-fire model
t̂
escape process
u(t)
t
)(t
))(()( tuftescape rate
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
))(()( tuftescape rate
u
Neuronal Dynamics – 8.4 mean waiting time
t
I(t)
( ) ( )restd u u u RI tdt
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
Biological Modeling of Neural Networks
Week 8 – Variability and Noise:AutocorrelationWulfram GerstnerEPFL, Lausanne, Switzerland
8.1 Variation of membrane potential - white noise approximation8.2 Autocorrelation of Poisson8.3 Noisy integrate-and-fire
- superthreshold and subthreshold8.4 Escape noise - stochastic intensity8.5 Renewal models
Week 8 – part 5 : Renewal model
Neuronal Dynamics – 8.5. Interspike Intervals
( ) ( )d u F u RI tdt
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
))(()( tuftescape rate
t
t
dttt^
)')'(exp()( )ˆ( ttPIInterval distribution
Survivor functionescape rate
)ˆ()()ˆ( ttStttS IIdtd
Examples now
u
Neuronal Dynamics – 8.5. Interspike Intervals
t̂
t̂
))ˆ(exp())ˆ(()ˆ( ttuttufttescape rate
)(t
Example: I&F with reset, constant input
t̂
Survivor function10
ˆ( )S t t )')ˆ'(exp()ˆ( ̂t
t
dtttttS
Interval distribution
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( ttuttufttescape rate
)(t
Example: I&F with reset, time-dependent input,
t̂
Survivor function1 ˆ( )S t t )')ˆ'(exp()ˆ( ̂t
t
dtttttS
Interval distribution
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( ) ut for u
neuron with relative refractoriness, constant input
t̂
t̂
Survivor function1 )ˆ(0 ttS 0ˆ( )S t t
Interval distribution
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 kt
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
Biological Modeling of Neural Networks
Week 8 – Noisy input models:Barrage of spike arrivalsWulfram 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 6 : Comparison of noise models
escape process (fast noise)
stochastic spike arrival (diffusive noise)
u(t)
^t ^tt
)(t
))(()( tuftescape rate
)(tRIudtdu
ii
noisy integration
Neuronal Dynamics – 8.6. Comparison of Noise Models
Assumption: stochastic spiking rate
Poisson spike arrival: Mean and autocorrelation of filtered signal
( ) ( )ff
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
Cq
ttCq
uuudtd
u0u
EPSC IPSC
Synaptic current pulses
Diffusive noise (stochastic spike arrival)
)()()( ttIRuuudtd
rest
Langevin equation,Ornstein Uhlenbeck process
Blackboard
Diffusive noise (stochastic spike arrival)
2)()()()()( tututututu
)(tu)(tu
)()()( ttIRuuudtd
rest
( ') ( ) ( ) ( ') ( ) ( ')u t u t u t u t u t u t
Math argument: - no threshold - trajectory starts at known value
Diffusive noise (stochastic spike arrival)
2)()()()()( tututututu
)(tu
0( ) ( )u t u t
)()()( ttIRuuudtd
rest
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
)(tRIudtdu
ii
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
)(tRIudtdu
ii
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 Gaussiansubthreshold 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)
stochastic spike arrival (diffusive 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
))(()( tuftescape rate
)(tRIudtdu
ii
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
stochastic spike arrival (diffusive noise)
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 tuftescape 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.