biological modeling of neural networks
DESCRIPTION
Week 8 – parts 4 -6 Noisy Output: Escape Rate and Soft Threshold. 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 - PowerPoint PPT PresentationTRANSCRIPT
Biological Modeling of Neural Networks
Week 8 – Noisy output models:
Escape rate and soft threshold
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 – parts 4-6 Noisy Output: Escape Rate and Soft Threshold
- 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
)(tRIudt
dui
i
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
( ) ( )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
))(()( 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)
( ) ( )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
Biological Modeling of Neural Networks
Week 8 – Variability and Noise:
Autocorrelation
Wulfram 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
( ) ( )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
))(()( tuftescape rate
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
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( 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
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( )u
t 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 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
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 6 : Comparison of noise models
escape process (fast noise)
stochastic spike arrival (diffusive noise)
u(t)
^t ^tt
)(t
))(()( tuftescape rate
)(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
Diffusive noise (stochastic spike arrival)
)()()( ttIRuuudt
drest
Langevin equation,Ornstein Uhlenbeck process
Blackboard
Diffusive noise (stochastic spike arrival)
2)()()()()( tututututu
)(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
Diffusive noise (stochastic spike arrival)
2)()()()()( tututututu
)(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)
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
)(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
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 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.