Biological Modeling of Neural Networks Week 8 – Noisy output models: Escape rate and soft threshold Wulfram Gerstner EPFL, Lausanne, Switzerland 8.1 Variation.

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Biological Modeling of Neural Networks Week 8 – Noisy output models: Escape rate and soft threshold Wulfram Gerstner EPFL, Lausanne, Switzerland 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 8.5 Renewal models 8.6 Comparison of noise models Week 8 – parts 4-6 Noisy Output: Escape Rate and Soft Threshold

- Intrinsic noise (ion channels) Na + K+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 escape ratenoisy integration Relation between the two models: later Now: Escape noise!

escape process u(t) t escape rate u Neuronal Dynamics – 8.4 Escape noise escape rate Example: leaky integrate-and-fire model

escape process u(t) t escape rate u Neuronal Dynamics – 8.4 stochastic intensity examples Escape rate = stochastic intensity of point process

u(t) t escape rate u Neuronal Dynamics – 8.4 mean waiting time t I(t) mean waiting time, after switch 1ms Blackboard, Math detour

u(t) t Neuronal Dynamics – 8.4 escape noise/stochastic intensity Escape rate = stochastic intensity of point process

Neuronal Dynamics – Quiz 8.4 Escape 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 Gerstner EPFL, Lausanne, Switzerland 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 8.5 Renewal models Week 8 – part 5 : Renewal model

Neuronal Dynamics – 8.5. Interspike Intervals Example: nonlinear integrate-and-fire model deterministic part of input noisy part of input/intrinsic noise  escape rate Example: exponential stochastic intensity t

escape process u(t) Survivor function t escape rate Neuronal Dynamics – 8.5. Interspike Interval distribution tt

escape process A u(t) Survivor function t escape rate Interval distribution Survivor function escape rate Examples now u Neuronal Dynamics – 8.5. Interspike Intervals

escape rate Example: I&F with reset, constant input Survivor function 1 Interval distribution Neuronal Dynamics – 8.5. Renewal theory

escape rate Example: I&F with reset, time-dependent input, Survivor function 1 Interval distribution Neuronal Dynamics – 8.5. Time-dependent Renewal theory

escape rate neuron with relative refractoriness, constant input Survivor function 1 Interval distribution Neuronal Dynamics – Homework assignement

Neuronal Dynamics – 8.5. Firing probability in discrete time Probability to survive 1 time step Probability to fire in 1 time step

Neuronal Dynamics – 8.5. Escape noise - experiments escape rate 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 Gerstner EPFL, Lausanne, Switzerland 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 8.5 Renewal models 8.6 Comparison of noise models Week 8 – part 6 : Comparison of noise models

escape process (fast noise) stochastic spike arrival (diffusive noise) u(t) t escape ratenoisy integration Neuronal Dynamics – 8.6. Comparison of Noise Models

Assumption: stochastic spiking rate Poisson spike arrival: Mean and autocorrelation of filtered signal mean Autocorrelation of output Autocorrelation of input Filter

Stochastic spike arrival: excitation, total rate R e inhibition, total rate R i u EPSC IPSC Synaptic current pulses Diffusive noise (stochastic spike arrival) Langevin equation, Ornstein Uhlenbeck process Blackboard

Diffusive noise (stochastic spike arrival) Math argument: - no threshold - trajectory starts at known value

Diffusive noise (stochastic spike arrival) Math argument

u u Membrane potential density: Gaussian p(u) constant input rates no threshold Neuronal Dynamics – 8.6. Diffusive noise/stoch. arrival A) No threshold, stationary input noisy integration

u u Membrane potential density: Gaussian at time t p(u(t)) Neuronal Dynamics – 8.6 Diffusive noise/stoch. arrival B) No threshold, oscillatory input noisy integration

u Membrane potential density u p(u) Neuronal Dynamics – 6.4. Diffusive noise/stoch. arrival C) With threshold, reset/ stationary input

Superthreshold vs. Subthreshold regime 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) : first passage time problem Interval distribution t Survivor function escape rate escape ratenoisy integration Neuronal Dynamics – 8.6. Comparison of Noise Models Stationary input: -Mean ISI -Mean firing rate Siegert 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 escape rate noisy integration

Comparison: diffusive noise vs. escape rates escape rate subthreshold potential Probability of first spike diffusive escape 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.