1. Biological Modeling of Neural Networks Week 11 – Variability and Noise: Autocorrelation Wulfram Gerstner EPFL, Lausanne, Switzerland 11.1 Variation of membrane potential - white noise approximation 11.2 Autocorrelation of Poisson 11.3 Noisy integrate-and-fire - superthreshold and subthreshold 11.4 Escape noise - stochastic intensity 11.5 Renewal models Wulfram Gerstner EPFL, Lausanne, Switzerland Reading for week 11: NEURONAL DYNAMICS Ch. 7.4-7.5.1 Ch. 8.1-8.3 + Ch. 9.1 Cambridge Univ. Press

2. Crochet et al., 2011 awake mouse, cortex, freely whisking, Spontaneous activity in vivo 11.1 Review from week 10 Variability - of membrane potential? - of spike timing?

3. 11.1 Review from week 10 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

4. - 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 11.1. Review from week 10

5. - Intrinsic noise (ion channels) Na + K+K+ -Network noise 11.1. Review from week 10 small contribution! big contribution! In vivo data  looks ‘noisy’ In vitro data  small fluctuations  nearly deterministic

6. 11.1 Review from week 10: Calculating the mean mean: assume Poisson process rate of inhomogeneous Poisson process use for exercises use for next slides

7. 11.1. Fluctuation of potential for a passive membrane, predict -mean -variance of membrane potential fluctuations Passive membrane =Leaky integrate-and-fire without threshold Passive membrane

8. EPSC Synaptic current pulses of shape  Passive membrane 11.1. Fluctuation of current/potential Blackboard, Math detour: White noise I(t) Fluctuating input current

9. 11.1 Calculating autocorrelations Autocorrelation Mean: Blackboard, Math detour I(t) Fluctuating input current

10. White noise: Exercise 1.1-1.2 now Expected voltage at time t Input starts here Assumption: far away from theshold Variance of voltage at time t Next lecture: 10:15 Report variance as function of time!

11. white noise (to mimic stochastic spike arrival) Math argument later 11.1 Calculating autocorrelations for stochastic spike arrival Image: Gerstner et al. (2014), Neuronal Dynamics

12. 11.1 Calculating autocorrelations rate of inhomogeneous Poisson process Autocorrelation Mean: Blackboard, Math detour

13. Assumption: stochastic spiking rate Poisson spike arrival: Mean and autocorrelation of filtered signal mean Autocorrelation of output Autocorrelation of input Filter 11.1 Mean and autocorrelation of filtered spike signal

14. Biological Modeling of Neural Networks Week 11 – Variability and Noise: Autocorrelation Wulfram Gerstner EPFL, Lausanne, Switzerland 11.1 Variation of membrane potential - white noise approximation 11.2 Autocorrelation of Poisson 11.3 Noisy integrate-and-fire - superthreshold and subthreshold 11.4 Escape noise - stochastic intensity 11.5 Renewal models

15. 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 Blackboard 11.2 Autocorrelation of Poisson (preparation)

16. Stochastic spike arrival: excitation, total rate Exercise 3 now: Poisson process in continuous time Show that autocorrelation for In each small time step Prob. Of firing Firing independent between one time step and the next Show that in a a long interval of duration T, the expected number of spikes is Next lecture: 10:40

17. Quiz – 1. Autocorrelation of Poisson spike train The Autocorrelation (continuous time) Has units [ ] probability (unit-free) [ ] probability squared (unit-free) [ ] rate (1 over time) [ ] (1 over time)-squred

18. Probability of spike in time step: 11.2. Autocorrelation of Poisson spike train math detour now! Probability of spike in step n AND step k Autocorrelation (continuous time)

19. Assumption: stochastic spiking (Poisson) rate Autocorrelation of output Autocorrelation of input (Poisson) We integrate twice! 11.2. Autocorrelation of Poisson: units

20. Stochastic spike arrival: excitation, total rate u Synaptic current pulses Exercise 2 Homework: stochastic spike arrival 1.Assume that for t>0 spikes arrive stochastically with rate -Calculate mean voltage 2. Assume autocorrelation - Calculate

21. Biological Modeling of Neural Networks Week 11 – Variability and Noise: Autocorrelation Wulfram Gerstner EPFL, Lausanne, Switzerland 11.1 Variation of membrane potential - white noise approximation 11.2 Autocorrelation of Poisson 11.3 Noisy integrate-and-fire - superthreshold and subthreshold 11.4 Escape noise - stochastic intensity 11.5 Renewal models

22. 11.3 Noisy Integrate-and-fire for a passive membrane, we can analytically predict the mean of membrane potential fluctuations Passive membrane =Leaky integrate-and-fire without threshold Passive membrane ADD THRESHOLD  Leaky Integrate-and-Fire

23. I effective noise current u(t) noisy input/ diffusive noise/ stochastic spike arrival LIF 11.3 Noisy Integrate-and-fire

24. I(t) fluctuating input current fluctuating potential Random spike arrival 11.3 Noisy Integrate-and-fire

25. stochastic spike arrival in I&F – interspike intervals I ISI distribution 11.3 Noisy Integrate-and-fire (noisy input) white noise Image: Gerstner et al. (2014), Neuronal Dynamics,

26. Superthreshold vs. Subthreshold regime 11.3 Noisy Integrate-and-fire (noisy input) Image: Gerstner et al. (2014), Neuronal Dynamics, Cambridge Univ. Press; See: Konig et al. (1996)

27. u(t) 11.3. Noisy integrate-and-fire (noisy input) noisy input/ diffusive noise/ stochastic spike arrival subthreshold regime: - firing driven by fluctuations - broad ISI distribution - in vivo like ISI distribution Image: Gerstner et al. (2014), Neuronal Dynamics,

28. Crochet et al., 2011 awake mouse, freely whisking, Spontaneous activity in vivo review- Variability in vivo Variability of membrane potential? Subthreshold regime Image: Gerstner et al. (2014), Neuronal Dynamics, Cambridge Univ. Press; Courtesy of: Crochet et al. (2011)

29. fluctuating potential Passive membrane for a passive membrane, we can analytically predict the amplitude of membrane potential fluctuations Leaky integrate-and-fire in subthreshold regime can explain variations of membrane potential and ISI Stochastic spike arrival: 11.3 Noisy Integrate-and-fire (noisy input)

30. Biological Modeling of Neural Networks Week 11 – Variability and Noise: Autocorrelation Wulfram Gerstner EPFL, Lausanne, Switzerland 11.1 Variation of membrane potential - white noise approximation 11.2 Autocorrelation of Poisson 11.3 Noisy integrate-and-fire - superthreshold and subthreshold 11.4 Escape noise - stochastic intensity 11.5 Renewal models

31. - 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 Review: Sources of Variability small contribution! big contribution! Noise models?

32. escape process, stochastic intensity stochastic spike arrival (diffusive noise) u(t) t escape ratenoisy integration Relation between the two models: see Ch. 9.4 of Neuronal Dynamics Now: Escape noise! 11.4 Noise models: Escape noise vs. input noise

33. escape process u(t) t escape rate u 11.4 Escape noise escape rate Example: leaky integrate-and-fire model

34. escape process u(t) t escape rate u 11.4 stochastic intensity examples Escape rate = stochastic intensity of point process

35. u(t) t escape rate u 11.4 mean waiting time t I(t) mean waiting time, after switch 1ms Blackboard, Math detour

36. u(t) t 11.4 escape noise/stochastic intensity Escape rate = stochastic intensity of point process

37. Quiz 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

38. Biological Modeling of Neural Networks Week 11 – Variability and Noise: Autocorrelation Wulfram Gerstner EPFL, Lausanne, Switzerland 11.1 Variation of membrane potential - white noise approximation 11.2 Autocorrelation of Poisson 11.3 Noisy integrate-and-fire - superthreshold and subthreshold 11.4 Escape noise - stochastic intensity 11.5 Renewal models

39. 11.5. Interspike Intervals for time-dependent input Example: nonlinear integrate-and-fire model deterministic part of input noisy part of input/intrinsic noise  escape rate Example: exponential stochastic intensity t

40. escape process u(t) Survivor function t escape rate 11.5. Interspike Interval distribution (time-dependent inp.) tt Blackboard

41. escape process A u(t) Survivor function t escape rate Interval distribution Survivor function escape rate Examples now u 11.5. Interspike Intervals

42. escape rate Example: I&F with reset, constant input Survivor function 1 Interval distribution 11.5. Renewal theory

43. escape rate Example: I&F with reset, time-dependent input, Survivor function 1 Interval distribution 11.5. Time-dependent Renewal theory

44. escape rate neuron with relative refractoriness, constant input Survivor function 1 Interval distribution Homework assignement: Exercise 4

45. 11.5. 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 THE END

46. Model of ‘Decoding’ Predict intended arm movement, given Spike Times Application: Neuroprosthetics frontal cortex Outlook: Helping Humans Many groups world wide work on this problem! motor cortex