Download presentation

Presentation is loading. Please wait.

Published byPenelope Black Modified over 4 years ago

1
Part II: Population Models BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapters 6-9 Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne Swiss Federal Institute of Technology Lausanne, EPFL

2
Chapter 6: Population Equations BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 6

3
10 000 neurons 3 km wires 1mm Signal: action potential (spike) action potential

4
Spike Response Model i j Spike reception: EPSP Spike emission: AP Firing: linear threshold Spike emission Last spike of i All spikes, all neurons

5
Integrate-and-fire Model i Spike reception: EPSP Fire+reset linear threshold Spike emission reset I j

6
escape process (fast noise) parameter changes (slow noise) stochastic spike arrival (diffusive noise) Noise models AB 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 rate stochastic reset Interval distribution Gaussian about noisy integration

7
Homogeneous Population

8
populations of spiking neurons I(t) ? population dynamics? t t population activity

9
Homogenous network (SRM) Spike reception: EPSP Spike emission: AP Last spike of i All spikes, all neurons Synaptic coupling potential fully connected N >> 1 external input

10
Last spike of i All spikes, all neurons potential external input potential input potential fully connected refractory potential

11
Homogenous network Response to current pulse Spike emission: AP potential input potential Last spike of i potential external input Population activity All neurons receive the same input

12
Assumption of Stochastic spike arrival: network of exc. neurons, total spike arrival rate A(t) u EPSC Synaptic current pulses Homogeneous network (I&F)

13
Density equations

14
Stochastic spike arrival: network of exc. neurons, total spike arrival rate A(t) u EPSC Synaptic current pulses Density equation (stochastic spike arrival) Langenvin equation, Ornstein Uhlenbeck process

15
u p(u) Density equation (stochastic spike arrival) u Membrane potential density Fokker-Planck drift diffusion spike arrival rate source term at reset A(t)=flux across threshold

16
Integral equations

17
Population Dynamics Derived from normalization

18
Escape Noise (noisy threshold) I&F with reset, constant input, exponential escape rate Interval distribution escape rate

19
Population Dynamics

20
Wilson-Cowan population equation

21
escape process (fast noise) Wilson-Cowan model h(t) t escape rate (i) noisy firing (ii) absolute refractory time population activity (iii) optional: temporal averaging escape rate

22
escape process (fast noise) Wilson-Cowan model h(t) t (i) noisy firing (ii) absolute refractory time population activity escape rate

23
Population activity in spiking neurons (an incomplete history) 1972 - Wilson&Cowan; Knight Amari 1992/93 - Abbott&vanVreeswijk Gerstner&vanHemmen Treves et al.; Tsodyks et al. Bauer&Pawelzik 1997/98 - vanVreeswijk&Sompoolinsky Amit&Brunel Pham et al.; Senn et al. 1999/00 - Brunel&Hakim; Fusi&Mattia Nykamp&Tranchina Omurtag et al. Fast transients Knight (1972), Treves (1992,1997), Tsodyks&Sejnowski (1995) Gerstner (1998,2000), Brunel et al. (2001), Bethge et al. (2001) Integral equation Mean field equations density (voltage, phase) Heterogeneous nets stochastic connectivity (Heterogeneous, non-spiking)

24
Chapter 7: Signal Transmission and Neuronal Coding BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 7

25
Coding Properties of Spiking Neuron Models Course (Neural Networks and Biological Modeling) session 7 and 8 Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne Swiss Federal Institute of Technology Lausanne, EPFL PSTH(t) 500 trials I(t) forward correlation fluctuating input I(t) reverse correlation Probability of output spike ? I(t) A(t)?

26
Theoretical Approach - population dynamics - response to single input spike (forward correlation) - reverse correlations A(t) 500 neurons PSTH(t) 500 trials I(t)

27
Population of neurons h(t) I(t) ? A(t) potential A(t) t population activity N neurons, - voltage threshold, (e.g. IF neurons) - same type (e.g., excitatory) ---> population response ?

28
Coding Properties of Spiking Neurons: Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne Swiss Federal Institute of Technology Lausanne, EPFL - forward correlations - reverse correlations 1. Transients in Population Dynamics - rapid transmission 2. Coding Properties

29
Example: noise-free Population Dynamics I(t) h’>0 h(t) T(t^) higher activity

30
noise-free Theory of transients I(t) h(t) I(t) ? potential input potential A(t) External input. No lateral coupling

31
Theory of transients A(t) no noise I(t) h(t) noise-free noise model B slow noise I(t) h(t) (reset noise)

32
u p(u) u Membrane potential density Hypothetical experiment: voltage step u p(u) Immediate response Vanishes linearly

33
Transients with noise

34
escape process (fast noise) parameter changes (slow noise) stochastic spike arrival (diffusive noise) Noise models AB C u(t) noise white (fast noise) synapse (slow noise) (Brunel et al., 2001) Interval distribution t Survivor function escape rate escape rate stochastic reset Interval distribution Gaussian about noisy integration

35
Transients with noise: Escape noise (noisy threshold)

36
linearize Theory with noise A(t) I(t) h(t)inverse mean interval low noise low noise: transient prop to h’ high noise: transient prop to h h: input potential high noise

37
Theory of transients A(t) low noise I(t) h(t) noise-free (escape noise/fast noise) noise model A low noise fast noise model A I(t) h(t) (escape noise/fast noise) high noise slow

38
Transients with noise: Diffusive noise (stochastic spike arrival)

39
escape process (fast noise) parameter changes (slow noise) stochastic spike arrival (diffusive noise) Noise models AB C u(t) noise white (fast noise) synapse (slow noise) (Brunel et al., 2001) Interval distribution t Survivor function escape rate escape rate stochastic reset Interval distribution Gaussian about noisy integration

40
u p(u) Diffusive noise u Membrane potential density p(u) Hypothetical experiment: voltage step Immediate response vanishes quadratically Fokker-Planck

41
u p(u) SLOW Diffusive noise u Membrane potential density Hypothetical experiment: voltage step Immediate response vanishes linearly p(u)

42
Signal transmission in populations of neurons Connections 4000 external 4000 within excitatory 1000 within inhibitory Population - 50 000 neurons - 20 percent inhibitory - randomly connected -low rate -high rate input

43
Population - 50 000 neurons - 20 percent inhibitory - randomly connected Signal transmission in populations of neurons 100 200 time [ms] Neuron # 32374 50 u [mV] 100 0 10 A [Hz] Neuron # 32340 32440 100 200 time [ms] 50 -low rate -high rate input

44
Signal transmission - theory - no noise - slow noise (noise in parameters) - strong stimulus - fast noise (escape noise) prop. h(t) (potential) prop. h’(t) (current) See also: Knight (1972), Brunel et al. (2001) fast slow

45
Transients with noise: relation to experiments

46
Experiments to transients A(t) V1 - transient response V4 - transient response Marsalek et al., 1997 delayed by 64 ms delayed by 90 ms V1 - single neuron PSTH stimulus switched on Experiments

47
input A(t) See also: Diesmann et al.

48
How fast is neuronal signal processing? animal -- no animal Simon Thorpe Nature, 1996 Visual processingMemory/associationOutput/movement eye Reaction time experiment

50
How fast is neuronal signal processing? animal -- no animal Simon Thorpe Nature, 1996 Reaction time # of images 400 ms Visual processingMemory/associationOutput/movement Recognition time 150ms eye

51
Coding properties of spiking neurons

52
Coding properties of spiking neurons - response to single input spike (forward correlations) A(t) 500 neurons PSTH(t) 500 trials I(t)

53
Coding properties of spiking neurons - response to single input spike (forward correlations) I(t) Spike ? Two simple arguments 1) 2) Experiments: Fetz and Gustafsson, 1983 Poliakov et al. 1997 (Moore et al., 1970) PSTH=EPSP (Kirkwood and Sears, 1978) PSTH=EPSP’

54
Forward-Correlation Experiments A(t) Poliakov et al., 1997 I(t) PSTH(t) 1000 repetitions noise high noise low noise prop. EPSP d dt

55
Population Dynamics h: input potential A(t) PSTH(t) I(t) full theory linear theory

56
Forward-Correlation Experiments A(t) Theory: Herrmann and Gerstner, 2001 high noise low noise Poliakov et al., 1997 high noise low noise blue: full theory red: linearized theory blue: full theory red: linearized theory

57
Forward-Correlation Experiments A(t) Poliakov et al., 1997 I(t) PSTH(t) 1000 repetitions noise high noise low noise prop. EPSP d dt prop. EPSP d dt

58
Reverse Correlations Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne Swiss Federal Institute of Technology Lausanne, EPFL fluctuating input I(t)

59
Reverse-Correlation Experiments after 1000 spikes

60
h: input potential Linear Theory Fourier Transform Inverse Fourier Transform

61
Signal transmission I(t) A(t) T=1/f (escape noise/fast noise) noise model A low noise high noise noise model B (reset noise/slow noise) high noise no cut-off low noise

62
Reverse-Correlation Experiments (simulations) after 1000 spikes theory: G(-s) after 25000 spikes

63
Laboratory of Computational Neuroscience, EPFL, CH 1015 Lausanne Coding Properties of spiking neurons I(t) ? - spike dynamics -> population dynamics - noise is important - fast neurons for slow noise - slow neurons for fast noise - implications for - role of spontaneous activity - rapid signal transmission - neural coding - Hebbian learning

64
Chapter 8: Oscillations and Synchrony BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 8

65
Stability of Asynchronous State

66
Search for bifurcation points linearize h: input potential A(t) fully connected coupling J/N

67
Stability of Asynchronous State A(t) delay period 0 for stable noise s

68
Stability of Asynchronous State s

69
Chapter 9: Spatially structured networks BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 9

70
Continuous Networks

71
Several populations Continuum

72
Continuum: stationary profile

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google