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Part II: Population Models BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapters 6-9 Laboratory of Computational.

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Presentation on theme: "Part II: Population Models BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapters 6-9 Laboratory of Computational."— Presentation transcript:

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

49

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

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