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Part II: Population Models

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

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Part II: Population Models

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

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

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

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

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

  6. noisy integration escaperate Interval distribution noise white (fast noise) synapse (slow noise) : first passage time problem Gaussian about (Brunel et al., 2001) escape rate Survivor function Noise models escape process (fast noise) parameter changes (slow noise) stochastic spike arrival (diffusive noise) A B C u(t) t stochasticreset Interval distribution

  7. Homogeneous Population

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

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

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

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

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

  13. Density equations

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

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

  16. Integral equations

  17. Derived from normalization Population Dynamics

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

  19. Population Dynamics

  20. Wilson-Cowan population equation

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

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

  23. Population activity in spiking neurons (an incomplete history) 1972 - Wilson&Cowan; Knight Amari Integral equation (Heterogeneous, non-spiking) 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. Mean field equations density (voltage, phase) Heterogeneous nets stochastic connectivity Fast transients Knight (1972), Treves (1992,1997), Tsodyks&Sejnowski (1995) Gerstner (1998,2000), Brunel et al. (2001), Bethge et al. (2001)

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

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

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

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

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

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

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

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

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

  33. Transients with noise

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

  35. Transients with noise: Escape noise (noisy threshold)

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

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

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

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

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

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

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

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

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

  45. Transients with noise: relation to experiments

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

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

  48. eye How fast is neuronal signal processing? Simon Thorpe Nature, 1996 animal -- no animal Reaction time experiment Visual processing Memory/association Output/movement

  49. eye How fast is neuronal signal processing? Simon Thorpe Nature, 1996 animal -- no animal # of images Reaction time Reaction time 400 ms Visual processing Memory/association Output/movement Recognition time 150ms

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