Biological Modeling of Neural Networks: Week 13 – Membrane potential densities and Fokker-Planck Wulfram Gerstner EPFL, Lausanne, Switzerland 13.1 Review:

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Presentation transcript:

Biological Modeling of Neural Networks: Week 13 – Membrane potential densities and Fokker-Planck Wulfram Gerstner EPFL, Lausanne, Switzerland 13.1 Review: Integrate-and-fire - stochastic spike arrival 13.2 Density of membrane potential - Continuity equation 13.3 Flux - jump flux - drift flux Fokker –Planck Equation - free solution Threshold and reset - time dependent activity - network states Week 13 – Membrane Potential Densities and Fokker-Planck Equation

Spike reception i j Spike emission -spikes are events -threshold -spike/reset/refractoriness Week 13-part 1: Review: integrate-and-fire-type models

i I j Week 13-part 1: Review: leaky integrate-and-fire model If firing:

LIF If firing: I=0 u I>0 u resting t u repetitive t flow (drift) towards threshold Week 13-part 1: Review: leaky integrate-and-fire model

I(t) Week 13-part 1: Review: microscopic vs. macroscopic

I(t) ? t t population activity Homogeneous network: -each neuron receives input from k neurons in network -each neuron receives the same (mean) external input Week 13-part 1: Review: homogeneous population

Stochastic spike arrival: excitation, total rate R e inhibition, total rate R i u EPSC IPSC Synaptic current pulses Langevin equation, Ornstein Uhlenbeck process Week 13-part 1: Review: diffusive noise/stochastic spike arrival

Biological Modeling of Neural Networks: Week 13 – Membrane potential densities and Fokker-Planck Wulfram Gerstner EPFL, Lausanne, Switzerland 13.1 Review: Integrate-and-fire - stochastic spike arrival 13.2 Density of membrane potential Flux Fokker –Planck Equation Threshold and reset - Week 13 part 2 – Membrane Potential Densities

EPSC IPSC For any arbitrary neuron in the population Blackboard: density of potentials? external current input excitatory input spikes Week 13-part 2: membrane potential density

Week 13-part 2: continuity equation

u p(u) Exercise 1: flux caused by stochastic spike arrival u Membrane potential density spike arrival rate Next lecture: 10h15 b) c) spike arrival rate a) Jump at time t Reference level u 0 What is the flux across u 0 ?

Biological Modeling of Neural Networks: Week 13 – Membrane potential densities and Fokker-Planck Wulfram Gerstner EPFL, Lausanne, Switzerland 13.1 Review: Integrate-and-fire - stochastic spike arrival 13.2 Density of membrane potential - continuity equation Flux - jump flux - drift flux Fokker –Planck Equation Threshold and reset - Week 13 – Membrane Potential Densities and Fokker-Planck Equation

Week 13-part 2: membrane potential density: flux by jumps

Week 13-part 2: membrane potential density: flux by drift

u p(u) flux – two possibilities u Membrane potential density spike arrival rate a) Jumps caused at Reference level u 0 What is the flux across u 0 ? b) flux caused by jumps due to stochastic spike arrival flux caused by systematic drift Blackboard: Slope and density of potentials

Biological Modeling of Neural Networks: Week 13 – Membrane potential densities and Fokker-Planck Wulfram Gerstner EPFL, Lausanne, Switzerland 13.1 Review: Integrate-and-fire - stochastic spike arrival 13.2 Density of membrane potential Flux - continuity equation Fokker –Planck Equation - source and sink Threshold and reset - Week 13 – Membrane Potential Densities and Fokker-Planck Equation

EPSC IPSC For any arbitrary neuron in the population external input Continuity equation: Flux: - jump (spike arrival) - drift (slope of trajectory) Blackboard: Derive Fokker-Planck equation Week 13-part 4: from continuity equation to Fokker-Planck

Fokker-Planck drift diffusion spike arrival rate Membrane potential density u p(u) Week 13-part 4: Fokker-Planck equation

u p(u) Exercise 2: solution of free Fokker-Planck equation u Membrane potential density: Gaussian Fokker-Planck drift diffusion spike arrival rate constant input rates no threshold Next lecture: 11h25

Biological Modeling of Neural Networks: Week 13 – Membrane potential densities and Fokker-Planck Wulfram Gerstner EPFL, Lausanne, Switzerland 13.1 Review: Integrate-and-fire - stochastic spike arrival 13.2 Density of membrane potential Flux - continuity equation Fokker –Planck Equation Threshold and reset - Week 13 – Membrane Potential Densities and Fokker-Planck Equation

u p(u) u Membrane potential density Fokker-Planck drift diffusion spike arrival rate blackboard Week 13-part 5: Threshold and reset (sink and source terms)

u p(u) u Membrane potential density blackboard Week 13-part 5: population firing rate A(t) Population Firing rate A(t): flux at threshold

I effective noise current frequency f with noise EPSC IPSC Synaptic current pulses Week 13-part 5: population firing rate A(t) = single neuron rate

Week 13-part 5: membrane potential density

Week 13-part 5: population activity, time-dependent Nykamp+Tranchina, 2000

Week 13-part 5: network states frequency f with noise EPSC IPSC mean I(T) depends on state Variance/noise depends on state

Week 13-part 5: network states Brunel 2000

u p(u) Exercise 3: Diffusive noise + Threshold u Membrane potential density Fokker-Planck A= f with noise - Calculate distribution p(u) - Determine population firing rate A Miniproject: 12h00

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