Exploring Membrane Potential Densities and the Fokker-Planck Equation in Neural Networks

Delve into the concepts of membrane potential densities and the Fokker-Planck Equation in neural networks, covering topics such as integrate-and-fire with stochastic spike arrival, continuity equation for membrane potential density, jump and drift flux, and the intriguing Fokker-Planck Equation.

• Neural Networks
• Membrane Potential
• Fokker-Planck Equation
• Stochastic Spike Arrival
• Continuity Equation

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1. Week 13 Membrane Potential Densities and Fokker-Planck Equation 13.1 Review: Integrate-and-fire - stochastic spike arrival 13.2 Density of membrane potential - Continuity equation 13.3 Flux - jump flux - drift flux 13.4. Fokker Planck Equation - free solution 13.5. Threshold and reset - time dependent activity - network states Biological Modeling of Neural Networks: Week 13 Membrane potential densities and Fokker-Planck Wulfram Gerstner EPFL, Lausanne, Switzerland

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

3. Week 13-part 1: Review: leaky integrate-and-fire model j i iu I d u u = eq+ If firing: ( ) ( ) u u u RI t reset dt d = + = ( ); ( ) V V RI t V u u eq dt

4. Week 13-part 1: Review: leaky integrate-and-fire model d LIF = eq + ( ) ( ) u u u RI t dt u u If firing: reset I=0 I>0 d d u u dt dt flow (drift) towards threshold u repetitive u u resting t t

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

6. Week 13-part 1: Review: homogeneous population t ? I(t) Homogeneous network: -each neuron receives input from k neurons in network -each neuron receives the same (mean) external input t + ( ; ) n t t t population activity = ( ) A t N t

7. Week 13-part 1: Review: diffusive noise/stochastic spike arrival Stochastic spike arrival: excitation, total rate Re inhibition, total rate Ri Synaptic current pulses ) ( f k d k ' f f = + ( ) ( ) u u u R q t t q t t eq e i ' k k dt , , ' ' f EPSC + IPSC ) t d mean = + ( ) ( ) ( u u u R I t eq dt u Langevin equation, Ornstein Uhlenbeck process 0u

8. Week 13 part 2 Membrane Potential Densities 13.1 Review: Integrate-and-fire - stochastic spike arrival 13.2 Density of membrane potential - 13.3 Flux - - 13.4. Fokker Planck Equation - - 13.5. Threshold and reset - Biological Modeling of Neural Networks: Week 13 Membrane potential densities and Fokker-Planck Wulfram Gerstner EPFL, Lausanne, Switzerland

9. Week 13-part 2: membrane potential density Blackboard: density of potentials? For any arbitrary neuron in the population f k dt d k ' f f = + ( ( ) ( )) u u R q t t q t t e i ' k k , , ' ' f EPSC u IPSC + q C d dt k f = + f ext ( ) ( ) t e u t t I k , external current input excitatory input spikes

10. Week 13-part 2: continuity equation d dt d du = ( , ) p u t ( , ) J u t

11. Exercise 1: flux caused by stochastic spike arrival Membrane potential density Next lecture: 10h15 u u p(u) Reference level u0 du dt a) Jump at time t = u u + f ( ) ( )} R q t t eq e f What is the flux across u0? b) c) spike arrival rate spike arrival rate k k

12. Week 13 Membrane Potential Densities and Fokker-Planck Equation 13.1 Review: Integrate-and-fire - stochastic spike arrival 13.2 Density of membrane potential - continuity equation - 13.3 Flux - jump flux - drift flux 13.4. Fokker Planck Equation - 13.5. Threshold and reset - Biological Modeling of Neural Networks: Week 13 Membrane potential densities and Fokker-Planck Wulfram Gerstner EPFL, Lausanne, Switzerland

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

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

15. du dt flux two possibilities = u u + + ( ) ext t f ( ) ( )} R I q t t eq e f Membrane potential density What is the flux across u0? u u p(u) Reference level u0 Jumps caused at a) flux caused by jumps due to stochastic spike arrival spike arrival rate Blackboard: Slope and density of potentials b) flux caused by systematic drift

16. Week 13 Membrane Potential Densities and Fokker-Planck Equation 13.1 Review: Integrate-and-fire - stochastic spike arrival 13.2 Density of membrane potential - 13.3 Flux - continuity equation 13.4. Fokker Planck Equation - source and sink - 13.5. Threshold and reset - Biological Modeling of Neural Networks: Week 13 Membrane potential densities and Fokker-Planck Wulfram Gerstner EPFL, Lausanne, Switzerland

17. Week 13-part 4: from continuity equation to Fokker-Planck Blackboard: Derive Fokker-Planck equation For any arbitrary neuron in the population q d u u dt C 1 C q C ', ' k f = + + ' f f ext ( ) ( ) ( ) t e i t t t t I ' k k , k f EPSC external input IPSC Continuity equation: Flux: - jump (spike arrival) - drift (slope of trajectory) = ( , ) p u t ( , ) J u t t u

18. Week 13-part 4: Fokker-Planck equation Membrane potential density u p(u) Fokker-Planck 2 = [ ( ) ( , )] u p u t + 2 ( , ) p u t ( , ) p u t 2 t u u diffusion 1 2 drift = u ) ( k + u kw = 2 2 k w k k k spike arrival rate

19. Exercise 2: solution of free Fokker-Planck equation Membrane potential density: Gaussian Next lecture: 11h25 constant input rates no threshold u u p(u) Fokker-Planck 2 = [ ( ) ( , )] u p u t + 2 ( , ) p u t ( , ) p u t 2 t u u diffusion 1 2 drift u = + + ( ) u ( ) w RI t = 2 2 k w k k k k k spike arrival rate

20. Week 13 Membrane Potential Densities and Fokker-Planck Equation 13.1 Review: Integrate-and-fire - stochastic spike arrival 13.2 Density of membrane potential - 13.3 Flux - continuity equation 13.4. Fokker Planck Equation - - 13.5. Threshold and reset - Biological Modeling of Neural Networks: Week 13 Membrane potential densities and Fokker-Planck Wulfram Gerstner EPFL, Lausanne, Switzerland

21. Week 13-part 5: Threshold and reset (sink and source terms) Membrane potential density u u blackboard p(u) Fokker-Planck 2 = [ ( ) ( , )] u p u t + + 2 ( , ) p u t ( , ) p u t ( ) ( A t ) u u reset 2 t u u diffusion 2 drift u 1 2 = + + ( ) u w RI = 2 k w k k k k k spike arrival rate

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

23. Week 13-part 5: population firing rate A(t) = single neuron rate Synaptic current pulses ) ( , ' k f k d ' f f = + ( ) ( ) u u u R q t t q t t eq e i ' k k dt , ' f EPSC IPSC I d mean (t ) I = + + ( ) ( ) ( ) u u u R I t t eq 0I dt d = + ( ) ( ) u u u R I t eq dt frequency = = + ( ) I g ( ) I t I I o noise f with noise effective noise current 0I

24. Week 13-part 5: membrane potential density

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

26. Week 13-part 5: network states frequency = ( ) I g f with noise mean I(T) depends on state Variance/noise depends on state 0I d k k ' f f = + ( ) ( ) ( ) u u u R q t t q t t eq e i ' k k dt , , ' ' f f EPSC IPSC du dt = + + mean ( ) ( ) t ( ) t u u RI eq

27. Week 13-part 5: network states Brunel 2000

28. Exercise 3: Diffusive noise + Threshold Membrane potential density A= u u Fokker-Planck f with noise = ( , ) p u t t p(u) [ ( ) ( , )] u p u t 0I u Miniproject: 12h00 2 + 1 + 2 ( , ) p u t source 2 2 u - Calculate distribution p(u) - Determine population firing rate A

29. THE END Sign up for: - miniproject fraud detection interview