Perspectives on Applications of a Stochastic Spiking Neuron Model to Neural Network Modeling
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Presented at Perspectives in Nonlinear Dynamics, Berlim, July 26, 2016.
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Perspectives on Applications of a Stochastic Spiking Neuron Model to Neural Network Modeling
- 9. The stochas'c model • Vi(t): time dependent membrane potential of neuron i at time t for i = 1, …, N; • t: discrete time given by integer multiples of constant step Δ small enough to exclude possibility of a neuron firing more than once during each step; • Xi(t): number of times neuron i fired between t and t+1, namely 0 or 1; • If neuron fires between t and t+1, its potential drops to VR by time t+1; • wij: weight of synapse from neuron j to neuron i; • µ: decay factor (in the interval [0, 1]) due to leakage during time step Δ; • Xi(t) = 1 with probability Φ(Vi(t)); • Φ(V) is assumed to be monotonically increasing and saturating at some saturation potential VS.
- 10. Comment • If Φ(V) = Θ(V−Vth), i.e. 0 for V<Vth and 1 for V>Vth, the model becomes the determinis'c discrete-'me leaky integrate-and-fire model (LIF). • Any other choice of Φ(V) gives a stochas'c neuron Vs
- 13. Macroscopic quan''es • Poten'al distribu'on: frac'on of neurons with poten'al in the range (V, V + dV) at 'me t • Network ac'vity: frac'on of neurons that fired between t and t + 1
- 15. Mean-field analysis • Fully connected network: each neuron receives inputs from all other N − 1 neurons; • VR = 0; • Uniform constant external input: Ii(t) = I; • All weights are equal:
- 17. • The amplitude is the frac'on of neurons with “age” k: neurons that fired between t – k – 1 and t – k and did not fire between t – k and t • For this type of distribu'on de network ac'vity ρ(t) is:
- 19. Examples of Φ(V) Satura'ng monomial func'on of degree r Ra'onal func'on Γ = 1; VT = 0NB.: The determinis'c LIF model corresponds to the monomial func'on with S
- 21. • In the case with µ = 0, neurons “forget” all previous input signals, except those received at t – 1. • P(V, t) contains only two peaks at poten'als: V0(t) = 0 and V1(t) = I + Wρ(t − 1) • Taking into account the normaliza'on condi'on, the frac'ons η0(t) and η1(t) evolve as:
- 22. • Assuming that neurons cannot fire at rest, Φ(0) = 0: • In a stationary regime, the recurrence equations reduce to: • Since Φ(V) ≤ 1, any stationary regime must have ρ ≤ 1/2
- 23. Φ(V) = (ΓV)r, I = 0; r = 1 Con'nuous phase transi'ons Absorbing State ρ = 0 Fixed point ρ > 0 2-cycle ρ1 = ½ − a ρ2 = ½ + a a ≤ ½ − Vs/W WC = 1/Γ Γ = 1 WB = 2/Γ Brochini et al., 2016
- 24. Φ(V) = (ΓV)r, I = 0; r > 1 Discon'nuous phase transi'ons r = 1.2 r = 2 ρ+ ρ− Non trivial solu'on ρ+ only for 1 ≤ r ≤ 2 For r = 2 this solu'on is a point at WC = 2/Γ The discon'nuity goes to zero for r = 1 W = WC(r) Γ = 1 Γ = 1 Brochini et al., 2016
- 25. Φ(V) = (ΓV)r, I = 0; r < 1 Ceaseless ac'vity No absorbing ρ = 0 solu'on Brochini et al., 2016 Γ = 1 ρ > 0 for any W > 0
- 26. Numerical solutions for µ > 0 Φ(V) = (ΓV)r, I = 0; r = 1 Brochini et al., 2016
- 27. Discon'nuous phase transi'ons for VT > 0 Φ(V) = (Γ(V-VT))r, I = 0, µ = 0 ; r = 1, Γ = 1 VT = 0 VT = 0.05 VT = 0.1 The discon'nuity ρC goes to zero for VT à 0 Brochini et al., 2016
- 29. Neuronal avalanches at the cri'cal point Φ(V) = (ΓV)r, I = 0, µ = 0; r = 1, ΓC = WC = 1 Brochini et al., 2016 Avalanche size sta's'cs
- 33. The Potjans-Diesmann Model 105 neurons (80% excit. 20% inhibit.) 109 synapses Available at www.opensourcebrain.org • Model for local cor'cal microcircuit • Integrates experimental data of more than 50 experimental papers • Excitatory and inhibitory neurons modeled by the same determinis'c LIF model • Asynchronous-irregular spiking of neurons • Higher spike rate of inhibitory neurons • Replicates well the distribu'on of spike rates across layers Potjans & Diesmann, 2014
- 35. −40 −35 −30 −25 μ = 0.9 Γ Γ
- 38. Conclusions • The stochas'c neuron model introduced by Galves and Löcherbach is an interes'ng element for studies of networks of spiking neurons • Enables exact analy'c results: – Phase transi'ons – Avalanches, SOC • Simple and efficient numerical implementa'on
- 39. Research Team USP, Sao Paulo USP, Ribeirão Preto Thanks! Unicamp, Campinas NeuroMat L. Brochini M. Abadi A. Galves A. Costa O. Kinouchi J. Stolfi R. Shimoura V. Cordeiro E. Löcherbach Univ. Cergy-Pontoise USP, Sao Paulo