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
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:
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