Escape noise

Two ways to introduce noise in formal spiking neuron models:
* noisy threshold(escape model or hazard model)
* noisy integration(stochastic spike arrival model or diffusion model)
In the escape model,
the neuron may fire when u<ϑ
the neuron may stay quiescent when u>ϑ

Escape rate and hazard function

In the escape model,
spikes can occur at any time with a probability density,

ρ=f(u−ϑ)

Since u is a function of time,ρ is also time dependent,

ρI(t|t^)=f[u(t|t^)−ϑ]

Required condition of function f,
when u→−∞, f→0

Example

f(u−ϑ)={0Δ−1forforu<ϑu≥ϑ

f(u−ϑ)=1τ0

f(u−ϑ)=β[u−ϑ]+={0β(u−ϑ)forforu<ϑu≥ϑ

f(u−ϑ)=12Δ[1+erf(u−ϑ2√σ)]

erf(x)=2π√∫x0exp(−y2)dy

Interval distribution and mean fire rate

the expect value of interval distribution = 1mean fire rate = mean period
use ρ we can get interval distribution,

PI(t|t^)=ρ exp[−∫tt^ρdt]

ρ=f[u(t|t^)−ϑ]

use SRM0,

u(t|t^)=η(t−t^)+h(t)

h(t)=∫∞0κ(s)I(t−s)ds

use non-leaky integrate-and-fire,

u(t|t^)=ur+1C∫tt^I(t′)dt′

use leaky integrate-and-fire,

u(t|t^)=RI0[1−e(−t−t^)/τm]

use SRM0 with periodic input,we get periodic response,

h(t)=h0+h1cos(Ωt+φ1)

η(s)={−∞−η0exp(−s−Δabsτ)forfors<Δabss>Δabs