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 $u<\vartheta$ the neuron may stay quiescent when $u>\vartheta$

Escape rate and hazard function

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

$$\rho=f(u-\vartheta)$$

Since $u$ is a function of time,$\rho$ is also time dependent,

$$\rho_{I}(t|\hat{t})=f[u(t|\hat{t})-\vartheta]$$

Required condition of function $f$, when $u\rightarrow-\infty$, $f\rightarrow0$

Example

$$f(u-\vartheta)=\begin{cases} 0 & for &u<\vartheta\
\Delta^{-1}& for &u\ge\vartheta \end{cases}$$

$$f(u-\vartheta)=\frac{1}{\tau_0}$$

$$f(u-\vartheta)=\beta[u-\vartheta]_{+}=\big{\begin{array}{lcc} 0 &for&u<\vartheta\
\beta(u-\vartheta)&for&u\ge\vartheta \end{array}$$

$$f(u-\vartheta)=\frac{1}{2\Delta}[1+erf(\frac{u-\vartheta}{\sqrt{2}\sigma})]$$

$$erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}\exp(-y^2)dy$$

Interval distribution and mean fire rate

the expect value of interval distribution = $\frac{1}{mean\space fire\space rate}$ = mean period

use $\rho$ we can get interval distribution,

$$P_{I}(t|\hat{t})=\rho\space \exp[-\int_{\hat{t}}^{t}\rho dt]$$

$$\rho=f[u(t|\hat{t})-\vartheta]$$

use $SRM_{0}$,

$$u(t|\hat{t})=\eta(t-\hat{t})+h(t)$$

$$h(t)=\int_{0}^{\infty}\kappa(s)I(t-s)ds$$

use non-leaky integrate-and-fire,

$$u(t|\hat{t})=u_r+\frac{1}{C}\int_{\hat{t}}^{t}I(t’)dt'$$

use leaky integrate-and-fire,

$$u(t|\hat{t})=RI_0[1-e^{(-t-\hat{t})/\tau_m}]$$

use $SRM_0$ with periodic input,we get periodic response,

$$h(t)=h_0+h_1cos(\Omega t+\varphi_1)$$

$$\eta(s)=\begin{cases} -\infty & for & s<\Delta^{abs}\
-\eta_0 \exp{\big(-\frac{s-\Delta^{abs}}{\tau}\big)} & for & s>\Delta^{abs} \end{cases}$$