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09.Escape Noise
2016-07-29
harryhare
Escape Noise
12
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 $u\rightarrow-\infty$ , $f\rightarrow0$
$$f(u-\vartheta)=\begin{cases}
0 & for &u<\vartheta\\
\Delta^{-1}& for &u\ge\vartheta
\end{cases}$$
$$f(u-\vartheta)=\frac{1}{\tau_0}$$
$$f(u-\vartheta)=\beta[u-\vartheta]_{+}=\big{\begin{array}{lcc}
0 &for&u<\vartheta\\
\beta(u-\vartheta)&for&u\ge\vartheta
\end{array}$$
$$f(u-\vartheta)=\frac{1}{2\Delta}[1+erf(\frac{u-\vartheta}{\sqrt{2}\sigma})]$$
$$erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}\exp(-y^2)dy$$
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,
$$h(t)=h_0+h_1cos(\Omega t+\varphi_1)$$
$$\eta(s)=\begin{cases}
-\infty & for & s<\Delta^{abs}\\
-\eta_0 \exp{\big(-\frac{s-\Delta^{abs}}{\tau}\big)} & for & s>\Delta^{abs}
\end{cases}$$