Noise Using Stochastic Parameter
For a short review of refractory kernel, please refer to page 114 of the textbook (4.2.3 Simplified model SRM0), i.e.,
For noise models, we define refractory kernel
where
which in turn is plugged back into the refractory kernel,
We require that $r$ to be a parameter with mean $\langle r \rangle = 0$.
We discussed threshold in 5.3.1 where we said spikes occur with probability density
in which $\theta$ is the threshold. For noise model the threshold is a noisy function.
“Noise reset” model
Spike occur at $t$ when the potential reaches threshold $u(t\vert \hat t,r)=\theta$, thus the interval of spike is given by
Interval distribution:
SRM0 model:
The stochastic parameter $r$ work as a shift of the spikes on time axis.
Diffusive Noise
Integrateandfire model:
 Membrane time constant $\tau_m$;
 Input resistance $R$;
 Input current $I$.
Introducing noise: add noise to the RHS,
where $\xi(t)$ is a stochastic term thus the equation becomes a stochastic differential equation.
Figure 5.12 is a very nice plot showing the effect of $\xi$ on threshold.
For a Gaussian white noise
 $\sigma$ amplitude of noise;
 $\tau_m$ membrane time constant.
c.f. OrnsteinUhlenbeck process.
Stochastic Spike Arrival
In a network, a integrateandfire neuron will take in
 input $I^{ext}(t)$,
 input spikes at $t^{(f)}_j$, where $j$ means the spike from neuron $j$,
 stochastic spikes (from the background of the brain that we are not really interested in for now) $t_k^{(f)}$,
so that
which is called Stein’s model.
The stochastic spike arrivals are Poissonian.
Example: Membrane Potential Fluctuations
 Poisson process with rate $\nu$

Input spike train
which has an average
and autocorrelation
$\nu_0^2$ is from the constant hazard $\rho_0(t\hat t) = \nu$ and Poisson has autocorrelation $C_{ii}(s) = \nu \delta(s) + \nu^2$.

Neglect both threshold and reset, which basically means weak input so that neuron doesn’t reach firing threshold.
Also neglect the term $u/\tau_m$?
 Average over time we have
 Variance of potential
Figure 5.14: We have equation 5.83 $\langle \delta u^2 \rangle = 0.5 \tau_m \sum_k w_k^2 \nu_k$, larger $w_k$ will give us larger variance of potential so that the spikes are more probable.
Diffusion Limit
Stein model
After each firing, probability density of membrane potential can be calculated.
Between $\Delta t$, the probability of firing is $\sum_k\nu_k \Delta t$. As a result, the probability of quite is
During this time the membrane potential will decay
Incoming spike at synapse $k$:
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