14.Noise in Refractory Kernel and Diffusive Noise

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.,

$$ u_i(t) = \sum_f \eta_0 (t-t_i^{(f)}) + \sum_j w_{ij} \sum_f \epsilon_0 (t-t_j^{(f)}) + \int_0^\infty \kappa_0 (s) I_i ^{ext} (t-s) ds. $$

For noise models, we define refractory kernel

$$ \eta(s) = \eta_0 e^{-s/\tau}, $$

where

$$ \eta_0 \equiv \eta_0(r) = \tilde \eta_0 e^{r/\tau}, $$

which in turn is plugged back into the refractory kernel,

$$ \eta(s) = \tilde \eta_0 e^{- (s-r)/\tau}. $$

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

$$ \rho = f(u-\theta), $$

in which $\theta$ is the threshold. For noise model the threshold is a noisy function.

“Noise reset” model

$$ \theta = u(t \vert \hat t, r) = \eta_0(r) e^{-(t-\hat t)/\tau} + \int_0^\infty \kappa (t-\hat t,s) I(t-s) ds. $$

Spike occur at $t$ when the potential reaches threshold $u(t\vert \hat t,r)=\theta$, thus the interval of spike is given by

$$ T(\hat t,r) = \mathrm{min}\left[ t -\hat t \vert u(t\vert \hat t,r)=\theta \right]. $$

Interval distribution:

$$ P_I(t\lvert \hat t) = \int dr \delta ( t-\hat t - T(\hat t, r) ) \mathscr{G}_0(r). $$

SRM0 model:

$$ \begin{align} u(t\lvert \hat t,r) &= \eta_r(t-\hat t) + h(t)\
&= \tilde \eta_0 e^{-(t-\hat t- r)/\tau} + h(t). \end{align} $$

The stochastic parameter $r$ work as a shift of the spikes on time axis.

Diffusive Noise

Integrate-and-fire model:

$$ \tau_m \frac{d}{d t} u = - u + R I(t). $$

• Membrane time constant $\tau_m$;
• Input resistance $R$;
• Input current $I$.

Introducing noise: add noise to the RHS,

$$ \tau_m \frac{d}{d t} u = - u + R I(t) + \xi (t), $$

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

$$ \langle \xi(t) \xi(t’) \rangle = \sigma^2 \tau_m \delta(t-t’). $$

• $\sigma$ amplitude of noise;
• $\tau_m$ membrane time constant.

c.f. Ornstein-Uhlenbeck process.

Stochastic Spike Arrival

In a network, a integrate-and-fire 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

$$ \frac{d}{dt} u = - \frac{u}{\tau_m} + \frac{1}{C}I^{ext}(t) + \sum_j \sum_{t_j^{(f)} > \hat t} w_j \delta(t- t_j^{(f)}) + \sum_k \sum_{t_k^{(f)} < \hat t} w_k \delta(t-t_k^{(f)}), $$

which is called Stein’s model.

The stochastic spike arrivals are Poissonian.

Example: Membrane Potential Fluctuations

• Poisson process with rate $\nu$

• Input spike train $$ S(t) = \sum_{k=1}^N \sum_{t_k^{(f)}} \delta(t-t_k^{(f)}), $$

  which has an average

  $$ \langle S(t) \rangle = \nu_0, $$

  and autocorrelation

  $$ \langle S(t) S(t’)\rangle - \nu_0^2 = N\nu_0 \delta(t-t’). $$

  $\nu_0^2$ 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. $$ u(t) = w_0 \int_0^\infty \epsilon_0(s) S(t-s) ds $$

  Also neglect the term $-u/\tau_m$?

• Average over time we have $$ u_0 \equiv \langle u(t) \rangle = w_0 \nu_0 \int_0^\infty \epsilon_0(s)ds. $$

• Variance of potential $$ \begin{align} \langle (u-u_0)^2 \rangle &= w_0^2 \left\langle \left( \int_0^\infty \epsilon_0(s) S(t-s) - w_0 \nu_0 \int_0^\infty \epsilon_0(s)ds \right)^2 \right\rangle \
  & = \left\langle w_0^2 \int_0^\infty \int_0^\infty \epsilon_0(s) \epsilon_0(s’) S(t-s) S(t-s’) ds’ ds - 2 u_0 w_0 \nu_0 \int_0^\infty \epsilon_0(s) S(t-s)ds + u_0^2 \right\rangle \
  & = \left\langle w_0^2 \int_0^\infty \int_0^\infty \epsilon_0(s) \epsilon_0(s’) S(t-s) S(t-s’) ds’ ds \right\rangle - 2 u_0 w_0 \left\langle \nu_0 \int_0^\infty \epsilon_0(s) S(t-s)ds \right\rangle + u_0^2 \
  & = \left\langle w_0^2 \int_0^\infty \int_0^\infty \epsilon_0(s) \epsilon_0(s’) S(t-s) S(t-s’) ds’ ds \right\rangle - 2 u_0^2 + u_0^2 \
  & = w_0^2 \nu_0 \int_0^\infty \epsilon_0(s)^2ds \end{align} $$

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

$$ \frac{d}{dt} u = - \frac{u}{\tau_m} + \frac{1}{C}I^{ext}(t) + \sum_j \sum_{t_j^{(f)} > \hat t} w_j \delta(t- t_j^{(f)}) + \sum_k \sum_{t_k^{(f)} < \hat t} w_k \delta(t-t_k^{(f)}), $$

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

$$ 1 - \sum_k\nu_k \Delta t. $$

During this time the membrane potential will decay

$$ u(t+\Delta t) = u(t) e^{-\Delta t/\tau_m}. $$

Incoming spike at synapse $k$:

$$ u(t+\Delta t) = u(t) e^{-\Delta t/\tau_m} + w_k. $$