Exercises

Integral equation of neurons with absolute refractory period .

(i) Apply the population equation ( 14.5 ) to SRM neurons with escape noise which have an absolute refractory period

(ii) Introduce the normalization condition Eq. ( 14.8 ) so as to arrive at the Wilson-Cowan integral equation ( 14.10 ).

(iii) Use Eq. ( 14.8 ) to show that the mean interspike interval of neurons firing stochastically with a rate $f[h_{0}]$

where $h_{0}$ is a constant input potential.

Gain function of SRM neurons

Consider SRM neurons with escape noise such that the hazard function is given by $\rho(s)=\overline{\rho}e^{h+\eta(s)}$ with $\eta(s)=\text{ln}\left[1-e^{-s/\tau}\right]$ .

(i) Show that the survivor function in the asynchronous state is

where $r=\tau\overline{\rho}e^{h}$ .

(ii) Using your results in (i), find the gain function $A_{0}=g(h_{0})$ for neurons.

Hint: Remember that the mean firing rate for fixed $h_{0}$ is the inverse of the mean interval. You will have to use the lower incomplete gamma function $\gamma(a,x)=\int_{0}^{x}t^{a-1}e^{-t}dt$ .

(iii) Suppose that you have SRM ${}_{0}$ neurons with an absolute and a relative refractory period as in Eq. ( 14.29 ). Calculate $A_{0}$ using your result from (ii) and compare with Eq. ( 14.30 ).

Linearization of the Wilson Cowan integral equation .

The aim is to find the frequency dependent gain $\hat{G}(\omega)$ for a population of neurons with absolute refractoriness.

(i) Start from the Wilson-Cowan integral equation and linearize around a stationary state $A_{0}$ .

(ii) Start with the filter in Eq. ( 14.60 ) and derive directly the filter $\hat{G}$ .

Slow noise in the parameters.

Consider a population of leaky integrate-and-fire neurons with time constant $\tau_{m}$ and resistance $R$ , driven by a constant super-threshold input $I_{0}$ . After each firing, the membrane potential is reset to $u_{r}$ which is chosen randomly from a distribution $P(u_{r})$ with mean $\langle u_{r}\rangle$ .

(i) Calculate the interspike interval $T_{0}$ for a neuron $i$ which was reset at time $t_{0}$ to a value $u_{i}(t_{0})=\langle u_{r}\rangle$ and that of another neuron $j$ which was reset at $t_{0}$ to $u_{j}(t_{0})=\langle u_{r}\rangle+\Delta u$ .

(ii) Suppose a Gaussian distribution of reset values with standard deviation $\sigma_{r}$ . Show that the standard deviation $\sigma_{\rm ISI}$ of the interval distribution is $\sigma_{\rm ISI}=\sigma_{r}/\dot{u}(T_{0})$ where $\dot{u}(T_{0})$ is the derivate of the membrane potential at the moment of threshold crossing.

Linear response filter with step-function escape rate .

Consider $f(u)=\rho\,{\mathcal{H}}(u-\vartheta)$ , i.e., a step-function escape rate. For $\rho\to\infty$ neurons fire immediately as soon as $u(t)>\vartheta$ and we are back to a noise-free sharp threshold. For finite $\rho$ , neurons respond stochastically with time constant $\rho^{-1}$ . We will show that

The neuron mode is a SRM ${}_{0}$ with arbitrary refractoriness $\eta(t-\hat{t})$ driven by a constant input $h_{0}$ and a time-dependent component $h_{1}(t)$ . The total membrane potential at time $t$ is $u(t)=\eta(t-\hat{t})+h_{0}+h_{1}(t)$ where $h_{1}(t)=\int_{0}^{\infty}\exp(-s/\tau_{m})\,I_{1}(t-s)\,{\text{d}}s$ .

(i) Show that the kernel ${\mathcal{L}}(x)$ for neurons with step-function escape rate is an exponential function.

[Hint: Denote by $T_{0}$ the time between the last firing time $\hat{t}$ and the formal threshold crossing, $T_{0}={\rm min}\left\{s\,|\,\eta(s)+h_{0}=\vartheta\right\}.$ The derivative of $f$ is a $\delta$ -function in time. Use a short-hand notation $\eta^{\prime}={{\text{d}}\eta(s)\over{\text{d}}s}|_{s=T_{0}}$ and exploit Eq. ( 14.58 ). ]

(ii) Calculate the liner filter $G(s)$ and the response to an input current $I_{1}(t)$ .