14 The Integral-equation Approach


  1. 1.

    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

    η(t)={-ift<Δabs0otherwise.\eta(t)=\left\{\begin{split}-\infty\;{\rm if}\,t<\Delta^{\rm abs}\\ 0\;{\rm otherwise}.\end{split}\right. (14.104)

    (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[h0]f[h_{0}]

    A0-1=Δabs+f(h0)-1A_{0}^{-1}=\Delta^{\rm abs}+f(h_{0})^{-1} (14.105)

    where h0h_{0} is a constant input potential.

  2. 2.

    Gain function of SRM neurons

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

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

    S0(t)=exp(-rtτ+r(1-e-t/τ)),S_{0}(t)=\exp\left(-\frac{rt}{\tau}+r(1-e^{-t/\tau})\right), (14.106)

    where r=τρ¯ehr=\tau\overline{\rho}e^{h} .

    (ii) Using your results in (i), find the gain function A0=g(h0)A_{0}=g(h_{0}) for neurons.

    Hint: Remember that the mean firing rate for fixed h0h_{0} is the inverse of the mean interval. You will have to use the lower incomplete gamma function γ(a,x)=0xta-1e-tdt\gamma(a,x)=\int_{0}^{x}t^{a-1}e^{-t}dt .

    (iii) Suppose that you have SRM 0{}_{0} neurons with an absolute and a relative refractory period as in Eq. ( 14.29 ). Calculate A0A_{0} using your result from (ii) and compare with Eq. ( 14.30 ).

  3. 3.

    Linearization of the Wilson Cowan integral equation .

    The aim is to find the frequency dependent gain G^(ω)\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 A0A_{0} .

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

  4. 4.

    Slow noise in the parameters.

    Consider a population of leaky integrate-and-fire neurons with time constant τm\tau_{m} and resistance RR , driven by a constant super-threshold input I0I_{0} . After each firing, the membrane potential is reset to uru_{r} which is chosen randomly from a distribution P(ur)P(u_{r}) with mean ur\langle u_{r}\rangle .

    (i) Calculate the interspike interval T0T_{0} for a neuron ii which was reset at time t0t_{0} to a value ui(t0)=uru_{i}(t_{0})=\langle u_{r}\rangle and that of another neuron jj which was reset at t0t_{0} to uj(t0)=ur+Δuu_{j}(t_{0})=\langle u_{r}\rangle+\Delta u .

    (ii) Suppose a Gaussian distribution of reset values with standard deviation σr\sigma_{r} . Show that the standard deviation σISI\sigma_{\rm ISI} of the interval distribution is σISI=σr/u˙(T0)\sigma_{\rm ISI}=\sigma_{r}/\dot{u}(T_{0}) where u˙(T0)\dot{u}(T_{0}) is the derivate of the membrane potential at the moment of threshold crossing.

  5. 5.

    Linear response filter with step-function escape rate .

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

    The neuron mode is a SRM 0{}_{0} with arbitrary refractoriness η(t-t^)\eta(t-\hat{t}) driven by a constant input h0h_{0} and a time-dependent component h1(t)h_{1}(t) . The total membrane potential at time tt is u(t)=η(t-t^)+h0+h1(t)u(t)=\eta(t-\hat{t})+h_{0}+h_{1}(t) where h1(t)=0exp(-s/τm)I1(t-s)   𝑑   sh_{1}(t)=\int_{0}^{\infty}\exp(-s/\tau_{m})\,I_{1}(t-s)\,{\text{d}}s .

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

    [Hint: Denote by T0T_{0} the time between the last firing time t^\hat{t} and the formal threshold crossing, T0=min{s| η(s)+h0=ϑ}.T_{0}={\rm min}\left\{s\,|\,\eta(s)+h_{0}=\vartheta\right\}. The derivative of ff is a δ\delta -function in time. Use a short-hand notation η=   𝑑   η(s)   𝑑   s|s=T0\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)G(s) and the response to an input current I1(t)I_{1}(t) .