The formulation of the dynamics of a population of integrate-and-fire neurons on the level of membrane potential densities has been developed by Abbott and van Vreeswijk (4), Brunel and Hakim (78), Fusi and Mattia (163), Nykamp and Tranchina (367), Omurtag et al. (371), and Knight (265), but the Fokker-Planck equation has been used much earlier for the probabilistic description of a single neuron driven by stochastic spike arrivals (243; 88; 430). The classic application of the Fokker-Planck approach to a network of excitatory and inhibitory leaky integrate-and-fire neurons is Brunel (79). For the general theory of Fokker-Planck equations see Risken (440).
Efficient numerical solutions of the Fokker-Planck equation, both for stationary input and periodic input have been developed by M.J.E. Richardson (2007,2009). These methods can be used to find activity in networks of nonlinear integrate-and-fire models, but also generalized neuron models with slow spike-triggered currents or conductances (435). For the treatment of colored noise, see Fourcaud and Brunel (154).
Diffusion limit in the Quadratic integrate-and-fire model . Consider a population of quadratic integrate-and-fire models. Assume that spikes from external sources arrive at excitatory and inhibitory synapses stochastically, and independently for different neurons, at a rate $\nu_{E}(t)$ and $\nu_{I}(t)$, respectively.
(i) Write down the membrane potential density equations assuming that each spike causes a voltage jump by an amount $\pm\Delta u$.
(ii) Take the diffusion limit so as to arrive at a Fokker-Planck equation.
Voltage distribution of the Quadratic integrate-and-fire model . Find the stationary solution of the membrane potential distribution for the quadratic integrate-and-fire model with white diffusive noise.
Non-leaky integrate-and-fire model Consider a non-leaky integrate-and fire model subject to stochastic spike arrival
${{\text{d}}u\over{\text{d}}t}={1\over C}I(t)={q\over C}\sum_{f}\delta(t-t^{(f)})$ | (13.76) |
where $q$ is the charge that each spike puts on the membrane and the spike arrival is constant and equal to $\nu$. At $u=\vartheta=1$ the membrane potential is reset to $u=u_{r}=0$.
(i) Formulate the continuity equation for the membrane potential density equation.
(ii) Make the diffusion approximation
(iii) Solve for the stationary membrane potential density distribution under the assumption that the flux through $u_{r}$ vanishes.
Linear response. A population is driven by a current $I_{0}+I_{1}(t)$. The response of the population is described by
$A(t)=A_{0}+A_{1}(t)=A_{0}+\int_{0}^{\infty}G(s)\,I_{1}(t-s)\,{\text{d}}s\,$ | (13.77) |
where $G$ is called the linear response filter.
(i) Take the Fourier transformation and show that the convolution with the filter $G$ turns into a simple multiplication:
$\hat{A}_{1}(\omega)=\hat{G}(\omega)\,\hat{I}_{1}(\omega)$ | (13.78) |
where the hats denote the Fourier transformed variable.
Hint: Replace $G(s)$ in Eq. (13.77) by a causal filter $G_{c}(s)=0$ for $s\leq 0$ and $G_{c}(s)=G(s)$ for $s>0$, extend the lower integral bound to $-\infty$, and apply standard rules for Fourier transforms.
(ii) The squared quantity $|\hat{I}_{1}(\omega)|^{2}$ is the power of the input at frequency $\omega$. What is the power $|\hat{A}_{1}(\omega)|^{2}$ of the population activity at frequency $\omega$?
(iii) Assume that the filter is given by $G(s)=\exp[-(s-\Delta)/\tau_{g}]$ for $s>\Delta$ and zero otherwise. Calculate $\hat{G}(\omega)$.
(iv) Assume that the input current has a power spectrum $|\hat{I}_{1}(\omega)|^{2}=(c/\omega)$ with $c>0$ for $\omega>\omega_{0}$ and zero otherwise.
What is the power spectrum of the population activity with a linear filter as in (iii)?
Stability of stationary state .
The response of the population is described by
$A(t)=A_{0}+A_{1}(t)=A_{0}+\int_{0}^{\infty}G(s)\,I_{1}(t-s)\,{\text{d}}s\,$ | (13.79) |
where $G$ is called the linear response filter. Set $G(s)=\exp[-(s-\Delta_{g})/\tau_{g}]$ for $s>\Delta_{g}$ and zero otherwise.
Assume that the input arises due to self-coupling with the population:
$I_{1}(t)=\int_{0}^{\infty}\alpha(s)\,A_{1}(t-s)\,{\text{d}}s\,$ | (13.80) |
Set $\alpha(s)=(\alpha_{0}/\tau_{\alpha})\,\exp[-(s-\Delta_{\alpha})/\tau_{\alpha}]$ for $s>\Delta_{\alpha}$ and zero otherwise.
(i) Search for solutions $A_{1}(t)\propto\exp[\lambda(\omega)t]\cos(\omega t)$. The stationary state $A_{0}$ is stable if $\lambda<0$ for all frequencies $\omega$.
(ii) Analyze the critical solutions $\lambda=0$ as a function of the delays $\Delta_{G}$ and $\Delta_{\alpha}$ and the feedback strength $\alpha_{0}$.
Conductance input .
Consider $N_{E}$ excitatory and $N_{I}$ inhibitory leaky integrate-and-fire neurons in the subthreshold regime
$C{du\over dt}=-g_{L}\,(u-E_{L})-g_{E}(t)\,(u-E_{E})-g_{I}(t)\,(u-E_{I})$ | (13.81) |
where $C$ is the membrane capacity, $g_{L}$ the leak conductance and $E_{L},E_{E},E_{I}$ are the reversal potentials for leak, excitation, and inhibition, respectively. Assume that input spikes at excitatory arrive at a rate $\nu_{E}$ and lead to a conductance change
$g_{E}(t)=\Delta g_{E}\,\sum_{j}\sum_{f}\exp[-(t-t_{j}^{(f)})/\tau_{E}]\quad{% \rm for~{}}t>t_{j}^{(f)}$ | (13.82) |
(and zero otherwise) with amplitude $\Delta g_{E}$ and decay time constant $\tau_{E}$. A similar expression holds for inhibition with $\Delta g_{I}=2\Delta g_{E}$ and $\tau_{I}=\tau_{E}/2$. Spike arrival rates are identical $\nu_{I}=\nu_{E}$.
(i) Determine the mean potential $\mu$.
(ii) Introduce
$\alpha_{E}(t-t_{j}^{(f)})=a_{E}\,\exp[-(t-t_{j}^{(f)})/\tau_{E}]{\mathcal{H}}(% t-t_{j}^{(f)})\,(\mu-E_{E})$ | (13.83) |
and an analogous expression for inhibitory input currents $\alpha_{I}$.
Show that the membrane with conductance-based synapses Eq. (13.81) can be approximated by a model with current-based synapses
$\tau_{\rm eff}{du\over dt}=-(u-E_{L})+\sum_{j\in N_{E}}\sum_{f}\alpha_{E}(t-t_% {j}^{(f)})+\sum_{j\in N_{I}}\sum_{f}\alpha_{I}(t-t_{j}^{(f)})$ | (13.84) |
where $E_{L}$ is the leak potential defined earlier in Eq. (13.81). Determine $a_{E},a_{I}$ and $\tau_{\rm eff}$. What are the terms that are neglected in this approximation? Why are they small?
(iii) Assume that the reversal potential for inhibition and leak are the same $E_{I}=E_{L}$. What is the mean potential $\mu$ in this case? How does inhibitory input manifest itself? What would change if you replaced inhibition by a constant current that sets the mean membrane potential (in the presence of the same amount of excitation as before) to $\mu$?
Firing rate of leaky integrate-and-fire neurons in the Brunel-Hakim formulation .
Show that the Siegert formula of Eq. ( 13.30 ) can be also written in the form ( 78 )
${1\over A_{0}\,\tau_{m}}=2\int_{0}^{\infty}{\text{d}}u\,e^{-u^{2}}\left[{e^{2y% _{2}u}-e^{2y_{1}u}\over u}\right]$ | (13.85) |
with $y_{2}=(\vartheta-h_{0})/\sigma$ and $y_{1}=(u_{r}-h_{0})/\sigma$ .
Hint: Use the definition of the error function ${\rm erf}$ given above Eq. ( 13.30 )
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