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Estimation of spike train statistics in spontaneously active biological neural networks
Henry Tuckwell and Jianfeng Feng
We consider the problem of theoretical determination of firing rates in some biological neural networks which consist of synaptically connected excitatory and inhibitory elements. A self-consistent argument is employed to write down equations satisfied by the firing times of the various cells in the network.
We first present results for networks composed of leaky integrate and fire model neurons in the case of impulsive currents representing synaptic inputs and an imposed threshold for firing. Solving a differential-difference equation with specified boundary conditions yields an estimate of the mean interspike interval of neurons in the network. Calculations with a diffusion approximation yield the following results for excitatory networks:
(i) for a given threshold for action potentials, there is a critical number of connections $n_c$ such that for $n < n_c$ there is no nontrivial solution whereas for $n > n_c$ there are three solutions. Of these, one is at zero activity, one is unstable and the other is asymptotically stable; (ii) the critical frequency of firing of neurons in the network is independent of the ratio ($\alpha$) of threshold voltage to excitatory postsynaptic potential amplitude and independent of the size of the network for $n>n_c$; (iii) the critical network size is proportional to $\alpha$.
We also consider a network of generalized Hodgkin-Huxley model neurons. Assuming a voltage threshold, which is a useful representation for slowly firing such nerve cells, a differential equation is obtained whose solution affords an estimate of the mean firing rate. Related differential equations enable one to estimate the second and higher order moments of the interspike interval.