When Is an Integrate-and-Fire Neuron like a Poisson Neuron?

Abstract

In the Poisson neuron model, the output is a rate-modulated Pois(cid:173) son process (Snyder and Miller, 1991); the time varying rate pa(cid:173) rameter ret) is an instantaneous function G[.] of the stimulus, ret) = G[s(t)]. In a Poisson neuron, then, ret) gives the instan(cid:173) taneous firing rate-the instantaneous probability of firing at any instant t-and the output is a stochastic function of the input. In part because of its great simplicity, this model is widely used (usu(cid:173) ally with the addition of a refractory period), especially in in vivo single unit electrophysiological studies, where set) is usually taken to be the value of some sensory stimulus. In the integrate-and-fire neuron model, by contrast, the output is a filtered and thresholded function of the input: the input is passed through a low-pass filter (determined by the membrane time constant T) and integrated un(cid:173) til the membrane potential vet) reaches threshold 8, at which point vet) is reset to its initial value. By contrast with the Poisson model, in the integrate-and-fire model the ouput is a deterministic function of the input. Although the integrate-and-fire model is a caricature of real neural dynamics, it captures many of the qualitative fea(cid:173) tures, and is often used as a starting point for conceptualizing the biophysical behavior of single neurons. Here we show how a slightly modified Poisson model can be derived from the integrate-and-fire model with noisy inputs yet) = set) + net). In the modified model, the transfer function G[.] is a sigmoid (erf) whose shape is deter(cid:173) mined by the noise variance /T~. Understanding the equivalence between the dominant in vivo and in vitro simple neuron models may help forge links between the two levels.