Continuum limit of discrete neuronal structures: is cortical tissue an "excitable" medium?

As a simple model of cortical tissue, we study a locally connected network of spiking neurons in the continuum limit of space and time. This is to be contrasted with the usual numerical simulations that discretize both of them. Refractoriness, noise, axonal delays, and the time course of excitatory and inhibitory postsynaptic potentials have been taken into account explicitly. We pose, and answer, the question of whether the continuum limit presents a full description of scenarios found numerically (the answer is no, not quite). In other words, can the numerics be reduced to a continuum description of a well-known type? As a corollary, we derive some classical results such as those of Wilson and Cowan (1973), thus indicating under what conditions they are valid. Furthermore, we show that spatially discrete objects may be fragile due to noise arising from the stochasticity of the individual neurons, whereas they are not once the continuum limit has been taken. This, then, resolves the above question. Finally, we indicate how one can directly incorporate orientation preference of the neurons.