The Cauchy problem for one-dimensional spiking neuron models

I consider spiking neuron models defined by a one-dimensional differential equation and a reset — i.e., neuron models of the integrate-and-fire type. I address the question of the existence and uniqueness of a solution on R for a given initial condition. It turns out that the reset introduces a countable and ordered set of backward solutions for a given initial condition. I discuss the implications of these mathematical results in terms of neural coding and spike timing precision.