A Contraction Argument for Two-Dimensional Spiking Neuron Models.
Description of Project: Two-dimensional spiking neuron models are a class of dynamical systems that combine continuous dynamics with a discrete reset. The models are capable of reproducing a variety of experimentally observed spiking patterns, and also have the advantage of being mathematically tractable. Previous work on these models has shown that the spiking dynamics are related to the orbits under a discrete map, the adaptation map, whose dynamics and bifurcations have been studied. For example, if the adaptation map is contracting, then all orbits converge to a unique fixed point, and this corresponds to a regular spiking behaviour. Here we derive an analytical method for estimating the contraction of the adaptation map, and we give simple conditions on the parameters of the model for regular spiking to occur. We illustrate the application of this method using three examples. In the first two examples, regular spiking follows directly. In the third example, with some additional quantitative arguments it is proved that regular spiking occurs.