Dynamics of a Piecewise-Linear Morris-Lecar Model: Bifurcations and Spike Adding

Multiple-timescale systems often display intricate dynamics, yet of great mathematical interest and well suited to model real-world phenomena such as bursting oscillations. In the present work, we construct a piecewise-linear version of the Morris-Lecar neuron model, denoted PWL-ML, and we thoroughly analyse its bifurcation structure with respect to three main parameters. Then, focusing on the homoclinic connection present in our PWL-ML, we study the slow passage through this connection when augmenting the original system with a slow dynamics for one of the parameters, thereby establishing a simplified framework for this slow-passage phenomenon. Our results show that our model exhibits equivalent behaviours to its smooth counterpart. In particular, we identify canard solutions that are part of spike-adding transitions. Focusing on the one-spike and on the two-spike scenarios, we prove their existence in a more straightforward manner than in the smooth context. In doing so, we present several techniques that are specific to the piecewise-linear framework and with the potential to offer new tools for proving the existence of dynamical objects in a wider context.