Utilizing discontinuous piecewise Lyapunov function for a biological PWA system and extending the analysis via input-to-state type stability

We study a piecewise affine (PWA) model of the spiking of neurons located in a subsystem of the olfactory system. Our long-term goal is to understand the stability properties of a network of such neurons and as a first step towards this goal we consider the simplest case of the interaction between two types of neurons (excitatory and inhibitory). Due to the discontinuous nature of the PWA model, it is challenging to find a continuous Lyapunov function, we therefore utilize a recently proposed constructive method to find a discontinuous Lyapunov function. In order to utilize this method, it is necessary to define a suitable polyhedral partition of the state-space and carefully investigate the dynamics at the boundaries. As an additional step, we propose extending the analysis by employing tools from the input-to-state (ISS) stability framework.