Linear-Threshold Dynamics for the Study of Epileptic Events

In this letter, we provide a detailed characterization of the equilibria and bifurcations of two-dimensional linear-threshold models. Using the input to the system as the bifurcation parameter, we characterize the location of the admissible equilibria, show that bifurcations can arise only when equilibria lie on the boundary of well-defined regions of the state space, and prove that (codimension-one) bifurcations can only be of three different types: persistent, non-smooth fold, and Hopf. We show how these bifurcations change the qualitative properties of the system trajectories, and how these behaviors resemble prototypical patterns of EEG activity observed before, during, and after seizure events in the human brain. Our findings suggest that low-dimensional linear threshold models can effectively be used to model, analyze, predict, and ultimately regulate the interactions of neuronal populations in the human brain.