Parametric Sensitivities of a Nonsmooth Wilson-Cowan Model with a Comparison to its Smooth Counterpart

The Wilson-Cowan neural mass model groups excitatory and inhibitory neurons and models their communication through a system of ODEs. The neural firing rate function is popularly approximated by a smooth sigmoid function and a discontinuous Heaviside function. The nondifferentiability of the latter function invalidates standard methods that require derivative information, such as stability and sensitivity theory. In this article, we directly analyze a Wilson-Cowan model with a piecewise linear firing rate function (modeling the "all-or-nothing" action potential) using a relatively new tool from generalized derivatives theory called the lexicographic directional derivative. Our contributions include establishing well-posedness of the nonsmooth Wilson-Cowan model and analyzing its stability and parametric sensitivities. In the smooth Wilson-Cowan model the limit cycle, which corresponds to spiking behavior, is globally attractive in the physically-meaningful domain, while the limit cycle is only locally attractive in the nonsmooth model because a locally attractive trivial equilibrium also exists in that case. A local and nonlocal sensitivity analysis of the nonsmooth model uncovers that the inhibitory neuron time scale is the most influential parameter in the nonsmooth case, dominating the effects from the other parameters. This differs significantly from the smooth model, which we also observe to be overall less sensitive to its parameters than the nonsmooth model, despite the state variable solutions in each model following similar spiking behaviors.