Nonsmooth Optimization for Synaptic Depression Dynamics

In the present paper, we present a non-smooth optimization problem for the synaptic depression model. It involves the Laplace operator defined on locally finite graphs and the local Lipschitz function's Clarke subdifferential. Our basic goal is to show the existence of a weak solution to this problem by using a Galerkin-like method involving an exhaustion procedure. On the multivalued nonmonotone and nonconvex part, we assume the so-called Rauch condition, which expresses the nonmonotonicity of the nonlinearities.