From a PDE model to an ODE model of dynamics of synaptic depression

Abstract

We provide a link between two recent models of dynamics of synaptic depression. To this end, we specify the missing transmission conditions in the PDE model of Bielecki and Kalita, and show that if diffusion is fast and communication between pools is slow, the PDE model is well approximated by the ODE model of Aristizabal and Glavinovic. From the mathematical point of view the ODE model is obtained as a singular perturbation of the PDE model with singularities both in the operator and in the boundary and transmission conditions. The result is put in the context of degenerate convergence of semigroups of operators, where a sequence of strongly continuous semigroups approaches a semigroup that is strongly continuous only on a subspace of the original Banach space. Biologically, our approach allows a new, natural interpretation of the ODE model’s parameters.