There are two main approaches to explain the differences between them. The first one relies on the role of the parasitic roots (this is what we usually teach). The second one is more indirect and based on the general definition of stability. Spijker was the first who presented a norm pair in which the midpoint method is not stable. This example can be extended to the general weakly stable case. Finally, we upgrade this latter approach keeping its advantages and eliminating its weak point.
Neural field equations are models that describe the spatio-temporal evolution of (spatially) coarse grained variables such as synaptic or firing rate activity in populations of neurons. We consider a single population of neurons, distributed over some bounded, connected, open region, whose state is described by their membrane potential. These potentials are assumed to evolve according to an integro-differential equation with space dependent delay.
Neural field models with transmission delay may be cast as abstract delay differential equations, which is the starting formulation for our numerical discretization. The numerical treatment of these systems is rare in the literature and has several restrictions on the space domain and the functions involved. The aim of this work is the development of an accurate numerical method without introducing limitations to its applicability. We present and analyze a novel time-discontinuous Galerkin finite element method. We give a theoretical analysis of the stability and order of accuracy of the numerical discretization and demonstrate the method on a number of neural field computations in one and two space dimensions.