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.
The inverse scattering is one of the most useful devices to obtain information about quantum mechanical forces. The solutions of the radial Schrödinger equation, describing mathematically the situation, behave asymptotically like sine functions with shifted phase. The sequence of these phase shifts is a central notion in scattering theory. In this talk some classical and recent results about phase shifts will be presented and some open questions will be discussed.