Analysis of dynamics of neural fields and neural networks

Abstract

We consider abstract delay differential equations which take values in arbitrary Banach spaces. Here the right-hand side can be broken down into a bounded, non-linear, compact operator which depends on the delay and a linear differential operator which does not need to be bounded. We extend the results of Janssens [2019] on the sun-star calculus for these equations, specifically we describe the generator of the semigroup corresponding to the linearisation of the delay equation. One key result is that the space of continuous functions on a bounded interval taking values in some arbitrary Banach space Y is sun-reflexive with respect to the shift semigroup if and only if Y itself is reflexive. Finally, we develop a method to analytically compute the normal form coefficients of certain bifurcations for these abstract delay differential equations.