Abstract
The first goal of this work is to study the solvability of the neural
field equation (known as ‘Amari equation’)
which is an integro‐differential equation in m+ 1 dimensions. In particular, we show the existence of global solutions for smooth activation functions f with values in [0, 1] and L^1^ kernels w via the Banach fixpoint theorem. We note that this setting is much more general than in most related studies, e.g. Ermentrout and McLeod (Proceedings of the Royal Society of Edinburgh 1993; 123A:461–478).
For a Heaviside‐type activation function f, we show that the approach above fails. However, with slightly more regularity on the kernel function w (we use Hölder continuity with respect to the argument x) we can employ compactness arguments, integral equation techniques and the results for smooth nonlinearity functions to obtain a global existence result in a weaker space.
Finally, general estimates on the speed and durability of waves are derived. We show that compactly supported waves with directed kernels (i.e. w(x, y)⩽0 for x⩽y ) decay exponentially after a finite time and that the field has a well‐defined finite speed. Copyright © 2009 John Wiley & Sons, Ltd.