by continuous mathematical equations often called neural fields, firing rate or mean-field models. They are integro-differential equations involving a differential operator for the synaptic processing, spatial convolution terms for the neuronal input and connectivity, and a nonlinear processing function converting the input firing rates to an output one. A variety of features could be added such as delays, adaptation, etc. Unfortunately, many of the studies of neural fields have concentrated on the biologically unrealistic local excitation-distal inhibition connectivity in order to obtain pattern solutions, an emphasis that we will try to balance.
The models described above show very rich behaviour of solutions - travelling waves and fronts, bumps, breathers and globally periodic patterns. One part of my research has been to study periodic solutions using the theory of regular pattern formation. It was originally developed in physics to study (for example) convection, and dealt with PDEs. The typical scenario is that we start with a stable homogeneous solution and look for parameters at which the linear analysis tells us it becomes unstable to periodic spatial perturbations, the so called Turing instability.