In 1992 Ermentrout and McLeod published in this journal a landmark study of travelling wavefronts for a differential–integral equation model of a neural network. Since then a number of authors have extended the model by adding an additional equation for a ‘recovery variable’, thus allowing the possibility of travelling-pulse-type solutions. In a recent paper, Faye gave perhaps the first rigorous proof of the existence (and stability) of a travelling-pulse solution for a model of this type, treating a simplified version of equations originally developed by Kilpatrick and Bressloff. The excitatory weight function J used in this work allowed the system to be reduced to a set of four coupled ordinary differential equations (ODEs), and a specific firing-rate function S, with parameters, was considered. The method of geometric singular perturbation was employed, together with blow-ups. In this paper we extend Faye's results on existence by dropping one of his key hypotheses, proving the existence of pulses at least two different speeds, and, in a sense, allowing a wider range of the small parameter in the problem. The proofs are classical and self-contained aside from standard ODE material.

The formation, persistence and movement of self-organised biological aggregations aremediated by signals (e.g., visual, acoustic or chemical) that organisms use to communicatewith each other. To investigate the effect that communication has on the movement ofbiological aggregations, we use a class of nonlocal hyperbolic models that incorporatesocial interactions and different communication mechanisms between group members. Weapproximate the maximum speed for left-moving and right-moving groups, and shownumerically that the travelling pulses exhibited by the nonlocal hyperbolic modelsactually travel at this maximum speed. Next, we use the formula for the speed of atravelling pulse to calculate the reversal time for the zigzagging behaviour, and showthat the communication mechanisms have an effect on these reversal times. Moreover, weshow that how animals communicate with each other affects also the density structure ofthe zigzags. These findings offer a new perspective on the complexity of the biologicalfactors behind the formation and movement of various aggregations.