Miriam Grace

• This thesis explores the effects of variability on spatiotemporal pattern formation. Using a variety of mathematical models, but mainly focussing on the FitzHugh-Nagumo model, we explore the ways in which variability in the properties of a group of diffusively coupled elements can determine the resulting pattern. We devise a series of distinct developmental paths, i.e. time-dependent changes of element properties, in lattices of FitzHugh-Nagumo oscillators. Our paths are modelled after the developmental paths created in previous works to explain spiral waves of cyclic adenosine monophosphate which lead to aggregation in populations of the slime mould Dictyostelium discoideum. These differ in the amount of element-to-element desycnhronisation in a specific parameter value, which varies as a function of time, as well as the rate of change of this property. The resulting paths display distinctions in the events which occur, the spiral and target wave numbers, and the relationships between the event locations and the element properties. We establish a general notion of diversity-induced resonance in the FitzHugh- Nagumo model in terms of the spiral tip and target wave numbers, using the standard deviation of an element property over the lattice of coupled elements as our measure of diversity. Exploring the mechanisms underlying this resonance, we observe that target-spiral competition appears to drive the resonance in these patterns. However, we find that the frequency of wave events, which in turn depends on the properties of the individual elements, is not sufficient in itself to determine the type of wave pattern which will eventually dominate the lattice of elements. Complex inter-wave interactions must be taken into account to arrive at more accurate predictions of the lattice state, while the number of elements in the oscillatory regime but close to the excitable border proved to accurately predict the values of diversity at which maximum spiral numbers were observed in simulated experiments. Based on the importance of understanding events occurring en route to spatiotemporal pattern formation in the discrimination of different pattern formation strategies, we advocate an event-based view of pattern formation. This view reflects the additional quantitative and qualitative information that can be obtained by exploring these pattern events, and could potentially allow the development of insights into mathematical models and real biological systems based on the observable properties of their specific pattern event sequences. In general, our results elucidate some of the mechanisms and diverse interactions which characterise the influence of variability on pattern formation in excitable systems.