Mr Minh-Son To

Honours thesis

The Morris-Lecar neuron model: from neuron to networks

Neuroscience, heralded by many as the last frontier of the biological sciences, is devoted to the study of the nervous system. At the centre lies the brain. Inherently linked to the brain is the mind, and the two have ba?ed scientists and philosophers alike for centuries. The brain is too profound and complex for us to fathom, and remains a black box to us, before and indeed after the completion of this work. Instead we set ourselves more humble goals, and so this study is concerned with the behaviour of single neurons and simple neural systems. Armed with a diverse, albeit unsophisticated, repertoire of mathematical tools, we seek to relate dynamical features with physiological phenomena that are observed in experimental settings. We hope that with this approach, we may develop some insight into the inner workings of the brain. Our strategy is one of a reductionist's, and so we begin by tackling single neurons, the building blocks of the nervous system. We elect the Morris-Lecar neuron model as the focus of our study, due to its simplicity and physiological relevance, and we describe various features of the model. In particular, we discuss certain bifurcations that occur in the model as well as its asymptotic structure. From there we connect neurons together, first a few, then many. We discuss synchronisation phenomena and we touch on chaotic dynamics in a simple network before exploring large-scale ensembles. Having found and characterised wave-like patterns in these larger networks, we conclude by finding explicitly travelling wave solutions to a related neuron model, the Fitzhugh-Nagumo model. Our work consists of a collation of past and present findings, but where possible new results were derived by applying existing techniques and ideas to the Morris-Lecar model. Despite not being a key feature, numerical computation and simulation played an instrumental role in the preparation of this work. The task of implementing various numerical algorithms was aptly handled by the software package MATLAB and for completeness, we have included, to serve as prime examples, several m-files that were developed. We also used the MATLAB packages MATCONT and MATDS for some of our analyses. Although this work was never intended to be a systematic review as such, an extensive literature search was conducted to gather material for discussion. The exercise of collecting literature was wholly unsupervised, and consumed a significant amount of time in the early phases of this work. While every effort has been made to acknowledge those authors who contributed to seminal works, we concede that there may be accidental omissions. i