| Summary: | The brain is the central hub regulating thought, memory, vision, and many other processes occurring within the body. Neural information transmission occurs through the firing of billions of connected neurons, giving rise to a rich variety of complex patterning. Mathematical models are used alongside direct experimental approaches in understanding the underlying mechanisms at play which drive neural activity, and ultimately, in understanding how the brain works.
This thesis focuses on network and continuum models of neural activity, and computational methods used in understanding the rich patterning that arises due to the interplay between non-local coupling and local dynamics. It advances the understanding of patterning in both cortical and sub-cortical domains by utilising the neural field framework in the modelling and analysis of thalamic tissue – where cellular currents are important in shaping the tissue firing response through the post-inhibitory rebound phenomenon – and of cortical tissue. The rich variety of patterning exhibited by different neural field models is demonstrated through a mixture of direct numerical simulation, as well as via a numerical continuation approach and an analytical study of patterned states such as synchrony, spatially extended periodic orbits, bumps, and travelling waves. Linear instability theory about these patterns is developed and used to predict the points at which solutions destabilise and alternative emergent patterns arise. Models of thalamic tissue often exhibit lurching waves, where activity travels across the domain in a saltatory manner. Here, a direct mechanism, showing the birth of lurching waves at a Neimark-Sacker-type instability of the spatially synchronous periodic orbit, is presented. The construction and stability analyses carried out in this thesis employ techniques from non-smooth dynamical systems (such as saltation methods) to treat the Heaviside nature of models. This is often coupled with an Evans function approach to determine the linear stability of patterned states.
With the ever-increasing complexity of neural models that are being studied, there is a need to develop ways of systematically studying the non-trivial patterns they exhibit. Computational continuation methods are developed, allowing for such a study of periodic solutions and their stability across different parameter regimes, through the use of Newton-Krylov solvers. These techniques are complementary to those outlined above. Using these methods, the relationship between the speed of synaptic transmission and the emergent properties of periodic and travelling periodic patterns such as standing waves and travelling breathers is studied. Many different dynamical systems models of physical phenomena are amenable to analysis using these general computational methods (as long as they have the property that they are sufficiently smooth), and as such, their domain of applicability extends beyond the realm of mathematical neuroscience.
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