## Achieving baseline states in sparsely connected spiking-neural networks: stochastic and dynamic approaches in mathematical neuroscience

Master Thesis

2015

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University of Cape Town

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Networks of simple spiking neurons provide abstract models for studying the dynamics of biological neural tissue. At the expense of cellular-level complexity, they are a frame-work in which we can gain a clearer understanding of network-level dynamics. Substantial insight can be gained analytically, using methods from stochastic calculus and dynamical systems theory. This can be complemented by data generated from computational simulations of these models, most of which benefit easily from parallelisation. One cubic millimetre of mammalian cortical tissue can contain between fifty and one-hundred thousand neurons and display considerable homogeneity. Mammalian cortical tissue (or grey matter") also displays several distinct firing patterns which are widely and regularly observed in several species. One such state is the "input-free" state of low-rate, stochastic firing. A key objective over the past two decades of modelling spiking-neuron networks has been to replicate this background activity state using "biologically plausible" parameters. Several models have produced dynamically and statistically reasonable activity (to varying degrees) but almost all of these have relied on some driving component in the network, such as endogenous cells (i.e. cells which spontaneously fire) or wide-spread, randomised external input (put down to background noise from other brain regions). Perhaps it would be preferable to have a model where the system itself is capable of maintaining such a background state? This a functionally important question as it may help us understand how neural activity is generated internally and how memory works. There has also been some contention as to whether driven" models produce statistically realistic results. Recent numerical results show that there are connectivity regimes in which Self-Sustained, Asynchronous, Irregular (SSAI) firing activity can be achieved. In this thesis, I discuss the history and analysis of the key spiking-network models proposed in the progression toward addressing this problem. I also discuss the underlying constructions and mathematical theory from measure theory and the theory of Markov processes which are used in the analysis of these models. I then present a small adjustment to a well known model and provide some original work in analysing the resultant dynamics. I compare this analysis to data generated by simulations. I also discuss how this analysis can be improved and what the broader future is for this line of research.

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Antrobus, A. D. (2015). Achieving baseline states in sparsely connected spiking-neural networks: stochastic and dynamic approaches in mathematical neuroscience (Unpublished masters thesis). University of Cape Town.