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Abstract

Understanding the dynamics of large-scale brain activity is a tough challenge. One reason for this is the presence of an incredible amount of complexity arising from having roughly 100 billion neurons connected via 100 trillion synapses. Because of the extremely high number of degrees of freedom in the nervous system, the question of how the brain manages to properly function and remain stable, yet also be adaptable, must be posed. Neuroscientists have identified many ways the nervous system makes this possible, of which synaptic plasticity is possibly the most notable one. On the other hand, it is vital to understand how the nervous system also loses stability, resulting in neuropathological diseases such as epilepsy, a disease which affects 1\% of the population. In the following work, we seek to answer some of these questions from two different perspectives. The first uses mean-field theory applied to neuronal populations, where the variables of interest are the percentages of active excitatory and inhibitory neurons in a network, to consider how the nervous system responds to external stimuli, self-organizes and generates epileptiform activity. The second method uses statistical field theory, in the framework of single neurons on a lattice, to study the concept of criticality, an idea borrowed from physics which posits that in some regime the brain operates in a collectively stable or marginally stable manner. This will be examined in two different neuronal networks with self-organized criticality serving as the overarching theme for the union of both perspectives. One of the biggest problems in neuroscience is the question of to what extent certain details are significant to the functioning of the brain. These details give rise to various spatiotemporal properties that at the smallest of scales explain the interaction of single neurons and synapses and at the largest of scales describe, for example, behaviors and sensations. In what follows, we will shed some light on this issue.

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