Collective Dynamics of Large-Scale Spiking Neural Networks by Mean-Field Theory

Abstract

The brain contains a large number of neurons, each of which typically has thousands of synaptic connections. Its functionality, whether function or dysfunction, depends on the emergent collective dynamics arising from the coordination of these neurons. Rather than focusing on large-scale realistic simulations of individual neurons and their synaptic coupling to understand these macroscopic behaviors, we emphasize the development of mathematically manageable models in terms of macroscopic observable variables. This approach allows us to gain insight into the underlying mechanisms of collective dynamics from a dynamical systems perspective. It is the central idea of this thesis.

We analytically reduce large-scale neural networks to low-dimensional mean-field mod- els that account for spike frequency adaptation, time delay between neuron communication, and short-term synaptic plasticity. These mean-field descriptions offer a precise correspondence between the microscopic dynamics of individual neurons and the macroscopic dynamics of the neural network, valid in the limit of infinitely many neurons in the network. Bifurcation analysis of the mean-field systems is capable of predicting net- work transitions between asynchronous and synchronous states, or different patterns of synchronization, such as slow-fast nested collective oscillations. We discuss how these dynamics are closely related to normal brain functions and neurological disorders. We also investigate the influence on these dynamic transitions induced by current heterogeneity, adaptation intensity, and delayed coupling. By integrating a kinetic model of synapses into the neural network, we describe calcium-dependent short-term synaptic plasticity in a relatively simple mathematical form. Through our mean-field modeling approach, we explore the impact of synaptic dynamics on collective behaviors, particularly the effect of muscarinic activation at inhibitory hippocampal synapses. Together, this thesis provides a tractable and reliable tool for model-based inference of neurological mechanisms from the perspective of theoretical neuroscience.