Path integral methods for correlated activity in neuronal networks

• Pfadintegralmethoden für korrelierte Aktivität in neuronalen Netzwerken

Kühn, Tobias; Helias, Moritz (Thesis advisor); Honerkamp, Carsten (Thesis advisor)

Aachen (2019, 2020)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2019

Abstract

Nervous systems of highly developed organisms consists of very many cells. The human brain, to name a very complex example, is composed of nearly 100 billion neurons, that are connected via up to thousand trillion synapses. It is an essential aim of theoretical neuroscience to discover the functioning of this complicated system on the basis of the interaction of its individual parts. Many methods for achieving this goal are borrowed from many particle physics - classical (non-quantum-mechanical) statistical physics, to be precise. An important common property of biological neuronal networks and the usual subjects of statistical physics is that both can be described by models of stochastic ("noisy") processes. The calculation of measurable quantities from these models is often difficult, for which reason approximate solutions are sought for. Here, the role of mean-field theory deserves to be emphasized, in which fluctuations are treated in a strongly simplified form. Even though in most cases many effects are neglected by this approach, it often yields quantitatively correct results in neuroscience. In this work, we use statistical field theory to derive this and related approximations for different systems, apply them to concrete problems and examine methods to improve them. To capture the interaction amongst different neurons, the description of the activity of a single neuron is often reduced to the question if it is active or not (binary model neuron). By means of its mean-field theory, we describe by which mechanisms the correlations between pairs of neurons change, when a network is driven by a stimulus varying in time. For inferences about the connections in an examined network from experimentally detected activity, the binary representation of neuronal activity is frequently used, as well. This method relies on the Ising model whose mean-field theory we derive using Feynman diagrams. We extend the formalism needed for this purpose to include expansions around non-Gaussian theories like the Ising model without coupling. Furthermore, we examine the statistics of the neuronal activity in an disordered network and its susceptibility to perturbation in mean-field theory. A generalized framework enables us to compare these results with the statistics and dynamics of networks consisting of rate model neurons. In the latter model, each nerve cell is solely characterized by the rate, which indicates how frequently it gets active. We use it in a different context to compare different path integral formalisms representing neuronal activity described by stochastic differential equations. Here we show how mean-field theory can be systematically corrected by the so called loop expansion and how the emerging correction terms can be interpreted in case mean-field theory should prove insufficient for a certain set of parameters. Another possibility to improve mean-field approximations is given be the functional Renormalization Group, whose application to simple models of biological networks we demonstrate for an example.