CALIFORNIA STATE UNIVERSITY, NORTHRIDGE Nonlinear Dynamics on a Novel Neural Topology A thesis submitted in partial fulfilment of the requirements For the degree of Master of Science in Physics By Alan Perstin May 2019 The thesis of Alan Perstin is approved: __________________________________ ___________ Dr. Eric Collins Date __________________________________ ___________ Dr. Tyler Luchko Date __________________________________ ___________ Dr. Yohannes Shiferaw, Chair Date California State University, Northridge ii Table of Contents Signature Page ii List of Figures v Abstract vii Chapter 1: Introduction 1 Chapter 2: Preliminary Biomedical Physics 5 2.1: Preliminaries Concerning Neurophysiology 5 2.2: Preliminaries Concerning Neural Diagnostics 11 2.3: Motivating the Island Network 15 Chapter 3: An Introduction to Graph Theory 20 3.1: Preliminaries Concerning Graphs 20 3.2: Graph Topology 21 3.2.1: The Adjacency Matrix 21 3.2.2: Degree Distribution and Edge Density 23 3.2.3: Average Shortest Path Length 25 3.2.4: Clustering Coefficient 27 3.2.5: Small-Worldness 29 3.2.6: Measures of Centrality 30 3.2.7: Hubs 34 3.2.8: Communities 37 3.2.9: Modularity 39 3.3: Topologies of Common Graphs 43 3.3.1: Random Networks 44 3.3.2: Small-World Networks 46 3.3.3: Scale-Free Networks 49 3.3.4: Closing Remarks 50 Chapter 4: Topology of the Island Network 52 4.1: Generating the Island Network 52 4.1.1: Properties of the Island Network 53 4.2: General Metrics 60 4.2.1 Degree Distribution and Edge Density 61 4.2.2 Small-Worldness 61 4.2.3 Hubs 63 4.2.4 Modular Communities 65 4.3 Discussion of Topology 66 Chapter 5: Dynamics of the Island Network 68 5.1: Preliminaries Concerning Connectome Dynamics 68 5.2: Nonlinear Stochastic Signal Spreading 69 5.3: Monte-Carlo Analysis 72 5.3.1: Dynamics as a Function of Community Structure 74 5.3.2: Dynamics as a Function of Hubs 80 5.4: Dynamics as a Function of Topology 82 iii Chapter 6: Conclusions 86 6.1: Discussion 86 6.2: Preliminaries Concerning Future Investigations 87 References 90 Appendix 1: List of Network Properties 95 Appendix 2: Algorithms 96 iv List of Figures 2.1: “Morphology of a Typical Neuron” 6 3.1: “A Human Connectome” 20 3.2: “A Graph-Theoretical Model of the Internet” 22 3.3: “A Graph and its Corresponding Adjacency Matrix” 23 3.4: “A Graph and its Corresponding Degree Distribution” 24 3.5: “Two Different Graphs with Identical Degree Distributions” 25 3.6: “Shortest Path Between Two Nodes” 27 3.7: “Clustering Coefficient: An Illustration” 28 3.8: “Centrality Measures: An Illustration” 32 3.9: “Connector Hubs and Provincial Hubs” 36 3.10: “Modularity: An Illustration” 39 3.11: “Modularity in a High School Social Network” 40 3.12: “An Erdos Reyni Graph” 45 3.13: “Clustering Coefficient against Average Path Length” 47 3.14: “Watts Strogatz Graph with β Modulated” 49 4.1: “Watts-Strogatz Compared to the Island Network” 53 4.2: “Degree Distributions: Watts-Strogatz Compared to the Island Network (1)” 56 4.3: “Island Network with Modulated” 59 4.4: “The Island Network and휅 an Equivalent Random Network” 62 4.5: “The Island Network Modulated Across Classes 1 and 2” 62 4.6: “Degree Distributions: Watts-Strogatz Compared to the Island Network (2)” 65 5.1: “Monte Carlo Plots” 72 5.2: “An Illustration of Individual Activation” 75 v 5.3: “Pearson Test of Individual Activations” 76 5.4: “An Illustration of Activation in Series” 77 5.5: “Pearson Test of Activations in Series” 78 5.6: “An Illustration of Activation in Parallel” 79 5.7: “Pearson Test of Activations in Parallel” 80 6.1: “A Graph with a Rich Club” 89 vi Abstract Nonlinear Dynamics on a Novel Neural Topology By Alan Perstin Master of Science in Physics In this study, we seek to develop a simple model of the human brain that reproduces certain characteristic structural and dynamical features. The structural features of interest, which are cast in terms of graph theory, are small-worldness, the presence of hub structures, and highly modular communities. The presence of these features in our model is motivated by the wealth of empirical research in diagnostic biomedical physics that reports these characteristics in real human brains. The dynamical feature of interest, which we acquire via a Monte Carlo analysis of the nonlinear stochastic signals that we spread across our model, is metastability as a function of community structures. The presence of this feature in our model is motivated by the metastability theory of the brain, which itself is a product of decades of research into the biophysics of neural oscillations. vii Chapter 1: Introduction The typical human brain contains approximately 85 billion neurons that interact with one another via electrochemical signals that pass through the tens of trillions of junctions where the neurons intersect[1]. Despite this extraordinarily large number of complexly- interacting components, it has nevertheless been a central aim of neuroscientists to develop a complete map (connectome) of the neural wirings of the human brain in order to better understand its structure and dynamics. The Human Connectome Project, analogously to the Human Genome Project, seeks to develop a better understanding of the underlying neural architecture of the human brain in order to facilitate advances in both the clinical and theoretical realms, with the former making use of this knowledge to diagnose and treat brain- related illnesses like Alzheimer’s and schizophrenia[2][3][4], and the latter making use of this knowledge to develop increasingly sophisticated artificial neural networks[5]. In both cases, a clearer picture of the structure and dynamics of the human brain is sorely needed, and though much progress has been made in identifying general structural and dynamical features of connectomes, no mathematical or algorithmic “recipe” for a generalized model of the human brain has been forthcoming. More specifically, the connectonomic literature returns no results for a generalized model of the human brain that captures its characteristic structural and dynamical features at every scale, and this study represents an attempt at beginning to remedy this deficit. Current applications of the widely used Watts-Strogatz[6] model to the clinical side of the connectome literature are generally restricted to a descriptive context; that is to say, imaging studies of the human brain consistently report on its small-world properties as well as the manner in which those properties are degraded in patients who present with neurophysiological syndromes[4][7][8][9], thus offering a description of the manner in which small-worldness contributes to the structure of the human brain. It has long been known, however, that small-world models alone are insufficient to adequately capture the structural features of the brain[10][11], which is why they are often used in tandem with other graph- 1 theoretical structures when appraising certain features of human brains[12][13]. That said, modifications to Watts-Strogatz graphs are typically restricted to mathematical and computational research adjacent to the connectonomic literature[14][15], and if the type of modification being introduced in this study has already been introduced in one of these adjacent fields, the current status of said model is that it has not captured the attention of the wider connectonomic research community. As far as may be reasonably ascertained via thorough engagement with the literature, modified Watts-Strogatz models of the sort being introduced in this study have not been employed in service of describing and explaining the structure of human brains, or the dynamics that follow from that structure. Our approach to the problem of brain structure is distinct in that we introduce a modification to a commonly used structural model that results in an improved graph- theoretical model that exhibits nontrivial structural properties that are not shared by its predecessor. In this study, we introduce a modified version of the Watts-Strogatz “small- world” network[6], which we have named the “Island Network,” that is sufficiently different from its precursor to warrant its characterization as a novel network structure. We aim to establish this distinction via several topological metrics, before proceeding to spread a nonlinear stochastic signal across the Island Network in order to simulate the flow of electrochemical signal across neuronal networks. As we shall see, under the metastability paradigm[16][17][18], the Island Network’s structure is particularly well-disposed to facilitating the kind of signal behavior that explains how large-scale brain dynamics can emerge from small-scale neuronal interactions. As an added bonus, this model’s structure and dynamics are consistently concordant with experimental results from diagnostic studies involving tomography[7][19][20], functional magnetic resonance imaging (fMRI)[11][8][9], magnetoencephalography (MEG)[22][23], and electroencephalography (EEG)[24][25]. The existing connectome literature on the topic of dynamics employs a plurality of approaches to modelling the biophysics of the human brain, including treatments of neuronal dynamics in terms of nonlinear coupled oscillators across graph-theoretical structures, as in 2 the case of Hodgkin-Huxley and Kuromoto models of neural interaction[26][27][28]; nonlinear Markov processes (without reference to graph-theoretical structures), as in the case of hybrid models that employ Kolmogorov relations[29]; and more simple linear threshold models across graph-theoretical structures[30]. As far as could be reasonably ascertained via thorough engagement with the literature, there appear to be no studies in which the elements of graph theory, stochasticity, and nonlinearity all simultaneously present in descriptions of neuronal dynamics. Our analysis of the dynamics, which does precisely this, is to the best of our knowledge the first of its kind within the context of human connectomes, and certainly with respect to the simple model being introduced.