University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2019-05-02 The Universal Critical Dynamics of Noisy Neurons Korchinski, Daniel James Korchinski, D. J. (2019). The Universal Critical Dynamics of Noisy Neurons (Unpublished master's thesis). University of Calgary, Calgary, AB. http://hdl.handle.net/1880/110325 master thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY The Universal Critical Dynamics of Noisy Neurons by Daniel James Korchinski A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE GRADUATE PROGRAM IN PHYSICS AND ASTRONOMY CALGARY, ALBERTA May, 2019 c Daniel James Korchinski 2019 Abstract The criticality hypothesis posits that the brain operates near a critical point. Typically, critical neurons are assumed to spread activity like a simple branching process and thus fall into the universality class of directed percolation. The branching process describes activity spreading from a single initiation site, an assumption that can be violated in real neurons where external drivers and noise can initiate multiple concurrent and independent cascades. In this thesis, I use the network structure of neurons to disentangle independent cascades of activity. Using a combination of numerical simulations and mathematical modelling, I show that criticality can exist in noisy neurons but that the presence of noise changes the underly- ing universality class from directed to undirected percolation. Directed percolation describes only small scale distributions of activity, on larger scales cascades can merge together and undirected percolation is the appropriate description. ii Preface This thesis is an original work by the author. No part of this thesis has been previously published. iii Acknowledgements This work would not be possible without the gracious financial support of the National Science and Engineering Research Council, Alberta Innovates, the University of Calgary's Faculty of Graduate Studies, Student Aid Alberta, and the Nathoo family. I would like to ex- press my gratitude to Professor J¨ornDavidsen, who captured my imagination by introducing criticality in the brain to me. He started me on this journey, and with his patience, support, and drive, has seen me through to completion. I'd also like to thank Dr. Seung-Woo, for the many fruitful conversations and suggestions made over coffee and while reviewing percolation theory. Javier Orlandi was also tremendously helpful with numerous technical details related to modelling biological neurons. Without these three, this thesis would be a shadow of its present state. I would also like to thank my parents for their support and gently prodding questions, and to Raelyn for her humour and cheer on days that mine lapsed. iv Table of Contents Abstract ii Preface iii Acknowledgements iv Table of Contents v List of Figures and Illustrations viii List of Tables xiv List of Symbols, Abbreviations and Nomenclature xv 1 Introduction 1 1.1 Complex systems . .1 1.2 Complex networks . .2 1.3 The brain as a complex system . .3 2 Criticality in Neural Systems 6 2.1 A brief review of criticality . .6 2.2 Experimental evidence of neural criticality . .9 2.3 Modelling criticality in the brain . 11 2.3.1 Hodgkin-Huxley and other \biological" dynamical neuron models . 12 2.3.2 Branching processes . 15 2.3.3 Contact processes . 17 2.4 Noise in the brain . 20 2.4.1 The effect of noise on observables . 20 2.4.2 Modelling noise in the brain . 24 2.5 Summary . 25 3 Mathematical Background 26 3.1 Random graphs and network theory . 26 3.1.1 k-ary trees . 28 3.1.2 k-regular graphs . 28 3.1.3 Erd¨os-R´enyi graphs . 30 v 3.1.4 Small-world graphs . 31 3.1.5 Power-law graphs . 34 3.1.6 Hierarchical modular graphs . 37 3.2 Percolation . 41 3.2.1 Percolation in 1-dimension . 42 3.2.2 Percolation on the Bethe lattice . 44 3.2.3 Percolation on other graphs . 47 3.3 Directed percolation . 49 3.3.1 Spreading processes . 51 3.4 Summary . 52 4 The Branching Process with Noise 54 4.1 Results for the branching process with noise on infinite k-regular graphs . 55 4.1.1 Active fraction . 56 4.1.2 Mean cluster size . 59 4.1.3 Phase diagram . 66 4.1.4 Mergeless cluster distribution . 72 4.1.5 Cluster size distribution . 74 4.1.6 Avalanche duration and scaling relations . 78 4.1.7 Correlation length . 82 4.1.8 Size of the giant component . 89 4.2 Numerical results for the branching process with noise on finite k-regular graphs 93 4.2.1 Avalanche distributions . 93 4.2.2 The giant component in finite graphs . 95 4.2.3 Mean cluster size . 96 4.3 Simulations on other finite networks . 99 4.3.1 Small-world graphs . 100 4.3.2 Power-law networks . 101 4.3.3 Hierarchical modular networks . 103 4.4 Thresholded avalanches . 108 4.5 Summary . 113 5 Quadratic Integrate-and-Fire neurons 116 5.1 The model . 116 5.2 Simulations on Erd¨os-R´enyi and hierarchical modular networks . 119 5.3 Summary . 121 6 Conclusions 124 6.1 Summary of results . 124 6.2 Outlook and future work . 126 Bibliography 129 A Supplementary Figures 141 vi B Numerical Methods 152 B.1 Simulation of infinite k-regular branching processes with spontaneous activity 152 vii List of Figures and Illustrations 2.1 Neuronal avalanches, reproduced from Beggs [Beggs and Plenz, 2003]. Top: each point indicates the detection of an action potential at that electrode label. Bottom: Detail showing the evolution of a single avalanche. 10 2.2 The basic anatomy of a pair of ideal neurons. The neuron outlines are repli- cated from [Mel, 1994]. 12 2.3 An example of a branching process on a simple linear bidirectional network (shown at the top). The dynamics consists of a single cascade initiated at node 1 at time t = 1. As connections here are recurrent, nodes can be reactivated, as occurs at node 1, at time t=3 and node 3 at time t=5. 16 2.4 The results of overlapping avalanches, when avalanches are initiated as a Pois- son process of various rates. Avalanche sizes are drawn from a pure power-law,p P (S) ∼ S−3=2, and avalanche durations are assumed to scale T ∼ S, with time rescaled so that the duration of a size 1 avalanche is T = 1. If another avalanche is triggered in the timespan of the first, their sizes are added and the length of the avalanche is potentially increased, possibly including another independent cascade. 21 2.5 Causal webs can be used to distinguish spatially distinct events, as well as the progenitor events in avalanches. On the left are the spike trains observed in the neurons on the right. There are two causal webs of size three, as well as a causal web of size 4. Under the traditional model of avalanches, with avalanches delineated by periods of silence, there would be two avalanches: one of size six and one of size four. 23 3.1 A demonstration of the Watts-Strogatz model. (a) A circulant graph, connect- ing the nearest two neighbours (giving each node a degree of four) is shown for N = 10. Of the 20 bonds, 4 are selected for rearrangement. (b) The bonds for rearrangement retain one end-point while the other is swapped for another at random. 33 3.2 Degree distribution for power-law networks with uncorrelated degree distri- butions generated via the configuration model, with λ = 3:5 and kmin = 5, averaged across 500 networks of size N = 105.................. 36 3.3 Degree distribution for power-law networks with uncorrelated degree distri- butions generated via the Goh model, with λ = 3:5 and hki = 10 averaged across 500 networks of size N = 105....................... 37 viii 3.4 The in/out-degree correlations resulting averaged from an ensemble of 500 networks, both with an asymptotic degree distribution of p(k) ∼ k−3:5. (a) In/out-degree correlations for power-law networks generated by the configura- tion algorithm, as in Figure-3.2. (b) In/out-degree correlations for power-law networks generated by the Goh algorithm as in Figure-3.3. 38 3.5 Base modules are represented by filled squares. Each base module might contain a dense network of neurons. Modules are wired into pairs { these pairs constitute a super module. Super-module pairings are indicated by a lighter shade of blue. Super-super modules are constructed from pairs of super- modules, and are indicated by the lightest shade of blue. During the formation of the super-super modules, a base module from each of the super-modules is selected, these two base-modules are then wired together as indicated with the lightest-blue edge. A single super3-module is constructed from the two super2 modules, and is indicated in green. Two base modules, one from each super2-module are wired together, this connection is indicated in green. 39 3.6 A simple example of the vertices and edges populating two base modules (coloured blue) connected together to form a super-module (coloured purple). Here the number of intra-vertices per module, NPN, is 5.