Statistical physics approaches to collective behavior in networks of neurons Xiaowen Chen A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics Adviser: Professor William Bialek November 2020 c Copyright by Xiaowen Chen, 2020. All rights reserved. Abstract In recent years, advances in experimental techniques have allowed for the first time si- multaneous measurements of many interacting components in living systems at almost all scales, making now an exciting time to search for physical principles of collective behavior in living systems. This thesis focuses on statistical physics approaches to collective behavior in networks of interconnected neurons; both statistical inference methods driven by real data, and analytical methods probing the theory of emergent behavior, are discussed. Chapter 3 is based on work with F. Randi, A. M. Leifer, and W. Bialek [Chen et al., 2019], where we constructed a joint probability model for the neural activity of the nematode, Caenorhabditis elegans. In particular, we extended the pairwise maximum entropy model, a statistical physics approach to consistently infer distributions from data that has successfully described the activity of networks with spiking neurons, to this very different system with neurons exhibiting graded potential. We discuss signatures of collective behavior found in the inferred models. Chapter 4 is based on work with W. Bialek [Chen and Bialek, 2020], where we examine the tuning condition for the connection matrix among neurons such that the resulting dynamics exhibit long time scales. Starting from the simplest case of random symmetric connections, we combine maximum entropy and random matrix theory methods to explore the constraints required from long time scales to become generic. We argue that a single long time scale can emerge generically from realistic constraints, but a full spectrum of slow modes requires more tuning. iii Acknowledgements Since the beginning of my graduate school training and perhaps even earlier, I have been enjoying reading the acknowledgement sections of doctoral theses and books, and was in awe of how a doctoral degree cannot be completed alone. My thesis is no different. Throughout the past five years, I have met many amazingly talented and friendly mentors and colleagues, and grown thanks to my interactions with them. First and foremost, I would like to thank my advisor Professor Bill Bialek. From our first meeting during the Open House, he has introduced me to the wonderland of theoretical biophysics; provided me with continuous support, encouragement and guidance; and allowed me sufficient freedom to explore my scientific interests. In ad- dition to his fine taste of choosing research questions and rigor in conducting research, I have also been influenced by his optimism amid scientific expeditions in the field of biophysics, and his leadership both as a scientist and as a citizen. I would like to thank Professor Andrew Leifer and Dr. Francesco Randi for our collaboration on the data-analysis project in this thesis; they have taught me to be true to the facts. I also appreciate Andy's enthusiasm for science, friendliness and continuous support throughout the years, advising my experimental project, inviting me to design followup experiments, and serving on my thesis committee. I also would like to thank Professor Michael Aizenman for serving on my thesis committee; and Professor Ned Wingreen for serving as a Second Reader of this thesis, providing much feedback over the past years, and allowing me to attend his group meetings. I consider myself very lucky to study biological physics at Princeton, especially while the NSF Center for the Physics of Biological Function (CPBF) was being es- tablished. The center offers a wonderful and unique community for collaborative science and learning, financial support for participation in conferences such as the APS March Meeting and the annual iPoLS meeting, and opportunities to give back to the community through the undergraduate summer school. I would like to thank iv the leadership of Bill and Prof. Josh Shaevitz, and the administrative support from Dr. Halima Chahboune and Svitlana Rogers. I also thank Halima for her friendliness and support throughout the years. I would like to thank the theory faculty, Pro- fessors Curt Callan, David Schwab, Stephanie Palmer, and Vijay Balasubramanian, who have given me many valuable pieces of advice during theory group meetings. I also would like to thank the experimental faculty, Professors Robert Austin, Thomas Gregor, and Josh Shaevitz, for learning, teaching opportunities and career advice. I learned a tremendous amount from many discussions with the postdocs, graduate students, and visiting students that I had the fortune to overlap with at the Center, including Vasyl Alba, Ricard Alert Zenon, Marianne Bauer, Farzan Beroz, Ben Brat- ton, Katherine Copenhagen, Yuval Elhanati, Amir Erez, Kamesh Krishnamurthy, Endao Han, Caroline Holmes, Daniel Lee, Zhiyuan Li, Andreas Mayer, Lauren Mc- Gough, Leenoy Meshulam, Luisa Fernanda Ram´ırezOchoa, Pierre Ronceray, Zachary Sethna, Ben Weiner, Jim Wu, Bin Xu, and Yaojun Zhang. I have also enjoyed many informal conversations and spontaneous Icahn lunches with many of you. I thank Cassidy Yang and Diana Valverde Mendez for being amazing office mates and for befriending a theorist; I miss seeing you in the office. I also would like to thank especially the members of the Leifer Lab, including Kevin Chen, Matthew Creamer, Kelsey Hallinen, Ashley Linder, Mochi Liu, Jeffery Nguyen, Francesco Randi, Anuj Sharma, Monika Scholz, and Xinwei Yu for all the discussions related to worms and experimental techniques, and for welcoming me in the lab meetings and the lab itself. I would like to thank the training and support provided by the Department of Physics. I thank Kate Brosowsky for her support to graduate students, Laurel Lerner for organizing the departmental recitals, and all of the very friendly and supportive administrators. My attendance in many conferences and summer schools were made possible by the Compton Fund. I also thank the Women in Physics groups for the effort in creating a more inclusive environment in the Department. At the University v level, I would like to thank the Graduate School and the Counseling & Psychological Services at University Health Services for support. My last five years would be less colorful without the friends I met through graduate school, including Trithep Devakul, Christian Jepsen, Ziming Ji, Du Jin, Rocio Kiman, Ho Tat Lam, Zhaoqi Leng, Xinran Li, Sihang Liang, Jingjing Lin, Jingyu Luo, Zheng Ma, Wenjie Su, Jie Wang, Wudi Wang, Zhenbin Yang, Zhaoyue Zhang, and many others. I value my friendship with Junyi Zhang and Jiaqi Jiang, which goes back all the way to attending the same high school in Shanghai to now being in the same cohort at Princeton Physics. I treasure my friendship with Xue (Sherry) Song and Jiaqi Jiang, who have been there for me through all the ups and downs of my graduate career. I would also like to thank Hanrong Chen for company, support, proofreading this thesis and many other things. Finally, I would like to thank my parents for their unwavering support and en- couragement. It was my father, Wei Chen, who bought me a frog to observe and learn swimming from, and my mother, Yanling Guo, who started a part-time PhD degree a few years before my graduate journey, who have kindled and cultivated my curiosity and courage for this scientific quest. vi To my parents. vii Contents Abstract . iii Acknowledgements . iv List of Tables . xi List of Figures . xii 1 Introduction 1 1.1 The nervous system and its collective behavior . .4 1.2 Key problems . .7 1.3 Thesis overview . .8 2 Mathematical and statistical physics methods 11 2.1 Maximum Entropy Principle . 11 2.2 Random Matrix Theory . 16 3 Collective behavior in the small brain of C. elegans 22 3.1 Introduction . 23 3.2 Data acquisition and processing . 25 3.3 Maximum Entropy Model . 31 3.4 Does the model work? . 35 3.5 What does the model teach us? . 40 3.5.1 Energy landscape . 40 3.5.2 Criticality . 42 viii 3.5.3 Network topology . 43 3.5.4 Local perturbation leads to global response . 44 3.6 Discussion . 46 4 Searching for long time scales without fine tuning 50 4.1 Introduction . 51 4.2 Setup . 53 4.3 Time scales for ensembles with different global constraints . 57 4.3.1 Model 1: the Gaussian Orthogonal Ensemble . 57 4.3.2 Model 2: GOE with hard stability threshold . 60 4.3.3 Model 3: Constraining mean-square activity . 63 4.4 Dynamic tuning . 70 4.5 Discussion . 75 5 Conclusion and Outlook 78 A Appendices for Chapter 3 81 A.1 Perturbation methods for overfitting analysis . 81 A.2 Maximum entropy model with the pairwise correlation tensor constraint 85 A.3 Maximum entropy model fails to predict the dynamics of the neural networks as expected . 86 B Dynamical inference for C. elegans neural activity 89 B.1 Estimate correlation time from the data . 90 B.2 Coupling the neural activity and its time derivative . 93 C Appendices for Chapter 4 97 C.1 How to take averages for the time constants? . 97 C.2 Finite size effect for Model 2 . 99 C.3 Derivation for the scaling of time constants in Model 3 . 100 ix C.4 Decay of auto-correlation coefficient . 106 C.5 Model with additional constraint on self-interaction strength . 107 Bibliography 110 x List of Tables C.1 Scaling of inverse slowest time scale (gap) g0, width of the support of spectral density l, and averaged norm per neuron x2 versus the h i i Lagrange multiplier ξ (to leading order) in different regimes. 102 xi List of Figures 3.1 Schematics of data acquisition and processing of C.