NOISE DECOMPOSITION FOR STOCHASTIC HODGKIN-HUXLEY MODELS by SHUSEN PU Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Dissertation Advisor: Dr. Peter J. Thomas Department of Mathematics, Applied Mathematics and Statistics CASE WESTERN RESERVE UNIVERSITY January, 2021 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis of Shusen Pu Candidate for the Doctor of Philosophy degree∗. Prof. Peter J. Thomas Committee Chair, Advisor Prof. Daniela Calvetti Professor of Mathematics Prof. Erkki Somersalo Professor of Mathematics Prof. David Friel External Faculty, Associate Professor of Neurosciences Date: October 2, 2020 *We also certify that written approval has been obtained for any proprietary material contained therein. i Table of Contents Table of Contents............................... ii List of Tables.................................v List of Figures................................. vi Acknowledgement.............................. viii Abstract....................................x I Introduction and Motivation 1 1 Physiology Background2 1.1 Single Cell Neurophysiology.........................2 1.2 Ion Channels.................................4 2 Foundations of the Hodgkin-Huxley Model6 2.1 Membrane Capacitance and Reversal Potentials...............7 2.2 The Membrane Current............................8 2.3 The Hodgkin-Huxley Model.........................9 3 Channel Noise 14 3.1 Modeling Channel Noise........................... 16 3.2 Motivation 1: The Need for Efficient Models................ 19 4 The Connection Between Variance of ISIs with the Random Gating of Ion Channels 22 4.1 Variability In Action Potentials........................ 22 4.2 Motivation 2: Understanding the Molecular Origins of Spike Time Variability 23 II Mathematical Framework 31 5 The Deterministic 4-D and 14-D HH Models 32 5.1 The 4D Hodgkin-Huxley Model....................... 35 5.2 The Deterministic 14D Hodgkin-Huxley Model............... 36 ii 5.3 Relation Between the 14D and 4D Deterministic HH Models........ 37 5.4 Local Convergence Rate........................... 47 6 Stochastic 14D Hodgkin-Huxley Models 51 6.1 Two Stochastic Representations for Stochastic Ion Channel Kinetics.... 52 6.2 Exact Stochastic Simulation of HH Kinetics................. 58 6.3 Previous Langevin Models.......................... 61 6.4 The 14 × 28D HH Model........................... 66 6.5 Stochastic Shielding for the 14D HH Model................. 71 III Model Comparison 78 7 Pathwise Equivalence for a Class of Langevin Models 79 7.1 When are Two Langevin Equations Equivalent?............... 80 7.2 Map Channel-based Langevin Models to Fox and Lu’s Model........ 84 8 Model Comparison 90 8.1 L1 Norm Difference of ISIs.......................... 91 8.2 Two-sample Kolmogorov-Smirnov Test................... 96 IV Applications of the 14 × 28D Langevin Model 102 9 Definitions, Notations and Terminology 103 9.1 The HH Domain............................... 103 9.2 Interspike Intervals and First Passage Times................. 106 9.3 Asymptotic phase and infinitesimal phase response curve.......... 114 9.4 Small-noise expansions............................ 116 9.5 Iso-phase Sections.............................. 118 10 Noise Decomposition of the 14-D Stochastic HH Model 122 10.1 Assumptions for Decomposition of the Full Noise Model.......... 124 10.2 Noise Decomposition Theorem........................ 127 10.3 Proof of Theorem4.............................. 131 11 Decomposition of the Variance of Interspike Intervals 139 2 2 11.1 Observations on σISI, and σIPI ......................... 142 11.2 Numerical Performance of the Decomposition Theorem........... 147 iii V Conclusions and Discussion 156 12 Conclusions 157 12.1 Summary................................... 157 13 Discussions 160 13.1 Discrete Gillespie Markov Chain Algorithms................ 160 13.2 Langevin Models............................... 161 13.3 Model Comparisons.............................. 163 13.4 Stochastic Shielding Method......................... 164 13.5 Which Model to Use?............................. 165 13.6 Variance of Interspike Intervals........................ 167 14 Limitations 173 A Table of Common Symbols and Notations 177 B Parameters and Transition Matrices 180 C Diffusion Matrix of the 14D Model 183 Bibliography.................................... 187 iv List of Tables 3.1 Number of Na+ and K+ channels....................... 15 5.1 Map from 14D to 4D HH model....................... 38 5.2 Map from 4D to 14D HH model....................... 39 8.1 Model comparison.............................. 94 A.1 Common symbols and notations in this thesis (part I)............. 177 A.2 Common symbols and notations in this thesis (part II)............ 178 A.3 Common symbols and notations in this thesis (part III)............ 179 B.1 Parameters used for simulations in this thesis................. 180 v List of Figures 1.1 Diagrams of neurons.............................3 1.2 Schematic plot of a typical action potential..................5 2.1 HH components................................ 10 2.2 Sample trace of the deterministic HH model................. 13 3.1 Hodgkin-Huxley kinetics........................... 17 4.1 Voltage recordings from intact Purkinje cells................. 24 4.2 Interdependent variables under current clamp................ 26 5.1 4D and 14D HH models........................... 34 5.2 Convergence to the 4D submanifold..................... 50 6.1 Stochastic shielding under voltage clamp................... 72 6.2 Stochastic shielding under current clamp................... 76 6.3 Robustness of edge-importance ordering under current clamp........ 77 7.1 Pathwise equivalency of 14D HH model and Fox and Lu’s model...... 89 8.1 Subunit model................................. 95 8.2 Relative computational efficiency....................... 97 8.3 Logarithm of the ratio of Kolmogorov-Smirnov test statistic........ 100 9.1 Example voltage trace for the 14-D stochastic HH model.......... 107 9.2 Sample trace of the 14 × 28D model..................... 109 9.3 Example trajectory and the partition of threshold crossing.......... 110 9.4 Sample trace on the limit cycle and its phase function............ 114 10.1 Approximate decomposition of interspike interval variance......... 124 11.1 Histogram of voltage recordings and number of spikes as a function of voltage threshold............................... 144 11.2 Standard deviation of the interspike intervals and iso-phase intervals.... 146 vi 2 2 11.3 Comparison of variance of ISIs (σISI) and IPIs (σIPI)............. 148 11.4 Numerical performance of the decomposition for the ISI variance for Na+ and K+ kinetics.................................. 152 11.5 Overall performance of the decomposition theorem on ISI variance..... 154 13.1 Variance of ISI using different threshold conditions............. 169 13.2 Variance of ISI using different thresholds for large noise.......... 171 vii Acknowledgement This thesis represents not only my development in applied mathematics, it is also a milestone in six years of study at Case Western Reserve University (CWRU). My experi- ence at CWRU has been a wonderful journey full of amazement. This thesis is the result of many experiences I have encountered at CWRU from dozens of remarkable individuals who I sincerely wish to acknowledge. First and foremost, I would like to thank my advisor, Professor Peter J. Thomas for his support in both my academic career and personal life. He has been supportive even before I officially registered at CWRU when we met on the 2014 SIAM conference on the life sciences. I still remember him sketching the neural spike train and explaining to me the mathematical models for action potentials, which turned out to be the main focus of my dissertation. Ever since, Professor Thomas has supported me not only by teaching me knowledge in computational neuroscience over almost six years, but also academically and emotionally through the rough road to finish this thesis. Thanks to his great effort in walking me through every detail of the field and proofreading line by line of all my papers and thesis. And during the most difficult times when my father left me, he gave me the moral support and the time that I needed to recover. There are so many things fresh in my mind and I cannot express how grateful I am for his supervision and support. I would also like to thank every professor who shared their knowledge and experience with me, Dr. Daniela Calvetti, Dr. Erkki Somersalo, Dr. Weihong Guo, Dr. Alethea Barbaro, Dr. Wanda Strychalski, Dr. Wojbor Woyczynski, and Dr. Longhua Zhao. viii Especially, I want to thank my committee members Dr. Daniela Calvetti, Dr. Erkki Somersalo in the Department of Mathematics, Applied Mathematics and Statistics at CWRU, and Dr. David Friel from the Department of Neurosciences at the Case School of Medicine. The book on mathematical modeling written by Dr. Daniela Calvetti and Dr. Erkki Som- ersalo is my first formal encounter with mathematical models for dynamic systems and stochastic processes. Moreover, I have taken a series of computational oriented courses offered by Dr. Daniela Calvetti, where she taught me many programming skills and how to elegantly present my work. I would like to thank Dr. David Friel for his inspiring dis- cussions during the past few years. All data recordings in this thesis come from Dr. David Friel’s laboratory, and he taught me a lot of the biophysical background of this dissertation.