DYNAMICAL ARCHITECTURES FOR CONTROLLING FEEDING IN APLYSIA CALIFORNICA by KENDRICK M. SHAW Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Dissertation Adviser: Dr. Hillel J. Chiel Dissertation Co-Adviser: Dr. Peter J. Thomas Department of Biology CASE WESTERN RESERVE UNIVERSITY January 2014 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of Kendrick Matthew Shaw Candidate for the Doctor of Philosophy degree1. Robbin E. Snyder Hillel J. Chiel Peter J. Thomas Dominique M. Durand Scott E. Cooper October 16th, 2013 1We also certify that written approval has been obtained for any proprietary information contained within. Contents Acknowledgements.......................9 Abstract............................ 11 1 Introduction.......................... 13 1.1 Background and history........................... 15 1.1.1 Chain reflexes and central pattern generators........... 15 1.1.2 Mathematical background..................... 17 1.1.3 Mathematical models of neurons.................. 25 1.2 The choice of Aplysia californica as a model system............ 26 1.3 Mathematical framework and central hypotheses............. 28 1.4 Outline of the remainder of the dissertation................ 31 2 Significance of Dynamical Architecture............. 33 2.1 Introduction................................. 34 2.2 Mathematical Framework.......................... 37 2.2.1 Limit cycles............................. 38 2.2.2 Destabilization of fixed points................... 39 2.2.3 Stable heteroclinic channels.................... 39 2.3 Model Description............................. 41 3 CONTENTS 4 2.3.1 Neural model............................ 41 2.3.2 Model of the periphery and load.................. 44 2.3.3 Proprioceptive input........................ 52 2.3.4 Noise................................ 52 2.3.5 Parameter changes used for the limit cycle simulations...... 54 2.3.6 Connection to mathematical framework.............. 55 2.4 Materials and Methods........................... 55 2.4.1 Intact animals............................ 56 2.4.2 Suspended buccal mass preparation................ 56 2.4.3 Isolated buccal ganglion...................... 57 2.4.4 Data analysis............................ 57 2.4.5 Numerical methods......................... 58 2.5 Results.................................... 59 2.5.1 Tuning the limit cycle....................... 59 2.5.2 Mechanisms of adaptation to load................. 65 2.6 Discussion.................................. 75 2.6.1 Limitation of the model and results................ 78 2.6.2 Larger implications for pattern generators............. 80 3 Phase Resetting in a Phaseless System.............. 84 3.1 Introduction................................. 85 3.2 The piecewise linear iris system...................... 95 3.3 Limit Cycles in the Iris System....................... 102 3.3.1 Dynamics Within A Linear Region................. 103 3.3.2 Dynamics Across Regions..................... 105 CONTENTS 5 3.4 Effects of a small instantaneous perturbation................ 109 3.4.1 Initial effects of a small perturbation................ 109 3.4.2 Subsequent effects of a perturbation................ 113 3.4.3 Infinitesimal phase response curve................. 117 3.5 Asymptotic phase resetting behavior as a 0............... 119 ! 3.6 Isochrons.................................. 124 3.7 Smooth System............................... 125 3.8 Discussion.................................. 127 3.8.1 Sensitivity and control....................... 129 3.8.2 Comparison to the PRC near a homoclinic bifurcation...... 131 3.8.3 Qualitative comparison with a biological model: iPRCs for the Morris–Lecar system........................ 134 3.8.4 Stability of synchronous solutions for two iris systems with diffu- sive coupling............................ 140 3.8.5 Generalization to higher dimensional systems........... 146 3.8.6 Phase resetting in the absence of an asymptotic phase....... 148 4 Conclusion and Future Directions................151 4.1 Review of previous chapters........................ 152 4.2 Future directions.............................. 156 4.3 Conclusion................................. 163 Bibliography..........................165 List of Tables 2.1 Model parameters.............................. 47 2.2 State variables................................ 47 2.3 Parameters used for the limit cycle simulations.............. 53 3.1 Comparison of iPRCs of homoclinic, QIF, and Iris systems........ 134 6 List of Figures 2.1 Isolated trajectories of the SHC and limit cycle.............. 43 2.2 Phases of swallowing behavior in the model................ 45 2.3 Model schematic.............................. 46 2.4 Timing dependence of the limit cycle.................... 60 2.5 Improved efficacy with increased maximum muscle activation...... 62 2.6 Metabolic cost of increased muscle activation............... 63 2.7 Mechanical efficiency of the limit cycle.................. 64 2.8 Effects of mechanical load on timing.................... 67 2.9 Effects of mechanical load on trajectory.................. 68 2.10 Simulation of held seaweed......................... 70 2.11 Effects of held seaweed in vivo ....................... 71 2.12 Simulation of reduced proprioception................... 72 2.13 Behavior of reduced preparations...................... 74 2.14 Skewness of retraction duration in simulations............... 76 2.15 Skewness of retraction duration in vivo ................... 77 3.1 Smooth system phase plot......................... 88 3.2 Smooth system time plot.......................... 90 7 LIST OF FIGURES 8 3.3 Iris system schematic............................ 95 3.4 Iris system phase plots........................... 96 3.5 Iris system time plots............................ 101 3.6 Iris square entry to exit map........................ 108 3.7 Iris bifurcation diagram........................... 110 3.8 Iris iPRCs.................................. 120 3.9 Iris isochrons................................ 126 3.10 Smooth system iPRCs............................ 128 3.11 Morris–Lecar time plots.......................... 137 3.12 Morris–Lecar iPRC............................. 139 3.13 Coupled Iris systems............................ 142 4.1 PRC for the limit cycle neuromechanical model.............. 157 4.2 PRC for the heteroclinic channel neuromechanical model......... 158 4.3 Model with biological noise........................ 160 4.4 Trade off between proprioception and noise................ 162 4.5 Saddle bypass diagram........................... 163 Acknowledgments I would like to thank my research advisor, Dr. Hillel Chiel, for his patience, friendship, and guidance throughout my time in the laboratory. His constant encouragement and relentless striving for perfection has helped me and many others accomplish what we did not think was possible. Without him this research would not have happened, and his wisdom will shape my future work. I would also like to thank my co-advisor, Dr. Peter Thomas, for his enthusiasm and guidance in introducing me to the world of mathematical biology, and his guidance on how to bound a problem well enough that it can be solved with rigor. I would like to thank Dr. Robbin Snyder, for her friendship and advice in navigating academia, Dr. Scott Cooper, for his careful clinical tutorship and advice on blending clinical work and basic research, and Dr. Dominique Durand, whose occasional skepticism has helped drive me to strengthen this work. I would like to thank my lab mates, Hui Lu, Jeffrey McManus, Miranda Cullins, Catherine Kehl, and Jeffery Gill for their friendship and useful feedback. Miranda Cullins, Jeffrey McManus, and Hui Lu also provided the in vivo and in vitro behavior and burst onset and offset times used in chapter2, which I am grateful for. I would also like to thank Andrew Horchler for his friendship, advice, and many good conversations on the behavior of heteroclinic channels and stochastic simulations. I am also indebted to my friends Barry Rountree, Eric Herman, and Stephanie Medlock. Their enthusiasm in tackling difficult problems together for the fun of it carried over into my work in the lab, and their encouragement and patience with me during difficult times Acknowledgments 10 in my research helped me have faith that each obstacle could be overcome. I would like to thank Dr. Cliff Harding, Dr. George Dubyak, and the other members of the MSTP steering committee for their advice and guidance of my training of a clinician– scientist, and how to develop this work in this larger context. I would also like to thank my parents, family, and close friends in Seattle for supporting me in pursing this work. Finally I would like to thank the various sources of funding that have supported this work, including the Case MSTP (NIH T32-GM007250), NIH NS047073, and the Case Innovation Achievement Award Fellowship. Dynamical Architectures for Controlling Feeding in Aplysia californica Abstract by KENDRICK M. SHAW For behaviors such as swallowing, walking, and swimming, the nervous system must reliably generate sequences of motor behavior. Two competing models have been proposed for how this task is accomplished - chain reflex theory and central pattern generator theory. Chain reflex theory posits that the nervous system contains a sequence of reflexes, so that the action of one reflex creates the sensory input required to trigger the next. In contrast, central pattern generator theory posits that the nervous system