Modeling the Dynamics of Central Pattern Generators and Anesthetic Action

v Acknowledgments vi Chapter 1. General Introduction 1 Part 1. Mechanisms Underlying Locomotion in the Crayfish Swimmeret System 3 Summary 4 Chapter 2. Introduction 6 2.1. Neural mechanisms generating locomotion are complex and largely unknown 6 2.2. The crayfish locomotor neural circuit 7 2.3. Phase Response Curve (PRC) 10 2.4. Theory of Weakly Coupled Oscillators (TWCO) 13 2.5. Previous modeling work 14 2.6. Aims of Part 1 18 Chapter 3. Phase Response Properties of Half-Center Oscillators using Morris-Lecar Intrinsic Dynamics 20 3.1. Summary 30 Chapter 4. Phase Response Properties of Half-Center Oscillators using Phase Models 31 4.1. The HCO phase model 31 4.2. Derivation of map for firing times 34 4.3. PRC of HCO 38 4.4. Application to sinusoidal PRCs 43 4.5. Summary 49 -iiChapter 5. Phase Response Properties of Half-Center Oscillators using Leaky Integrate-and-Fire Models 52 5.1. General form of PRC for HCO 52 5.2. δ-function synapses 58 5.3. Exponential synapses 61 5.4. Examples 62 5.5. Summary 65 Chapter 6. Phase Locking Dynamics in a Chain of Half-Center Oscillators 66 6.1. Intersegmental Connectivity 67 6.2. Numerical simulations in a chain of HCOs 67 6.3. Applying the theory of weakly coupled oscillators (TWCO) 71 6.4. Summary 77 Chapter 7. Addendum: Analysis of HCO phase models 79 7.1. HCO phase model with standard δ-function synapses 79 7.2. HCO phase model with exponential synapses 82 7.3. The correct limit to synaptic currents with fast decay 84 7.4. Summary 87 Chapter 8. Crayfish Conclusion 88 8.1. Summary 88 8.2. Future Modeling and Experimental Work 89 Appendix A. 91 A.1. Morris Lecar Model 91 A.2. Linearization of Firing Map 93 A.3. iPRC for LIF model 94 Part 2. Dynamical and Biophysical Mechanisms of Anesthetic Action 96 Summary 97 Chapter 9. Physiological Background 98 9.1. Volatile anesthesia in the lamprey spinal cord 98 -iii9.2. Neural network underlying fictive locomotion in lamprey spinal cord 99 9.3. Isoflurane action on the isolated spinal cord preparation 102 9.4. Previous work on anesthetics 105 9.5. Isoflurane targets 108 Chapter 10. Mathematical Models and the Mechanism of Bursting Using Bifurcation Diagrams 109 10.1. The Brodin et al. model is a biophysical model of the lamprey EIN 109 10.2. The Butera et al. model is a minimal bursting model. 110 10.3. Hodgkin-Huxley formalism for neuronal models. 110 10.4. Bursting mechanisms 111 10.5. Understanding bursting through bifurcation analysis 113 Chapter 11. Results 117 11.1. Model for TREK and TASK 117 11.2. The Modified Butera et al. Model 121 11.3. The Reduced and Modified Brodin et al. Model 126 Chapter 12. Conclusion 133 12.1. Summary 133 12.2. Future Experimental Work 134 12.3. Future Modeling Work 136 Appendix B. 139 B.1. Modified Butera et al. model 139 B.2. Reduced and modified Brodin et al. model 142 Bibliography 145