Understanding Neuronal Dynamics by Geometrical Dissection of Minimal Models

Contents 1. Introduction 5 1.1. Nonlinear behaviors, time scales, our approach 5 1.2. Electrical activity of cells 6 2. Revisiting the Hodgkin-Huxley equations 10 2.1. Background and formulation 10 2.2. Hodgkin-Huxley gating equations as idealized kinetic models 12 2.3. Dissection of the action potential 13 2.3.1. Current-voltage relations 13 2.3.2. Qualitative view of fast-slow dissection 15 2.3.3. Stability of the fast subsystem's steady states 17 2.4. Repetitive firing 18 2.4.1. Stability of the four-variable model's steady state 18 2.4.2. Stability of periodic solutions 19 2.4.3. Bistability 20 3. Morris-Lecar model 23 3.1. Excitable regime 26 3.2. Post-inhibitory rebound 27 3.3. Single steady state. Onset of repetitive firing, type II 28 3.4. Three steady states 31 3.4.1. Large φ. Bistability of steady states 32 3.4.2. Small φ. Onset of repetitive firing, Type I 32 3.4.3. Intermediate φ. Bistability of rest state and a depolarized oscillation 33 3.5. Similar phenomena in the Hodgkin-Huxley model 35 3.6. Summary: onset of repetitive firing, Types I and II 35 4. Bursting, cellular level 37 4.1. Geometrical analysis and fast-slow dissection of bursting dynamics 38 4.2. Examples of bursting behavior 39 4.2.1. Square wave bursting 39 4.2.2. Parabolic bursting 40 4.2.3. Elliptic bursting 41 4.2.4. Other types of bursting 42 5. Bursting, network generated. Episodic rhythms in the developing spinal cord 44 5.1. Experimental background 44 5.2. Firing rate model 45 5.2.1. Basic recurrent network 45 5.2.2. Full model 46 5.3. Predictions of the model 47 6. Chapter summary 50 6.1. Appendix A. Mathematical formulation of fast-slow dissection. 51 6.2. Appendix B. Stability of periodic solutions. 53 References 55 3