Cover -- PREFACE -- CONTENTS -- PART I: BURSTING AT THE SINGLE CELL LEVEL -- CHAPTER 1 THE DEVELOPMENT OF THE HINDMARSH-ROSE MODEL FOR BURSTING -- 1.1. Introduction -- 1.2. Tail Current Reversal -- 1.3. The 1982 Model -- 1.4. The 1984 Model -- 1.5. Subthreshold Oscillations -- 1.6. A Bifurcation Theorem -- References -- CHAPTER 2 NEGATIVE CALCIUM FEEDBACK: THE ROAD FROM CHAY-KEIZER -- 2.1. Introduction -- 2.2. Before the Beginning -- 2.3. The Beginning -- 2.4. The Demise of K(Ca) -- 2.5. The Return of K(Ca): Help from the Endoplasmic Reticulum -- 2.6. Further Modifications to the Model -- 2.7. Discussion -- Acknowledgements -- References -- CHAPTER 3 AUTOREGULATION OF BURSTING OF AVP NEURONS OF THE RAT HYPOTHALAMUS -- 3.1. Introduction -- 3.2. Electrical Properties of AVP Cells -- 3.3. Mathematical Model -- 3.4. Firing Patterns -- 3.5. Burst Structure -- 3.6. The Role of Calcium -- 3.7. The Action of Dynorphin -- 3.8. The Bursting Mechanism -- 3.9. The Dynamics of Dynorphin -- 3.10. Analysis of Bursting -- 3.11. Discussion -- Acknowledgements -- References -- CHAPTER 4 BIFURCATIONS IN THE FAST DYNAMICS OF NEURONS: IMPLICATIONS FOR BURSTING -- 4.1. Introduction -- 4.2. A TWO Dimensional Model of Spiking Sodium Currents -- 4.3. Fast-Slow Analysis of Bursting -- 4.4. Discussion -- References -- CHAPTER 5 BURSTING IN 2-COMPARTMENT NEURONS: A CASE STUDY OF THE PINSKY-RINZEL MODEL -- 5.1. Introduction -- 5.2. The Pinsky-Rinzel Model -- 5.3. Dynamics of the Pinsky-Rinzel Model -- 5.4. Morris-Lecar Two-Compartment Models -- 5.5. Discussion -- Acknowledgments -- References -- CHAPTER 6 GHOSTBURSTING: THE ROLE OF ACTIVE DENDRITES IN ELECTROSENSORY PROCESSING -- 6.1. Introduction -- 6.2. Bursting Mechanism -- 6.3. Ghostburster Dynamics -- 6.4. Unique Features -- 6.5. Extensions and Other Work -- 6.6. Parallel Processing with Bursts and Isolated Spikes -- 6.7. Summary -- Acknowledgements -- References -- PART II: BURSTING AT THE NETWORK LEVEL -- CHAPTER 7 ANALYSIS OF CIRCUITS CONTAINING BURSTING NEURONS USING PHASE RESETTING CURVES -- 7.1. Introduction -- 7.2. Stability Analysis for Two Coupled Oscillators -- 7.3. Analysis of a Circuit of Two Model Neurons -- 7.4. Stability Analysis for a Three Neuron Ring Circuit -- 7.5. Analysis of a Circuit of Three Model Neurons -- 7.6. Analysis of a Two Neuron Hybrid Circuit -- 7.7. Effect of Changing Burst Durations in the Two Neuron Circuit -- 7.8. Phenomenology of Resetting in a Biological Bursting Neuron -- 7.9. Significance -- Acknowledgments -- References -- CHAPTER 8 BURSTING IN COUPLED CELL SYSTEMS -- 8.1. Introduction -- 8.2. Unfolding Theory and Bursting in Fast-Slow Systems -- 8.3. Bursting in Two Coupled Cells -- 8.4. Za-Equivariant Bifurcations -- 8.5. Pitchfork Bifurcation -- 8.6. Hopf / Hopf Mode Interactions -- 8.7. Takens-Bogdanov Bifurcation with 22 Symmetry -- 8.8. Conclusion -- Acknowledgments -- References -- CHAPTER 9 MODULATORY EFFECTS OF COUPLING ON BURSTING MAPS -- 9.1. Introduction -- 9.2. Examples of Bursting Maps -- 9.3. Effects of Coupling -- 9.4. Rulkov's First Bursting Map: Explaining the Effect of Coupling -- 9.5. Discussion -- Acknowledgments -- References -- CHAPTER 10 BEYOND SYNCHRONIZATION: MODULAT.