Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting

Transcription

1 Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting Eugene M. Izhikevich The MIT Press Cambridge, Massachusetts London, England

2 Contents Preface xv 1 Introduction Neurons What Is a Spike? Where Is the Threshold? Why Are Neurons Different, and Why Do We Care? Building Models Dynamical Systems Phase Portraits Bifurcations Hodgkin Classification Neurocomputational properties Building Models (Revisited) 20 Review of Important Concepts 21 Bibliographical Notes 21 2 Electrophysiology of Neurons Ions Nernst Potential s Ionic Currents and Conductances Equivalent Circuit Resting Potential and Input Resistance Voltage-Clamp and I-V Relation Conductances Voltage-Gated Channels Activation of Persistent Currents Inactivation of Transient Currents Hyperpolarization-Activated Channels The Hodgkin-Huxley Model Hodgkin-Huxley Equations Action Potential Propagation of the Action Potentials 42 vii

3 viii CONTENTS Dendritic Compartments Summary of Voltage-Gated Currents 44 Review of Important Concepts 49 Bibliographical Notes 50 Exercises 50 3 One-Dimensional Systems Electrophysiological Examples I-V Relations and Dynamics Leak + Instantaneous /N a, P Dynamical Systems Geometrical Analysis Equilibria Stability Eigenvalues Unstable Equilibria Attraction Domain Threshold and Action Potential Bistability and Hysteresis Phase Portraits Topological Equivalence Local Equivalence and the Hartman-Grobman Theorem Bifurcations Saddle-Node (Fold) Bifurcation Slow Transition ' Bifurcation Diagram Bifurcations and I-V Relations Quadratic Integrate-and-Fire Neuron.> Review of Important Concepts 82 Bibliographical Notes 83 Exercises 83 4 Two-Dimensional Systems Planar Vector Fields Nullclines Trajectories Limit Cycles Relaxation Oscillators Equilibria Stability Local Linear Analysis Eigenvalues and Eigenvectors Local Equivalence 103

4 CONTENTS ix Classification of Equilibria Example: FitzHugh-Nagumo Model Phase Portraits Bistability and Attraction Domains Stable/Unstable Manifolds Homoclinic/Heteroclinic Trajectories Ill Saddle-Node Bifurcation Andronov-Hopf Bifurcation 116 Review of Important Concepts 121 Bibliographical Notes 122 Exercises Conductance-Based Models and Their Reductions Minimal Models Amplifying and Resonant Gating Variables / NajP +/ K -Model /Na.t-model / Na^p+/ h -Model / h +/ Ki r-model / K +/ K ir-model / A -Model Ca 2+ -Gated Minimal Models Reduction of Multidimensional Models Hodgkin-Huxley model Equivalent Potentials Nullclines and I-V Relations Reduction to Simple Model 153 Review of Important Concepts 156 Bibliographical Notes 156 Exercises Bifurcations Equilibrium (Rest State) Saddle-Node (Fold) Saddle-Node on Invariant Circle Supercritical Andronov-Hopf Subcritical Andronov-Hopf Limit Cycle (Spiking State) Saddle-Node on Invariant Circle Supercritical Andronov-Hopf Fold Limit Cycle Homoclinic Other Interesting Cases 190

5 CONTENTS Three-Dimensional Phase Space Cusp and Pitchfork Bogdanov-Takens Relaxation Oscillators and Canards Bautin Saddle-Node Homoclinic Orbit Hard and Soft Loss of Stability 204 Bibliographical Notes 205 Exercises 210 Neuronal Excitability Excitability Bifurcations Hodgkin's Classification Classes 1 and Class Ramps, Steps, and Shocks ' Bistability Class 1 and 2 Spiking Integrators vs. Resonators Fast Subthreshold Oscillations Frequency Preference and Resonance Frequency Preference in Vivo Thresholds and Action Potentials Threshold manifolds Rheobase Postinhibitory Spike : Inhibition-Induced Spiking Spike Latency Flipping from an Integrator to a Resonator Transition Between Integrators and Resonators Slow Modulation! Spike Frequency Modulation I-V Relation Slow Subthreshold Oscillation Rebound Response and Voltage Sag AHP and ADP 260 Review of Important Concepts 264 Bibliographical Notes 264 Exercises 265

6 CONTENTS xi 8 Simple Models Simplest Models Integrate-and-Fire Resonate-and-Fire Quadratic Integrate-and-Fire Simple Model of Choice Canonical Models Cortex Regular Spiking (RS) Neurons Intrinsically Bursting (IB) Neurons Multi-Compartment Dendritic Tree Chattering (CH) Neurons Low-Threshold Spiking (LTS) Interneurons Fast Spiking (FS) Interneurons Late Spiking (LS) Interneurons Diversity of Inhibitory Interneurons Thalamus Thalamocortical (TC) Relay Neurons Reticular Thalamic Nucleus (RTN) Neurons Thalamic Interneurons Other Interesting Cases Hippocampal CA1 Pyramidal Neurons Spiny Projection Neurons of Neostriatum and Basal Ganglia Mesencephalic V Neurons of Brainstem Stellate Cells of Entorhinal Cortex Mitral Neurons of the Olfactory Bulb 316 Review of Important Concepts 319 Bibliographical Notes 319 Exercises Bursting Electrophysiology Example: The / Nai p+/ K +/ K (M)-Model Fast-Slow Dynamics., Minimal Models Central Pattern Generators and Half-Center Oscillators Geometry Fast-Slow Bursters Phase Portraits Averaging Equivalent Voltage Hysteresis Loops and Slow Waves Bifurcations "Resting <-> Bursting <-> Tonic Spiking" 344

7 xii CONTENTS 9.3 Classification Fold/Homoclinic Circle/Circle SubHopf/Fold Cycle ' Fold/Fold Cycle Fold/Hopf Fold/Circle Neurocomputational Properties How to Distinguish? Integrators vs. Resonators Bistability Bursts as a Unit of Neuronal Information Chirps. ' Synchronization 373 Review of Important Concepts 375 Bibliographical Notes 376 Exercises Synchronization 385 Solutions to Exercises 387 References 419 Index Synchronization ( Pulsed Coupling Phase of Oscillation \ Isochrons ' ; PRC Type 0 and Type 1 Phase Response Poincare Phase Map Fixed points Synchronization " Phase-Locking Arnold Tongues Weak Coupling Winfree's Approach Kuramoto's Approach Malkin's Approach Measuring PRCs Experimentally Phase Model for Coupled Oscillators Synchronization 467

8 CONTENTS xiii Two Oscillators Chains Networks Mean-Field Approximations Examples Phase Oscillators SNIC Oscillators Homoclinic Oscillators Relaxation Oscillators and FTM Bursting Oscillators 486 Review of Important Concepts 488 Bibliographical Notes 489 Solutions 497