| |
| |
| Preface | |
| |
| |
| |
| 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) | |
| |
| |
| Review of Important Concepts | |
| |
| |
| Bibliographical Notes | |
| |
| |
| |
| Electrophysiology of Neurons | |
| |
| |
| |
| Ions | |
| |
| |
| |
| Nernst Potential | |
| |
| |
| |
| 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 | |
| |
| |
| |
| Dendritic Compartments | |
| |
| |
| |
| Summary of Voltage-Gated Currents | |
| |
| |
| Review of Important Concepts | |
| |
| |
| Bibliographical Notes | |
| |
| |
| Exercises | |
| |
| |
| |
| One-Dimensional Systems | |
| |
| |
| |
| Electrophysiological Examples | |
| |
| |
| |
| I-V Relations and Dynamics | |
| |
| |
| |
| Leak + Instantaneous INa, p | |
| |
| |
| |
| Dynamical Systems | |
| |
| |
| |
| Geometrical Analysis | |
| |
| |
| |
| Equilibria | |
| |
| |
| |
| Stability | |
| |
| |
| |
| Eigenvalues | |
| |
| |
| |
| Unstable Equilibria | |
| |
| |
| |
| Attraction Domain | |
| |
| |
| |
| Threshold and Action Potential | |
| |
| |
| |
| 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 | |
| |
| |
| Bibliographical Notes | |
| |
| |
| Exercises | |
| |
| |
| |
| Two-Dimensional Systems | |
| |
| |
| |
| Planar Vector Fields | |
| |
| |
| |
| Nullclines | |
| |
| |
| |
| Trajectories | |
| |
| |
| |
| Limit Cycles | |
| |
| |
| |
| Relaxation Oscillators | |
| |
| |
| |
| Equilibria | |
| |
| |
| |
| Stability | |
| |
| |
| |
| Local Linear Analysis | |
| |
| |
| |
| Eigenvalues and Eigenvectors | |
| |
| |
| |
| Local Equivalence | |
| |
| |
| |
| Classification of Equilibria | |
| |
| |
| |
| Example: FitzHugh-Nagumo Model | |
| |
| |
| |
| Phase Portraits | |
| |
| |
| |
| Bistability and Attraction Domains | |
| |
| |
| |
| Stable/Unstable Manifolds | |
| |
| |
| |
| Homoclinic/Heteroclinic Trajectories | |
| |
| |
| |
| Saddle-Node Bifurcation | |
| |
| |
| |
| Andronov-Hopf Bifurcation | |
| |
| |
| Review of Important Concepts | |
| |
| |
| Bibliographical Notes | |
| |
| |
| Exercises | |
| |
| |
| |
| Conductance-Based Models and Their Reductions | |
| |
| |
| |
| Minimal Models | |
| |
| |
| |
| Amplifying and Resonant Gating Variables | |
| |
| |
| |
| INa,p+IK -Model | |
| |
| |
| |
| INa,t -Model | |
| |
| |
| |
| INa, p+Ih -Model | |
| |
| |
| |
| Ih+IKir -Model | |
| |
| |
| |
| IK+IKir -Model | |
| |
| |
| |
| IA -Model | |
| |
| |
| |
| Ca2+ -Gated Minimal Models | |
| |
| |
| |
| Reduction of Multidimensional Models | |
| |
| |
| |
| Hodgkin-Huxley model | |
| |
| |
| |
| Equivalent Potentials | |
| |
| |
| |
| Nullclines and I-V Relations | |
| |
| |
| |
| Reduction to Simple Model | |
| |
| |
| Review of Important Concepts | |
| |
| |
| Bibliographical Notes | |
| |
| |
| 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 | |
| |
| |
| |
| Three-Dimensional Phase Space | |
| |
| |
| |
| Cusp and Pitchfork | |
| |
| |
| |
| Bogdanov-Takens | |
| |
| |
| |
| Relaxation Oscillators and Canards | |
| |
| |
| |
| Bautin | |
| |
| |
| |
| Saddle-Node Homoclinic Orbit | |
| |
| |
| |
| Hard and Soft Loss of Stability | |
| |
| |
| Bibliographical Notes | |
| |
| |
| Exercises | |
| |
| |
| |
| Neuronal Excitability | |
| |
| |
| |
| Excitability | |
| |
| |
| |
| Bifurcations | |
| |
| |
| |
| Hodgkin's Classification | |
| |
| |
| |
| Classes 1 and 2 | |
| |
| |
| |
| Class 3 | |
| |
| |
| |
| 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 | |
| |
| |
| Review of Important Concepts | |
| |
| |
| Bibliographical Notes | |
| |
| |
| Exercises | |
| |
| |
| |
| 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 Brainstream | |
| |
| |
| |
| Stellate Cells of Entorhinal Cortex | |
| |
| |
| |
| Mital Neurons of the Olfactory Bulb | |
| |
| |
| Review of Important Concepts | |
| |
| |
| Bibliographical Notes | |
| |
| |
| Exercises | |
| |
| |
| |
| Bursting | |
| |
| |
| |
| Electrophysiology | |
| |
| |
| |
| Example: The INa,p+IK+IK(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� | |
| |
| |
| |
| Classification | |
| |
| |
| |
| Fold/Homoclinic | |
| |
| |
| |
| Circle/Circle | |
| |
| |
| |
| 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 | |
| |
| |
| Review of Important Concepts | |
| |
| |
| Bibliographical Notes | |
| |
| |
| Exercises | |
| |
| |
| |
| Synchronization | |
| |
| |
| Solutions to Exercises | |
| |
| |
| References | |
| |
| |
| Index | |
| |
| |
| Synchronization (www.izhikevich.com) | |
| |
| |
| |
| 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 | |
| |
| |
| |
| Two Oscillators | |
| |
| |
| |
| Chains | |
| |
| |
| |
| Networks | |
| |
| |
| |
| Mean-Field Approximations | |
| |
| |
| |
| Examples | |
| |
| |
| |
| Phase Oscillators | |
| |
| |
| |
| SNIC Oscillators | |
| |
| |
| |
| Homoclinic Oscillators | |
| |
| |
| |
| Relaxation Oscillators and FTM | |
| |
| |
| |
| Bursting Oscillators | |
| |
| |
| Review of Important Concepts | |
| |
| |
| Bibliographical Notes | |
| |
| |
| Solutions | |