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Preface | |
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Introduction | |
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Neurons | |
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What Is a Spike? | |
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Where Is the Threshold? | |
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Why Are Neurons Different, and Why Do We Care? | |
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Building Models | |
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Dynamical Systems | |
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Phase Portraits | |
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Bifurcations | |
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Hodgkin Classification | |
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Neurocomputational properties | |
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Building Models (Revisited) | |
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Review of Important Concepts | |
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Bibliographical Notes | |
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Electrophysiology of Neurons | |
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Ions | |
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Nernst Potential | |
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Ionic Currents and Conductances | |
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Equivalent Circuit | |
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Resting Potential and Input Resistance | |
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Voltage-Clamp and I-V Relation | |
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Conductances | |
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Voltage-Gated Channels | |
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Activation of Persistent Currents | |
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Inactivation of Transient Currents | |
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Hyperpolarization-Activated Channels | |
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The Hodgkin-Huxley Model | |
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Hodgkin-Huxley Equations | |
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Action Potential | |
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Propagation of the Action Potentials | |
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Dendritic Compartments | |
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Summary of Voltage-Gated Currents | |
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Review of Important Concepts | |
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Bibliographical Notes | |
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Exercises | |
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One-Dimensional Systems | |
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Electrophysiological Examples | |
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I-V Relations and Dynamics | |
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Leak + Instantaneous INa, p | |
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Dynamical Systems | |
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Geometrical Analysis | |
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Equilibria | |
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Stability | |
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Eigenvalues | |
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Unstable Equilibria | |
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Attraction Domain | |
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Threshold and Action Potential | |
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Threshold and Action Potential | |
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Bistability and Hysteresis | |
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Phase Portraits | |
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Topological Equivalence | |
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Local Equivalence and the Hartman-Grobman Theorem | |
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Bifurcations | |
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Saddle-Node (Fold) Bifurcation | |
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Slow Transition | |
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Bifurcation Diagram | |
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Bifurcations and I-V Relations | |
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Quadratic Integrate-and-Fire Neuron | |
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Review of Important Concepts | |
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Bibliographical Notes | |
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Exercises | |
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Two-Dimensional Systems | |
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Planar Vector Fields | |
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Nullclines | |
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Trajectories | |
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Limit Cycles | |
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Relaxation Oscillators | |
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Equilibria | |
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Stability | |
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Local Linear Analysis | |
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Eigenvalues and Eigenvectors | |
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Local Equivalence | |
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Classification of Equilibria | |
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Example: FitzHugh-Nagumo Model | |
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Phase Portraits | |
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Bistability and Attraction Domains | |
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Stable/Unstable Manifolds | |
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Homoclinic/Heteroclinic Trajectories | |
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Saddle-Node Bifurcation | |
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Andronov-Hopf Bifurcation | |
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Review of Important Concepts | |
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Bibliographical Notes | |
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Exercises | |
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Conductance-Based Models and Their Reductions | |
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Minimal Models | |
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Amplifying and Resonant Gating Variables | |
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INa,p+IK -Model | |
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INa,t -Model | |
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INa, p+Ih -Model | |
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Ih+IKir -Model | |
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IK+IKir -Model | |
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IA -Model | |
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Ca2+ -Gated Minimal Models | |
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Reduction of Multidimensional Models | |
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Hodgkin-Huxley model | |
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Equivalent Potentials | |
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Nullclines and I-V Relations | |
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Reduction to Simple Model | |
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Review of Important Concepts | |
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Bibliographical Notes | |
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Exercises | |
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Bifurcations | |
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Equilibrium (Rest State) | |
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Saddle-Node (Fold) | |
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Saddle-Node on Invariant Circle | |
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Supercritical Andronov-Hopf | |
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Subcritical Andronov-Hopf | |
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Limit Cycle (Spiking State) | |
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Saddle-Node on Invariant Circle | |
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Supercritical Andronov-Hopf | |
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Fold Limit Cycle | |
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Homoclinic | |
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Other Interesting Cases | |
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Three-Dimensional Phase Space | |
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Cusp and Pitchfork | |
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Bogdanov-Takens | |
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Relaxation Oscillators and Canards | |
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Bautin | |
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Saddle-Node Homoclinic Orbit | |
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Hard and Soft Loss of Stability | |
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Bibliographical Notes | |
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Exercises | |
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Neuronal Excitability | |
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Excitability | |
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Bifurcations | |
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Hodgkin's Classification | |
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Classes 1 and 2 | |
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Class 3 | |
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Ramps, Steps, and Shocks | |
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Bistability | |
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Class 1 and 2 Spiking | |
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Integrators vs. Resonators | |
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Fast Subthreshold Oscillations | |
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Frequency Preference and Resonance | |
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Frequency Preference in Vivo | |
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Thresholds and Action Potentials | |
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Threshold Manifolds | |
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Rheobase | |
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Postinhibitory Spike | |
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Inhibition-Induced Spiking | |
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Spike Latency | |
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Flipping from an Integrator to a Resonator | |
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Transition Between Integrators and Resonators | |
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Slow Modulation | |
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Spike Frequency Modulation | |
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I-V Relation | |
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Slow Subthreshold Oscillation | |
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Rebound Response and Voltage Sag | |
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AHP and ADP | |
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Review of Important Concepts | |
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Bibliographical Notes | |
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Exercises | |
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Simple Models | |
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Simplest Models | |
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Integrate-and-Fire | |
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Resonate-and-Fire | |
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Quadratic Integrate-and-Fire | |
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Simple Model of Choice | |
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Canonical Models | |
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Cortex | |
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Regular Spiking (RS) Neurons | |
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Intrinsically Bursting (IB) Neurons | |
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Multi-Compartment Dendritic Tree | |
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Chattering (CH) Neurons | |
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Low-Threshold Spiking (LTS) Interneurons | |
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Fast Spiking (FS) Interneurons | |
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Late Spiking (LS) Interneurons | |
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Diversity of Inhibitory Interneurons | |
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Thalamus | |
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Thalamocortical (TC) Relay Neurons | |
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Reticular Thalamic Nucleus (RTN) Neurons | |
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Thalamic Interneurons | |
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Other Interesting Cases | |
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Hippocampal CA1 Pyramidal Neurons | |
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Spiny Projection Neurons of Neostriatum and Basal Ganglia | |
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Mesencephalic V Neurons of Brainstream | |
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Stellate Cells of Entorhinal Cortex | |
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Mital Neurons of the Olfactory Bulb | |
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Review of Important Concepts | |
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Bibliographical Notes | |
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Exercises | |
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Bursting | |
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Electrophysiology | |
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Example: The INa,p+IK+IK(M)-Model | |
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Fast-Slow Dynamics | |
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Minimal Models | |
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Central Pattern Generators and Half-Center Oscillators | |
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Geometry | |
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Fast-Slow Bursters | |
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Phase Portraits | |
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Averaging | |
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Equivalent Voltage | |
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Hysteresis Loops and Slow Waves | |
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Bifurcations �Resting Bursting Tonic Spiking� | |
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Classification | |
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Fold/Homoclinic | |
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Circle/Circle | |
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Fold/Fold Cycle | |
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Fold/Hopf | |
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Fold/Circle | |
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Neurocomputational Properties | |
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How to Distinguish? | |
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Integrators vs. Resonators | |
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Bistability | |
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Bursts as a Unit of Neuronal Information | |
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Chirps | |
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Synchronization | |
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Review of Important Concepts | |
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Bibliographical Notes | |
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Exercises | |
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Synchronization | |
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Solutions to Exercises | |
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References | |
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Index | |
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Synchronization (www.izhikevich.com) | |
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Pulsed Coupling | |
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Phase of Oscillation | |
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Isochrons | |
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PRC | |
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Type 0 and Type 1 Phase Response | |
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Poincare Phase Map | |
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Fixed Points | |
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Synchronization | |
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Phase-Locking | |
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Arnold Tongues | |
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Weak Coupling | |
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Winfree's Approach | |
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Kuramoto's Approach | |
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Malkin's Approach | |
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Measuring PRCs Experimentally | |
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Phase Model for Coupled Oscillators | |
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Synchronization | |
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Two Oscillators | |
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Chains | |
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Networks | |
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Mean-Field Approximations | |
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Examples | |
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Phase Oscillators | |
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SNIC Oscillators | |
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Homoclinic Oscillators | |
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Relaxation Oscillators and FTM | |
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Bursting Oscillators | |
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Review of Important Concepts | |
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Bibliographical Notes | |
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Solutions | |