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| The Hodgkin-Huxley Equations | |
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| The Resting Potential | |
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| The Nernst Equation | |
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| The Goldman-Hodgkin-Katz Equation | |
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| Equivalent Circuits: The Electrical Analogue | |
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| The Membrane Time Constant | |
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| The Cable Equation | |
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| The Squid Action Potential | |
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| Voltage-Gated Channels | |
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| Hodgkin-Huxley Model | |
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| The Action Potential Revisited | |
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| Bibliography | |
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| Exercises | |
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| Dendrites | |
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| Multiple Compartments | |
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| The Cable Equation | |
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| The Infinite Cable | |
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| Finite and Semi-infinite Cables | |
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| Branching and Equivalent Cylinders | |
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| An Isolated Junction | |
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| Dendrites with Active Processes | |
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| Concluding Remarks | |
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| Bibliography | |
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| Exercises | |
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| Dynamics | |
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| Introduction to Dynamical Systems | |
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| The Morris-Lecar Model | |
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| The Phase Plane | |
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| Stability of Fixed Points | |
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| Excitable Systems | |
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| Oscillations | |
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| Bifurcation Analysis | |
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| The Hopf Bifurcation | |
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| Saddle-Node on a Limit Cycle | |
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| Saddle-Homoclinic Bifurcation | |
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| Class I and Class II | |
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| Bifurcation Analysis of the Hodgkin-Huxley Equations | |
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| Reduction of the Hodgkin-Huxley Model to a Two-Variable Model | |
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| FitzHugh-Nagumo Equations | |
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| Bibliography | |
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| Exercises | |
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| The Variety of Channels | |
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| Overview | |
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| Sodium Channels | |
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| Calcium Channels | |
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| Voltage-Gated Potassium Channels | |
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| A-Current | |
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| M-Current | |
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| The Inward Rectifier | |
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| Sag | |
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| Currents and Ionic Concentrations | |
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| Calcium-Dependent Channels | |
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| Calcium Dependent Potassium: The Afterhyperpolarization | |
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| Calcium-Activated Nonspecific Cation Current | |
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| Bibliography | |
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| Exercises | |
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| Projects | |
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| Bursting Oscillations | |
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| Introduction to Bursting | |
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| Square-Wave Bursters | |
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| Elliptic Bursting | |
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| Parabolic Bursting | |
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| Classification of Bursters | |
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| Chaotic Dynamics | |
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| Chaos in Square-Wave Bursting Models | |
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| Symbolic Dynamics | |
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| Bistability and the Blue-Sky Catastrophe | |
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| Bibliography | |
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| Exercises | |
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| Propagating Action Potentials | |
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| Traveling Waves and Homoclinic Orbits | |
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| Scalar Bistable Equations | |
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| Numerical Shooting | |
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| Singular Construction of Waves | |
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| Wave Trains | |
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| Dispersion Relations | |
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| Dispersion Kinematics | |
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| Morris-Lecar Revisited and Shilnikov Dynamics | |
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| Class II Dynamics | |
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| Class I Dynamics | |
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| Stability of the Wave | |
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| Linearization | |
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| The Evans Function | |
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| Myelinated Axons and Discrete Diffusion | |
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| Bibliography | |
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| Exercises | |
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| Synaptic Channels | |
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| Synaptic Dynamics | |
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| Gluiamate | |
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| �-Aminobutyric Acid | |
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| Gap or Electrical Junctions | |
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| Short-Term Plasticity | |
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| Other Models | |
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| Long-Term Plasticity | |
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| Bibliography | |
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| Exercises | |
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| Neural Oscillators: Weak Coupling | |
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| Neural Oscillators, Phase, and Isochrons | |
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| Phase Resetting and Adjoints | |
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| The Adjoint | |
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| Examples of Adjoints | |
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| Bifurcations and Adjoints | |
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| Spike-Time Response Curves | |
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| Who Cares About Adjoints? | |
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| Relationship of the Adjoint and the Response to Inputs | |
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| Forced Oscillators | |
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| Coupled Oscillators | |
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| Other Map Models | |
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| Weak Coupling | |
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| Geometric Idea | |
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| Applications of Weak Coupling | |
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| Synaptic Coupling near Bifurcations | |
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| Small Central Pattern Generators | |
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| Linear Arrays of Cells | |
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| Two-Dimensional Arrays | |
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| All-to-All Coupling | |
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| Pulse-Coupled Networks: Solitary Waves | |
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| Integrate-and-Fire Model | |
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| Stability | |
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| Bibliography | |
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| Exercises | |
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| Projects | |
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| Neuronal Networks: Fast/Slow Analysis | |
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| Introduction | |
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| Mathematical Models for Neuronal Networks | |
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| Individual Cells | |
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| Synaptic Connections | |
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| Network Architecture | |
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| Examples of Firing Patterns | |
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| Singular Construction of the Action Potential | |
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| Synchrony with Excitatory Synapses | |
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| Postinhibitory Rebound | |
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| Two Mutually Coupled Cells | |
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| Clustering | |
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| Dynamic Clustering | |
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| Antiphase Oscillations with Excitatory Synapses | |
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| Existence of Antiphase Oscillations | |
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| Stability of Antiphase Oscillations | |
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| Almost-Synchronous Solutions | |
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| Almost Synchrony with Inhibitory Synapses | |
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| Almost Synchrony with Excitatory Synapses | |
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| Synchrony with Inhibitory Synapses | |
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| Slow Inhibitory Synapses | |
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| Fast/Slow Decomposition | |
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| Antiphase Solution | |
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| Suppressed Solutions | |
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| Propagating Waves | |
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| Bibliography | |
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| Exercises | |
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| Noise | |
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| Stochastic Differential Equations | |
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| The Wiener Process | |
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| Stochastic Integrals | |
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| Change of Variables: It�'s Formula | |
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| Fokker-Planck Equation: General Considerations | |
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| Scalar with Constant Noise | |
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| First Passage Times | |
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| Firing Rates of Scalar Neuron Models | |
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| The Fokker-Planck Equation | |
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| First Passage Times | |
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| Interspike Intervals | |
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| Colored Noise | |
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| Nonconstant Inputs and Filtering Properties | |
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| Weak Noise and Moment Expansions | |
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| Poisson Processes | |
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| Basic Statistics | |
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| Channel Simulations | |
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| Stochastic Spike Models: Beyond Poisson | |
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| Bibliography | |
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| Exercises | |
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| Projects | |
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| Firing Rate Models | |
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| A Number of Derivations | |
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| Heuristic Derivation | |
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| Derivation from Averaging | |
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| Populations of Neurons | |
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| Population Density Methods | |
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| The Wilson-Cowan Equations | |
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| Scalar Recurrent Model | |
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| Two-Population Networks | |
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| Excitatory-Inhibitory Pairs | |
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| Generalizations of Firing Rate Models | |
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| Beyond Mean Field | |
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| Some Methods for Delay Equations | |
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| Exercises | |
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| Projects | |
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| Spatially Distributed Networks | |
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| Introduction | |
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| Unstructured Networks | |
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| McCulloch-Pitts | |
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| Hopfield's Model | |
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| Designing Memories | |
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| Waves | |
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| Wavefronts | |
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| Pulses | |
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| Bumps | |
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| The Wilson-Cowan Equations | |
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| Stability | |
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| More General Stability | |
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| More General Firing Rates | |
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| Applications of Bumps | |
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| Spatial Patterns: Hallucinations | |
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| Exercises | |
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| References | |
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| Index | |