Mathematical Foundations of Neuroscience

ISBN-10: 038787707X

ISBN-13: 9780387877075

Edition: 2010

List price: $53.99
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Description:

This book is motivated by a perceived need for an overview of how dynamical systems and computational analysis have been used in understanding the types of models that come out of neuroscience.The book arose out of several courses that the authors have taught including a graduate course in computational neuroscience with students from psychology, mathematics, computer science, physics and neuroscience backgrounds.The book begins with bio-physics of the cell membrane and from this introduces compartmental models, continuum limits and cable theory and active ion channels.Prior to the work on active channels, all equations are linear and in theory completely solvable in closed form.
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Book details

List price: $53.99
Copyright year: 2010
Publisher: Springer London, Limited
Publication date: 7/8/2010
Binding: Hardcover
Pages: 422
Size: 6.00" wide x 9.25" long x 1.25" tall
Weight: 1.628
Language: English

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