Computational Neuroscience: An Essential Guide to Membrane Potentials, Receptive Fields, and Neural Networks

Preface
Contents
About the Author
1 Excitable Membranes and Neural Conduction
	1.1 Membrane Potentials
		1.1.1 Equilibrium Potentials
		1.1.2 Resting Potential
		1.1.3 The Action Potential
	1.2 The Hodgkin–Huxley Theory
		1.2.1 The Total Current Equation
		1.2.2 Modeling Conductance Change with DifferentialEquations
		1.2.3 The Potassium Current
		1.2.4 The Sodium Current
		1.2.5 Combining the Conductances in Space Clamp
	1.3 Approximations
		1.3.1 Integrate-and-Fire
		1.3.2 State Space Analysis
		1.3.3 The FitzHugh–Nagumo Equations
	1.4 Passive Conduction
		1.4.1 Core Conductors
		1.4.2 The Cable Equation
	1.5 Propagating Action Potentials
		1.5.1 The Fuse Analogy
		1.5.2 The Spatiotemporal Theory
		1.5.3 The Speed of Neural Conduction
	1.6 Summary and Further Reading
		Texts
		Suggested Original Papers for Classroom Seminars
	References
2 Receptive Fields and the Specificity of Neuronal Firing
	2.1 Specificity and Reverse Correlation
	2.2 Linear Shift-Invariant (LSI) Systems
		2.2.1 Correlation and Linear Spatial Summation
			The Superposition Principle
			The Receptive Field Function
		2.2.2 Lateral Inhibition and Convolution
		2.2.3 A Formulation with a Differential Operator
		2.2.4 Correlation and Convolution
		2.2.5 Convolution and Linear Shift-Invariant (LSI) Systems
		2.2.6 Temporal and Spatiotemporal Summation
	2.3 Nonlinearities in Receptive Fields
		2.3.1 Point Nonlinearity
		2.3.2 Nonlinearity as Interaction
			Volterra Kernels
			Gain Control
	2.4 Summary and Further Reading
		Texts
		Suggested Original Papers for Classroom Seminars
	References
3 Functional Models of Receptive Fields
	3.1 Retinal Ganglion Cells: Isotropic Center-Surround Organization
		3.1.1 Difference of Gaussians
		3.1.2 Dynamic Model
		3.1.3 Why ON–OFF Channels?
	3.2 Primary Visual Cortex: Edge Orientation
		3.2.1 Orientation Specificity
		3.2.2 Gabor Function in One and Two Dimensions
	3.3 Simple and Complex Cells: The ``Energy'' Model
		3.3.1 Response Properties
		3.3.2 Model
	3.4 Motion Detection
		3.4.1 Motion and Flicker
		3.4.2 Coincidence Detector
		3.4.3 Correlation Detector
		3.4.4 Motion as Orientation in Space-Time
	3.5 Summary and Further Reading
		Texts
		Suggested Original Papers for Classroom Seminars
	References
4 Fourier Analysis for Neuroscientists
	4.1 Examples
		4.1.1 Light Spectra
		4.1.2 Acoustics
		4.1.3 Spatial Vision
		4.1.4 Magnetic Resonance Tomography
	4.2 Why Are Sinusoidals Special?
		4.2.1 Eigenfunctions
		4.2.2 The Eigenfunctions of Convolution: Real Notation
		4.2.3 Complex Numbers
		4.2.4 The Eigenfunctions of Convolution: Complex Notation
		4.2.5 Example: Gaussian Convolution Kernels
	4.3 Fourier Decomposition
		4.3.1 Basic Theory
			Periodic Functions: Fourier Series
			Gaussian High- and Low-Pass: A Preview of the Convolution Theorem
			Finding the Coefficients
		4.3.2 Generalizations
			Nonperiodic Functions
			Fourier Transforms in Two and More Dimensions
	4.4 The Convolution Theorem
	4.5 Facts on Fourier Transforms
	4.6 Summary and Further Reading
		Texts
		Suggested Original Papers for Classroom Seminars
	References
5 Artificial Neural Networks and Classification
	5.1 Elements of Neural Networks
		5.1.1 Background
		5.1.2 Model
		5.1.3 Activation Dynamics
			Activation Vector
			Activation Function and Synaptic Weights
			The Activation Function as a Measure of Similarity
			The Weight Matrix
		5.1.4 Weight Dynamics (``Learning Rules'')
	5.2 Classification
		5.2.1 The Perceptron
		5.2.2 Linear Classification
			Decision Boundary
			Optimal Stimulus
		5.2.3 Limitations
			Linear Separability
			Locality
	5.3 Supervised Learning and Error Minimization
		5.3.1 Two-Layer Perceptron
		5.3.2 Gradient Descent
		5.3.3 The δ-Rule
		5.3.4 Multilayer Perceptrons: Backpropagation
		5.3.5 Deep Neural Networks
	5.4 The Perceptron and the Brain
		5.4.1 Feedback and Feedforward
		5.4.2 Hierarchy and Processing Steps
		5.4.3 The Role of Single Neurons
	5.5 Summary and Further Reading
		Texts
		Suggested Original Papers for Classroom Seminars
	References
6 Artificial Neural Networks with Interacting Output Units
	6.1 Tasks of Neural Information Processing
	6.2 Associative Memory
		6.2.1 The Feedforward Associator
			Example: A 2 3 Associator
		6.2.2 The Outer Product Rule
		6.2.3 General Least Square Solution
		6.2.4 Applications
			Memory
			Autoassociation and Attractor Neural Networks
			Neuroprostheses
	6.3 Self-Organization and Competitive Learning
		6.3.1 Exponential Weight Growth in Simple Hebbian Learning
		6.3.2 The Oja Learning Rule
		6.3.3 Self-Organizing Feature Map (Kohonen Map)
		6.3.4 Applications
			Decorrelation
			Feature Maps in the Brain
			Adult Plasticity
	6.4 Sparse Coding
	6.5 Continuous-Field Attractor
	6.6 Summary and Further Reading
		Texts
		Suggested Original Papers for Classroom Seminars
	References
7 Coding and Representation
	7.1 Specificity Revisited
		Tuning Curves
		Coding Schemes
		Rate Coding vs. Spike Time Coding
	7.2 Population Code
		7.2.1 Information Content of Population Codes
			Shannon Entropy
			The Entropy of Overlapping Channels
			Mutual Information and the Case of Graded Tuning Curves
		7.2.2 Reading a Population Code
			The Center-of-Gravity Estimator
			Least Squares, Maximum Likelihood, and Bayesian Estimators
		7.2.3 Examples and Further Properties
			Hyperacuity (Sub-pixel Resolution)
			Aftereffects and Working Range Adjustment
			Vector-Valued Parameters and Interpolation
	7.3 Topological Maps
		7.3.1 Locality and Ordered Maps
		7.3.2 Retinotopic Maps in the Visual Cortex
		7.3.3 Mathematical Descriptions of Retinotopic Maps
			Areal Magnification
			Log-Polar Mapping
		7.3.4 Functional Relevance
	7.4 Summary and Further Reading
		Texts
		Suggested Original Papers for Classroom Seminars
	References
Index