Theoretical neuroscience

432 Pages·2001·5.078 MB·English

by  Peter Dayan, L. F. Abbott

THEORETICAL NEUROSCIENCE
THEORETICAL NEUROSCIENCE
Peter Dayan and L.F. Abbott
Preface
PART I - ANALYZING AND MODELING NEURAL RESPONSES
Chapter 1 - Neural Encoding I: Firing Rates and Spike Statistics
Introduction
Properties of Neurons
Recording Neuronal Responses
From Stimulus to Response
Spike Trains and Firing Rates
Measuring Firing Rates
Tuning Curves
Spike-Count Variability
What Makes a Neuron Fire?
Describing the Stimulus
The Spike-Triggered Average
White-Noise Stimuli
Multiple-Spike-Triggered Averages and Spike-Triggered Correlations
Spike Train Statistics
The Homogeneous Poisson Process
The Spike-Train Autocorrelation Function
The Inhomogeneous Poisson Process
The Poisson Spike Generator
Comparison with Data
The Neural Code
Independent-Spike, Independent Neuron and Correlation Codes
Temporal Codes
Chapter Summary
Appendices
A) The Power Spectrum of White Noise
B) Moments of the Poisson Distribution
D) Inhomogeneous Poisson Statistics
Annotated Bibliography
Chapter 2 - Neural Encoding II: Reverse Correlation and Receptive Fields
Introduction
Estimating Firing Rates
The Most Effective Stimulus
Static Nonlinearities
Introduction to the Early Visual System
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THEORETICAL NEUROSCIENCE
The Retinotopic Map
Visual Stimuli
The Nyquist Frequency
Reverse Correlation Methods - Simple Cells
Spatial Receptive Fields
Temporal Receptive Fields
Response of a Simple Cell to a Counterphase Grating
Space-Time Receptive Fields
Nonseparable Receptive Fields
Static Nonlinearities - Simple Cells
Static Nonlinearities - Complex Cells
Receptive Fields in the Retina and LGN
Constructing V1 Receptive Fields
Chapter Summary
Appendices
A) The Optimal Kernel
B) The Most Effective Stimulus
C) Bussgang's Theorem
Annotated Bibliography
Chapter 3 - Neural Decoding
Encoding and Decoding
Discrimination
ROC Curves
ROC Analysis of Motion Discrimination
The Likelihood Ratio Test
Population Decoding
Encoding and Decoding Direction
Optimal Decoding Methods
Fisher Information
Optimal Discrimination
Spike Train Decoding
Chapter Summary
Appendices
A) The Neymann-Pearson Lemma
B) The Cramér-Rao Bound
C) The Optimal Spike-Decoding Filter
Annotated Bibliography
Chapter 4 - Information Theory
Entropy and Mutual Information
Entropy
Mutual Information
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THEORETICAL NEUROSCIENCE
Entropy and Mutual Information for Continuous Variables
Information and Entropy Maximization
Entropy Maximization for a Single Neuron
Populations of Neurons
Application to Retinal Ganglion Cell Receptive Fields
The Whitening Filter
Filtering Input Noise
Temporal Processing in the LGN
Cortical Coding
Entropy and Information for Spike Trains
Chapter Summary
Appendix
Positivity of the Kulback-Leibler Divergence
Annotated Bibliography
PART II - MODELING NEURONS AND NETWORKS
Chapter 5 - Model Neurons I: Neuroelectronics
Levels of Neuron Modeling
Electrical Properties of Neurons
Intracellular Resistance
Membrane Capacitance and Resistance
Equilibrium and Reversal Potentials
The Membrane Current
Single-Compartment Models
Integrate-and-Fire Models
Spike-Rate Adaptation and Refractoriness
Voltage-Dependent Conductances
Persistent Conductances
Transient Conductances
Hyperpolarization-Activated Conductances
The Hodgkin-Huxley Model
Modeling Channels
Synaptic Conductances
The Postsynaptic Conductance
Release Probability and Short-Term Plasticity
Synapses on Integrate-and-Fire Neurons
Regular and Irregular Firing Modes
Chapter Summary
Appendices
A) Integrating the Membrane Potential
B) Integrating the Gating Variables
Annotated Bibliography
Chapter 6 - Model Neurons II: Conductances and Morphology
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THEORETICAL NEUROSCIENCE
Levels of Neuron Modeling
Conductance-Based Models
The Connor-Stevens Model
Postinhibitory Rebound and Bursting
The Cable Equation
Linear Cable Theory
An Infinite Cable
An Isolated Branching Node
The Rall Model
The Morphoelectrotonic Transform
Multi-Compartment Models
Action Potential Propagation Along an Unmyelinated Axon
Propagation Along a Myelinated Axon
Chapter Summary
Appendices
A) Gating Functions for Conductance-Based Models
Connor-Stevens Model
Transient Ca2+ Conductances
Ca2+-dependent K+ Condutances
B) Integrating Multi-Compartment Models
Annotated Bibliography
Chapter 7 - Network Models
Introduction
Firing-Rate Models
Feedforward and Recurrent Networks
Continuously Labelled Networks
Feedforward Networks
Neural Coordinate Transformations
Recurrent Networks
Linear Recurrent Networks
Selective Amplification
Input Integration
Continuous Linear Recurrent Networks
Nonlinear Recurrent Networks
Nonlinear Amplification
A Recurrent Model of Simple Cells in Primary Visual Cortex
A Recurrent Model of Complex Cells in Primary Visual Cortex
Winner-Take-All Input Selection
Gain Modulation
Sustained Activity
Maximum Likelihood and Network Recoding
Network Stability
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THEORETICAL NEUROSCIENCE
Associative Memory
Excitatory-Inhibitory Networks
Homogeneous Excitatory and Inhibitory Populations
Phase-Plane Methods and Stability Analysis
The Olfactory Bulb
Oscillatory Amplification
Stochastic Networks
Chapter Summary
Appendix
Lyapunov Function for the Boltzman Machine
Annotated Bibliography
PART III - PLASTICITY AND LEARNING
Chapter 8 - Plasticity and Learning
Introduction
Stability and Competition
Synaptic Plasticity Rules
The Basic Hebb Rule
The Covariance Rule
The BCM Rule
Synaptic Normalization
Subtractive Normalization
Multiplicative Normalization and the Oja Rule
Timing-Based Rules
Unsupervised Learning
Single Postsynaptic Neuron
Principal Component Projection
Hebbian Development and Ocular Dominance
Hebbian Development of Orientation Selectivity
Temproal Hebbian Rules and Trace Learning
Multiple Postsynaptic Neurons
Fixed Linear Recurrent Connections
Competitive Hebbian Learning
Feature-Based Models
Anti-Hebbian Modification
Timing-Based Plasticity and Prediction
Supervised Learning
Supervised Hebbian Learning
Classification and the Perceptron
Function Approximation
Supervised Error-Correcting Rules
The Perceptron Learning Rule
The Delta Rule
Contrastive Hebbian Learning
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THEORETICAL NEUROSCIENCE
Chapter Summary
Appendix
Convergence of the Perceptron Learning Rule
Annotated Bibliography
Chapter 9 - Classical Conditioning and Reinforcement Learning
Introduction
Classical Conditioning
Predicting Reward - The Rescola-Wagner Rule
Predicting Reward Timing - Temporal-Difference Learning
Dopamine and Prediction of Reward
Static Action Choice
The Indirect Actor
The Direct Actor
Sequential Action Choice
The Maze Task
Policy Evaluation
Policy Improvement
Generalizations of Actor-Critic Learning
Learning the Water Maze
Chapter Summary
Appendix - Markov Decision Problems
The Bellman Equation
Policy Iteration
Annotated Bibliography
Chapter 10 - Representational Learning
Introduction
Density Estimation
Factor Analysis
Principal Components Analysis
Clustering
Sparse Coding
Independent Components Analysis
Multi-Resolution and Wavelet Models
The Helmholtz Machine
Chapter Summary
Annotated Bibliography
Appendix - Mathematical Methods
Introduction
Linear Algebra
Differential Equations
Probability Theory
Fourier Transforms
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THEORETICAL NEUROSCIENCE
Electrical Circuits
d
The Function
Lagrange Multipliers
Annotated Bibliography
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Preface
Theoretical analysis and computational modeling are important tools for
characterizingwhatnervoussystemsdo,determininghowtheyfunction,
and understanding why they operate in particular ways. Neuroscience
encompasses approaches ranging from molecular and cellular studies to
humanpsychophysicsandpsychology. Theoreticalneuroscienceencour-
agescross-talkamongthesesub-disciplinesbyconstructingcompactrep-
resentationsofwhathasbeenlearned,buildingbridgesbetweendifferent
levelsofdescription,andidentifyingunifyingconceptsandprinciples. In
thisbook,wepresentthebasicmethodsusedforthesepurposesanddis-
cuss examples in which theoretical approaches have yielded insight into
nervoussystemfunction.
Thequestionswhat, how, andwhyareaddressedbydescriptive, mecha-
nistic,andinterpretivemodels,eachofwhichwediscussinthefollowing
chapters. Descriptive models summarize large amounts of experimental descriptivemodels
data compactly yet accurately, thereby characterizing what neurons and
neural circuits do. These models may be based loosely on biophysical,
anatomical,andphysiologicalfindings,buttheirprimarypurposeistode-
scribephenomenanottoexplainthem. Mechanisticmodels,ontheother mechanisticmodels
hand, address the question of how nervous systems operate on the ba-
sisofknownanatomy,physiology,andcircuitry. Suchmodelsoftenform
a bridge between descriptive models couched at different levels. Inter-
pretivemodelsusecomputationalandinformation-theoreticprinciplesto interpretivemodels
explorethebehavioralandcognitivesignificanceofvariousaspectsofner-
voussystemfunction,addressingthequestionofwhynervoussystemop-
erateastheydo.
Itisoftendifficultoidentifytheappropriatelevelofmodelingforapartic-
ularproblem. Afrequentmistakeistoassumethatamoredetailedmodel
is necessarily superior. Because models act as bridges between levels of
understanding, they must be detailed enough to make contact with the
lowerlevelyetsimpleenoughtoyieldclearresultsatthehigherlevel.
Draft: December17,2000 TheoreticalNeuroscience
2
OrganizationandApproach
This book is organized into three parts on the basis of general themes.
Part I (chapters 1-4) is devoted to the coding of information by action
potentialsandtherepresentionofinformationbypopulationsofneurons
withselectiveresponses. Modelingofneuronsandneuralcircuitsonthe
basis of cellular and synaptic biophysics is presented in part II (chapters
5-7). The role of plasticity in development and learning is discussed in
Part III (chapters 8-10). With the exception of chapters 5 and 6, which
jointlycoverneuronalmodeling,thechaptersarelargelyindependentand
canbeselectedandorderedinavarietyofwaysforaone-ortwo-semester
courseateithertheundergraduateorgraduatelevel.
Although we provide some background material, readers without previ-
ousexposuretoneuroscienceshouldrefertoaneurosciencetextbooksuch
as Kandel, Schwartz & Jessell (2000); Nicholls, Martin & Wallace (1992);
Bear,Connors&Paradiso(1996);Shepherd(1997);Zigmond,Bloom,Lan-
dis&Squire(1998);Purvesetal(2000).
Theoreticalneuroscienceisbasedonthebeliefthatmethodsofmathemat-
ics,physics,andcomputersciencecanelucidatenervoussystemfunction.
Unfortunately,mathematicscansometimesseemmoreofanobstaclethan
an aid to understanding. We have not hesitated to employ the level of
analysisneededtobepreciseandrigorous. Attimes,thismaystretchthe
tolerance of some of our readers. We encourage such readers to consult
themathematicalappendix,whichprovidesabriefreviewofmostofthe
mathematicalmethodsusedinthetext,butalsotopersevereandattempt
tounderstandtheimplicationsandconsequencesofadifficultderivation
evenifitsstepsareunclear.
Theoretical neuroscience, like any skill, can only be mastered with prac-
tice. Wehaveprovidedexercisesforthispurposeonthewebsiteforthis
book and urge the reader to do them. In addition, it will be highly in-
structive for the reader to construct the models discussed in the text and
exploretheirpropertiesbeyondwhatwehavebeenabletodointheavail-
ablespace.
Referencing
In order to maintain the flow of the text, we have kept citations within
the chapters to a minimum. Each chapter ends with an annotated bib-
liography containing suggestions for further reading (which are denoted
byaboldfont),informationaboutworkcitedwithinthechapter,andref-
erences to related studies. We concentrate on introducing the basic tools
ofcomputationalneuroscienceanddiscussingapplicationsthatwethink
besthelpthereadertounderstandandappreciatethem. Thismeansthat
anumberofsystemswherecomputationalapproacheshavebeenapplied
PeterDayanandL.F.Abbott Draft: December17,2000
3
with significant success are not discussed. References given in the anno-
tated bibliographies lead the reader toward such applications. In most
of the areas we cover, many people have provided critical insights. The
books and review articles in the further reading category provide more
comprehensive references to work that we apologetically have failed to
cite.
Acknowledgments
We are extremely grateful to a large number of students at Brandeis,
the Gatsby Computational Neuroscience Unit and MIT, and colleagues
at many institutions, who have painstakingly read, commented on, and
criticized, numerous versions of all the chapters. We particularly thank
Bard Ermentrout, Mark Kvale, Mark Goldman, John Hertz, Zhaoping
Li, Eve Marder, and Read Montague for providing extensive discussion
and advice on the whole book. A number of people read significant
portions of the text and provided valuable comments, criticism, and in-
sight: Bill Bialek, Pat Churchland, Nathanial Daw, Dawei Dong, Peter
Fo¨ldia´k, Fabrizio Gabbiani, Zoubin Ghahramani, Geoff Goodhill, David
Heeger,GeoffHinton,KenMiller,TonyMovshon,PhilNelson,SachaNel-
son, Bruno Olshausen, Mark Plumbley, Alex Pouget, Fred Rieke, John
Rinzel, Emilio Salinas, Sebastian Seung, Mike Shadlen, Satinder Singh,
Rich Sutton, Nick Swindale, Carl Van Vreeswijk, Chris Williams, David
Willshaw,CharlieWilson,AngelaYu,andRichZemel. Wehavereceived
significant additional assistance and advice from: Greg DeAngelis, Matt
Beal, Sue Becker, Tony Bell, Paul Bressloff, Emery Brown, Matteo Caran-
dini, FrancesChance, YangDan, KenjiDoya, EdErwin, JohnFitzpatrick,
David Foster, Marcus Frean, Ralph Freeman, Enrique Garibay, Frederico
Girosi,CharlieGross,MikeJordan,ShamKakade,SzabolcsKa´li,Christof
Koch,SimonLaughin,JohnLisman,ShawnLockery,GuyMayraz,Quaid
Morris,RandyO’Reilly,MaxRiesenhuber,SamRoweis,SimonOsindero,
TomasoPoggio,ClayReid,DarioRingach,HoracioRotstein,LanaRuther-
ford,KenSagino,ManeeshSahani,AlexeiSamsonovich,IdanSegev,Terry
Sejnowski, Haim Sompolinksy, Fiona Stevens, David Tank, Alessandro
Treves, Gina Turrigiano, David Van Essen, Martin Wainwright, Xiao-Jing
Wang,MaxWelling,MattWilson,DannyYoung,andKetchenZhang. We
apologisetoanyonewemayhaveinadvertentlyomittedfromtheselists.
Karen Abbott provided valuable help with the figures. From MIT Press,
wethankMichaelRutterforhispatienceandconsistentcommitment,and
SaraMeirowitzandLarryCohenforpickingupwhereMichaelleftoff.
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