Methods and Models in Neurophysics

, by Chow; Gutkin; Hansel; Meunier; Dalibard

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ISBN: 9780444517920 | 0444517928
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Copyright: 2/9/2005

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Neuroscience is an interdisciplinary field that strives to understand the functioning of neural systems at levels ranging from biomolecules and cells to behaviour and higher brain functions (perception, memory, cognition). Neurophysics has flourished over the past three decades, becoming an indelible part of neuroscience, and has arguably entered its maturity. It encompasses a vast array of approaches stemming from theoretical physics, computer science, and applied mathematics. This book provides a detailed review of this field from basic concepts to its most recent development.

Experimenting with theory

(16)

E. Marder

Overcoming communication barriers

(2)

Modeling with biological neurons-the dynamic clamp

(1)

The traps inherent in building conductance-based models

(3)

Theory can drive new experiments

(1)

Conclusions

(4)

References

(3)

Understanding neuronal dynamics by geometrical dissection of minimal models

(56)

A. Borisyuk

J. Rinzel

Introduction

(4)

Nonlinear behaviors, time scales, our approach

(1)

Electrical activity of cells

(3)

Revisiting the Hodgkin-Huxley equations

(13)

Background and formulation

(3)

Hodgkin-Huxley gating equations as idealized kinetic models

(1)

Dissection of the action potential

(1)

Current-voltage relations

(1)

Qualitative view of fast-slow dissection

(2)

Stability of the fast subsystem's steady states

(1)

Repetitive firing

(1)

Stability of the four-variable model's steady state

(1)

Stability of periodic solutions

(1)

Bistability

(3)

Morris-Lecar model

(13)

Excitable regime

(1)

Post-inhibitory rebound

(1)

Single steady state. Onset of repetitive firing, type II

(3)

Three steady states

(1)

Large φ. Bistability of steady states

(1)

Small φ. Onset of repetitive firing, Type I

(3)

Intermediate φ. Bistability of rest state and a depolarized oscillation

(1)

Similar phenomena in the Hodgkin-Huxley model

(1)

Summary: onset of repetitive firing, Types I and II

(1)

Bursting, cellular level

(7)

Geometrical analysis and fast-slow dissection of bursting dynamics

(1)

Examples of bursting behavior

(1)

Square wave bursting

(2)

Parabolic bursting

(1)

Elliptic bursting

(1)

Other types of bursting

(1)

Bursting, network generated. Episodic rhythms in the developing spinal cord

(7)

Experimental background

(1)

Firing rate model

(1)

Basic recurrent network

(1)

Full model

(2)

Predictions of the model

(2)

Chapter summary

(8)

Appendix A. Mathematical formulation of fast-slow dissection

(2)

Appendix B. Stability of periodic solutions

(2)

References

(4)

Geometric singular perturbation analysis of neuronal dynamics

(50)

D. Terman

Introduction

(1)

Introduction to dynamical systems

(12)

First order equations

(1)

Bifurcations

(3)

Bistability and hysteresis

(1)

The phase plane

(1)

Oscillations

(1)

Local bifurcations

(2)

Global bifurcations

(1)

Geometric singular perturbation theory

(2)

Properties of a single neuron

(7)

Reduced models

(1)

Response to injected current

(2)

Traveling wave solutions

(3)

Two mutually coupled cells

(11)

Introduction

(1)

Synaptic coupling

(1)

Geometric approach

(2)

Synchrony with excitatory synapses

101

(4)

Desynchrony with inhibitory synapses

105

(3)

Excitatory-inhibitory networks

108

(7)

Introduction

108

(3)

Synchronous solution

111

(3)

Clustered solutions

114

(1)

Activity patterns in the basal ganglia

115

(8)

The basal ganglia

115

(2)

The model

117

(2)

Activity patterns

119

(1)

Concluding remarks

119

(2)

References

121

(2)

Theory of neural synchrony

123

(56)

G. Mato

Introduction

127

(1)

Weakly coupled oscillators

128

(25)

Stability of cluster states

131

(1)

One-cluster state

131

(1)

Two-cluster states

132

(1)

N-cluster state

132

(2)

Stochastic dynamics

134

(1)

Evaluation of the phase interaction function

135

(1)

Synaptic interactions

136

(1)

Conductance based neurons

136

(8)

Integrate-and-fire neurons

144

(2)

Gap junctions

146

(1)

Numerical simulations of conductance based models

146

(3)

Phase interaction for electrotonic interactions

149

(4)

Strongly coupled oscillators: mechanisms of synchrony

153

(16)

The two-population QIF model

153

(3)

Stability of the asynchronous state

156

(3)

Mechanisms of synchrony

159

(1)

The symmetric case

159

(5)

The general case

164

(3)

Application: persistent states

167

(2)

Conclusion

169

(10)

Appendix A. Hodgkin-Huxley and Wang-Buszaki models

172

(2)

Appendix B. Measure of synchrony and variability in numerical simulations

174

(1)

Appendix C. Reduction of a conductance-based model to the QIF model

175

(2)

References

177

(2)

Some useful numerical techniques for simulating integrate-and-fire networks

179

(18)

M. Shelley

Introduction

183

(1)

The conductance-based I&F model

184

(1)

Modified time-stepping schemes

185

(5)

Synaptic interactions

190

(2)

Simulating a V1 model

192

(5)

References

195

(2)

Propagation of pulses in cortical networks: the single-spike approximation

197

(48)

D. Golomb

Introduction

202

(1)

Propagating pulses in networks of excitatory neurons

203

(14)

The model

203

(2)

Continuous and lurching pulses

205

(1)

Existence, stability and velocity of continuous pulses

206

(6)

Velocity of lurching pulses

212

(3)

The nature of lurching pulses

215

(1)

Finite axonal conduction velocity

216

(1)

Propagating pulses in networks of excitatory and inhibitory neurons

217

(20)

The model

217

(2)

Fast and slow pulses

219

(1)

Analysis of traveling pulse solutions

220

(1)

Volterra representation

220

(1)

Existence of traveling pulses

220

(2)

Stability of traveling pulses

222

(1)

Voltage profile

223

(1)

Theory of propagation of fast and slow pulses

223

(1)

Effects of I-to-E inhibition and slow E-to-E excitation

224

(3)

Response to shock initial conditions

227

(2)

Effects of strength of fast E-to-E excitation

229

(2)

I-to-I conductance and irregular pulses

231

(2)

Lurching pulses

233

(1)

Finite axonal conduction velocity

234

(1)

Traveling pulse solution with finite axonal conduction velocity

234

(2)

Pulse velocity may decrease with c

236

(1)

Discussion

237

(8)

Types of propagating pulses

237

(1)

Effects of approximations

238

(1)

Application to biological pulses

239

(1)

Network without inhibition

239

(1)

Network with inhibition

240

(1)

Appendix A. Stability of the lower branch

241

(2)

References

243

(2)

Activity-dependent transmission in neocortical synapses

245

(22)

M. Tsodyks

Introduction

249

(1)

Phenomenological model of synaptic depression and facilitation

250

(3)

Synaptic depression

250

(1)

Modeling synaptic facilitation

251

(2)

Dynamic synaptic transmission on the population level

253

(3)

Recurrent networks with synaptic depression

256

(7)

Conclusion

263

(4)

References

264

(3)

Theory of large recurrent networks: from spikes to behavior

267

(74)

H. Sompolinsky

O.L. White

Introduction

271

(1)

From spikes to rates I: rates in asynchronous states

272

(12)

Introduction: rate models

272

(1)

Synchronous and asynchronous states

273

(3)

Synchronous and asynchronous states in highly connected networks

276

(3)

Rate equations for asynchronous networks

279

(5)

From spikes to rates II: dynamics and conductances

284

(11)

Introduction

284

(1)

Dynamics of firing rates---linear response theory

285

(4)

Synaptic inputs as conductances

289

(3)

Rate dynamics for slow synapses

292

(1)

Summary

293

(2)

Persistent activity and neural integration in the brain

295

(17)

Introduction

295

(2)

Dynamics of linear recurrent networks

297

(7)

Recurrent network model of persistence and integration

304

(3)

Single neuron model for integration and persistence

307

(5)

Feature selectivity in recurrent networks---the ring model

312

(12)

Introduction

312

(1)

The ring network

313

(2)

Stationary solution of the ring model

315

(1)

Symmetry breaking in the ring model

316

(8)

Models of associative memory

324

(14)

Introduction

324

(1)

Binary networks

324

(3)

The Hopfield model

327

(5)

Networks with sparse memories

332

(6)

Concluding remarks

338

(3)

References

339

(2)

Irregular activity in large networks of neurons

341

(66)

C. van Vreeswijk

H. Sompolinsky

Introduction

345

(2)

A simple binary model

347

(19)

The model

348

(1)

Population averaged activity

349

(4)

Heterogeneity of the firing rate.

353

(4)

Temporal correlations

357

(3)

Dependence on initial conditions

360

(2)

Non-linearity of response of individual cells

362

(4)

A memory model

366

(6)

A modified Willshaw model

367

(1)

Mean field equations

368

(2)

Multiple solutions

370

(2)

A model of visual cortex hypercolumn

372

(12)

The model

373

(1)

Population averaged response

374

(3)

Quenched disorder

377

(3)

Contrast invariance

380

(4)

Adding realism: integrate-and-fire network

384

(16)

Current based synapses

384

(4)

Conductance based synapses

388

(2)

Networks with two populations

390

(2)

A model of a hypercolumn

392

(6)

Beyond the Poisson assumption

398

(2)

Discussion

400

(7)

References

402

(5)

Network models of memory

407

(70)

N. Brunel

Introduction

411

(1)

Persistent neuronal activity during delayed response experiments

412

(9)

Discrete working memory: persistent activity in the delayed match to sample experiments

412

(2)

Plasticity of persistent activity patterns induced by learning

414

(3)

Spatial short-term memory: persistent activity in the oculomotor delayed response task

417

(1)

Parametric short-term memory

418

(2)

Persistent activity in slices

420

(1)

Scenarios for multistability in neural systems

421

(2)

Single cell

421

(1)

Local network

422

(1)

Networks involving several areas

422

(1)

Networks of binary neurons with discrete attractors

423

(16)

The Hopfield model

423

(2)

Signal-to-noise ratio (SNR) analysis

425

(1)

Statistical mechanics of the Hopfield model

426

(3)

Critiques of the Hopfield model

429

(1)

Robustness to perturbations of the synaptic matrix

430

(2)

Models with 0,1 neurons and low coding levels

432

(1)

Summary---models with binary neurons and analog synapses

433

(1)

A model with binary synapses: the Willshaw model

434

(2)

Optimal capacity of networks of binary neurons

436

(3)

Other generalizations of the Hopfield model

439

(1)

Learning

439

(6)

Supervised learning: the perceptron learning algorithm

440

(2)

Unsupervised learning---palimpsest models

442

(1)

Palimpsest models with analog synapses

442

(1)

Unsupervised learning in networks with discrete synapses

443

(2)

Summary

445

(1)

Networks of spiking neurons with discrete attractors

445

(14)

A cortical network model

447

(1)

Dynamics of networks of spiking neurons---single population analysis

448

(3)

Two population networks

451

(1)

Storing binary non-overlapping patterns in the cortical network model

452

(4)

Spontaneous and persistent activity in the large C, sparse coding limit

456

(2)

Capacity

458

(1)

Stability of persistent states vs synchronized oscillations

458

(1)

Plasticity of persistent activity

459

(6)

Learning correlations between patterns separated by delay periods

459

(2)

Network subjected to DMS task with a fixed sequence of samples

461

(2)

Network subjected to pair-associate tasks

463

(1)

Transitions between states in the absence of external inputs in networks of spiking neurons

463

(2)

Models with continuous attractors

465

(3)

Conclusions

468

(9)

References

470

(7)

Pattern formation in visual cortex

477

(98)

P.C. Bressloff

Introduction

481

(4)

The functional architecture of V1

485

(8)

Retinotopic map

485

(3)

Feature maps

488

(3)

Long-range horizontal and feedback connections

491

(2)

Large-scale models of V1

493

(18)

Planar model of V1

493

(2)

Receptive fields and the feedforward input h

495

(6)

Orientation pinwheels as organizing centers for feature map F(r)

501

(2)

Long-range connections and the weight distribution w

503

(3)

Coupled hypercolumn model of V1

506

(5)

Pattern formation in a single hypercolumn

511

(17)

Orientation tuning in a one-population ring model

511

(2)

Derivation of amplitude equation using the Fredholm alternative

513

(4)

Bifurcation theory and O(2) symmetry

517

(2)

Two-population model

519

(2)

Amplitude equation for oscillatory patterns

521

(3)

Mean-field theory

524

(4)

Pattern formation in a coupled hypercolumn model of V1

528

(19)

Linear stability analysis

529

(3)

Marginal stability

532

(3)

Doubly-periodic planforms

535

(3)

Amplitude equation

538

(2)

Bifurcation theory and shift-twist symmetry

540

(2)

Selection and stability of patterns

542

(3)

From cortical patterns to geometric visual hallucinations

545

(2)

Pattern formation in a planar model of V1

547

(14)

Homogeneous weights

547

(3)

Pinning of cortical patterns by spatially periodic horizontal connections

550

(6)

Ginzburg--Landau equation

556

(2)

Commensurate-incommensurate transitions in cortex

558

(3)

Pattern formation in a model of cortical development

561

(7)

Future directions

568

(7)

References

570

(5)

Symmetry breaking and pattern selection in visual cortical development

575

(66)

F. Wolf

Introduction

579

(3)

The pattern of orientation preference columns

582

(1)

Symmetries in the development of orientation columns

583

(4)

From learning to dynamics

587

(1)

Generation and motion of pinwheels

588

(9)

Random orientation maps

588

(5)

Gaussian fields from a dynamic instability

593

(3)

Predicting pinwheel annihilation

596

(1)

The problem of pinwheel stability

597

(1)

Weakly nonlinear analysis of pattern selection

598

(16)

Amplitude equations

599

(2)

Fixed points of the amplitude equations and their stability

601

(5)

Linking amplitude equations and field dynamics

606

(6)

Extrinsic stability

612

(1)

Swift-Hohenberg models

613

(1)

A Swift-Hohenberg model with stable pinwheel patterns

614

(21)

Construction of the model

615

(5)

Essentially complex planforms

620

(5)

Stability ranges and phase diagram

625

(8)

The spectrum of pinwheel densities

633

(2)

Discussion

635

(6)

References

638

(3)

Of the evolution of the brain

641

(50)

A. Treves

Y. Roudi

Introduction and summary

645

(1)

The phase transition that made us mammals

645

(5)

An information-theoretical advantage in the hippocampus

646

(1)

An information-theoretical hypothesis about layers and maps

647

(3)

Maps and patterns of threshold-linear units

650

(9)

A model with geometry in its connections

651

(1)

Retrieval states

652

(3)

The network without structure

655

(2)

Appearance of bumps of activity

657

(2)

The main points

659

(1)

Validation of the lamination hypothesis

659

(4)

Differentiation among isocortical layers

661

(2)

What do we need DG and CA1 for?

663

(5)

Distinguishing storage from retrieval

664

(2)

CA1 in search of a role

666

(2)

Infinite recursion and the origin of cognition

668

(6)

Infinite recursion and its ambiguities

669

(1)

Memory---statics and dynamics

670

(2)

Memory latching as a model of recursive dynamics

672

(2)

Reducing local networks to Potts units

674

(17)

A discrete-valued model

675

(1)

Storage capacity

676

(1)

Sparse coding

677

(1)

A Potts model with graded response

678

(1)

Correlated patterns

679

(2)

Scheme of the simulations

681

(3)

Conclusions

684

(1)

References

685

(6)

Theory of point processes for neural systems

691

(38)

E.N. Brown

Neural spike trains as point processes

694

(1)

Integrate and fire models and interspike interval distributions

695

(6)

Non-leaky integrator with excitatory Poisson inputs

695

(2)

Non-leaky integrator with excitatory and inhibitory Poisson inputs

697

(1)

Non-leaky integrator with random walk inputs

698

(2)

Remarks

700

(1)

The conditional intensity function and interevent time probability density

701

(3)

Joint probability density of a point process

704

(4)

Derivation of the joint probability density

704

(3)

An alternative derivation of the joint probability density of a spike train

707

(1)

Special point process models

708

(7)

Poisson processes

709

(1)

Axioms for a Poisson process

709

(1)

Interevent and waiting time probability densities for a Poisson process

709

(2)

Inhomogeneous Poisson process

711

(2)

Renewal processes

713

(1)

Counting process associated with a renewal process

713

(1)

Asymptotic distribution of N(t) for large t

714

(1)

Stationary process

714

(1)

Covariance function

714

(1)

Spectral density function

715

(1)

The time-rescaling theorem

715

(5)

An elementary proof of the time-rescaling theorem

716

(2)

The time-rescaling theorem: assessing model goodness-of-fit

718

(1)

Kolmogorov-Smirnov test

719

(1)

Quantile-quantile plot

719

(1)

Normalized point process residuals

720

(1)

Simulation of point processes

720

(4)

Thinning algorithm 1

720

(1)

Simulating a multivariate point process model

721

(2)

Simulating a univariate point process by time-rescaling

723

(1)

Poisson limit theorems

724

(1)

Poisson approximation to the binomial probability mass function

724

(1)

Superposition of point processes

725

(1)

Problems

725

(4)

References

726

(3)

Technique(s) for spike-sorting

729

(58)

C. Pouzat

Introduction

733

(1)

The problem to solve

734

(1)

Two features of single neuron data we would like to include in the spike-sorting procedure

735

(4)

Spike waveforms from a single neuron are usually not stationary on a short time-scale

735

(1)

An experimental illustration with cerebellar Purkinje cells

735

(1)

A phenomenological description by an exponential relaxation

735

(1)

Neurons inter-spike interval probability density functions carry a lot of information we would like to exploit

736

(1)

An experimental illustration from projection neurons in the locust antennal lobe

736

(1)

A phenomenological description by a log-normal pdf

737

(1)

Noise properties

738

(1)

Noise whitening

739

(2)

Probabilistic data generation model

741

(1)

Model assumptions

741

(8)

Likelihood computation for single neuron data

741

(4)

Complications with multi-neuron data

745

(1)

Notations for the multi-neuron model parameters

745

(1)

Configuration and data augmentation

745

(2)

Posterior density

747

(1)

Remarks on the use of the posterior

747

(1)

The normalizing constant problem and its solution

748

(1)

The problem

748

(1)

Its solution

748

(1)

Markov chains

749

(8)

Some notations and definitions

750

(2)

The fundamental theorem and the ergodic theorem

752

(5)

The Metropolis--Hastings algorithm and its relatives

757

(5)

The Metropolis--Hastings algorithm

757

(1)

Second fundamental theorem

757

(1)

Metropolis--Hastings and Gibbs algorithms for multi-dimensional spaces

758

(1)

An example: the Gibbs sampler for the parameters of the ISI density

759

(2)

Generation of the amplitude parameters of our model

761

(1)

Generation of the configuration

762

(1)

Our complete MC step

762

(1)

Priors choice

762

(2)

The good use of the ergodic theorem. A warning

764

(1)

Autocorrelation functions and confidence intervals

764

(1)

Initialization bias

765

(1)

Slow relaxation and the replica exchange method

765

(3)

An Example from a simulated data set

768

(10)

Data properties

769

(1)

Algorithm dynamics and parameters estimates without the REM

769

(1)

Initialization

769

(1)

Energy evolution

770

(1)

Model parameters evolution

771

(1)

Model parameters estimates

771

(1)

Algorithm dynamics and parameters estimates with the REM

772

(1)

Making sure the REM ``works''

773

(1)

Posterior estimates with the REM

774

(1)

Configuration estimate

775

(2)

A more detailed illustration of the REM dynamics

777

(1)

Conclusions

778

(2)

Exercises solutions

780

(7)

Cholesky decomposition (or factorization)

780

(1)

Bayesian posterior densities for the parameters of a log-normal pdf

781

(1)

Stochastic property of a Markov matrix

782

(1)

Detailed balance

783

(1)

References

783

(4)

The emergence of relevant data representations: an information theoretic approach

787

N. Tishby

Part I: the fundamental dilemma

792

Fitting models to data---what is the goal?

792

Accuracy versus complexity

793

Quantifying complexity and accuracy

794

Part II: Shannon's information theory---a new perspective

795

Formalizing the problem: distortion versus cost

797

Solving the cost-distortion tradeoff: the emergence of mutual information

798

The rate-distortion function

801

Example: the Gaussian source and channel

802

Perfect matching of source and channel

803

Part III: relevant data representation

804

The ``inverse Shannon problem''

805

Looking into the black box: the information bottleneck method

806

IB as perfectly matched source-channel

809

Alternating projections and the IB algorithm

810

Part IV: applications and extensions

812

IB clustering

812

IB dimensionality reduction: the Gaussian IB

813

Towards multivariate network IB

817

Bayesian networks and multi-information

817

Network information bottleneck principle

820

Characterization of the IB fix-points

823

Deterministic annealing algorithm

825

References

827

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Methods and Models in Neurophysics

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