Methods and Models in Neurophysics

by Chow; Gutkin; Hansel; Meunier; Dalibard

• ISBN13: 9780444517920
• ISBN10: 0444517928
• Edition: 1st
• Format: Hardcover
• Copyright: 2005-02-09
• Publisher: Elsevier Science

Summary

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.

Table of Contents

Experimenting with theory	1	(16)
E. Marder
Overcoming communication barriers	6	(2)
Modeling with biological neurons-the dynamic clamp	8	(1)
The traps inherent in building conductance-based models	9	(3)
Theory can drive new experiments	12	(1)
Conclusions	13	(4)
References	14	(3)
Understanding neuronal dynamics by geometrical dissection of minimal models	17	(56)
A. Borisyuk
J. Rinzel
Introduction	21	(4)
Nonlinear behaviors, time scales, our approach	21	(1)
Electrical activity of cells	22	(3)
Revisiting the Hodgkin-Huxley equations	25	(13)
Background and formulation	25	(3)
Hodgkin-Huxley gating equations as idealized kinetic models	28	(1)
Dissection of the action potential	29	(1)
Current-voltage relations	29	(1)
Qualitative view of fast-slow dissection	30	(2)
Stability of the fast subsystem's steady states	32	(1)
Repetitive firing	33	(1)
Stability of the four-variable model's steady state	33	(1)
Stability of periodic solutions	34	(1)
Bistability	35	(3)
Morris-Lecar model	38	(13)
Excitable regime	41	(1)
Post-inhibitory rebound	42	(1)
Single steady state. Onset of repetitive firing, type II	43	(3)
Three steady states	46	(1)
Large φ. Bistability of steady states	46	(1)
Small φ. Onset of repetitive firing, Type I	47	(3)
Intermediate φ. Bistability of rest state and a depolarized oscillation	50	(1)
Similar phenomena in the Hodgkin-Huxley model	50	(1)
Summary: onset of repetitive firing, Types I and II	51	(1)
Bursting, cellular level	51	(7)
Geometrical analysis and fast-slow dissection of bursting dynamics	52	(1)
Examples of bursting behavior	53	(1)
Square wave bursting	53	(2)
Parabolic bursting	55	(1)
Elliptic bursting	56	(1)
Other types of bursting	57	(1)
Bursting, network generated. Episodic rhythms in the developing spinal cord	58	(7)
Experimental background	58	(1)
Firing rate model	59	(1)
Basic recurrent network	60	(1)
Full model	61	(2)
Predictions of the model	63	(2)
Chapter summary	65	(8)
Appendix A. Mathematical formulation of fast-slow dissection	65	(2)
Appendix B. Stability of periodic solutions	67	(2)
References	69	(4)
Geometric singular perturbation analysis of neuronal dynamics	73	(50)
D. Terman
Introduction	77	(1)
Introduction to dynamical systems	78	(12)
First order equations	78	(1)
Bifurcations	79	(3)
Bistability and hysteresis	82	(1)
The phase plane	83	(1)
Oscillations	84	(1)
Local bifurcations	85	(2)
Global bifurcations	87	(1)
Geometric singular perturbation theory	88	(2)
Properties of a single neuron	90	(7)
Reduced models	91	(1)
Response to injected current	92	(2)
Traveling wave solutions	94	(3)
Two mutually coupled cells	97	(11)
Introduction	97	(1)
Synaptic coupling	98	(1)
Geometric approach	99	(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