Parameter Estimation for Spatio-Temporal Maximum Entropy Distributions: Application to Neural Spike Trains

We propose a numerical method to learn maximum entropy (MaxEnt) distributions with spatio-temporal constraints from experimental spike trains. This is an extension of two papers, [10] and [4], which proposed the estimation of parameters where only spatial constraints were taken into account. The extension we propose allows one to properly handle memory effects in spike statistics, for large-sized neural networks.

[1]  A. V. Skorohod,et al.  The theory of stochastic processes , 1974 .

[2]  William Bialek,et al.  Entropy and Information in Neural Spike Trains , 1996, cond-mat/9603127.

[3]  Gerhard Stock,et al.  Maximum caliber inference of nonequilibrium processes. , 2010, The Journal of chemical physics.

[4]  Michael J. Berry,et al.  Gibbs distribution analysis of temporal correlations structure in retina ganglion cells , 2011, Journal of Physiology - Paris.

[5]  F. Papangelou GIBBS MEASURES AND PHASE TRANSITIONS (de Gruyter Studies in Mathematics 9) , 1990 .

[6]  Shun-ichi Amari,et al.  Information-Geometric Decomposition in Spike Analysis , 2001, NIPS.

[7]  Konrad P Kording,et al.  How advances in neural recording affect data analysis , 2011, Nature Neuroscience.

[8]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[9]  Bruno Cessac,et al.  Spatio-temporal spike train analysis for large scale networks using the maximum entropy principle and Monte Carlo method , 2012, 1209.3886.

[10]  Hilbert J. Kappen,et al.  Boltzmann Machine Learning Using Mean Field Theory and Linear Response Correction , 1997, NIPS.

[11]  Eero P. Simoncelli,et al.  Spatio-temporal correlations and visual signalling in a complete neuronal population , 2008, Nature.

[12]  Daniel N Hill,et al.  Quality Metrics to Accompany Spike Sorting of Extracellular Signals , 2011, The Journal of Neuroscience.

[13]  E J Chichilnisky,et al.  A simple white noise analysis of neuronal light responses , 2001, Network.

[14]  R. Segev,et al.  The Architecture of Functional Interaction Networks in the Retina , 2011, The Journal of Neuroscience.

[15]  R. Segev,et al.  Sparse low-order interaction network underlies a highly correlated and learnable neural population code , 2011, Proceedings of the National Academy of Sciences.

[16]  Michael J. Berry,et al.  Mapping a Complete Neural Population in the Retina , 2012, The Journal of Neuroscience.

[17]  Ronald Rosenfeld,et al.  Adaptive Statistical Language Modeling; A Maximum Entropy Approach , 1994 .

[18]  E. T. Jaynes,et al.  Where do we Stand on Maximum Entropy , 1979 .

[19]  Imre Csiszár,et al.  On the computation of rate-distortion functions (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[20]  J. Besag Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[21]  Shun-ichi Amari,et al.  Information geometry on hierarchy of probability distributions , 2001, IEEE Trans. Inf. Theory.

[22]  Adam L. Berger,et al.  A Maximum Entropy Approach to Natural Language Processing , 1996, CL.

[23]  A.M. Litke,et al.  What does the eye tell the brain?: Development of a system for the large scale recording of retinal output activity , 2003, 2003 IEEE Nuclear Science Symposium. Conference Record (IEEE Cat. No.03CH37515).

[24]  G. Keller Equilibrium States in Ergodic Theory , 1998 .

[25]  G. Keller,et al.  Pressure and Equilibrium States in Ergodic Theory , 2008, Encyclopedia of Complexity and Systems Science.

[26]  D. Ruelle Statistical Mechanics: Rigorous Results , 1999 .

[27]  A. Maccione,et al.  Large-scale, high-resolution electrophysiological imaging of field potentials in brain slices with microelectronic multielectrode arrays , 2012, Front. Neural Circuits.

[28]  Robert E. Schapire,et al.  Faster solutions of the inverse pairwise Ising problem , 2008 .

[29]  Michael J. Berry,et al.  The simplest maximum entropy model for collective behavior in a neural network , 2012, 1207.6319.

[30]  Lide Wu,et al.  A Fast Algorithm for Feature Selection in Conditional Maximum Entropy Modeling , 2003, EMNLP.

[31]  Michael J. Berry,et al.  Spin glass models for a network of real neurons , 2009, 0912.5409.

[32]  E. Jaynes Information Theory and Statistical Mechanics , 1957 .

[33]  B. Cessac,et al.  Estimating maximum entropy distributions from periodic orbits in spike trains , 2013 .

[34]  Ronald Rosenfeld,et al.  Efficient sampling and feature selection in whole sentence maximum entropy language models , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[35]  J. Donoghue,et al.  Collective dynamics in human and monkey sensorimotor cortex: predicting single neuron spikes , 2009, Nature Neuroscience.

[36]  R. Bowen Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms , 1975 .

[37]  DANNY CALEGARI,et al.  Thermodynamic Formalism , 2021, Lecture Notes in Mathematics.

[38]  R. Quian Quiroga,et al.  Unsupervised Spike Detection and Sorting with Wavelets and Superparamagnetic Clustering , 2004, Neural Computation.

[39]  Sami El Boustani,et al.  Prediction of spatiotemporal patterns of neural activity from pairwise correlations. , 2009, Physical review letters.

[40]  Rob Koeling Chunking with Maximum Entropy Models , 2000, CoNLL/LLL.

[41]  Thermodynamical Formalism and Multifractal Analysis for Meromorphic Functions of Finite Order , 2007, math/0701275.

[42]  Michael J. Berry,et al.  Weak pairwise correlations imply strongly correlated network states in a neural population , 2005, Nature.

[43]  E. Jaynes The Minimum Entropy Production Principle , 1980 .

[44]  Simon R. Schultz,et al.  The Ising decoder: reading out the activity of large neural ensembles , 2010, Journal of Computational Neuroscience.

[45]  Roberto Fernández,et al.  Chains with Complete Connections: General Theory, Uniqueness, Loss of Memory and Mixing Properties , 2003, math/0305026.

[46]  David R. Cox,et al.  The Theory of Stochastic Processes , 1967, The Mathematical Gazette.

[47]  Miroslav Dudík,et al.  Performance Guarantees for Regularized Maximum Entropy Density Estimation , 2004, COLT.

[48]  John M. Beggs,et al.  A Maximum Entropy Model Applied to Spatial and Temporal Correlations from Cortical Networks In Vitro , 2008, The Journal of Neuroscience.

[49]  E. T. Jaynes,et al.  Macroscopic Prediction , 1996 .

[50]  Xiaoli Li,et al.  Estimating Temporal Causal Interaction between Spike Trains with Permutation and Transfer Entropy , 2013, PloS one.

[51]  Yoram Singer,et al.  Logistic Regression, AdaBoost and Bregman Distances , 2000, Machine Learning.