Robust Estimation of Self-Exciting Generalized Linear Models With Application to Neuronal Modeling

We consider the problem of estimating self-exciting generalized linear models from limited binary observations, where the history of the process serves as the covariate. We analyze the performance of two classes of estimators, namely, the $\ell _1$-regularized maximum likelihood and greedy estimators, for a canonical self-exciting process and characterize the sampling trade-offs required for stable recovery in the non-asymptotic regime. Our results extend those of compressed sensing for linear and generalized linear models with independent identically distributed covariates to those with highly interdependent covariates. We further provide simulation studies as well as application to real spiking data from the mouse's lateral geniculate nucleus and the ferret's retinal ganglion cells, which agree with our theoretical predictions.

[1]  Luca Citi,et al.  Point-Process Nonlinear Models With Laguerre and Volterra Expansions: Instantaneous Assessment of Heartbeat Dynamics , 2013, IEEE Transactions on Signal Processing.

[2]  Uri T Eden,et al.  A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects. , 2005, Journal of neurophysiology.

[3]  Lingjiong Zhu.  Nonlinear Hawkes Processes , 2013, 1304.7531.

[4]  Esko Valkeila,et al.  An Introduction to the Theory of Point Processes, Volume II: General Theory and Structure, 2nd Edition by Daryl J. Daley, David Vere‐Jones , 2008 .

[5]  Liam Paninski,et al.  Model-Based Decoding, Information Estimation, and Change-Point Detection Techniques for Multineuron Spike Trains , 2011, Neural Computation.

[6]  M. S. Bartlett,et al.  STATISTICAL ESTIMATION OF DENSITY FUNCTIONS , 2016 .

[7]  Riccardo Barbieri,et al.  Statistical Inference for Assessing Functional Connectivity of Neuronal Ensembles With Sparse Spiking Data , 2011, IEEE Transactions on Neural Systems and Rehabilitation Engineering.

[8]  Min Wu,et al.  Sampling Requirements for Stable Autoregressive Estimation , 2016, IEEE Transactions on Signal Processing.

[9]  P. Reynaud-Bouret,et al.  Some non asymptotic tail estimates for Hawkes processes , 2007 .

[10]  D. Linden,et al.  Regulation of the rebound depolarization and spontaneous firing patterns of deep nuclear neurons in slices of rat cerebellum. , 1999, Journal of neurophysiology.

[11]  Venkatesh Saligrama,et al.  Sparse signal recovery under Poisson statistics , 2013, 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[12]  R. Kass,et al.  Multiple neural spike train data analysis: state-of-the-art and future challenges , 2004, Nature Neuroscience.

[13]  Liam Paninski,et al.  Statistical models for neural encoding, decoding, and optimal stimulus design. , 2007, Progress in brain research.

[14]  C. Shatz,et al.  Transient period of correlated bursting activity during development of the mammalian retina , 1993, Neuron.

[15]  Tong Zhang,et al.  Sparse Recovery With Orthogonal Matching Pursuit Under RIP , 2010, IEEE Transactions on Information Theory.

[16]  M. Bartlett.  The Spectral Analysis of Point Processes , 1963 .

[17]  S. Sombati,et al.  Recurrent spontaneous seizure activity in hippocampal neuronal networks in culture. , 1995, Journal of neurophysiology.

[18]  Deanna Needell,et al.  CoSaMP: Iterative signal recovery from incomplete and inaccurate samples , 2008, ArXiv.

[19]  Min Wu,et al.  Sparse estimation of self-exciting point processes with application to LGN neural modeling , 2014, 2014 IEEE Global Conference on Signal and Information Processing (GlobalSIP).

[20]  B. Mandelbrot,et al.  RANDOM WALK MODELS FOR THE SPIKE ACTIVITY OF A SINGLE NEURON. , 1964, Biophysical journal.

[21]  Emery N. Brown,et al.  A Granger Causality Measure for Point Process Models of Ensemble Neural Spiking Activity , 2011, PLoS Comput. Biol..

[22]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[23]  G L GERSTEIN,et al.  An approach to the quantitative analysis of electrophysiological data from single neurons. , 1960, Biophysical journal.

[24]  Emery N. Brown,et al.  The Time-Rescaling Theorem and Its Application to Neural Spike Train Data Analysis , 2002, Neural Computation.

[25]  R. Desimone,et al.  Neural mechanisms of spatial selective attention in areas V1, V2, and V4 of macaque visual cortex. , 1997, Journal of neurophysiology.

[26]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[27]  D. Pinault,et al.  Intracellular recordings in thalamic neurones during spontaneous spike and wave discharges in rats with absence epilepsy , 1998, The Journal of physiology.

[28]  Sarah M. N. Woolley,et al.  A Generalized Linear Model for Estimating Spectrotemporal Receptive Fields from Responses to Natural Sounds , 2011, PloS one.

[29]  M. Feller,et al.  Mechanisms underlying spontaneous patterned activity in developing neural circuits , 2010, Nature Reviews Neuroscience.

[30]  K. Ramanan,et al.  Concentration Inequalities for Dependent Random Variables via the Martingale Method , 2006, math/0609835.

[31]  M. Quirk,et al.  Construction and analysis of non-Poisson stimulus-response models of neural spiking activity , 2001, Journal of Neuroscience Methods.

[32]  Bruno A Olshausen,et al.  Spontaneous Activity of Morphologically Identified Ganglion Cells in the Developing Ferret Retina , 2003, The Journal of Neuroscience.

[33]  Michael Elad,et al.  From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images , 2009, SIAM Rev..

[34]  D. Vere-Jones.  Stochastic Models for Earthquake Occurrence , 1970 .

[35]  E. Brown,et al.  A point-process model of human heartbeat intervals: new definitions of heart rate and heart rate variability. , 2005, American journal of physiology. Heart and circulatory physiology.

[36]  L. Paninski.  Maximum likelihood estimation of cascade point-process neural encoding models , 2004, Network.

[37]  G Mann,et al.  ON THE THALAMUS * , 1905, British medical journal.

[38]  E N Brown,et al.  An analysis of neural receptive field plasticity by point process adaptive filtering , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[39]  Roummel F. Marcia,et al.  Compressed Sensing Performance Bounds Under Poisson Noise , 2009, IEEE Transactions on Signal Processing.

[40]  Yosihiko Ogata,et al.  Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes , 1988 .

[41]  Y. C. Pati,et al.  Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition , 1993, Proceedings of 27th Asilomar Conference on Signals, Systems and Computers.

[42]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[43]  Thomas Jackson,et al.  A data repository and analysis framework for spontaneous neural activity recordings in developing retina , 2013, bioRxiv.

[44]  N. Tritsch,et al.  The origin of spontaneous activity in the developing auditory system , 2007, Nature.

[45]  Nicholas J. Priebe,et al.  Emergence of Orientation Selectivity in the Mammalian Visual Pathway , 2013, The Journal of Neuroscience.

[46]  Gautam B Awatramani,et al.  An Intrinsic Neural Oscillator in the Degenerating Mouse Retina , 2011, The Journal of Neuroscience.

[47]  A. Grinvald,et al.  Linking spontaneous activity of single cortical neurons and the underlying functional architecture. , 1999, Science.

[48]  Venkatesh Saligrama,et al.  Minimax Optimal Sparse Signal Recovery With Poisson Statistics , 2015, IEEE Transactions on Signal Processing.

[49]  M. Crair,et al.  An Instructive Role for Patterned Spontaneous Retinal Activity in Mouse Visual Map Development , 2011, Neuron.

[50]  Emery N. Brown,et al.  Estimating a State-space Model from Point Process Observations Emery N. Brown , 2022 .

[51]  E.J. Candes.  Compressive Sampling , 2022 .

[52]  Martin J. Wainwright,et al.  A unified framework for high-dimensional analysis of $M$-estimators with decomposable regularizers , 2009, NIPS.

[53]  廣瀬雄一,et al.  Neuroscience , 2019, Workplace Attachments.

[54]  S. Geer.  On Hoeffding's Inequality for Dependent Random Variables , 2002 .

[55]  A. Hawkes.  Spectra of some self-exciting and mutually exciting point processes , 1971 .

[56]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[57]  J. Neuman,et al.  Statistical and Stochastic Modeling of Gang Rivalries in Los Angeles , 2010 .