Statistical analysis of neural data

Spring 2007


This is a Ph.D.-level topics course in statistical analysis of neural data. Students from statistics, neuroscience, and engineering are all welcome to attend.

Time: W 2:10-4
Place: Rm 903, 1255 Amsterdam Ave (Stat Dept building)
Professor: Liam Paninski; Office: 1255 Amsterdam Ave, Rm 1028. Email: liam at stat dot columbia dot edu

Prerequisite: A good working knowledge of basic statistical concepts (likelihood, Bayes' rule, Poisson processes, Markov chains, Gaussian random vectors), including especially linear-algebraic concepts related to regression and principal components analysis, is necessary. No previous experience with neural data is required.
Evaluation: Final grades will be based on class participation and a student project. The project can involve either the implementation and justification of a novel analysis technique, or a standard analysis applied to a novel data set. Students can work in pairs or alone (if you work in pairs, of course, the project has to be twice as impressive). Projects are due May 9 (no extensions, since I have to turn in grades shortly after this date). See the Courseworks page for some project ideas.
Course goals: We will introduce a number of advanced statistical techniques relevant in neuroscience. Each technique will be illustrated via application to problems in neuroscience. The focus will be on the analysis of single and multiple spike train data, with a few applications to analyzing intracellular voltage and dendritic imaging data; however, we will not cover any topics in the analysis of fMRI data (for those interested in fMRI data, see this course by Martin Lindquist instead). A brief list of statistical concepts and corresponding neuroscience applications is below.

Statistical concept / technique Neuroscience application
Point processes; conditional intensity functions Neural spike trains; Poisson images
Time-rescaling theorem for point processes Fast simulation of network models
Bias, consistency, principal components Spike-triggered averaging; spike-triggered covariance
Generalized linear models Neural encoding models including spike-history effects; inferring network connectivity
Regularization; shrinkage estimation Maximum a posteriori estimation of high-dimensional neural encoding models
State-space models; sequential Monte Carlo / particle filtering Decoding spike trains; optimal voltage smoothing
Optimization and convexity techniques Spike-train decoding; ML estimation of encoding models
Quadratic programming / nonnegative least-squares Biophysical model fitting; inference of subthreshold voltage given spike trains
Markov chain Monte Carlo: Metropolis-Hastings and hit-and-run algorithms Firing rate estimation and spike-train decoding
Markov processes; first-passage times; Fokker-Planck equation Integrate-and-fire-based neural models
Kalman filter; extended/unscented Kalman filter; EM algorithm Inferring common-input from multineuronal spike-train data; analysis of behavioral learning experiments
Mixture models Spike-sorting / clustering
Laplace approximation; Fisher information Model-based decoding and information estimation; adaptive design of optimal stimuli

For those new to neuroscience: While we will cover all the necessary background as we go, for those who want to explore the material in greater depth, there are a bunch of good computational neuroscience resources. Four good books (each of which emphasize different topics, albeit with some overlap) are: Theoretical Neuroscience, by Dayan and Abbott; Biophysics of Computation: information processing in single neurons, by Koch; Spiking Neuron Models, by Gerstner and Kistler; and Spikes: exploring the neural code, by Rieke et al. The first chapter of the Spikes book has been kindly made available online - this makes a nice overview of some of the questions we will address in this course. Even more kindly, the full text of the Gerstner-Kistler book is online; for our purposes, the key sections are Chapter 1 and Part I. Another good online tutorial is available here. Finally, many of our examples will be drawn from the visual system: see the book Eye, Brain, and Vision (again, very kindly provided online) by David Hubel for an excellent introduction.

See this page for some introductory notes on probability and statistics (central limit theorem, Neyman-Pearson hypothesis testing, asymptotic theory for maximum likelihood estimation, sufficient statistics, etc.).

Schedule

Date Topic Reading Notes
Jan 17 Introduction Spikes introduction; Kass et al '05; Brown et al. '04
Jan 24-31 Classification approaches: spike-triggered averaging; Volterra-Wiener series; spike-triggered covariance; logistic regression; semiparametric regression Simoncelli et al. '04; Chichilnisky '01; Paninski '03; Sharpee et al. '04; Weisberg and Welsh, '94 notes, revised 5/6/07
Feb 7 Point processes: Poisson process, renewal process, self-exciting process, Cox process; time-rescaling: goodness-of-fit, fast simulation of network models Brown et al. '01 Brown's point process notes; supplementary notes
Feb 14-21 Generalized linear models (GLM); Poisson regression; log-concavity; spike-history effects; regularization/penalized likelihood; low-rank approximation; Fisher information; optimal experimental design for GLMs / optimal stimulus design Paninski et al. '07; Paninski '04; Truccolo et al '05; Kass et al. '01; Li and Duan '89; Duan and Li '91; Ahrens et al '06; Lewi et al '06 notes, revised 5/15/07
Feb 28 - Mar 7 Optimization: Newton-Raphson, conjugate gradients, trust-region, backtracking line search; bound optimization (auxiliary functions); convexity; maximum a posteriori (MAP) decoding of spike trains; estimation of Shannon information Pillow+Paninski '06; Boyd convexity notes; Lee et al '06; Krishnapuram et al '05; Fortin '01; Shewchunk conjugate gradient notes; Bialek et al '91; Warland et al '97; Paninski '03b; Paninski '04b; Beirlant et al '97; Nemenman et al '04 notes, revised 5/20/07
Mar 14 Spring break
Mar 21 Bayesian decoding of spike trains; Markov chain Monte Carlo (MCMC): Metropolis-Hastings and hit-and-run algorithms Metropolis/MCMC notes by Walsh; Survey on geometric random walks (including hit-and-run) by Vempala notes
Mar 28 The expectation-maximization (EM) algorithm for maximum likelihood given indirect measurements / hidden data; mixture models; spike sorting Two intros to EM, by Bilmes and Tagare; Neal and Hinton '98; Lewicki '98; Salakhutdinov et al '03; Shoham et al '03; Pouzat et al '04 notes, updated 5/21/07
Apr 4 Hidden Markov models (HMM) in discrete space; multistate GLMs for neurons with bistable firing properties; ion channel models Rabiner tutorial; Jordan review of graphical models; Gat et al '97; Escola multistate GLM notes; Sinclair's notes on continuous-time Markov chains; Colquhoun and Hawkes '82 notes
Apr 11-18 State space models; autoregressive models; Kalman filter; extended Kalman filter; particle filter; stratified resampling; applications in neural prosthetics, optimal smoothing of voltage/calcium traces, fitting common-input models for population spike train data, and analysis of nonstationary spike train data Kalman filter notes by Minka and Welling; Roweis and Ghahramani '99; Huys et al '06; Paninski et al '04; Jolivet et al '04; Beeman's notes on conductance-based neural modeling; Brown et al '98; Doucet et al '00; Douc et al '05; Huys voltage/calcium smoothing notes; Brockwell et al '04; Smith and Brown '03; Smith et al '04; Yu et al '05; Kulkarni and Paninski '06; Vogelstein calcium imaging notes; Wu et al '05. Additional useful papers collected by Minka here. notes
Apr 25 Continuous-time state-space models; stochastic differential equations; forward (Fokker-Planck) and backward equations; integrate-and-fire models; Gaussian models for neural prosthetics Paninski et al '04b; Paninski '04c; Pillow et al '05; Toyoizumi GLM mean field notes; Nykamp and Tranchina '00; Spiegelman notes on diffusion equation; Kulkarni notes on Gaussian forward-backward method; Paninski '06; Paninski '06b, Nikitchenko integrate-and-fire notes notes
May 2 No class (study period)
May 9 Final projects due (email me your report as a pdf file).

Extra topics

Some important topics we didn't get to...

Topic Reading
PSTH estimation / smoothing Kass et al '03, Kaufman et al '04
Fourier domain / multitaper methods Mitra and Pesaran '99
Analysis of cross-correlations / temporal precision; jitter-based methods Harrison and Geman '04, Amarasingham et al, '06