Statistical analysis of neural data

Fall 2013

This was a Ph.D.-level topics course in statistical analysis of neural data. Students from statistics, neuroscience, and engineering are all welcome to attend. A link to this year's course is here.

Time: Th 10-12
Place: Rm 1025 School of Social Work building, 1255 Amsterdam Ave (Stat dept conference room)
Professor: Liam Paninski; Office: 1255 Amsterdam Ave, Rm 1028. Email: liam at stat dot columbia dot edu. Hours by appointment.
T.A.: Josh Merel. Email: jsm2183 at Hours: Tues 11:30-12:30, 9th floor stat lounge or room 930, 1255 Amsterdam Ave.

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, three short problem sets, 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). See this page for some links to available datasets; or talk to other students in the class, many of whom have collected their own datasets. Some project ideas are listed on the courseworks page.
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. 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; photon-limited image data
Time-rescaling theorem for point processes Fast simulation of network models; goodness-of-fit tests for spiking 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
Laplace approximation; Fisher information Model-based decoding and information estimation; adaptive design of optimal stimuli
Mixture models; EM algorithm; Dirichlet processes Spike-sorting / clustering
Optimization and convexity techniques Spike-train decoding; ML estimation of encoding models
Markov chain Monte Carlo: Metropolis-Hastings and hit-and-run algorithms Firing rate estimation and spike-train decoding
State-space models; sequential Monte Carlo / particle filtering Decoding spike trains; optimal voltage smoothing
Fast high-dimensional Kalman filtering Optimal smoothing of voltage and calcium signals on large dendritic trees
Markov processes; first-passage times; Fokker-Planck equation Integrate-and-fire-based neural models

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.

More recently, a couple good online courses in computational neuroscience have appeared: one directed by Raj Rao and Adrienne Fairhall, another by Wulfram Gerstner, and another by Idan Segev.

For those new to statistics: 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.). Also, here is an excellent online book on convex optimization.

Finally, Cox and Gabbiani have written a nice Matlab-based book on Mathematics for Neuroscientists, available online here if your library has access. A lot of very useful background material, along with some more advanced ideas. See also the Paninski group page here for more background and some possible project ideas.


Date Topic Reading Notes
Sep 5 No class due to graphical models workshop
Sep 12,19 Introduction; background on neuronal biophysics, regression, MCMC Spikes introduction; Kass et al '05; Brown et al. '04 Neuroscience review by Josh Merel. Regression notes
Sep 26 Estimating time-varying firing rates Kass et al (2003), Wallstrom et al (2008) Generalized linear model notes
Oct 3 Linear-nonlinear Poisson cascade models: spike-triggered averaging; Poisson regression Simoncelli et al. '04; Chichilnisky '01; Paninski '03; Sharpee et al. '04; Paninski '04; Weisberg and Welsh '94; Williamson et al '13 Try these practice problems, courtesy of Dayan and Abbott; any problem in chapter 1; also problems 2-3 in chapter 2.
Oct 10 Expected log-likelihood; quadratic models; spike-triggered covariance; sparsity-promoting and rank-penalizing priors; hierarchical models Park and Pillow '11, Ramirez and Paninski, '13, Field, Gauthier, Sher et al '10, Ahrens et al '08
Oct 17 No class due to Grossman workshop Hope to see you there.
Oct 24 The expectation-maximization (EM) algorithm for maximum likelihood given indirect measurements / hidden data; mixture models; spike sorting Neal and Hinton '98; Lewicki '98; Salakhutdinov et al '03; Shoham et al '03; Pouzat et al '04, Pillow et al `13, Carlson et al '13 Guest lecture by David Carlson; EM notes
Oct 31, Nov 7 Experimental design. Point processes: Poisson process, renewal process, self-exciting process, Cox process; time-rescaling: goodness-of-fit, fast simulation of network models Lewi et al '09; Shababo et al '13; Keshri et al '13; Brown et al. '01 Uri Eden's point process notes; supplementary notes. Do inhomogenous Poisson problem set, available on courseworks.
Nov 7 Presentations of project ideas
Nov 14 No class
Nov 21 State space models; autoregressive models; Kalman filter; extended Kalman filter; fast tridiagonal methods. 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; Roweis and Ghahramani '99; Huys et al '06; Paninski et al '04; Jolivet et al '04; Beeman's notes on conductance-based neural modeling; Wu et al '05; Brown et al '98; Smith et al '04; Yu et al '05; Kulkarni and Paninski '08; Calabrese and Paninski '11; Paninski et al '10, Vogelstein et al '10. Additional useful papers collected by Minka here. notes. Do state-space problem set, on courseworks.

Thanks to the NSF for support.