Statistical analysis of neural data

Fall 2015

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. A link to last year's course is here.

Time: Tu 2:40-4:25
Place: Rm 903, 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.: Ella Batty. Email: erb2180 at Hours by appointment.

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. Additional informal exercises will be suggested, but not required. 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.
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 and calcium imaging data, with a few applications to analyzing intracellular voltage and dendritic imaging data. Note that this class will not focus on MRI or EEG 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. A very non-exhaustive list of useful books (each of which emphasize different topics, albeit with some overlap): Theoretical Neuroscience, by Dayan and Abbott; Spiking Neuron Models, by Gerstner et al; 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. The full text of the Gerstner et al book is online. 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: The new book by Kass et al is an excellent introduction to statistics, illustrated with a number of neural examples; Columbia e-link here. 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.


Date Topic Reading Notes
Sep 8,15 Introduction; background on neuronal biophysics, regression, MCMC Spikes introduction; Kass et al '05; Brown et al. '04 Neuroscience review by Ella Batty. Regression notes
Sep 22 Signal acquisition: spike sorting, calcium imaging Lewicki '98; Shoham et al '03; Pouzat et al '04, Pillow et al `13, Carlson et al '13, Pnevmatikakis et al '14 EM notes
Sept 29 No class
Oct 6 Estimating time-varying firing rates Kass et al (2003), Wallstrom et al (2008) Generalized linear model notes
Oct 13 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 20 Expected log-likelihood; quadratic models; spike-triggered covariance; sparsity-promoting and rank-penalizing priors; hierarchical models. Experimental design. Park and Pillow '11, Ramirez and Paninski, '13, Field, Gauthier, Sher et al '10, Ahrens et al '08, Lewi et al '09, Shababo et al '13, Soudry et al '15
Oct 27 Presentations of project ideas Just two minutes each
Nov 3 No class (University holiday)
Nov 10 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, Mena and Paninski '14 Uri Eden's point process notes; supplementary notes.
Nov 17 - Dec 1 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 HMM tutorial by Rabiner; 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; Paninski et al '10, Calabrese and Paninski '11; Vogelstein et al '10, Buesing et al '12, Vidne et al '12, Pfau et al '13. state-space notes (need updating)
Dec 8 No class (office hours) Stop by if you want to discuss your project.
Dec 15 Project presentations E-mail me your report as a pdf by Dec 18.

Thanks to the NSF for support.