Home Page: Mathematical Tools for Neural and Cognitive Science

PSYCH-GA.2211 / NEURL-GA.2201, Fall Semester 2022

Mathematical Tools for Neural and Cognitive Science

Instructors:	Mike Landy & Eero Simoncelli
Teaching Assistants:	Marissa Fassold (mhe229 AT nyu DOT edu), Jordan Lei (hl3976 AT nyu DOT edu), Srinidhi Venkatesan Kalavai (Srinidhi.VenkatesanKalavai@nyulangone.org AT nyulangone DOT org)
Time:	Lectures: Tuesday/Thursday, 10:00-12:00 Labs: selected Fridays, 9:30-12:00
Location:	Lectures: Meyer 636 Labs: Meyer 636
TA Office hours:	Tuesdays 12-1 (Meyer 631) and occasional Thursdays 12-1 (location TBD)

Description: A graduate lecture course covering fundamental mathematical methods for visualization, analysis, and modeling of neural and cognitive data and systems. The course was introduced in Spring of 1999, became a requirement for Neural Science doctoral students in 2000, and for Psychology doctoral students in the Cognition and Perception track in 2008. The course covers a foundational set of mathematical and statistical tools, providing assumptions, motivation, logical and geometric intuition, and simple derivations for each. Concepts are reinforced with extensive computational exercises. The goal is for students to be able to understand, use and interpret these tools.

Topics include: Linear algebra, least-squares and total-least-squares regression, eigen-analysis and PCA, linear shift-invariant systems, convolution, Fourier transforms, Nyquist sampling, basics of probability and statistics, hypothesis testing, model comparison, bootstrapping, estimation and decision theory, signal detection theory, linear discriminants, classification, clustering, simple models of neural spike generation, white noise (reverse-correlation) analysis.

Prerequisites: Algebra, trigonometry, and calculus. Some experience with matrix algebra and/or computer programming is helpful, but not required. The real prerequisites are an aptitude for logical and geometric reasoning, and a willingness to work hard!

Announcements: We use Piazza for online class announcements, questions, and discussion: piazza.com/nyu/fall2022/psychga2211neurlga2201/ . Rather than emailing the instructors or TAs, we encourage you to post your questions/comments there, where they can be discussed and/or answered by any of us or your fellow classmates.

Schedule: (Notes: labs are in green, content will appear incrementally, for a preview see last year's course page)

Date	Topic	Handouts	Homework
Thu, Sep 1	Introduction to the course Linear algebra I: vectors, operations, vector spaces Zoom recording, whiteboard (pdf)	Course description (pdf) Slides: Course intro (pdf - updated 20sep) Slides: Linear algebra (pdf - updated 13sep)
Fri, Sep 2	Matlab I: Environment, basic data types and operations, plotting	Matlab installation (txt) Matlab primer (pdf) Lab1 (m), Lab1_solutions (m) Lab1 (Python code or direct access to the notebook here), Lab1_solutions (p)	HW submission instructions (pdf)
Tue, Sep 6	Linear algebra II: inner products, projection, coordinate systems Zoom recording, whiteboard (pdf)	Notes: Linear Algebra (pdf)	Homework 0 (pdf - "due" 9/9)
Thu, Sep 8	Linear algbra III: linear systems, orthogonal/diagonal matrices, geometry Zoom recording, whiteboard (pdf)
Fri, Sep 9	Matlab II: Scripts, conditionals, iteration, functions	Lab 2 files (zip) Lab 2 solutions (zip)
Tue, Sep 13	Linear algebra IV: orthogonal matrices, singular value decomposition Zoom recording, whiteboard (pdf)	[note: linear Algebra slides (linked above) have been updated to match lectures]	Homework 1 (pdf - due 9/23) Data for Question 6 (.mat)
Thu, Sep 15	Linear algebra V: Null/Range spaces, Trichromacy example Zoom recording, whiteboard (pdf)	Slides: trichromacy (pdf)
Fri, Sep 16	[no lab]
Tue, Sep 20	Regression I: regression, multiple regression, partitioning variance Zoom recording, whiteboard (pdf)	Slides: Least squares optimization (pdf)
Thu, Sep 22	Regression II: choosing regressors, weighting, outliers, overfitting Zoom recording, whiteboard (pdf)
Fri, Sep 23	Lab: Regression	Lab 3 files (Matlab Python) Lab 3 solutions Data file mult_linreg.mat
Tue, Sep 27	Regression III: incorporating linear constraints, TLS regression Zoom recording I, Zoom recording II, whiteboard (pdf)		Homework 2 (pdf - due 10/11) Files needed (.zip)
Thu, Sep 29	Regression IV: Summary statistics, PCA, eigenvalues/eigenvectors Zoom recording, whiteboard (pdf)
Fri, Sep 30	[no lab]
Tue, Oct 4	PCA wrapup. LinSys I: Linear shift-invariant systems, convolution Zoom recording, whiteboard(pdf)	Notes: Least Squares (pdf)
Thu, Oct 6	LinSys II: Sinusoids and LSI systems, Discrete Fourier transform Zoom recording, whiteboard (pdf)	Slides: Linear shift-invariant systems (pdf)
Fri, Oct 7	Lab: Convolution - 1D and 2D + PCA	Lab files (zip)
Tue, Oct 11	[No class: NYU "Legislative Day"]
Thu, Oct 13	LinSys III: Fourier Transforms+LSI Zoom recording, whiteboard (pdf)	Notes: Linear Shift-invariant Systems (pdf)	Homework 3 (zip - due 10/25)
Fri, Oct 14	Lab: Fourier transforms	Lab files (zip convolution zip) Convolution slides Fourier slides Solutions (zip)
Tue, Oct 18	LinSys IV: Fourier examples, indexing/plotting, sampling Zoom recording, whiteboard (pdf)
Thu, Oct 20	LinSys V: Extended example: Sound, filtering, and the cochlea Zoom recording, whiteboard (pdf)
Tue, Oct 25	Stats intro: summary stats, central tendency, disperson, multi-D Zoom recording, whiteboard (pdf)		Homework 4 (pdf - due 11/8) File needed (.mat)
Thu, Oct 27	Stats: Multi-D, correlation, regression. Intro probability Zoom recording, whiteboard (pdf)	Slides: Probability and statistics (pdf
Fri, Oct 28	Lab: Probability/sampling	Lab files (zip) Lab solutions (zip)
Tue, Nov 1	Probability II: joint/marginal/conditional densities, Bayes Rule, independence Zoom recording, whiteboard (pdf)
Thu, Nov 3	Probability III: Gaussians: marginals, conditionals, dependency. correlation mis-interpretations Zoom recording , whiteboard (pdf)
Fri, Nov 4	Lab: Bayes	Lab files (MatLab, Python ) Solutions (MatLab, Python )
Tues, Nov 8	Inference I. Averages, 1/N convergence, CLT, significance tests, p-values Zoom recording, whiteboard (pdf)	Slides: Statistical inference (pdf)	Homework 5 (pdf - due 11/29 File needed (.mat)
Thur, Nov 10	Inference II: Statistical inference, MLE, examples Zoom recording, whiteboard (pdf)
Tue, Nov 15	Inference III: bias-variance tradeoff, confidence intervals, bootstrapping Zoom recording, whiteboard (pdf)
Thu, Nov 17	Inference IV: Estimation Example: Signal Detection theory: ML/MAP/Bayes, d', ROC Zoom recording, whiteboard (pdf)	Slides: Signal detection theory (pdf)
Fri, Nov 18	Lab: Simulations, bootstrapping, cross-validation	Lab files (zip)
Tue, Nov 22	Inference V: Multi-D Decision: FLD, SVM, QDA Zoom recording, whiteboard (pdf)
Thu, Nov 24	[No class: Thanksgiving]
Tue, Nov 29	Inference V: Discriminability, Fisher Information Zoom recording, whiteboard (pdf)		Homework 6 (pdf) - due 12/13 File needed (.zip)
Thu, Dec 1	Model fitting: errors, optimization, overfitting, ridge (L2) and LASSO (L1) regression Zoom recording, whiteboard (pdf)	Slides: model Fitting (pdf)
Fri, Dec 2	Lab: Regularization, classification, clustering	Lab files (zip), Solutions (zip)
Tue, Dec 6	Clustering: K-means, Hierarchical (mixture) models, soft K-means Zoom recording, Whiteboard (pdf)
Thu, Dec 8	Model fitting: Example: Fitting an LNP model to neural responses via STA Zoom recording, Whiteboard (pdf)
Tue, Dec 13	Encoding/decoding: Spike-triggered analysis, population decoding Zoom recording	Slides: Neural encoding and decoding (pdf)

Resources (electrons):

Carlos Fernandez-Granda's Math Tools for Data Science course (here at NYU)!
Ella Batty's new Math Tools for Neuroscience course at Harvard (follows a similar syllabus)
Online matlab help at The MathWorks | Tutorial at the MathWorks v | Intro video at MIT | Antonia Hamilton's Tutorial | at U. Utah | on reddit
Richard Johnson's Matlab Style Handbook (lots of helpful tips, if a bit idiosyncratic)
Linear algebra Appendix from PDP series, by Michael Jordan. (pdf)
Online lecture videos from Gilbert Strang's linear algebra course at MIT
Online YouTube BlueBrown Linear Algebra videos, YouTube Essence of Linear Algebra videos
Todd Will's Interactive Intro to the SVD
The Elements of Statistical Learning, Hastie, Tibshirani and Friedman - Excellent textbook on regression, decision/classification, clustering, and many advanced topics in data fitting and analysis. Available online (pdf)
Convex Optimization, Boyd and Vandenberg - Excellent textbook on the formulation of, and algorithms for, optimization of convex functions. Available online (pdf)
Thomas Minka's On-line Glossary of Statistical Pattern Recognition
Wolfram Research World of Mathematics
History of various topics in mathematics

Resources (dead trees):

Matlab:
Getting Started with MATLAB; A Quick Introduction for Scientists and Engineers, R. Pratap, Oxford U. Press, 2009.
Matlab for Neuroscientists. An introduction to scientific computing in Matlab, P. Wallisch, M. Lusignan, M. Benayoun, T. Baker, A. Dickey & N. Hatsopoulos, Elsevier Press, 2008.
Mastering MATLAB, B. L. Littlefield & D. C. Hanselman, Prentice-Hall, 2011.
Linear algebra / Least squares:
Linear Algebra and Its Applications, Gilbert Strang. Academic Press, 1980.
Introduction to Applied Linear Algebra, Stephen Boyd & Lieven Vandenberghe. Cambridge U. Press, 2018.
Linear (shift-invariant) Systems:
Discrete-time Signal Processing, A. Oppenheim & R. Schafer. Prentice Hall, 1989.
The Fourier transform and its applications, R. Bracewell, McGraw Hill Science, 1999.
Fast Fourier transform and its applications, E. Brigham, Prentice Hall, 1988.
Probability/Statistics:
Statistics, Freedman, Pisani, Purves, Norton, 2007 (4th ed.)
Mathematical statistics, J. E. Freund, Prentice Hall, 1992.
Bootstrap/Resampling:
An Intoduction to the Bootstrap, by Bradley Efron and Robert Tibshirani. Chapman & Hall, 1998.
Resampling Methods: A practical guide to data analysis, by Phillip Good. Birkhäuser, 1999.
Decision Theory:
Biology: Elementary Signal Detection Theory, by Thomas D. Wickens. Oxford University Press, 2001.
Signal Detection Theory and Psychophysics, by David Green & John Swets. Peninsula Publishing, 1988.
Math: Statistical Decision Theory, by James O. Berger. Springer-Verlag, 1980.
Chapter 2 of Pattern Classification, by Duda, Hart and Storck. Wiley, 2001.
Computational/Theoretical Neuroscience:
Theoretical Neuroscience , by Peter Dayan and Larry Abbott. MIT Press, 2001.
Spikes: Exploring the Neural Code, by Fred Rieke, David Warland, Rob De Ruyter, & Bill Bialek. MIT Press, 1997.

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