Home Page: Mathematical Tools for Neural and Cognitive Science

PSYCH-GA.2211 / NEURL-GA.2201, Fall Semester 2024

Mathematical Tools for Neural and Cognitive Science

Instructors:	Mike Landy & Eero Simoncelli
Teaching Assistants:	Jessica Chen (xc2780 AT nyu DOT edu), Hanzhi (Phoebe) Chen (hc2896 AT nyu DOT edu), Robert Woodry (rfw256 AT nyu DOT edu)
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:	Tue 12-2pm room 307; Thur 4-6pm room 207

Description: A graduate lecture course covering fundamental mathematical methods for analysis, modeling, and visualization 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, 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: This year, we will use brightspace for class announcements and online questions/discussions: https://brightspace.nyu.edu/d2l/home/391464 . 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
Tue, Sep 3	Introduction to the course Linear algebra I: vectors, operations, vector spaces Zoom recording, whiteboard (pdf)	Course description (pdf) Notes: Linear Algebra (pdf) Slides: Linear algebra (pdf)
Thu, Sep 5	Linear algebra II: inner products, projection, coordinate systems Zoom recording, whiteboard (pdf)
Fri, Sep 6	Lab: linear algebra basics. Homework preparation	Matlab installation (txt) Matlab primer (pdf) Lab1 (zip) - includes matlab and python.	Homework submission instructions (pdf)
Tue, Sep 10	Linear algbra III: linear systems, orthogonal/diagonal matrices, geometry Zoom recording (sorry, only 2nd half), whiteboard (pdf)
Thu, Sep 12	Linear algebra IV: orthogonal matrices, singular value decomposition Zoom recording, whiteboard (pdf)		Homework 1 (pdf due 9/26) File for question 6: mtxExamples2024.mat Files for question 3 (Matlab, Python)
Fri, Sep 13	[no lab]
Tue, Sep 17	Linear algebra V: Null/Range spaces, (Pseudo-)Inverses, Trichromacy example Zoom recording, whiteboard (pdf)	Slides: Trichromacy (pdf)
Thu, Sep 19	Regression I: regression, multiple regression, partitioning variance Zoom recording whiteboard (pdf)	Slides: Least squares (pdf)
Fri, Sep 20	Lab: Regression	Lab2 (zip) - includes matlab and python.
Tue, Sep 24	Regression II: choosing regressors, weighting, outliers, overfitting Zoom recording, whiteboard (pdf)
Thu, Sep 26	Regression III: incorporating linear constraints, TLS regression Zoom recording, whiteboard (pdf)
Fri, Sep 27	[no lab]		Homework 2 (pdf - due ~~oct 11~~ oct 13) Data and code files (zip)
Tue, Oct 1	Regression IV: TLS regression, PCA, eigenvalues/eigenvectors Zoom recording, whiteboard (pdf)
Thu, Oct 3	LinSys I: Linear shift-invariant systems, convolution Zoom recording, whiteboard(pdf)	Slides: Linear systems theory (pdf) Notes: Linear systems theory (pdf)
Fri, Oct 4	Lab: PCA + Convolution - 1D and 2D
Tue, Oct 8	[no class - SfN meeting]
Thu, Oct 10	LinSys II: Sinusoids and LSI systems, Discrete Fourier transform Zoom recording, whiteboard (pdf)
Fri, Oct 11, [10:00 - 12:00]	LinSys III: Fourier Transforms+LSI Zoom recording, whiteboard (pdf)
Thu, Oct 17	LinSys IV: Fourier examples, indexing/plotting, sampling Zoom recording, whiteboard(pdf)		Homework 3 (pdf - due 10/31) Files needed: hw3-files.zip
Fri, Oct 18	Lab: Fourier Transform	Lab 4 materials (zip)
Tue, Oct 22	LinSys V: Extended example: Sound, filtering, and the cochlea Zoom recording, whiteboard (pdf)	Slides: Example: Auditory system (pdf)
Thu, Oct 24	Stats intro: summary stats, central tendency, dispersion, regression redux Zoom recording, whiteboard (pdf)	Slides: Probability and statistics (pdf)
Fri, Oct 25	[no lab]
Tue, Oct 29	Stats intro (continued) Zoom recording, whiteboard (pdf)
Thu, Oct 31	Probability I: histograms, distributions, expectations, moments, cumulatives, transformations Zoom recording, whiteboard (pdf)
Fri, Nov 1	[no lab]
Tue, Nov 5	Prob II: drawing samples, joint distributions, marginals, conditionals, Bayes Rule, independence Zoom recording, whiteboard (pdf)		Homework 4 (pdf - due 11/17) Files needed: experimentData.mat
Thu, Nov 7	Prob III: Gaussians, marginals, conditionals, dependency. Correlation mis-interpretations, introduction to inference Zoom recording (screen share failed; no slides nor whiteboard visible), whiteboard (pdf)	Slides: Inference (pdf)
Fri, Nov 8	[no lab]
Tues, Nov 12	Inference II: Statistical inference, MLE, examples Zoom recording, whiteboard (pdf)
Thur, Nov 14	Inference III: MAP, bias-variance tradeoff, Bayes estimators Zoom recording, whiteboard (pdf)
Fri, Nov 15	Lab: Bayesian inference Zoom recording
Tue, Nov 19	Inference IV: Signal Detection Theory Zoom recording, whiteboard (pdf)	Slides: SDT (pdf) Notes: SDT (pdf)	Homework 5 (pdf - due 12/1)
Thu, Nov 21	Multidimensional Decision Zoom recording, whiteboard (pdf)
Fri, Nov 22	[no lab]
Tue, Nov 26	Fisher Information and Discriminability Zoom recording, whiteboard (pdf)
Thu, Nov 28	[no class - Thanksgiving break]
Tues, Dec 3	Model fitting: error sources, overfitting, cross-validation, regularization (Ridge, LASSO) Zoom recording, whiteboard (pdf)	Slides: Model fitting (pdf)
Thur, Dec 5	Clustering: K-means, Hierarchical (mixture) Zoom recording, whiteboard (pdf)
Fri, Dec 6	Lab: Regularization, classification, clustering Zoom recording		Homework 6 (pdf - due 12/20) hw6 files (zip)
Tues, Dec 10	Spike-triggered average/Classification images, LNP models of neural responses Zoom recording, whiteboard (pdf)	Slides: STA (pdf)
Thur, Dec 12	LNP models, population decoding Zoom recording, whiteboard (pdf)
Fri, Dec 13	[no lab]

Resources (electrons):

Carlos Fernandez-Granda's Math Tools for Data Science course (here at NYU)!
Jonathan Pillow's Math Tools course at Princeton (Jonathan was TA for our course in 2000 :)
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 at MathWorks or Cambridge Press (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
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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