Home Page: Mathematical Tools for Neural and Cognitive Science

PSYCH-GA.2211 / NEURL-GA.2201, Fall Semester 2025

Mathematical Tools for Neural and Cognitive Science

Instructors:	Mike Landy & Eero Simoncelli
Teaching Assistants:	Isabel Garon (isabelgaron AT nyu DOT edu) Luhe Li (luhe.li AT nyu DOT edu) Timothy Ma (timothy.ma 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:	Meyer 635 Tuesday/Thursday, 2:00-3:00

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: We use brightspace for class announcements and online questions/discussions: https://brightspace.nyu.edu/d2l/home/499154 . 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 2	Introduction to the course Linear algebra I: vectors, operations, vector spaces Zoom recording, whiteboard (pdf)	Course description (pdf) Slides: Linear algebra (pdf) Notes: Linear Algebra (pdf)
Thu, Sep 4	Linear algebra II: inner products, projection, coordinate systems Zoom recording, whiteboard (pdf)
Fri, Sep 5	Lab: linear algebra basics in matlab/python. Homework preparation	Lab1 (zip) - includes matlab and python
Tue, Sep 9	Linear algebra III: linear systems, matrix multiplication Zoom recording, whiteboard (pdf)
Thu, Sep 11	Linear algebra IV: orthogonal/diagonal matrices, singular value decomposition Zoom recording, whiteboard (pdf)		Homework 1 (pdf, due 9/25) Submission Instructions
Fri, Sep 12	[no lab]
Tue, Sep 16	Extended example: Color vision and trichromacy Zoom recording, whiteboard (pdf)	Slides: Color vision and trichromacy (pdf)
Thu, Sep 18	Regression I: regression, multiple regression via linear algebra, partitioning variance Zoom recording, whiteboard (pdf)	Slides: Least Squares regression (pdf)
Fri, Sep 19	Lab: linear regression	Lab2 (zip) - includes matlab and python
Tue, Sep 23	Regression II: Regression example, choosing regressors, weighting, outliers Zoom recording, whiteboard (pdf)
Thu, Sep 25	Regression III: Linear/quadratic constraints, Total least squares Zoom recording, whiteboard (pdf)
Fri, Sep 26	[no lab]		Homework 2 (pdf, due 10/10) Data files and functions, trichromacy.py
Tue, Sep 30	Regression IV: TLS regression LinSys I: Linear shift-invariant systems, convolution Zoom recording, whiteboard (pdf)	Slides: Linear shift-invariant systems (pdf)
Thu, Oct 2	[no class - Yom Kippur]
Fri, Oct 3 10am	LinSys II: Sinusoids and LSI systems, Discrete Fourier transform Zoom recording (missing last few minutes), whiteboard (pdf)
Tue, Oct 7	LinSys III: Fourier Transforms + LSI Zoom recording (missing last 25 minutes or so), whiteboard (pdf)
Thu, Oct 9	LinSys IV: Fourier examples, indexing/plotting, sampling Zoom recording, whiteboard (pdf)
Fri, Oct 10	Lab: Convolution - 1D and 2D

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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