Home Page: Mathematical Tools for Neural and Cognitive Science

PSYCH-GA.2211 / NEURL-GA.2201, Fall Semester 2020

Mathematical Tools for Neural and Cognitive Science

Instructors:	Mike Landy & Eero Simoncelli
Teaching Assistants:	Aaron Lanz (ajl787 AT nyu DOT edu), Hope Lutwak (hlutwak AT nyu DOT edu), Thomas (Teddy) Yerxa (tey214 AT nyu DOT edu)
Time:	Lectures: Tuesday/Thursday, 10:00-12:00 Labs: selected Fridays, 9:30-12:00
Location:	Lectures: Silver 405 (moved to Meyer 636, as of 9/22) Labs: online

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: https://piazza.com/nyu/fall2020/psychga2211neurlga2201/home . 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 3	Introduction to the course Linear algebra I: vectors, vector spaces, inner products Zoom recording, whiteboard (pdf), videos from 2019	Course description (pdf) Slides: Linear algebra (pdf)
Fri, Sep 4	Matlab I: Environment, basic data types and operations, plotting zoom recording	Lab1 materials (zip) Matlab installation (txt) Matlab primer (pdf)	Homework 0 (pdf, due 9/11) HW submission guidelines (pdf)
Tue, Sep 8	Linear algebra II: projection, coordinate systems, linear systems zoom recording, whiteboard (pdf)
Thu, Sep 10	Linear algbra III: linear systems, orthogonal/diagonal matrices, geometry zoom recording, whiteboard (pdf)
Fri, Sep 11	Matlab II: scripts, conditionals, iteration, functions zoom recording	Lab2 materials (zip)	Homework 1 (due 9/24) - (pdf) mtxExamples (mat), system p-files (zip)
Tue, Sep 15	Lab: review of linear algebra concepts zoom recording	Lab3 slides (pdf)
Thu, Sep 17	Linear algebra IV: singular value decomposition zoom recording, whiteboard (pdf)
Fri, Sep 18	Linear algebra V: SVD, trichromacy example zoom recording, whiteboard (pdf), 2019 lecture (youTube)
Tue, Sep 22	Regression I: regression, multiple regression via linear algebra zoom recording, whiteboard (pdf)	Slides: Trichromacy (pdf) Notes: Linear Algebra (pdf)
Thu, Sep 24	Regression II: choosing regressors, weighting, outliers, overfitting zoom recording, whiteboard (pdf)	Slides: Least Squares (pdf)
Tue, Sep 29	Regression III: incorporating linear/quadratic constraints zoom recording 1, zoom recording 2, whiteboard (pdf)		Homework 2 (due 10/11) - (pdf) Data and code files (zip) Python trichromacy code (py)
Thu, Oct 1	Regression IV: TLS regression, PCA, eigenvalues/eigenvectors zoom recording, whiteboard (pdf)
Fri, Oct 2	Lab: Regression/PCA zoom recording	lab4 files
Tue, Oct 6	LinSys I: Linear shift-invariant systems, convolution zoom recording, whiteboard (pdf)
Thu, Oct 8	LinSys II: Sinusoids and LSI systems, Discrete Fourier transform zoom recording, whiteboard (pdf)
Fri, Oct 9	Lab: Convolution in 1 and 2 dimensions zoom recording	lab5 files
Tue, Oct 13	LSI systems: review and examples zoom recording, whiteboard (pdf)
Thu, Oct 15	LSI Systems: sampling and aliasing. zoom recording, whiteboard (pdf)	Slides: LSI systems (pdf)	Homework 3 (due 10/27) - (pdf) Data and code files (zip) Python unknownSystems code (py)
Fri, Oct 16	Lab: Fourier Transform zoom recording	lab6 files
Tue, Oct 20	LSI Systems: Extended example: Sound, filtering, and the cochlea. zoom recording, whiteboard (pdf)	Slides: Sound, filtering, and the cochlea (pdf)
Thu, Oct 22	Stats intro: summary stats, central tendency, disperson, multi-D. zoom recording file1, file2, whiteboard (pdf)
Tue, Oct 27	Stats: Multi-D, correlation, regression. Intro probability. zoom recording, whiteboard (pdf)
Thu, Oct 29	Probability: expectations, moments, cumulatives, transformations, drawing samples zoom recording, whiteboard (pdf)	Slides: Probability and statistics (pdf)	Homework 4 (due 11/12) - (pdf) Data file (mat)
Fri, Oct 30	Lab: Probability/sampling zoom recording	lab7 files
Tue, Nov 3	Probability: Gaussians, marginals, conditionals, dependency, correlation mis-interpretations zoom recording, whiteboard (pdf)
Thu, Nov 5	Inference I: Averages: 1/N convergence to mean, CLT, significance tests,p-values, z-test, t-test, permutation tests zoom recording, whiteboard (pdf)	Slides: Statistical inference (pdf)
Fri, Nov 6	Lab: Bayes zoom recording	lab8 files
Tue, Nov 10	Inference II: estimation, the MLE, examples zoom recording, whiteboard (pdf),
Thu, Nov 12	Inference III: confidence, MAP, sequential updating, bias-variance tradeoff, Bayes estimators zoom recording, whiteboard (pdf)		Homework 5 (due 11/25) - (pdf)
Tue, Nov 17	Inference IV: Bayes. Signal Detection theory: ML/MAP/Bayes, d', ROC zoom recording, whiteboard (pdf)	Slides: Bayesian estimation and signal detection theory (pdf)
Thu, Nov 19	Inference V: decision theory, Multi-D decisions: prototype classifier, FLD. zoom recording, whiteboard (pdf)
Fri, Nov 20	Lab: Simulations, bootstrapping, cross-validation zoom recording	lab9 files
Tue, Nov 24	Inference VI: Multi-D decisions: FLD, SVM, QDA. zoom recording, whiteboard (pdf)
Thu, Nov 26	[No class: Happy Thanksgiving!]
Tue, Dec 1	Inference VII: Fisher information, regularization zoom recording, whiteboard (pdf)
Thu, Dec 3	modelFitting I: Regularization, model comparison zoom recording, whiteboard (pdf)	Slides: Model fitting and comparison (pdf)	Homework 6 (due 12/15) - (pdf) Data and code files (zip)
Fri, Dec 4	Lab: Classification, regularization, clustering zoom recording	lab10 files
Tue, Dec 8	Fitting LNP models to data zoom recording, whiteboard (pdf)	Slides: Fitting neural models to data (pdf)
Thu, Dec 10	Spike-triggered covariance, population decoding zoom recording, whiteboard (pdf)

Resources (Electrons):

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 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.
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.
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.
Spikes, Neural Coding, Reverse Correlation:
Spikes: Exploring the Neural Code, by Fred Rieke, David Warland, Rob De Ruyter, & Bill Bialek. MIT Press, 1997.
Computational/Theoretical Neuroscience:
Theoretical Neuroscience , by Peter Dayan and Larry Abbott. MIT Press, 2001.

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