PSYCHGA.2211 / NEURLGA.2201 Mathematical Tools for Neural and Cognitive Science
Instructors: Mike Landy & Eero Simoncelli Teaching Assistants: TBD
Time: Lectures: Tuesday/Thursday, 10:0012:00 Labs: selected Fridays, 9:3012:00
Location: Meyer 636
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, leastsquares and totalleastsquares regression, eigenanalysis and PCA, linear shiftinvariant 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 (reversecorrelation) 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.

Topic 



Introduction to the course Linear algebra I: vectors, vector spaces, inner products 



Matlab I: Environment, basic data types and operations, plotting 



Linear algebra II: projection, coordinate systems, linear systems 



Linear algbra III: linear systems, orthogonal/diagonal matrices, geometry 



Matlab II: scripts, conditionals, iteration, functions 



Lab: review of linear algebra concepts 



Linear algebra IV: singular value decomposition 



Linear algebra V: SVD, trichromacy example 



Regression I: regression, multiple regression via linear algebra 



Regression II: choosing regressors, weighting, outliers, overfitting 



Regression III: incorporating linear/quadratic constraints 



Regression IV: TLS regression, PCA, eigenvalues/ eigenvectors




Lab: Regression/PCA 



LinSys I: Linear shiftinvariant systems, convolution




LinSys II: Sinusoids and LSI systems, Discrete Fourier transform 



Lab: Convolution in 1 and 2 dimensions 



LSI systems: review and examples 





LSI Systems: sampling and aliasing. 


Lab: Fourier Transform 


LSI Systems: Extended example: Sound, filtering, and the cochlea. 


Stats intro: summary stats, central tendency, disperson, multiD. 


Stats: MultiD, correlation, regression. Intro probability. 


Probability: expectations, moments, cumulatives, transformations, drawing samples 


Lab: Probability/samplinling 


Probability: Gaussians, marginals, conditionals, dependency, correlation misinterpretations 


Inference I: Averages: 1/N convergence to mean, CLT, significance tests,pvalues, ztest, ttest, permutation tests 


Lab: Bayes 


Inference II: estimation, the MLE, examples 


Inference III: confidence, MAP, sequential updating, biasvariance tradeoﬀ, Bayes estimators 


Inference IV: Bayes. Signal Detection theory: ML/MAP/Bayes, d', ROC 


Inference V: decision theory, MultiD decisions: prototype classifier, FLD. 


Lab: Simulations, bootstrapping, crossvalidation 


Inference VI: MultiD decisions: FLD, SVM, QDA. 


[No class: Happy Thanksgiving!] 


Inference VII: Fisher information, regularization 


modelFitting I: Regularization, model comparison 


Lab: Classification, regularization, clustering 


Fitting LNP models to data 


Spiketriggered covariance, population decoding 

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 Online 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, PrenticeHall, 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 (shiftinvariant) Systems:
Discretetime 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. SpringerVerlag, 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.