PSYCH-GA.2211 / NEURL-GA.2201, Fall Semester 2019

Mathematical Tools for Neural and Cognitive Science

Instructors: Mike Landy  &  Eero Simoncelli
Teaching Assistants: Lyndon Duong (lyndon DOT duong AT nyu DOT edu),
Ionatan Kuperwajs (ik1125 AT nyu DOT edu),
Zhiwei Li (zhiwei DOT li AT nyu DOT edu)
Time: Lectures: Tuesday/Thursday, 10:00-12:00
Labs: selected Fridays, 9:30-12:00
Location: Lectures: Meyer Hall, rm 636
Labs: Meyer Hall, rm 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 in the Matlab programming language. 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, classification, linear discriminants, 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/fall2019/psychga2211neurlga2207/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:   (Note: labs in green, content will appear incrementally, for a preview see last year's course page)

Date Topic (Videos) Handouts Homework
Tue, Sep 3 Introduction to the course
Linear algebra I: vectors, vector spaces, inner products
(video, video from 2018)
Handout: Course description (pdf)  
Thu, Sep 5 Linear algebra II: projection, coordinate systems, linear systems
(video)
Notes: Linear Algebra review (pdf)
Fri, Sep 6 Matlab I: Environment, basic data types and operations, plotting Matlab installation (txt)
Matlab primer (pdf)
lab1Files (dir)
Homework 0 (due 9/13) - (pdf)
HW submission guidelines (pdf)
Tue, Sep 10 MatLab II: scripts, conditionals, iteration Slides: Linear Algebra (pdf)
lab2Files (dir)
Homework 1 (due 9/24) - (pdf)
mtxExamples (mat)
Thu, Sep 12 Linear Algbra III: orthogonal/diagonal matrices, geometry
(video)
Fri, Sep 13 Matlab III: functions, debugging [also, SVD example]
lab3Files (dir)
Tue, Sep 17 LinAlg IV: SVD, geometry, nullspaces, inverses
(video)
Thu, Sep 19 LinAlg extended example: Trichromacy
(video)
Fri, Sep 20 Regression I: regression, multiple regression via linear algebra
(video)
Slides: Trichromacy (pdf)
Tue, Sep 24 Regression II: choosing regressors, weighting, outliers, overfitting
(video)
Notes: Least Squares review (pdf)
Thu, Sep 26 Regression III: incorporating linear/quadratic constraints
(video)
Homework 2 (due 10/10) - (pdf)
colMatch (mat), regress1 (mat),
humanColorMatcher (p), altHumanColorMatcher (p),
constrainedLS (mat), windowedSpikes (mat)
trichromacy (py)
Tue, Oct 1 Regression IV: TLS regression, PCA, eigenvalues/eigenvectors
(video)
Slides: Least Squares (pdf)
Thu, Oct 3 LinSys I: LSI systems, convolution
(video)
Fri, Oct 4 Lab: Regression/PCA Lab4 matlab file (m),
Lab4 data (csv)
Tue, Oct 8 LinSys II: Sinusoids, Discrete Fourier transforms
(video)
TA Review: Constrained LS regression / PCA
(video)
Thu, Oct 10 LinSys III: Fourier Transforms+LSI
(video)
Fri, Oct 11 Lab: Convolution - 1D and 2D Slides: LSI Systems (pdf) Homework 3 (due 10/25) - (pdf)
unknownSystem1 (p), unknownSystem2 (p)
unknownSystem3 (p), unknown_systems2019 (py),
hrfDeconv (mat), myMeasurements (mat),
Tue, Oct 15 [no class - NYU classes meet on Monday schedule]
Thu, Oct 17 LinSys IV: Fourier examples, indexing/plotting, sampling
(video)
Fri, Oct 18 Lab: Fourier Transform
(Sorry, video failed!)
Tue, Oct 22 [no class - SfN meeting]
Thu, Oct 24 LinSys extended example: Sound, filtering, and the cochlea
(video)
Notes: LSI Systems review (pdf)
Tue, Oct 29 Stats Intro: Summary stats, central tendency, disperson.
(video)
Thu, Oct 31 Stats: regression and correlation revisited.
(video)
Homework 4 (due 11/12) - (pdf)
experimentData (mat)
Fri, Nov 1 Lab: Probability/Sampling Slides: Auditory filtering (pdf)
Tue, Nov 5 Prob II: joint distributions, marginals, conditionals, Bayes Rule, independence
(video)
Thu, Nov 7 Prob III: Gaussians, marginals, conditionals, dependency. Correlation mis-interpretations.
(video)
Fri, Nov 8 Lab: Bayes rule
Tue, Nov 12 Stats I: Averages: 1/N convergence, CLT, significance tests,p-values, z-test, t-test, permutation tests
(video)
Slides: Summary Stats + Probability (pdf) Homework 5 (due 11/26) - (pdf)
psychopathy (mat)
Thu, Nov 14 Stats II: the MLE, examples, confidence (SEM, Hessian of LL, simulation, bootstrapping)
(video)
Tue, Nov 19 Stats III: bootstrapping, MAP, sequential updating,
(video)
Thu, Nov 21 Stats IV: Regression to the mean, Bayes estimators, signal detection theory
(video)
Slides: Statistical Inference (pdf)
Fri, Nov 22 Lab: resampling, bootstrap, cross-validation
Tue, Nov 26 Stats V: Psychophysics: psychometric functions, d', ROC; Multi-D decision: FLD, SVM, QDA.
(video)
Slides: Signal Detection Theory (pdf)
Thu, Nov 28 [no class - Happy Thanksgiving!] Homework 6 (due 12/13, 12/16) - (pdf)
newMeasurements (mat),
fisherData (mat)
Tue, Dec 3 Model fitting: error sources, optimization, model comparison
(video)
Slides: Statistical model fitting (pdf)
Thu, Dec 5 Regularization, cross-validation, L2 (ridge) regression , L1 regularization (LASSO)
video)
Fri, Dec 6 Lab: signal detection theory / classification
Tue, Dec 10 Classification/clustering. Fitting an LNP model to neural responses via STA
(video)
Slides: Fitting an LNP model (pdf)
Thu, Dec 12 LNP neural models, population codes
(video)

Resources (Electrons):

Resources (Dead Trees):

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