PSYCH-GA.2211 / NEURL-GA.2201, Fall Semester 2023

Mathematical Tools for Neural and Cognitive Science

Instructors: Mike Landy  &  Eero Simoncelli
Teaching Assistants: Sarah Master    (sm4937 AT nyu DOT edu),
Ajay Subramanian    (as15003 AT nyu DOT edu),
Xinyuan Zhao    (xz2556 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: Monday, 4-5pm, Meyer 760
Friday, 1-2pm, Meyer 635

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: This year, we will use brightspace for class announcements and online questions/discussions: https://brightspace.nyu.edu/d2l/home/315787 . 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 5 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 7 Linear algebra II: inner products, projection, coordinate systems
Zoom recording (sorry, only first 1 hr 20 min, whiteboard (pdf)
Fri, Sep 8 Programming Lab: Scripts, conditionals, iteration, functions, maninpulating matrices and vectors
Matlab installation (txt)
Matlab primer (pdf)
Lab1 (zip) - includes matlab and python.
Tue, Sep 12 Linear algbra III: linear systems, orthogonal/diagonal matrices, geometry
Zoom recording (sorry, no audio for first half), whiteboard (pdf)
 
Thu, Sep 14 Linear algebra IV: orthogonal matrices, singular value decomposition
Zoom recording, whiteboard (pdf)
Fri, Sep 15 [no lab]
Tue, Sep 19 Linear algebra V: Null/Range spaces, Trichromacy example
Zoom recording, whiteboard (pdf)
Slides: Trichromacy (pdf) Homework 1 (pdf - due 10/3)
mat file for question 6
submission instructions
Thu, Sep 21 Regression I: regression, multiple regression, partitioning variance
Zoom recording: part1, part2. whiteboard (pdf)
Slides: Least squares (pdf)  
Fri, Sep 22 Lab: Regression
Tue, Sep 26 Regression II: choosing regressors, weighting, outliers, overfitting
Zoom recording, whiteboard (pdf)
Thu, Sep 28 Regression III: incorporating linear constraints, TLS regression
Zoom recording I, whiteboard (pdf)
Fri, Sep 29 [no lab] Notes: Least Squares (pdf)
Tue, Oct 3 Regression IV: TLS regression, PCA, eigenvalues/eigenvectors
Zoom recording, whiteboard (pdf)
Homework 2 (pdf - due 10/19)
Data and code files (zip)
Thu, Oct 5 LinSys I: Linear shift-invariant systems, convolution
Zoom recording, whiteboard(pdf)
Slides: Linear systems theory (pdf)
Fri, Oct 6 Lab: Convolution - 1D and 2D + PCA
Tue, Oct 10 [no class - NYU legislative day]
Thu, Oct 12 LinSys II: Sinusoids and LSI systems, Discrete Fourier transform
Zoom recording, whiteboard (pdf)
Fri, Oct 13 [no lab]
Tue, Oct 17 LinSys III: Fourier Transforms+LSI
Zoom recording,
Thu, Oct 19 LinSys IV: Fourier examples, indexing/plotting, sampling
Zoom recording, whiteboard(pdf)
Fri, Oct 20 Lab: Fourier Transform Homework 3 (pdf - due 11/2)
Data and code files (zip)
Tue, Oct 24 LinSys V: Extended example: Sound, filtering, and the cochlea
Zoom recording, whiteboard (pdf)
Slides: Auditory example (pdf)
Thu, Oct 26 Stats intro: summary stats, central tendency, disperson, multi-D
Zoom recording, whiteboard (pdf)
Slides: Probability and statistics (pdf)
Fri, Oct 27 [no lab]
Tue, Oct 31 Prob I: histograms, distributions, expectations, moments, cumulatives, transformations, sampling
Zoom recording (no screenshare), ``live'' whiteboard with audio (mov), whiteboard (pdf)
Thu, Nov 2 Prob II: joint distributions, marginals, conditionals, Bayes Rule, independence
Zoom recording, whiteboard (pdf)
Fri, Nov 3 Lab: Probability/sampling Homework 4 (pdf - due 11/16)
Tue, Nov 7 Prob III: Gaussians: marginals, conditionals, dependency. correlation mis-interpretations
Zoom recording , whiteboard (pdf)
Thu, Nov 9 Inference I: Averages: 1/N convergence to mean, CLT, significance tests,p-values, z-test, t-test, permutation tests
Zoom recording, whiteboard (pdf)
Fri, Nov 10 [no lab]
Tues, Nov 14 Inference II: Statistical inference, MLE, examples
Zoom recording, whiteboard (pdf)
Slides: Statistical Inference (pdf)
Thur, Nov 16 Inference III: MAP, bias-variance, Bayes estimators
Zoom recording, whiteboard (pdf)
Fri, Nov 17 Lab: Bayes
Zoom recording
Homework 5 (pdf - due 12/1)
Tue, Nov 21 Inference IV: Signal Detection theory: ML/MAP/Bayes, d', ROC
Zoom recording, whiteboard (pdf)
Slides: Signal detection theory (pdf)
Thu, Nov 23 [no class - Thanksgiving]
Fri, Nov 24 [no lab - Thanksgiving break]
Tue, Nov 28 Inference V: Multi-D decision: FLD, SVM, QDA
Zoom recording, whiteboard (pdf)
Thu, Nov 30 Discriminability, Fisher Information. Model fitting
Zoom recording, whiteboard (pdf)
Slides: model fitting (pdf)
Fri, Dec 1 Lab: Simulations, bootstrapping, cross-validation Homework 6 (pdf - due 12/19)
Mat files: pychopathy, fisherData
Tue, Dec 5 Model fitting: errors, optimization, overfitting, ridge (L2) and LASSO (L1) regression
Zoom recording, whiteboard (pdf)
Thu, Dec 7 Clustering: K-means, Hierarchical (mixture) models, soft K-means
Zoom recording, whiteboard (pdf)
Fri, Dec 8 Lab: Regularization, classification, clustering
Tue, Dec 12 Model fitting: Example: Fitting an LNP model to neural responses via STA
Zoom recording, whiteboard (pdf)
Slides: model fitting examples (pdf)
Thu, Dec 14 Encoding/decoding: Spike-triggered analysis, population decoding
Zoom recording, whiteboard (pdf)

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