Mathematical Tools for Neuroscience
Princeton University, NEU 314, Fall 2022
Time: Tu/Th 1:30–2:50 pm
Location: Jadwin Hall A06
Instructor: Sam Nastase (snastase@princeton.edu)
AIs: Rober Boshra, Zaid Zada, Wayan Gauthey, David Allen
Syllabus: Syllabus
This course introduces students to the mathematical tools at the core of computational neuroscience research. The course aims to familiarize students with topics in linear algebra, statistics, and machine learning, with a heavy emphasis on applications to neurobiology. Lectures on each topic will develop the relevant mathematical background with links to foundational applications in the field. Coursework will focus primarily on problem sets requiring the implementation of models and analyses in Python. The course will equip students with a practical proficiency in various computational methods, including programming skills in data analysis and visualization that are increasingly important to scientific inquiry in general, and neuroscience in particular.
Lecture schedule
| Date | Topic | Slides/code | Homework | Optional reading |
|---|---|---|---|---|
| Tu 9/6 | Introduction to computational neuroscience | Lecture 0 | Homework 0Q Homework 0A | Marr, 1982 |
| Th 9/8 | Introduction to linear algebra: vectors | Lecture 1 Code 1 | ||
| Tu 9/13 | Linear combinations and vector spaces | Lecture 2 Code 2 | Homework 1 | |
| Th 9/15 | Applications: trichromatic vision | Lecture 3 Code 3 | Maxwell, 1857 Kuffler, 1953 | |
| Tu 9/20 | Matrix multiplication, outer product, transpose | Lecture 4 Code 4 | ||
| Th 9/22 | Row/column/null spaces, linear systems | Lecture 5 Code 5 | ||
| Tu 9/27 | Applications: neural population codes | Lecture 6 | Homework 2 | Georgopoulos et al., 1986 Haxby et al., 2001 |
| Th 9/29 | Singular value decomposition (SVD) | Lecture 7 Code 7 | ||
| Tu 10/4 | Principal component analysis (PCA) | Lecture 8 Code 8 | ||
| Th 10/6 | Applications: dimensionality reduction | Lecture 9 Code 9 | ||
| Tu 10/11 | Least-squares regression | Lecture 10 Code 10 | Homework 3 | |
| Th 10/13 | Applications: univariate regression in fMRI | Lecture 11 Code 11 | ||
| Tu 10/18 | No class (fall recess) | |||
| Th 10/20 | No class (fall recess) | |||
| Tu 10/25 | Introduction to probability and Bayes’ rule | Lecture 12 Code 12 | ||
| Th 10/27 | Covariance and correlation | Lecture 13 Code 13 | ||
| Tu 11/1 | Applications: representational geometry and functional connectivity | Lecture 14 Code 14 | Homework 4 | Kriegeskorte & Kievit, 2013 Yeo et al., 2011 |
| Th 11/3 | Permutation tests and bootstrapping | Lecture 15 Code 15 | ||
| Tu 11/8 | Information theory and efficient coding | Lecture 16 Code 16 | Homework 5 | |
| Th 11/10 | Overfitting and out-of-sample prediction | Lecture 17 Code 17 | ||
| Tu 11/15 | Regularized regression | Lecture 18 Code 18 | ||
| Th 11/17 | Applications: encoding models in fMRI | Lecture 19 Code 19 | Huth et al., 2016 | |
| Tu 11/22 | No class (Thanksgiving recess) | |||
| Th 11/24 | No class (Thanksgiving recess) | |||
| Tu 11/29 | Classification and confusion matrices | Lecture 20 Code 20 | Homework 6 | |
| Th 12/1 | Applications: neural decoding | Lecture 21 Code 21 | Norman et al., 2006 | |
| Tu 12/6 | Clustering and mixture models | Lecture 22 Code 22 | ||
| Th 12/8 | Artificial neural networks | Lecture 23 Code 23 |
The content of this course is inspired by related courses designed by Jonathan Pillow, Eero Simoncelli, Michael Landy, Ella Batty, and Tal Yarkoni.
Resources and references
“Welcome to Colaboratory” introduction to browser-based interactive Jupyter Notebooks hosted by Google Colab: Welcome to Colaboratory
Princeton Neuroscience Institute’s handbook for best practices in scientific computing and reproducible neuroscience: Princeton Handbook for Reproducible Neuroimaging
NumPy’s User Guide “Quickstart,” “Absolute Basics,” and “Fundamentals” sections: NumPy User Guide
Pyplot tutorial and Matplotlib’s “Getting Started,” “Tutorials,” and “Examples” documentation: Pyplot tutorial Matplotlib Documentation
Scikit-learn’s User Guide and Examples: User Guide Examples
Managing Python environments and installing software using conda: Conda Documentation PHRN conda guide
Jupyter ecosystem documentation: Jupyter Documentation
Jupyter Notebook introductory documentation: Jupyter Notebook Documentation
Guide to Markdown Cells in Jupyter Notebooks: Jupyter Markdown Cells
Neuromatch Academy course in Computational Neuroscience: Neuromatch
Neurohackademy summer school lectures: Neurohackademy
“Theoretical Neuroscience” textbook by Peter Dayan and Larry Abbott: PDF
— Dayan, P., & Abbott, L. F. (2001). Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. MIT Press.
“Introduction to Linear Algebra” textbook materials available online at Gilbert Strang’s website: website
— Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press.
“Statistics” textbook by Freedman, Pisani, and Purves:
— Freedman, D., Pisani, F., & Purves, R. (2007). Statistics (4th ed.). Norton.
“Permutation Tests” textbook by Phillip Good:
— Good, P. (2000). Permutation Tests: A Practical Guide to Resampling Methods for Testing Hypotheses (2nd ed.). Springer.
“An Introduction to the Bootstrap” textbook by Efron and Tibshirani:
— Efron, B., & Tibshirani, R. J. (1994). An Introduction to the Bootstrap. Chapman & Hall/CRC.
“The Elements of Statistical Learning” textbook available online at Trevor Hastie’s website: website PDF
— Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction (2nd ed.). Springer.