Mathematical Tools for Neuroscience (Neurobio 212 at Harvard)

Developed and taught by Ella Batty, Lucy Lai, Alex Chen, and John Assad

Course contact: [email protected]

Scroll down to bottom for the materials

Description of course

Numerical data analysis has become a nearly indispensable tool in modern neuroscience. This course aims to equip graduate students with the fundamental mathematical skills in quantitative modeling and data analysis necessary for neuroscience research (and for further computational neuroscience courses). This course is essentially organized into three sections: one on linear algebra, one on probabiliy & statistics, and one on the basics of machine learning. We will also cover some additional topics such as differential equations and dynamical systems).

One goal in formulating this course was to alleviate the need for taking multiple undergraduate-level courses in each of the stated topics (which may be cumbersome due to back-and-forth commute, inconvenient scheduling, or just an excess of materical with no clear applicability to neuroscience research). Our goal is to make this as fun, approachable, and applicable as possible. We would like to build mathematical intuition for these essential topics.

Course Prerequisites

You will not need any math experience beyond high school calculus. Python will be used in this class so some knowledge of it is necessary, although this course will also serve as an opportunity to practice those skills.

Description of materials

There are three components to the course: video lectures, comprehension questions, and tutorials.

Video lectures: The course consists of short video lectures each structured around a specific topic (10-20 minutes per video). We have created most of these videos using a Khan academy inspired style. We don't want to reinvent the wheel so when great videos already exist on a specific topic, we link to those instead, but this will mostly occur in the linear algebra section.

Comprehension questions: These are short questions designed to be looked at right after watching the videos to consolidate your knowledge and try out quick computations. Both questions and answers are provided below.

Tutorials: Each week has 1 - 2 tutorials, or problem sets. These are Google colab notebooks with exercises where you might be asked to do computations by hand, engage with interactive demos, or code. They are designed to review the video material and in some cases introduce new concepts. They are not designed for repetitive problem solving (i.e. you will not be asked to solve a matrix equation by hand 100 times...).

Other resources

Helpful for before this course: Basic Introduction to Maths and Python for Neuroscience by John Butler

Similar course: Mathematical Tools for Neural and Cognitive Science by Eero Simoncelli & Mike Landy

A good follow-up to learn computational neuroscience: Neuromatch Academy

The materials

Video links below will take you to Youtube, tutorial links will open Google colab notebooks.

We welcome constructive criticism given via opening a git issue. Note that the Week 1 videos created by me are a little rough right now due a steep learning curve (and a delay in acquiring appropriate audio equipment) but I promise they get better so bear with it.

Topic	Content	Content description
Week 1: Linear Algebra I, Vectors	Video 1.1: What is a vector? Notes	Why learn linear algebra, intuition behind vectors, definition of a vector
	Video 1.2: Vector properties & operations Notes	Vector length, unit & zero vectors, scalar multiplication, vector addition, dot product, neuroscience example
	Video 1.3: Vector spaces Notes	Linear combinations, linear independence, spanning vectors, basis, vector spaces
	Tutorial 1	Geometry of the dot product, neuron optimal stimuli, correlation coefficient
	Tutorial 2	Vector sets
Week 2: Linear Algebra II, Matrices	Video 2.1: Linear transformations and matrices (3Blue1Brown Essence of Linear Algebra Chapter 3)	Properties of linear transformations, how matrices represent them,
This week videos are all from the 3Blue1Brown Essence of Linear Algebra series	Video 2.2: Matrix multiplication as composition (3Blue1Brown Chapter 4)	Intuition behind matrix multiplication as composition of linear transformations, properties of matrix multiplication
	Video 2.3: Three dimensional linear transformations (3Blue1Brown Chapter 5)	Brief review of transformations in 3D
	Video 2.4: The determinant (3Blue1Brown Chapter 6)	Intuition behind determinants, computation of 2x2 matrix determinant
	Video 2.5: Inverse matrices, column space, and null space (3Blue1Brown Chapter 7)	Systems of linear equations, solving with inverse matrices, when do solutions exist, definition of rank, how to think of column & null space geometrically
	Video 2.6: Nonsquare matrices as transformations between dimensions (3Blue1Brown Chapter 8)	Brief review of transformations in relation to nonsquare matrices
	Comprehension Questions
	Tutorial 1	Matrix multiplication by hand, thinking about transformations, encoding model matrices in the context of the invertible matrix theorem
	Tutorial 2	Properties of nonsquare matrices, changing bases
Week 3: Linear Algebra III, Eigenstuff	Video 3.1: Eigenvectors and eigenvalues (3Blue1Brown Chapter 14)	Definition of eigenvectors/eigenvalues, how to find eigenvalues of a matrix, brief intro to matrix diagonalization
	Video 3.2: Eigenstuff in neural circuits Notes	Outline of using eigenvalues/vectors to understand dynamics of a small neural circuit
	Comprehension Questions
	Tutorial 1	Describing transformations with eigenvectors, looking at eigenstuff of a squared matrix, complex eigenvalues, understanding neural circuit discrete dynamics using eigenvalues/eigenvectors
Week 4: Linear Algebra IV, Matrix Decomposition & Dimensionality Reduction	Video 4.1: Special Matrices Notes	Covers diagonal, orthogonal, and symmetric matrices
	Video 4.2: Matrix decomposition & SVD Notes	Covers what matrix decomposition is/what we can use it for/and types, especially eigendecomposition and SVD
	Video 4.3: Dimensionality Reduction in Neuroscience (NMA W1D5 Intro lecture)	Byron Yu gives overview of using dimensionality reduction on neural population responses, discusses specific research, highlights different methods and when you might use each
	Video 4.4: PCA	What is PCA, how do we compute it, how does it relate to SVD
	Comprehension Questions
	Tutorial 1	Delving into SVD, implementing PCA and exploring correlation effects, PCA on MNIST images
	Tutorial 2	Delving into SVD as low rank receptive field approximation
Week 5: Dynamical Systems & Differential Equations	Video 5.1: Intro to Dynamical Systems Notes	What is a dynamical system and what are the types
	Video 5.2: Solving Differential Equations Notes	Covers analytical, numerical, and graphical solutions for differential equations
	Video 5.3: Systems of Differential Equations Notes	Continuous dynamical systems, phase portraits, eigenvalue dependence
	Comprehension Questions
	Tutorial 1	Dynamical system exploration concerning rate of ion channels opening
	Tutorial 2	Exploration of FitzHugh-Nagumo Model
Week 6: Probability & Statistics I, Intro to Probability	Read Chapters 1, 2, 3, and 5 of https://seeing-theory.brown.edu/ (they're brief)	Covers basic probability, probability distributions, and Bayesian inference
	Tutorial 1	Explore Poisson distribution, compute marginal distributions
	Tutorial 2	Compute probabilities for 2 armed bandit task, code simulations
Week 7: Probability & Statistics II, Intro to Statistics	Video 7.1: Descriptive Statistics Notes	Descriptive vs inferential statistics, measures of central tendency, dispersion, and shape
	Video 7.2: Overview of Statistical Inference Notes	Covers probability vs statistics, estimation vs hypothesis testing
	Video 7.3: Point Estimators Examples & Goodness	Point estimators, bias, variance, consistency, population mean & variance estimators
	Video 7.4: Maximum Likelihood Estimation	Likelihood defintion, maximum likelihood estimator of firing rate, numerical methods
	Video 7.5: Bayesian Inference Notes	Frequentist vs Bayesian probability, priors, Bayes estimators & risk
	Comprehension Questions
	Tutorial 1	Clarify correlation, Bayesian decoding, MLE derivation
	Tutorial 2	Normal distribution point estimators
Week 8: Probability & Statistics III, Statistical Encoding Models	Video 8.1: What are encoding & decoding?	Broad intro to encoding/decoding
	Video 8.2: Statistical Encoding Models	Spike triggered averages, linear-nonlinear-Poisson models, gradient descent
	Comprehension Questions
	Tutorial 1	Compute STA of retinal data, implement gradient descent by hand to compute LNP parameters
	Tutorial 2	Play with LNP filters and see resulting dynamics