November 7, 2020

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Materials for Mathematical Tools for Neuroscience course at Harvard (Neurobio 212)

repo name ebatty/MathToolsforNeuroscience
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language Jupyter Notebook
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created 2020-09-30
license MIT License

Mathematical Tools for Neuroscience (Neurobio 212 at Harvard)

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

Course contact:

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…).

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.

The materials

Topic Content Content description
Week 1: Linear Algebra I, Vectors Video 1.1: What is a vector? Why learn linear algebra, intuition behind vectors, definition of a vector
Video 1.2: Vector properties & operations Vector length, unit & zero vectors, scalar multiplication, vector addition, dot product, neuroscience example
Video 1.3: Vector spaces 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
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 Outline of using eigenvalues/vectors to understand dynamics of a small neural circuit
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 Covers diagonal, orthogonal, and symmetric matrices
Video 4.2: Matrix decomposition & SVD 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
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 What is a dynamical system and what are the types
Video 5.2: Solving Differential Equations Covers analytical, numerical, and graphical solutions for differential equations
Video 5.3: Systems of Differential Equations Continuous dynamical systems, phase portraits, eigenvalue dependence
Tutorial 1 Dynamical system exploration concerning rate of ion channels opening
Tutorial 2 Exploration of FitzHugh-Nagumo Model
Linear Algebra Review Review Video: Coming soon Brief recap of key concepts of the last month
Linking to Neuro Video: Coming soon Goes through a neuroscience paper that uses linear algebra heavily
Week 6: Probability & Statistics I, Intro to Probability Coming soon
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