Faculty: Larry Abbott, Stefano Fusi, Ashok Litwin Kumar, Ken Miller
TAs: Matteo Alleman, Dan Biderman, Salomon Muller, Amin Nejatbakhshesfahani, Marjorie Xie
Meetings: Tuesdays & Thursdays, JLGSC L5-084, Lecture 2.00 - 3.30pm
Text: Theoretical Neuroscience by P. Dayan and L.F. Abbott (MIT Press)
January
21 (Larry) Introduction to the Course and to Theoretical Neuroscience
23 (Larry) Mathematics Review
28 (Larry) Electrical Properties of Neurons, Integrate-and-Fire Model (Assignment 1)
30 (Larry) Adaptation, Synapses, Spiking Networks
February
4 (Larry) Numerical Methods, Filtering (Assignment 2)
5 Assignment 1 Due
6 (Larry) The Hodgkin-Huxley Model
11 (Larry) Types of Neuron Models and Networks (Assignment 3)
12 Assignment 2 Due
13 (Ken) Linear Algebra I
18 (Ken) Linear Algebra II (Assignment 4)
19 Assignment 3 Due
20 (Ken) Linear Algebra III
25 (Ken) PCA and Dimensionality Reduction (Assignment 5)
26 Assignment 4 Due
27 COSYNE
March
3 COSYNE
5 (Ken) Rate Networks/E-I networks I
10 (Ken) Rate Networks/E-I networks II (Assignment 6)
11 Assignment 5 Due
12 (Ken) Unsupervised/Hebbian Learning, Developmental Models
17 Spring Break
19 Spring Break
24 (Ashok) – Introduction to Probability, Encoding, Decoding
25 Assignment 6 Due
26 (Ashok) – GLMs
31 (Ashok) – Decoding, Fisher Information (Assignment 7)
April
2 (Ashok) – Decoding, Fisher Information II
7 (Ashok) – Information Theory (Assignment 8)
8 Assignment 7 Due
9 (Ashok) – Optimization
14 (Ashok) – Optimization II (Assignment 9)
15 Assignment 8 Due
16 Research Topic
21 (Stefano) Perceptron (Assignment 10)
22 Assignment 9 Due
23 (Stefano) Multilayer Perceptrons and Mixed Selectivity
28 (Stefano) – Deep Learning I (backpropagation) (Assignment 11)
29 Assignment 10 Due
30 (Stefano) – Deep Learning II (convolutional networks)
May
5 (Stefano) Learning in Recurrent Networks (Assignment 12)
6 Assignment 11 Due
7 (Stefano) Continual Learning and Catastrophic Forgetting
12 (Stefano) Reinforcement Learning
13 Assignment 12 Due
14 Research Topic
Meetings: Wednesdays, 10.15 - 11.45am
Location – Jerome L Greene Science Center, L6-087
See course website for updated schedule
Control Theory and Reinforcement Learning
Jan 29 Examples of control theory in neuroscience (Bettina Hein, Laureline Logiaco)
Feb 5 Methods in control theory (Bettina Hein, Laureline Logiaco)
Feb 12 Reinforcement learning (James Murray)
Feb 19 From optimal control theory to reinforcement learning (Samuel Muscinelli)
Feb 26 Cosyne break
Mar 4 Cosyne break
Latent Variable Models
Mar 11 Hidden Markov Models (Matt Whiteway)
Mar 18 Latent Dynamical Systems (Josh Glaser)
Mar 25 Gaussian Processes (Rainer Engelken)
Apr 1 Modeling of Behavior (Juri Minxha)
Apr 8 Hackathon on Latent Variable Models
Miscellaneous
Apr 15 Phenomenological Renormalization Group (Serena Di Santo)
Apr 22 Additional and diverse perspectives in neuroscience (Laureline Logiaco, Matt Whiteway, Juri Minxha)
Apr 29 Introduction to replica theory (SueYeon Chung)
May 6 Classification and Geometry of General Perceptual Manifolds (SueYeon Chung)
Mathematical Tools for Theoretical Neuroscience (NBHV GU4359)
Faculty: Ken Miller (kdm2103@columbia.edu)
Instructor: Dan Tyulmankov (dt2586@columbia.edu)
Teaching Assistant: Amin Nejatbakhsh (mn2822@columbia.edu)
Time: Tuesdays, Thursdays 4:10pm-5:25pm (tentative, may change based on demand)
Place: JLGSC L5-084 (Jan 21, 23, and 28 only, permanent room TBA)
Webpage: https://ctn.zuckermaninstitute.columbia.edu/courses
Credits: 3 credits, pass/fail only (registration details TBA)
Description: An introduction to mathematical concepts used in theoretical neuroscience aimed to give a minimal requisite background for NBHV G4360, Introduction to Theoretical Neuroscience. The target audience is students with limited mathematical background who are interested in rapidly acquiring the vocabulary and basic mathematical skills for studying theoretical neuroscience, or who wish to gain a deeper exposure to mathematical concepts than offered by NBHV G4360. Topics include single- and multivariable calculus, linear algebra, differential equations, dynamical systems, and probability. Examples and applications are drawn primarily from theoretical and computational neuroscience.
Prerequisites: Basic prior exposure to trigonometry, calculus, and vector/matrix operations at the high school level
Grading: 100% attendance-based
Readings and exercises: Lecture notes and optional practice exercises will be provided for each lecture, and supplementary readings will be assigned from various textbooks.
Topics:
1. Vocabulary (sets, functions, limits, define e=lim(1+1/n)^n, complex numbers, Euler’s)
2. Calculus (derivatives, integrals, fundamental theorem)
3. ODEs (first-order)
4. Taylor series (prove Euler’s)
5. Linear algebra 0 (vectors, matrices, add/multiply/inverse)
6. Linear algebra 1 (vector spaces, independence, span, basis, orthonormal, linear transformations, projections)
7. Linear algebra 2 (rank, range, nullspace, SVD)
8. Linear algebra 3 (change of basis, trace, determinant, eigenvector primer)
9. Linear system dynamics (phase plane, linear systems, eigenvectors)
10. Nonlinear system dynamics 1 (fixed points, stability, linearization, nullclines)
11. Nonlinear system dynamics 2 (bifurcations, chaos)
12. Combinatorics (set algebra, permutations, combinations)
13. Probability 1 (probability space, random vars)
14. Probability 2 (pdfs, expectation, variance, conditioning)
15. Probability 3 (joints, multivariate Gaussian)
16. Multivariable calculus 1 (partial derivative, gradient)
17. Multivariable calculus 2 (Hessian, multivar Taylor series)
18. Multivariable calculus 3 (chain rule, Lagrange Multipliers)
19. Convolution (signals & systems, impulse response, LTI systems, filters)
20. Fourier series
21. Discrete Fourier Transform (filters, linear algebra notation)
22. Random signals (stationarity, cross-correlation, auto-correlation, signal detection theory, ROC)
23. Stats 1 (summary stats, convergence, MLE, MAP)
24. Stats 2 (regression, classification)
25. Stats 3 (machine learning basics)
26. Information Theory (entropy, KL, mutual info)
Schedule:
Tue Jan 21 Vocabulary
Thu Jan 23 Single-variable calculus
Tue Jan 28 ODEs, Amin substituting (Dan away)
Thu Jan 30 Taylor series, Amin substituting (Dan away)
Tue Feb 4 Linear algebra 0
Thu Feb 6 Linear algebra 1
Tue Feb 11 Linear algebra 2
Thu Feb 13 Linear algebra 3
Tue Feb 18 Linear dynamics
Thu Feb 20 Nonlinear dynamics 1
Tue Feb 25 Nonlinear dynamics 2
Thu Feb 27 No class, Cosyne (Dan, Amin away)
Tue Mar 3 No class, Cosyne (Dan, Amin away)
Thu Mar 5 Combinatorics, Amin substituting (Dan away)
Tue Mar 10 Probability 1
Thu Mar 12 Probability 2
Tue Mar 17 No class, Spring break
Thu Mar 19 No class, Spring break
Tue Mar 24 No class, NAISys (Dan, Amin away)
Thu Mar 26 No class, NAISys (Dan, Amin away)
Tue Mar 31 Probability 3
Thu Apr 2 Multivariable calculus 1
Tue Apr 7 Multivariable calculus 2
Thu Apr 9 Multivariable calculus 3
Tue Apr 14 Convolution
Thu Apr 16 Fourier series
Tue Apr 21 Discrete Fourier Transform
Thu Apr 23 Random signals
Tue Apr 28 Statistics 1
Thu Apr 30 Statistics 2