Ma 191 b Topics Course: Geometry of Neuroscience
Winter 2017: taught jointly by Matilde Marcolli and Doris Tsao. The class is cross listed as CNS/Bi 286b. The class meets Tuesday-Thursday 10:30-noon, ROOM CHANGE: new room: Sloan 151
Brief Course Description
This class will cover topics in geometry and topology applied
to neuroscience, with particular focus on vision and language.
Topics covered will include:
- Geometry and topology of the visual cortex
- Geometry of segmentation and invariance
- Neural codes and neural rings
- Deep learning neural networks: mathematical aspects,
and applications to vision and language
- Mathematics and neuroscience of language and syntax
Suggested readings
The class does not have an official textbook, but the following
references will be useful.
Books:
- G. Bard Ermentrout, David H. Terman, "Mathematical Foundations of
Neuroscience", Springer, 2010.
- Jean Petitot, "Neurogeometrie de la vision", Les Editions de l'Ecole
Polytecnique, 2008. ( pdf )
- Giovanna Citti, Alessandro Sarti, "Neuromathematics of Vision",
Springer, 2014
- Robert Ghrist, "Elementary Applied Topology", Createspace Independent,
2014.
- David Mumford, Agnes Desolneux, "Pattern Theory: The Stochastic Analysis of Real-World Signals", CRC Press, 2010.
- Derek Bickerton and Eors Szathmary, "Biological Foundations and
Origin of Syntax", MIT Press, 2009
- David Marr, "Vision: A Computational Investigation into the Human Representation and Processing of Visual Information", MIT Press, 2010.
- Peter Dayan, "Theoretical Neuroscience", MIT Press, 2005.
- David Spivak, "Category Theory for the Sciences" MIT Press 2014,
html
Some Articles: (more will be added)
General overview:
- Misha Gromov, "Structures Learning and Ergosystems",
pdf
- Misha Gromov, "Ergostructures, Ergologic and the Universal
Learning Problem"
pdf (*)
- Felipe Cucker and Steve Smale, "On the Mathematical
Foundations of Learning"
pdf
- Yuri Manin, Error-correcting codes and neural networks
pdf (*)
Dynamical models of the neuron:
- Ryan Siciliano, "The Hodgkin-Huxley Model"
pdf (*)
- Tanya Kostova, Renuka Ravindran, Maria Schonbek,
FitzHugh-Nagumo Revisited: Types of Bifurcations, Periodical
Forcing and Stability Regions by a Lyapunov Functional,
Internat. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004), no. 3, 913-925
pdf (*)
Receptor fields and Gabor frames:
- Karlheinz Gròˆchenig, "Multivariate Gabor frames and sampling of entire functions of several variables"
pdf
- Kristian Seip, "Density theorems for sampling and interpolation in
the Bargmann-Fock space I"
pdf
(*)
- Kristian Seip, "Density theorems for sampling and interpolation in
the Bargmann-Fock space"
pdf
- Bruce MacLennan, "Gabor Representations"
pdf (*)
Conformal Geometry:
- Peter Olver "Complex Analysis and Conformal Mapping"
pdf (*)
- F. Helein and J.C. Wood, "Harmonic Maps"
pdf (*)
- Yalin Wang, Xianfeng Gu, Tony Chan, Paul Thompson, Shing-Tung Yau,
"Intrinsic Brain Surface Conformal Mapping using a Variational Method",
pdf
- D.Ta, J.Shi, B.Barton, A.BRewer, Z.L.Lu, Y.Wang, "Characterizing human
retinotopic mapping with conformal geometry: A preliminary study"
pdf
- P. Koehl, J. Hass, "Automatic Alignment of Genus-Zero Surfaces"
pdf (*)
- S.J.Gortler, C.Gotsman, D.Thurston "Discrete one-forms on meshes
and applications to 3D mesh parameterization"
pdf (*)
- N.Aigerman, Y.Lipman, "Orbifold Tutte Embeddings"
pdf (*)
- M.Hurdal, P.Bowers, K.Stephenson, D.Sumners, K.Rehm, K.Schaper, D.Rottenberg,
"Quasi-conformal flat mapping the human cerebellum"
pdf (*)
Contact Geometry:
- John B. Entyre, "Introductory Lectures on Contact Geometry",
pdf
- William C. Hoffman, "The Visual Cortex is a Contact Bundle"
pdf
- Alessandro Sarti, Giovanna Citti, Jean Petitot, "Functional geometry of the horizontal connectivity in the primary visual cortex"
pdf (*)
Segmentation and tracking:
- David Mumford, Jayant Shah, " Optimal Approximations by Piecewise Smooth
Functions and Associated Variational Problems"
pdf
- Laurent Younes, Peter W. Michor, Jayant Shah, David Mumford, A metric
on shape space with explicit geodesics, Rend. Lincei Mat. Appl. 19 (2008) 25-57
pdf (*)
- Mumford, D.; Kosslyn, S. M.; Hillger, L. A.; Herrnstein, R. J.
Discriminating figure from ground: the role of edge detection and
region growing, Proc. Nat. Acad. Sci. U.S.A. 84 (1987), no. 20, 7354-7358
pdf (*)
- Leah Bar et al. Mumford and Shah Model and its Applications to
Image Segmentation and Image Restoration, in Handbook of Mathematical Methods in Imaging, Springer 2011, 1095-1157
pdf (*)
- Thomas J. Tsao, Doris Y. Tsao, "Mathematical analysis of the general
conditions of vision reveals a new solution to segmentation and invariance",
pdf
- M. Belkin, P. Niyogi, "Laplacian Eigenmaps and
Spectral Techniques for Embedding and Clustering
pdf
(*)
- Frank Sottile, Thorsten Theobald, "Line problems in nonlinear
computational geometry", arXiv:math/0610407
- Marco Pellegrini, "Ray shooting and lines in space",
Handbook of discrete and computational geometry, 599-614
pdf
- Thorsten Theobald "An enumerative geometry framework for algorithmic line
problems in R3", SIAM J. Comput. 31 (2002), no. 4, 1212-1228
pdf
Bistable images:
- Emily J. Wald, Brian J. Scholl, "Stochastic or systematic? Seemingly random perceptual switching in bistable events triggered by transient unconscious cues" pdf (*)
- N. Rubin, M.C. Pugh, "Global effects in Figure/Ground segregation by
a model with only local interactions",
pdf (*)
Neural Codes and Neural Rings:
- Yuri Manin, "Neural codes and homotopy types: mathematical models
of place field recognition", arXiv:1501.00897 (*)
- Carina Curto, Vladimir Itskov, Alan Veliz-Cuba, Nora Youngs,
"The neural ring: an algebraic tool for analyzing the intrinsic structure
of neural codes", arXiv:1212.4201
- Nora Youngs, "The neural ring: using algebraic geometry to analyze
neural codes", arXiv:1409.2544 (*)
- Carina Curto, "What can topology tell us about the neural code?",
arXiv:1605.01905
- Carina Curto, Vladimir Itskov, Katherine Morrison, Zachary Roth,
Judy L. Walker, "Combinatorial neural codes from a mathematical coding
theory perspective", arXiv:1212.518 (*)8
- Carina Curto, Anda Degeratu, Vladimir Itskov, "Encoding binary
neural codes in networks of threshold-linear neurons", arXiv:1212.0031 (*)
Neural Networks, Deep Learning:
- Fabio Anselmi, Joel Z. Leibo, Lorenzo Rosasco, Jim Mutch, Andrea Tacchetti and Tomaso Poggio, "Unsupervised learning of invariant representations with low sample complexity: the magic of sensory cortex or a new framework for machine learning?"
pdf
- Thomas Wiatowski, Helmut Boelcskei, "A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction", arXiv:1512.06293 (*)
- Arnab Paul and Suresh Venkatasubramanian, "Why does unsupervised
learning work? A perspective from group theory"
pdf
- Cedric Beny, "Deep learning and the renormalization group",
arXiv:1301.3124
- Pankal Metha, David J. Schwab, "An exact mapping between the Variational Renormalization Group and Deep Learning", arXiv:1410.3831
- Yann Ollivier, "Riemannian metrics for neural networks I: feedforward networks", arXiv:1303.0818 (*)
- Yann Ollivier, "Riemannian metrics for neural networks II: recurrent networks and learning symbolic data sequences", arXiv:1306.0514 (*)
Notes of classes
Notes and slides of the classes will be posted here. A rough outline
of the lectures:
- Thursday January 5: Gromov's Ergobrain program (Doris/Matilde)
- Tuesday January 10: An introduction to neurons and the brain (Doris)
- Thursday January 12: Mathematical models of neurons via nonlinear
dynamics (Matilde)
- Tuesday January 17: An introduction to the neuroscience of vision (Doris)
- Thursday January 19: Receptor fields and Gabor frames analysis (Matilde)
- Tuesday January 24: Conformal Geometry
of the primary visual cortex (Matilde)
- Thursday January 26: Contact Geometry
of the primary visual cortex (Matilde)
- Tuesday January 31: Topological approach to segmentation
and invariance (Doris)
- Thursday February 2:
Variational problems in image segmentation,
the Mumford-Shah functional (Matilde)
- Tuesday February 7:
Segmentation and tracking with computational
and enumerative algebraic geometry (Matilde)
- Thursday February 9:
Neural codes (Doris/Matilde)
- Tuesday February 14:
Neural codes and neural rings, topology and
algebraic geometry (Matilde)
- Thursday February 16:
Bistable images and dynamical systems (Doris)
- Tuesday February 21:
Brain Networks and Topology (Matilde)
- Thursday February 23:
Mathematics and Language: generative grammars
(Matilde)
- Tuesday February 28:
Language, syntax and neuroscience (Matilde)
- Thursday March 2: Deep learning (Doris)
- Tuesday March 7: Deep learning and vision (Doris)
- Thursday March 9: Learning, Vision, and Deep Networks (Matilde)
Slides
Slides of classes available here:
Syllabus
The course is offered Pass/Fail. Requirements: attendance
of (most) lectures and one seminar presentation.
In the bibliographical list of papers linked above, all papers marked
with (*) are available for student presentations (more will be added shortly).
Choose an article on a topic that is most interesting to you (first come
first serve) and prepare a 30min seminar presentation about it.
The final student presentations will take place Tuesday, March 14
and Thursday, March 16 in Room SLN 151 (same room where the class
takes place), starting at 9am. The tentative schedule of
presentations is given below.
Student Presentations
- Tuesday March 14, 9:00am: Jane Panangaden, Functional geometry of the horizontal connectivity in the primary visual cortex
- Tuesday March 14, 9:30am: Arjun Bose, the Hodgkin-Huxley model
- Tuesday March 14, 10:00am: Aaron Anderson, Neural codes and homotopy types
- Tuesday March 14, 10:30am: Jeremy Bernstein, Gardner analysis of the statistical mechanics of neural networks
- Tuesday March 14, 11:30am: Matthew Rosenberg, Unsupervised learning of invariant representations with low sample complexity
- Thursday March 16, 9:00am: John Thompson, Ergostructures, Ergologic and the Universal Learning Problem
- Thursday March 16, 9:30am: Preston Rasmussen, Complex Analysis and Conformal Mapping
- Thursday March 16, 10:00am: Jialing Song, Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering
- Thursday March 16, 10:30am: Evan Davis, Error-correcting codes and neural networks
- Thursday March 16, 11:00am: Alvita Tran, Discriminating figure from ground