Ma191b Winter 2026: Geometry of Neuroscience

Caltech, Linde Hall Room 255, Tuesday-Thursday 1:00-2:30pm

Brief Course Description

This class will present a broad overview of mathematical methods for the modeling of neuroscience. We will focus in particular on geometric and topological models. The content of the class is articulated in three parts. A first part focuses on structures in the brain, from single neurons to large scale connectivity, including neural codes and models of learning, and will show how topological methods have come to play an important role in describing these structures and arguing about functionality. A second part will discuss the visual system, and show how conformal geometry, harmonic analysis, and contact geometry play a role in modeling the visual cortex, and how this suggests new relations between these different fields of mathematics. This part also covers the problem of segmentation and tracking of images by the visual system and how differential topology, calculus of variations, and algebraic geometry interact in addressing this problem. The last part focuses on language and its embodiment in the brain and how that differs from current artificial models of language.

Workload

Studente are required to give a final presentation on a paper selected from the reading material for the class in agreement with the instructor. Participation in (most) classes is expected (consult the instructor about a reasonable arrangement in case of scheduling conflicts). Students are also expected to read and provide feedback on notes (from a book draft) that will be circulated to the class by the instructor. The class is offered P/F only.

Slides of Lectures

Slides of lectures will be posted here as the class progresses

Summary of lectures

Tuesday January 6: General introduction: Gromov's ergobrain project, overview of the course, structures in the brain, vision, language; brief introduction to main mathematical tools
Thursday January 8: Introduction to the structure of the brain; single neuron and synaptic connections, resting potential at the membrane and Nerst-Planck equation, voltage-gated ion channels and Hodgkin-Huxley equations, numerical results, FitzHugh-Nagumo model, discretization and Coupled Lattice Maps, behavior of critical points and periodic orbits, period doubling cascade and transition to chaos
Tuesday January 13: discrete combinatorial neural codes, simplicial complex of the code, receptive fields, convexity, simplicial nerve of open coverings, reconstruction of homotopy type of stimulus space, simplicial complexes and homology, embedding dimension, neural rings
Thursday January 15: presentations of neural rings, coding theory, code parameters, Gilbert-Varshamov curve and random codes, asymptotic bound and good codes, neural codes as error correcting codes, codes and expander graphs, binary Hopfield networks, Hebbian rule
Tuesday January 20: expander graphs, Tanner codes, higher Hopfield networks and linear codes, Chaudhuri-Fiete construction of expander codes via Hopfield networks, general facts on large networks and random graphs: randomness (regular, small-world, random), connectivity, connection density, degree distribution (Erdos-Renyi graphs, scale-free networks, broad-scale network), notions of centrality
Thursday January 22: transition to connectedness for Erdos-Renyi graphs, transition to growth of a giant component, robusteness to lesion in networks, k-core decompositions, walks, trails, paths, global efficiency index, spectral properties, graph Laplacian, expansion constant and spectrum, quantum graphs and quantum chaos, zeta function of graphs, Ramanujan graphs and the Riemann Hypothesis for graph zeta functions, non-regular case, Ruelle zeta function, Hashimoto determinant formula, Kotani-Sunada spectrum and tests of expander property
Tuesday January 27:
Thursday January 29:
Tuesday February 3:
Thursday February 5:
Tuesday February 10:
Thursday February 12:
Tuesday February 17: (lecture cancelled due to traveling: will make it up at the end of the class)
Thursday February 19:
Tuesday February 24:
Thursday February 26:
Tuesday March 3:
Thursday March 5:
Tuesday March 10: final presentationa
Thursday March 12: (additional time) final presentations

Reading Materials

There is no specific textbook for the class, though some notes will be made available to the students. Reading material and suggested material for presentation will be posted here as the class progresses.

General references:
- Jean Petitot, "Neurogeometrie de la vision", Les Editions de l'Ecole Polytecnique, 2008. ( pdf )
- David Spivak, "Category Theory for the Sciences" MIT Press 2014, html
- Eugene M. Izhikevich, "Dynamical systems in neuroscience", MIT Press 2007 ( pdf )
- Alex Fornito, Andrew Zalesky and Edward T. Bullmore, "Fundamentals of Brain Network Analysis", Elsevier 2016
- Mikhail Gromov, "Structures, Learning and Ergosystems" pdf
- Mikhail Gromov, "Ergostructures, Ergologic and the Universal Learning Problem" pdf
Structures in the Brain:
- 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
- Yakov Pesin, Vaugham Climenhaga, Lectures on Fractal Geometry and Dynamical Systems, American Mathematical Society 2009
- Carina Curto, "What can topology tell us about the neural code?", arXiv:1605.01905
- 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, Vladimir Itskov, Katherine Morrison, Zachary Roth, Judy L. Walker, "Combinatorial neural codes from a mathematical coding theory perspective", arXiv:1212.5188
- Carina Curto, Anda Degeratu, Vladimir Itskov, "Encoding binary neural codes in networks of threshold-linear neurons", arXiv:1212.0031
- Elizabeth Gross, Nida Kazi Obatake, Nora Youngs, Neural ideals and stimulus space visualization, arXiv:1607.00697
- Yuri Manin, "Neural codes and homotopy types: mathematical models of place field recognition", arXiv:1501.00897
- Yuri Manin, "Error-correcting codes and neural networks", pdf
- Rishidev Chaudhuri and Ila Fiete, "Bipartite expander Hopfield networks as self-decoding high-capacity error correcting codes" pdf
- Uzy Smilansky, Quantum chaos on discrete graphs, arXiv:0704.3525
- Audrey Terras, "Zeta functions and chaos" pdf
- M.W.Reimann, M.Nolte, M.Scolamiero, K.Turner, R.Perin, G.Chindemi, P.Dlotko, R.Levi, K.Hess, H.Markram, Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function pdf
- Eric Jonas, Konrad Paul Kording, Could a Neuroscientist Understand a Microprocessor? pdf
- Richard Steiner, "The algebra of the nerves of omega-categories", arXiv:1307.4236
- Philippe Gaucher, "Homotopy invariants of higher dimensional categories and concurrency in computer science", arXiv:math/9902151
- Glynn Winskel, Mogens Nielsen "Models of Concurrency" pdf
- Joshua Lieber, "Comparison between different models of concurrency", arXiv:2012.04246
- Yuri Manin, Matilde Marcolli, "Homotopy Theoretic and Categorical Models of Neural Information Networks", arXiv:2006.15136
- Matilde Marcolli, "Categorical Hopfield Networks", arXiv:2201.02756
- B. Coecke, T. Fritz, R.W. Spekkens, "A mathematical theory of resources", arXiv:1409.5531
- Carina Curto, Katherine Morrison, "Graph rules for recurrent neural network dynamics: extended version", arXiv:2301.12638
- Caitlyn Parmelee, Samantha Moore, Katherine Morrison, Carina Curto, "Core motifs predict dynamic attractors in combinatorial threshold-linear networks", arXiv:2109.03198
- D. Beniaguev, I. Segev, M. London, Single cortical neurons as deep artificial neural networks, biorxiv:10.1101/613141
- David Balduzzi, Giulio Tononi, "Qualia: the geometry of integrated information" pdf
- M. Oizumi, N. Tsuchiya, S. Amari, Unified framework for information integration based on information geometry, pdf
Visual system:
- Karlheinz Gröchenig, "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
- Vasiliki Liontou, Matilde Marcolli, "Gabor frames from contact geometry in models of the primary visual cortex" pdf
- Alessandro Sarti, Giovanna Citti, Jean Petitot, "Functional geometry of the horizontal connectivity in the primary visual cortex" pdf
- William C. Hoffman, "The Visual Cortex is a Contact Bundle" pdf
- John B. Entyre, "Introductory Lectures on Contact Geometry", pdf
- 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
- 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 Tsao, Doris Tsao, "A topological solution to object segmentation and tracking" 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