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

Language in the Brain
Mathematical Model of Language

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: the problem of reverse engineering in neuroscience, the Atari chip, the Blue Brain project and simulations of the cortex microcircuitry, topological analysis, cliques, growth of homology during stimulus processing, directed flag complexes, algebraic topology of distributed computing and the role of nontrivial homology, directed algebraic topology, directed complexes
Thursday January 29: Neural codes and learning, neural networks, McCulloch-Pitts artificial neuron, perceptron, universal approximation theorem, back propagation, Kadanoff's renormalization and restricted Boltzmann machines
Tuesday February 3: Cucker-Smale mathematical theory of learning, bias-variance trade off, Vapnik's statistical learning theory, expansion in Laplace eigenfunctions, sample error estimate, approximation error estimate
Thursday February 5: mathematical theory of resources in symmetric monoidal categories, assignments of resources via summing functors, category of summing functors, summing functors on networks (convervation laws at vertices and equalizers)
Tuesday February 10: categorical Hopfield equations, examples of neural networks and automata, integrated information, Phi function; introduction to the visual system, receptor profiles and Gabor filters
Thursday February 12: Gabor frames, lattices and the frame condition for signal analysis; conformal geometry of the retinotopic map: harmonic maps, energy minimization, conformal structures on Riemann surfaces and conformal maps, harmonic maps and Riemann surfaces, genus zero case, algorithmic construction of conformal maps, compression via spherical harmonics projections, test of conformality via Beltrami differentials.
Tuesday February 17: (lecture cancelled due to traveling: will make it up at the end of the class)
Thursday February 19: Contact geometry of the V1 cortex: fiber bundles, contact structures on 3-manifolds, Darboux theorem, fillability, contact and symplectic geometry, tight and overtwisted, receptor fields in V1 and contact planes, contact form and path lifting, role of non-integrability, circle bundle of contact elements on the complex projective line with tautological Liouville form and twist by the complex structure, dual contact structures and respective Reeb fields, bundle of signal planes, injectivity radius of the exponential map, associated lattices and Gabor system, frame condition, symplectization and contactization.
Tuesday February 24: Differential topology of image segmentation and tracking: perspective projection, ray spaces, perspective transition groupoid and stereo projection, occluding contours, Whitney singularities (folds and cusps), ownership of contours and accretion, diffeomorphisms and manifolds with boundary, correspondences and transversality, Whitney's theorem, correspondences and visibility, semigroupoid of visibility.
Thursday February 26: Image segmentation through the Mumford-Shah functional: segmentation problem, the functional, two related functionals, relation to Ising model, variational equation, elliptic problem, the role of singularities, contour variation, curvature, special points, elliptic boundary value problems with corners, approximation by locally constant functions and isoperimetric constant, minimizers and geometric measure theory approach, relation to Lorentzian geodesic equation.
Tuesday March 3: Introduction to language in the brain, levels of structure in language, early observations (Broca, Wernicke), imaging and signal detection methods, tonotopy and periodotopy, phonological structure, words, semantics distributed theory, syntax, P600 signal versus N400 signal, fronto-temporal combinatoric network, compositional processes, long distance dependencies, dual stream model; Mathematical model of language (syntax): Minimalism model of syntax, Merge, magma of syntactic objects, monoid of workspaces, accessible terms and coproduct operation.
Thursday March 5: Hopf algebras, workspaces and Hopf algebra of binary rooted trees and forests, Merge as Hopf algebra Markov chain, dynamical properties of Merge, limiting distributions, Perron Frobenius normalization and Merge as maximal entropy random walk, weight by cost functions and Ruelle free energy optimization, interfaces (sensory-motor and semantic-conceptual), externalization as planarization and filtering, planarizations, associahedra and moduli spaces of trees, minimal semantic properties for a syntax-first interface, the idea of Birkhoff factorization for semantic parsing.
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
- pdf U. Grenander, Patterns in Mathematical Semantics
Language:
- pdf Anna Mai, Andrea E. Martin, "Linguistic structure as a guiding principle for human neuroscience" Neuroscience Biobehavioral Reviews
- pdf Coopmans, C. W., Mai, A., Slaats, S., Weissbart, H., Martin, A. E. (2023) What oscillations can do for syntax depends on your theory of structure building. Letter in Nature Reviews Neuroscience
- pdf Martin, A. E. (2020). A compositional neural architecture for language. Journal of Cognitive Neuroscience
- pdf Weissbart, H., Martin, A. E. (2024). The structure and statistics of language jointly shape cross-frequency neural dynamics during spoken language comprehension. Nature Communications
- pdf Junyuan Zhao, Ruimin Gao, Jonathan R. Brennan, "Decoding the Neural Dynamics of Headed Syntactic Structure Building"
- pdf Michael Wolfman, Donald Dunagan, Jonathan Brennan, John Hale, "Hierarchical syntactic structure in human-like language models"
- pdf Elliot Murphy, ROSE: A Neurocomputational Architecture for Syntax
- pdf M.Marcolli, R.C.Berwick, "Encoding syntactic objects and Merge operations in function spaces"

Schedule of Final Presentations

Presentations will take place March 10, Room LH 387 from 9am-2:30pm and March 12 Room LH 187 from 9am-1pm and Room 255 from 1-2:30

March 10, 9:00am Aamina
March 10, 9:30am Zeyu
March 10, 10:00am Ishita
March 10, 10:30am Ruoxi
March 10, 11:00am Enoch
March 10, 11:30am Alex
March 10, 12:00pm Max
March 10, 12:30pm Shrujana
March 10, 1:00pm Peicong
March 10, 1:30pm Raaghav
March 10, 2:00pm Tejas
March 12, 9:00am Arnauld
March 12, 9:30am Guanxi
March 12, 10:00am Marisa
March 12, 10:30am Austin
March 12, 11:00am Bhargav
March 12, 11:30am Leo
March 12, 12:00pm Tianyi
March 12, 12:30pm Rosa
March 12, 1:00pm Mahider

Zeyu, Categorical Hopfield Networks
Shrujana, Hierarchical syntactic structure in human-like language models
Ruoxi, What can topology tell us about the neural code?
Max, Toroidal topology of population activity in grid cells
Ishita, The Structure and Statistics of Language Jointly Shape Cross-frequency Neural Dynamics During Spoken Language Comprehension
Guanxi, An Invitation to Neuroalgebraic Geometry
Aamina, Intrinsic Brain Surface Conformal Mapping using a Variational Method
Arnauld, On the nature and use of models in network neuroscience
Leo, The Visual Cortex is a Contact Bundle
Enoch, An enumerative geometry framework for algorithmic line problems in R3
Alex, The information bottleneck method
Tejas, Orbifold Tutte Embeddings
Marisa, A compositional neural architecture for language
Peicong, The Mumford-Shah functional
Mahider, Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function
Rosa, Bipartite expander Hopfield networks as self-decoding high-capacity error correcting codes
Austin, The neural ring
Bhargav, Patterns in Mathematical Semantics
Tianyi, Combinatorial neuralcodes from a mathematical coding theory perspective
Raaghav, Graph rules for recurrent neural network dynamics