Ma191b Winter 2026: Geometry of Neuroscience
Caltech, Linde Hall Room 255, Tuesday-Thursday 1:00-2:30pm
Instructor:
Matilde Marcolli
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:
- Thursday January 8:
- Tuesday January 13:
- Thursday January 15:
- Tuesday January 20:
- Thursday January 22:
- 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.
Papers and other reading material
- 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:
Schedule of Final Presentations
Schedule of presentations
-