S. D. V. Stephens

notes / Basic Neuroscience and Geometry of Neuroscience / chapter 6: Introduction

G1 A Mathematical Promenade and Gromov’s Ergobrain Program

This is the first lecture of Ma 191b: Geometry of Neuroscience (Caltech, Winter 2026). It has two parts: a motivational introduction built around Misha Gromov’s idea of “ergosystems”, and a quick promenade through the kinds of mathematics that will come up during the course.

About this class

The class is based on a course designed and taught jointly by Matilde Marcolli (mathematics) and Doris Tsao (neuroscience) in 2017, later taught by Marcolli at the University of Toronto in 2018. It is also the basis for an ongoing book project, Geometry of Neuroscience, with Doris Tsao (UC Berkeley).

The class will cover three parts.

  1. Structures in the Brain: Neurons, Networks, Codes
    • Single neuron as a dynamical system
    • Hopfield equation and neural computation
    • Neural codes and homotopy types
    • Brain networks and random graphs
    • Directed algebraic topology and microcircuitry
    • Models of learning
    • Artificial and brain networks
    • Category theory of networks and resources
    • Integrated Information
  2. Vision: Visual Cortex, Segmentation/Tracking
    • The Visual System and deep networks
    • Gabor frames and signal analysis
    • Conformal geometry of the visual cortex
    • Contact geometry of the visual cortex
    • Gabor frames and contact geometry, Gabor bundles
    • Segmentation and tracking: differential topology
    • Segmentation: variational analysis
    • Tracking: algebraic geometry
    • Pattern Theory
  3. Language: Syntax as Brain Structure
    • Universal Grammar hypothesis
    • Merge and syntax
    • Syntax in the brain

Gromov’s ergobrain

The motivational part of this lecture follows two long, provocative, programmatic papers by Misha Gromov about the goal of a mathematical understanding of learning and of structures in the biological brain:

  • Misha Gromov, Structures, Learning, and Ergosystems (2011);
  • Misha Gromov, Ergostructures, Ergologic and the Universal Learning Problem (2013).

Gromov conjectures the existence of a mathematical structure implementing the transformation of incoming signals into representations in the brain. He calls it the ergobrain. It is a dynamic entity, continuously built by the brain through (goal-free) structure learning, with constraints coming from the architecture of the network.

Definition G1.1 (Gromov’s terminology).

  • The neuro-brain is a (mathematical) model of the physiology of the brain: chemical, electrical, connectivity, and so on.
  • The ergo-brain is a “dynamical reduction” of the neuro-brain (a quotient of it).
  • An ergo-system is like an ergobrain, but not necessarily derived from a neurobrain.
  • The ego-mind consists of the mental processes and the interactions of organisms with the outside world.
  • Ergo-learning is the spontaneous process of building structure.

Why “ergosystem”?

The name plays on Descartes’s cogito ergo sum, “I think, therefore I am”. The slide varies it:

  • I see therefore I am
  • I envy therefore I am
  • I love therefore I am
  • I dream therefore I am

and reads the “therefore” (ergo) as the hope of understanding consciousness through reasoning and mathematics.

Definition G1.2 (Ergosystem). An ergosystem is a machine for extracting structure from incoming signals, through a process of goal-free learning. This is in contrast to reinforcement learning, where learning is driven by rewards for reaching a goal.

An analogy with the biology of cells

Gromov draws an analogy with the biology of cells, where there are different levels of understanding:

  1. interaction with the environment (cytology);
  2. selection of beneficial mutations (molecular genetics);
  3. the biochemistry of proteins and cell structures.

An example of the third level is a protein interaction network, which carries both structure and information. The slide shows one: several dozen human proteins (nodes labelled with names such as TIP49a, H2AZ, ARP5 and PFDN2) joined by a dense web of dashed edges for detected interactions, clustering into groups labelled by the protein complexes Prefoldin, SRCAP, hINO80 and TRRAP/TIP60.

In neuroscience there are analogs of the first two levels:

  1. psychology: observation of the output signals that follow interaction with input signals;
  2. neurophysiology: the function of the nervous system, from molecules and cells to systems.

What is missing is the analog of the “biochemistry level” for cells: an understanding of structures and of the mechanisms that govern their formation and functionality. “Structures” is meant here in the mathematical sense, since mathematics is the study of structures. Language is an example of a structure in the human brain, but there are many open questions about how its computational and mathematical structure is realized in the physiology of the brain.

Language and vision as model ergosystems

Gromov singles out language and vision as model ergosystems. He describes language as a model ergosystem as follows:

Linguistic information entering the ergobrain does not much depend on the physical carrier of this information. This suggest a universal class of structures encoding this information; our main objective is describing these structures, which we call syntactic ergo-structures.

Such a structure is a combinatorial object , a kind of a network made of finitely many “atomic units = ergo-ideas”. This network structure generalizes/refines that of a graph, where some patterns are similar to those found in the mathematical theory of -categories.

The combinatorial structure is intertwined with a geometric one; both an individual network and the totality of these, say , carry natural distance-like geometries which are derived from combinatorics and, at the same time, are used for constructing combinatorial relations.

The learning process is modeled by some transformation(s) in the space ; the resulting “educated learner” appears as an attractive fixed point (or a class of points) under this transformation(s).

The (ergo)learning process (transformation) starts from a space of signals and eventually compresses (folds) the information they carry by some coclustering algorithms into our .

For vision: the input of the visual system amounts, roughly, to arrays of pixels changing over time. How does the brain segment groups of pixels into objects, and learn that these objects are invariant under transformations (the rotations and reflections of three-dimensional space)? Gromov again:

“The mathematics of building/identifying the -symmetry of the visual perception field is similar to but more complicated than how Alfred Sturtevant reconstructed in 1913 linearity of the gene arrangements on the basis of distributions of phenotype linkages long before the advent of the molecular biology and discovery of DNA.”

The role of mathematics

There is a long history of very successful interactions between mathematics and physics.

  • “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” (Eugene Wigner, 1960) refers to the physical sciences, and to how mathematics can drive new advances in physics.
  • In more recent years there has also been an “unreasonable effectiveness of physics in the mathematical sciences”, with new progress in mathematics driven by physics.
  • There is also an ongoing “unreasonable effectiveness of mathematics in linguistics”, which suggests a broader role for mathematics in brain structures and ergosystems.

Biology has traditionally not put much emphasis on theoretical ideas as a guiding principle for progress in the field; interactions between mathematics and biology are recent and still largely underdeveloped.

Gromov’s speculation on mathematics and neuroscience is that “basic mental processes can be meaningfully described, if at all, only in a broader mathematical context and this mathematics does not exist at the present day”. Or does it?

Example: symmetry

As an example of the mathematical viewpoint, consider symmetry.

  • Several molecules occur with symmetries: the icosahedral symmetry of viruses, the helix and double helix symmetries of proteins and DNA, and so on.
  • Energy and symmetry. Let be a configuration space of molecules with an action of a group , and let be a -invariant energy functional. Typically, local minima of over the -symmetric configurations are local minima over all configurations.
  • Information and symmetry. A symmetric form is specified by fewer parameters, which makes it preferable from the viewpoint of Shannon information.

Symmetry constrains and simplifies the space of possibilities. Gromov uses viruses to ask what the architecture of the ergobrain might be (the bold marks the words highlighted in red on the slide):

“In the end of the day, the symmetry of viruses depends on the structural constrains imposed by the geometry of the physical space which allows the existence of such improbable objects as icosahedra.

Similarly, we expect that the constrains of the mathematical space (which we have not defined yet) where ergosystems reside will have a strong structural imprint on a possible architecture of the ergobrain. For example, we want to understand what in the (ergo)brain is responsible for the unreasonable power of our visual system which allowed a discovery of icosahedra by humans (at least) as early as −400.”

Example: the Hardy–Weinberg principle

Algebra can constrain the behaviour of a system and reveal universal laws. A biology example is the Hardy–Weinberg principle: allele and genotype frequencies in a population would remain constant through the generations in the absence of other evolutionary influences (mutation, selection, etc.). Here an allele is one of the variant forms of a gene, and an individual’s genotype is the pair of alleles it carries.

Take two alleles and with frequencies (the case of alleles is similar). The expected genotype proportions in the next generation are The Punnett square shows where these come from: an offspring receives one allele from each parent, each allele independently being with probability and with probability .

Punnett square for two alleles: male gametes A and a (frequencies p and q) along the bottom, female gametes A and a along the side; the four cells give genotype AA with frequency p squared, aa with frequency q squared, and Aa twice with frequency pq each.

These proportions still satisfy which is fine since . As functions of they look as follows.

Plot of the Hardy-Weinberg genotype frequencies against p, with q = 1 - p: the curve q squared for aa falls from 1 to 0, the curve p squared for AA rises from 0 to 1, and the curve 2pq for Aa is an arch peaking at 0.5 when p = 1/2.

The new distribution of alleles depends on the distribution of genotypes: at the -th generation, So in the first new generation

Proposition G1.3 (Hardy–Weinberg). The genotype frequencies of generation depend only on the allele frequencies of generation : Consequently, for all ,

Proof. To compute , consider the possible combinations of parental genotypes. The probability of, say, the combination – is , and it always results in ; write this result as . Doing the analogous count for all six possible combinations:

parents probability offspring
–
–
–
–
–
–

and summing gives Now need not be the same as . But since , and the same computation shows that the allele frequencies are preserved at every step, all later generations have ∎

Everything stabilizes by the second generation. So gene recombination alone is not sufficient to explain evolution, in the absence of additional phenomena like gene mutation.

What kind of mathematics?

The general framework involves:

  1. Combinatorial objects: from graphs to -categories (networks and relations);
  2. Transformations and symmetries: groups and their generalizations (groupoids, categories, Hopf algebras, operads, etc.);
  3. Probabilities and information/entropy and complexity: algebraic structures with superimposed probabilistic and thermodynamical (statistical physics) structures;
  4. Topological and metric structures: proximity, similarity, deformation, homotopy, distance, measurement, manifolds, shapes;
  5. Grammars: formal languages, logic, types, compositional structures.

We will meet several of these along the way. For now, just a quick glimpse of some notions.

Categories

The traditional setting of mathematics is based on set theory, and the first example of a category is Sets (or Finite Sets).

Definition G1.4 (Category). A category has objects and, for any two objects, a set of morphisms . Morphisms compose, and the composition is associative: for , Each object has an identity , which is a unit for composition.

Associativity says that the square commutes: both ways around from to give the same morphism.

Definition G1.5 (Functor). A functor acts on objects, , and on morphisms, sending to , with This is a covariant functor. A contravariant functor reverses arrows, , so that .

Definition G1.6 (Natural transformation). Given functors , a natural transformation assigns to every object a morphism such that for every morphism ; that is, the square commutes.

Other examples are the categories of vector spaces, topological spaces, smooth manifolds, groups, rings, etc. This is how category theory captures the concept of a mathematical structure.

Example G1.7 (Directed graphs are functors).

A directed graph on seven vertices, with one vertex sending edges to three others, a pair of opposite edges between two vertices, and a loop at one vertex.

Let be the category with two objects and two non-identity morphisms . A functor consists of sets and (vertices and edges) and maps of sets , the source and target maps of the directed edges. So it is exactly a directed graph, such as the one pictured (loops and pairs of opposite edges are allowed).

A natural transformation is the usual notion of a map of directed graphs: maps preserving the incidence relations

Higher categorical structures

Definition G1.8 (2-category). A 2-category is a category in which the sets are themselves the objects of a category; that is, there are morphisms between morphisms, called 2-morphisms.

In a 2-category the objects are drawn as vertices, the morphisms as 1-cells (arrows between vertices) and the 2-morphisms as 2-cells (double arrows filling the region between two parallel 1-cells). 2-morphisms can be composed in two ways: vertically, along a common 1-cell (written ), and horizontally, along a common vertex (written ). The two compositions satisfy the exchange relation which says that the pasting diagram in the middle of the picture can be evaluated either way.

Pasting diagrams for the exchange law in a 2-category: two horizontal composites of 2-cells alpha, beta and gamma, delta stacked and composed vertically equal the single four-cell diagram, which equals the horizontal composite of the vertical composites of alpha with gamma and of beta with delta.

Remark (Homotopy and higher category theory). In a 2-category, composition of 1-morphisms is associative; in a bicategory it is associative only up to 2-isomorphism.

Example: take as objects the points of an open set in the plane, as morphisms the oriented paths connecting points, and as 2-morphisms the homotopies of paths (continuous deformations of one path into another with the same endpoints). Composition of paths is associative only up to homotopy. One can also compose homotopies, up to homotopy, and so on. This leads to higher -categories, with different levels of “isomorphism”: 0-isomorphism is equality, 1-isomorphism is realized by invertible 1-morphisms, 2-isomorphism by 2-morphisms, etc., and in the limit to -categories.

The contemporary mathematical viewpoint is shifting more and more towards these higher (or ) structures and homotopy.

Groups, semigroups, groupoids and their algebras

Below, a small category is one whose objects and morphisms form sets.

Definition G1.9 (Group and group algebra). A group is a small category with one object in which all morphisms are invertible (product, associativity, unit). Groups describe symmetries, acting by automorphisms.

For a discrete group , the group algebra consists of the finitely supported functions , with convolution product and involution .

Definition G1.10 (Semigroup and semigroup algebra). A semigroup is a small category with one object, whose morphisms need not have inverses. Semigroups act by endomorphisms.

The semigroup algebra consists of functions with convolution It is no longer necessarily involutive.

Definition G1.11 (Groupoid and groupoid algebra). A groupoid is a small category in which all morphisms are invertible. The product of two arrows is defined only when the target of the first arrow is the source of the second. Groupoids describe another type of symmetry.

Write a groupoid as : objects, morphisms, and the source and target maps. The groupoid algebra is the algebra of functions with convolution and involution .

Definition G1.12 (Semigroupoid and semigroupoid algebra). A semigroupoid is a small category (an associative composition of morphisms, not necessarily invertible). The semigroupoid algebra consists of functions of the morphisms with convolution

Topology and invariants

Topology is the study of topological spaces and continuous functions: the study of shapes up to continuous deformations. Invariants are ways of distinguishing between topological spaces, and invariants are functors from the category of topological spaces to another category (vector spaces, groups, rings, etc.).

Definition G1.13 (Homology). Homology is a functor to -graded vector spaces, given by where is the abelian group spanned by the -simplices of a triangulation of , and is the (oriented) boundary map.

Homology is independent of the choice of triangulation. It measures “holes” and “connectivity” in various dimensions. The slide illustrates this with a triangulation of a surface of genus 3: a closed surface with three holes through it, covered by a fine mesh of small triangles.

Metric spaces and Riemannian manifolds

Metric spaces are topological spaces with a quantitative measure of proximity: the topology is induced by a metric (the open sets are generated by the open balls of the metric).

Definition G1.14 (Metric). A distance function (or metric) on is a function with

  • for all ;
  • if and only if ;
  • the triangle inequality for all .

Definition G1.15 (Smooth and Riemannian manifolds). A smooth manifold is covered by local charts homeomorphic to , with changes of coordinates where charts overlap. It has tangent spaces at all points, which make up the tangent bundle .

A Riemannian manifold is a smooth manifold with a metric structure determined by a metric tensor : a symmetric, positive section of .

On a Riemannian manifold the length of a curve is and the geodesic distance is the infimum over curves from to .

Probabilities and entropy

  • Mathematical structures, especially algebraic structures, can be endowed with probabilities.
  • This has been a successful approach in formal languages and generative grammars (systems of rules that generate the sentences of a language): probabilistic grammars, in which the generative rules are applied with certain probabilities.
  • Other algebraic structures can be made probabilistic too: groups, semigroups, groupoids, semigroupoids. These are all like directed graphs: assign probabilities to each successive choice of the next oriented edge in a path, as in Markov processes (random processes whose next step depends only on the current state).
  • One can also attach information measures (entropy) to algebraic structures, with the operations weighted by entropy functionals: information algebras.

Remark (The key idea). In applications to biology, all mathematical structures should be endowed with probabilistic weights and entropy/information weights.

Complexity and patterns

Definition G1.16 (Kolmogorov complexity). The Kolmogorov complexity of a given output is the shortest length of a program required to compute it (theory of computation, Turing machines).

Definition G1.17 (Gell-Mann effective complexity). The effective complexity of an object, in the sense of Gell-Mann, is the description length of the “regularities” (structured patterns) contained in the object.

Ambiguity, Galois symmetry and category theory

Mathematics offers examples of an interplay between properties of categories and geometric symmetries or ambiguities.

  • Symmetries describe ambiguities: up to isomorphism, up to invertible transformations, up to homotopy, etc.
  • The categorical viewpoint on symmetry needs categories with good properties, very much like the category of vector spaces: abelian (kernels and cokernels), tensor, rigid (duality, internal Hom).
  • One then needs a fiber functor to vector spaces, preserving all these properties (tensor, etc.). The symmetries are , the invertible natural transformations of the fiber functor.

Definition G1.18 (Tannakian category). A pair as above is a Tannakian category, with Galois group .

This includes usual Galois theory, motives, regular-singular differential systems, symmetries of quantum field theory, etc.

What interplay between categorical compositional structures and symmetries is there in neuroscience?

Mathematical structures in neuroscience

Structures in the brain

Mathematical tools have expanded and evolved as more levels of structure in the brain are studied, from small scales (single neurons, microcircuitry) to large scales (the connectome), and in modeling neural codes:

  • differential equations, dynamical systems: the functioning of the single neuron, ion channels, synaptic connections;
  • topology: random graphs, connectivity, directed algebraic topology, concurrent and distributed computing, simplicial complexes (activation complex, microcircuitry);
  • homotopy: neural codes, patterns of brain activity and reconstruction of stimulus space;
  • category theory: allocation of resources to networks, interactions between different types of resources;
  • information structures: integrated information, quantitative approaches to consciousness.

Visual system

Modeling the structure of the V1 visual cortex involves a novel interplay of different mathematical tools: signal analysis (Gabor frames), contact geometry (modeling the connectivity of the V1 cortex) and conformal geometry of the retinal map. The main problem of segmentation and tracking in vision also involves an interplay of very different areas of mathematics: differential topology, variational analysis, algebraic geometry.

  • signal analysis: receptive fields of V1 neurons;
  • conformal geometry: retinotopic mapping to the V1 cortex;
  • contact geometry: connectivity of the V1 cortex;
  • differential topology: segmentation and tracking;
  • variational analysis: image segmentation;
  • algebraic geometry: image tracking.

Language system

A good mathematical model of language as a computational system can be obtained by algebraic methods. The problem of identifying neurocomputational mechanisms that realize this computational model is still developing, and involves an interplay of algebra, analysis and topology.

  • algebra: Hopf algebras, magmas, semirings and operads play a role in modeling the syntax of human language;
  • topology: semantic spaces rely on notions of proximity and similarity;
  • representations: realizations of the algebra of syntax in neurocomputational settings.

References

  • Misha Gromov, Structures, Learning and Ergosystems, 2011.
  • Misha Gromov, Ergostructures, Ergologic and the Universal Learning Problem, 2013.
  • Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, 1960.
  • Alfred Sturtevant, 1913: the reconstruction of the linear order of genes from linkage data (mentioned in Gromov’s quotation on vision).
  • Matilde Marcolli and Doris Tsao, Geometry of Neuroscience, book project in progress.