G2 Geometry of Neuroscience: the 2017 Opening Lecture
This was the first lecture of the Spring 2017 course Geometry of Neuroscience, taught at Caltech by Matilde Marcolli and Doris Tsao. The course page was http://www.its.caltech.edu/~matilde/GeomNeuroClass.html.
Overview
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
Gromov’s ergosystems
The lecture is built around Mikhail Gromov’s papers Structures, Learning, and Ergosystems (2011) and Ergostructures, Ergologic and the Universal Learning Problem (2013); the title slide shows a photograph of Gromov. The quotations below are from these papers unless another author is named.
The tail of the lion
“Nature shows us only the tail of the lion. But there is no doubt in my mind that the lion belongs with it even if he cannot reveal himself to the eye all at once because of his huge dimension.” (Albert Einstein)
The first version of this slide shows only a lion’s tail curling in from the edge of a photograph of a sandy riverbank. The next one reveals the whole lion walking along the bank.
Consciousness
The slide titled “Consciousness” shows the brains of several mammals side by side, all at the same scale (with a 5 cm scale bar): human, elephant, dolphin, gorilla, dog, cat, macaque and mouse. They range from the elephant’s, the largest, down to the mouse’s, which is barely visible.
Precursors
The slide “Precursors” pairs the title page of Sigmund Freud’s Die Traumdeutung (The Interpretation of Dreams; the eighth edition, Leipzig and Vienna, 1930) with the cover of Julian Jaynes’s The Origin of Consciousness in the Breakdown of the Bicameral Mind.
- Freud: dreams, slips of the tongue, jokes, etc. led to a theory of subconscious mental processes, and to psychoanalysis as a means to treat mental disorders caused by these processes.
- Jaynes: the Iliad, hypnosis, schizophrenia, the anatomy of language pathways, the inordinate power of music led to a theory of how modern consciousness arose from an early pre-conscious state in which the left brain obeyed orders given by the right brain.
The organization of Gromov’s text
Gromov’s text has five chapters:
- free-form speculation about why the ergosystem should exist and what its nature should be;
- a mathematical promenade;
- unwritten;
- language as ergosystem;
- a postmodern AI-inspired short story.
The lecture first takes up Chapter 1.
Chapter 1: structures, learning, and ergosystems
The existence of Ramanujan, the activity of mathematics, children learning natural languages, the play of dholes (Asian wild dogs), the evolution of peacock feathers, the icosahedral shape of viruses, … all lead Gromov to a hypothesis.
Remark (Gromov’s hypothesis). The interface between the brain and the mind must consist of a yet unknown mathematical entity, the ergosystem, capable of goal-free structure learning.
The hypothesis comes with many implications.
A deep critique of the short-sightedness of our society’s focus on the ego:
“It takes 5-10 mental regurgitations over a several months time interval to assimilate an entirely new non-trivial idea, be it a mathematician’s or non-mathematician’s ergo brain, while keeping your smart ego-mind at bay.”
A deep critique of our educational system:
“Several months old babies start playing with: ‘PA PA BA BA’. Their auditory system records these ‘PA PA BA BA’ with an amazing correlation between the sounds and the somatosensory and tactual perceptions. Deaf children start ‘PA PA BA BA’ at the same age but stop doing it sooner: there is less to be amazed with. Eventually, babies get bored with this and start producing more interesting/meaningful sequences of sounds, unless taught that BLAH-BLAH-BLAH is an acceptable adult speech.”
A glimpse into a future in which humans co-exist with intelligent robots (an attempt to understand the last chapter…).
What an ergosystem is
The brain transforms signals: incoming signals are turned into outgoing signals. The structure mediating this transformation is the “ergosystem”.
Definition G2.1 (Ergosystem). An ergosystem is a machine for extracting structure from incoming signals, through a process of goal-free learning. This is in contrast with reinforcement learning, where learning is driven by rewards for reaching a goal. The ergosystem is a “dynamical reduction (quotient)” of the neuro-brain, the brain as described by its physiology.
It contrasts with the ego-mind: the pragmatic mental processes serving survival needs.
Why “ergosystem”?
The name plays on Descartes’s cogito ergo sum, “I think, therefore I am”:
- I see therefore I am
- I envy therefore I am
- I love therefore I am
- I dream therefore I am
“Therefore” (ergo) stands for the hope of understanding consciousness through reasoning and mathematics.
What is the nature of the ergosystem?
The slide asking this question brings back the lion’s tail: so far only a small part of the animal is in view.
Analogy to the cell
- Interaction with the environment, the “ego-activity” of the cell: studied by the cytologist for cells, by the psychologist for the brain.
- Selection of beneficial DNA mutations, which corresponds to the neurophysiology of learning: studied by the molecular geneticist for cells, by the neurophysiologist for the brain.
- What is missing for the brain is the analog of the biochemistry of proteins.
The next slide shows a protein interaction network: 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.
What is the architecture of the ergobrain? An analogy to viruses
“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.”
The power of the ergobrain
The existence of Ramanujan
The slide shows a photograph of Srinivasa Ramanujan, seated, next to his series for
“What makes a star (of a given mass) a supernova is (essentially) the value of a single parameter, the age of the star. What made Ramanujan, I believe, is a minor (on the brain scale) increment of something like the ‘conductivity parameter’ of the communication line from his (ergo)brain to his conscious mind or a decrement in translucency of the ‘brain/mind window’.”
Listening to Bach
The slide links a video of Bach’s crab canon (a canon in which one voice plays the melody forwards while the other plays it backwards) drawn on a Möbius strip: https://vimeo.com/69715960.
Imaginative non-rigorous mathematics
“The motto of ergosystems is that of a practicing theoretical physicist: imaginative non-rigorous mathematics”
Example G2.2 (The bird and the two trains). A bird flies back and forth between two trains, one travelling at 40 mph, the other at 60 mph, towards each other. The trains are separated by 100 km. The bird travels at 100 km/hour. What is the distance covered by the bird before the trains meet?
Show solution
Summing the infinite series of the bird’s back-and-forth legs is the hard way. The trains close the gap at
The slide ends with the lesson the lecture draws next: seeking symmetries.
Is there a mathematical rule for “symmetry searching”?
- Galois theory: “a mathematical framework for the study of symmetry, arbitrariness and ambiguity”, based on the “Galois group”, the permutations of the roots of a polynomial which preserve rational relations (polynomial relations with rational coefficients) among them.
- Category theory: a very abstract way to formulate all of mathematics.
“The ideas of the category theory show that there are certain (often non-obvious) rules for generating proper concepts.”
Language as a model ergosystem
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 .
Vision as a model ergosystem
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
“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.”
Back to the outline
The lecture closes by returning to the five chapters of Gromov’s text, now with Chapter 2 (mathematical promenade) and Chapter 4 (language as ergosystem) highlighted.
References
- Mikhail Gromov, Structures, Learning and Ergosystems, 2011.
- Mikhail Gromov, Ergostructures, Ergologic and the Universal Learning Problem, 2013.
- Sigmund Freud, Die Traumdeutung, eighth edition, Franz Deuticke, Leipzig and Vienna, 1930.
- Julian Jaynes, The Origin of Consciousness in the Breakdown of the Bicameral Mind.
- Johann Sebastian Bach Crab Canon on a Möbius Strip, video,
https://vimeo.com/69715960. - Alfred Sturtevant, 1913: the reconstruction of the linear order of genes from linkage data (mentioned in Gromov’s quotation on vision).
- Course page:
http://www.its.caltech.edu/~matilde/GeomNeuroClass.html.