More on category theory and the brain

I’ve referred to category theory on more than one occasion (particularly with respect to physicist Bob Coecke’s graphical language).  Not too long ago, Ronald Brown, at Bangor University, brought my attention to the work that he and colleagues have been doing to investigate the kind of mathematics that could be used to model the complexity of neural activity (and the products of this activity).  Their success would suggest to me again that there is a correspondence between mathematics and how we perceive or come to know anything.  I looked at some of his papers and have chosen some excerpts from one of them because they serve, I believe, to illustrate some of the ideas and their motivation.  Brown co-authored the paper with Timothy Porter (also at Bangor).  It has the title:  Category Theory and Higher Dimensional Algebra: potential descriptive tools in neuroscience Here is an early observation:

One may presume that the power of abstraction, in some sense of making maps, must be deeply encoded in evolutionary history as a technique for encouraging survival, since a map gives a small and manipulable model of the environment.

The idea of ‘mapping,’ or making a correspondence, is one of the most fundamental ideas in mathematics.  Mathematics could not proceed without it.   But it has also been used to describe how the brain functions.  For example, the brain performs what have been called computations to construct a visual image from visual input. The computations involve actions referred to as ‘mappings’ (in the mathematical sense) of aspects of the world onto elements of the brain.  It makes sense that one would expect this kind of neural activity to be deeply encoded in our evolutionary history.  And, with this in mind,  one could argue that mathematics may be built on underlying, and mostly unconscious, biological action.

Approaching it from the other direction.

A study of why and how mathematics works could be useful for making models for neurological functions involving maps of the environment. Mathematics may also possibly provide a comprehensible case study of the evolution of complex interacting structures, and so may yield analogies helpful for developing and evaluating models of brain activity, in order to derive better models, and so better understanding. We expect to need a new language, a new mathematics, for describing brain activity. To see what is involved in this search, it is reasonable to study the evolution of mathematics, and of particular branches such as category theory.

This look at the evolution of abstractions is an interesting way to proceed.  The authors provide a nice analogy for understanding the value of the idea of a colimit in category theory.

To focus on a common example, consider the process of sending an email document, call it E. To send this we need a server S, which breaks down the document E into many parts Ei for i in some indexing set I, and labels each part Ei so that it becomes E′ i. The labelled parts E′i are then sent to various servers Si which then send these as messages E′′ i to a server SC for the receiver C. The server SC combines the E′′i to produce the received message ME at C. Notice also that there is an arbitrariness in breaking the message down, and in how to route through the servers Si, but the system is designed so that the received message ME is independent of all the choices that have been made at each stage of the process. A description of the email system as a colimit may be difficult to realise precisely, but this analogy does suggest the emphasis on the amalgamation of many individual parts to give a working whole, which yields exact final output from initial input, despite choices at intermediate stages.

One question for neuroscientists is: does the brain use analogous processes for communication between its various structures? What we can say is that this general colimit notion represents a general mathematical process which is of fundamental importance in describing and calculating with many algebraic and other structures.

…Thus a conjecture as far as biological processes are concerned, is that this notion of colimit may give useful analogies to the way complex systems operate. More generally, it seems possible that this particular concept in category theory, seeing how a big object is built up of smaller related pieces, may be useful for the mathematics of processes.

The paper goes on to describe the value of higher dimensional algebra in putting the pieces together:

Information is often ‘subdivided’ by the sensory organs and is reintegrated by the brain. To enable different parts of that information to be integrated, there must be some ‘glue’, some inter-relational information available. If we are given arrows a, b, c, d with no information on where they start or end, then we could form combinations
which make no geometric sense. The colimit/composition process makes sense only where the inter-relations are also such as to enable the ‘integration’ to be well defined. Higher dimensional algebra allows more complex notions of ‘well formed composition’, and ones more adapted to geometry.

The authors do a nice job of describing the strength higher dimensional algebra and I recommend taking a look.