Category Theory and Higher Dimensional Algebra: potential descriptive tools in neuroscience

We explain the notion of colimit in category theory as a potential tool for describing structures and their communication, and the notion of higher dimensional algebra as potential yoga for dealing with processes and processes of processes. Introduction There does seem to be a problem in neuroscience in finding a language suitable for describing brain activity in a way which could lead to deduction, evaluation of theories, and even perhaps calculation. This is especially so when dealing with questions of global activity, as against activity of individual organs or cells. How can we bridge the gap between neuronal activity and what are variously described in the literature as percepts, concepts, thoughts, emotions, ideas, and so on? What are the relations between the meanings of these various words? When should we use one rather than another? One hope is that a mathematics will arise which could help in these problems. To encourage such a mathematics, we need a dialogue between mathematicians and neuroscientists. Mathematicians can contribute by showing the way mathematics works, describing processes such as abstraction, concept refinement, etc., explaining what is currently available, and analysing the deficiences of current mathematics in helping with these problems. It is likely to be a difficult process to move towards such new mathematics, since life has evolved for a long period, whereas language and mathematics are relatively recent. Yet the ability to do mathematics is itself a result of evolution, and mathematics has a good track record in scientific discovery. This is an extended account of a presentation given at the International Conference on Theoretical Neurobiology, Delhi, Feb 24-26, 2003, by the first author, who would like to thank the National Institute for Brain Research for support. Both authors would like to thank Posina Rayadu for correspondence. Mathematics Division, School of Informatics, University of Wales, Bangor, Gwynedd LL57 1UT, U.K.. email: {r.brown,t.porter}@bangor.ac.uk http://www.bangor.ac.uk/∼{mas010,mas013}