PUBLIC LECTURE: How to think about the structure and dynamics of the brain using mathematics

This talk will be a brief introduction to two topics on which it has proved possible to learn about how the brain may work using methods of mathematical physics. It has some relevance to the way in which theorists approach the problems of the shape, structure, and distribution of galaxies, and of stellar and galactic dynamics. The first topic will be how cooperative activity in millions of interacting nerve cells in the brain leads to the formation of stable patterns of activity. The mechanism underlying such pattern formation will be used to explain the seeing of hallucinations, near-death experiences and migraines, and the generation of epileptic seizures. The second topic will be the effects of fluctuations and correlations of brain activity, on the large-scale dynamics. It will be shown how non-equilibrium phase transitions occur in brain dynamics, and how this gives rise to scaling laws that allow the interpretation of various experimental measures of large-scale brain activity such as local field potentials, EEG and fMRI signals, and the spiking activity of single nerve cells. This leads to a comprehensive overview of how brain states are generated, and a theory for the origins of both random and rhythmic brain activity.