Some Aspects of "Mean Field" Models for the Electrocortical Activity of the Brain.

Abstract:
The human brain is, for various aspects, the most complex system in nature, and it has been the subject of a number of relevant, mathematically-inclined studies since the 50s. Among the existing approaches, mean field models have produced interesting results, allowing the association of fundamental concepts in dynamical systems theory to some features typical of the electrocortical activity of the brain.
In particular, using mathematical models that describe neuronal populations as a whole rather than single neurons organised in networks, it has been possible to quantitatively investigate a number of neurologically relevant problems.
These include the electrical activity of the brain at rest, the response to external sensorial stimuli and learning, traits of the typical dynamics associated to neurological dysfunctions such as epilepsy and encephalopathy, and some characteristic aspects of anaesthesia and sedation.
In this seminar, I will try and convey some of the fundamental aspects of the anatomical and functional complexity of the cerebral cortex, advocating for a mean field model approach in describing brain dynamics. With the aid of a specific model, i.e. Liley's theory of the EEG (electroencephalogram), I will discuss some interesting mathematical results that capture brain's complexity and try to quantitatively explain fundamental concepts in electroencephalography, such as the ubiquituous alpha rhythm and the emergence of 40 Hz activity. Finally, I will illustrate some recent results that bear some mathematical and physiological interest, such as: the role of multistability in memory formation, the dynamical character of the thalamus as an active agent in the processes involved in sensations, the value of the slow-fast manifolds for wave generation and mixed-mode oscillations, intermittency, high order chaos and inverse period-doubling cascades in the resting brain.