# Mathematical models of large scale brain activity

A mathematical breakthrough developed by a collaboration between mathematics, medicine, and the Sir Peter Mansfield Imaging Centre, has been a model for large scale brain activity.

The use of mathematics has many historical successes, especially in the fields of physics and engineering, where mathematical concepts have been put to good use to address challenges far beyond the context in which they were originally developed. Physicists in particular are well aware of the "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Indeed, modern society either knowingly or unknowingly makes everyday use of deep mathematical ideas when it uses technologies like those of mobile phone networks, secure banking, and genetic data analysis. Mathematics is equally at home in a Healthcare setting, and has a number of applications in Precision Imaging, ranging from the use of statistics to find patterns in complex medical datasets, through to mechanistic modelling of tumour growth.

One recent mathematical breakthrough developed by a collaboration between mathematics, medicine, and the Sir Peter Mansfield Imaging Centre, under the umbrella of the Beacon of Excellence in Precision Imaging, has been a model for large scale brain activity. This mathematical model traces its roots back to models of spiking neurons arranged in a network with interactions that describe the chemical process of neurotransmission. Given the billions of neurons in even a small region of brain tissue a mathematical challenge has been to reduce the model complexity so that it can be efficiently simulated on our High Performance Computing cluster.

Borrowing ideas from statistical physics, whereby a gas filled room is adequately described by a handful of macroscopic variables for pressure, volume, and temperature, we have managed a similar reduction that describes a vast array of neurons in terms of their population firing rate and their degree of network synchrony. Synchrony is a very important descriptor of neural tissue dynamics, as a more coherent firing pattern generates a larger electromagnetic field, and this change in power is the one readily detected in electro- and magneto-encephalography (EEG/MEG) imaging.

A first major success of the model has been in explaining the phenomenon of beta-rebound, with the model matching date from the MEG group led by Dr Matt Brookes. Here a sharp decrease in neural oscillatory power in the 15 Hz beta band is observed during movement followed by an increase above baseline on movement cessation. Subsequent work with Professor Peter Liddle from the Institute of Mental Health has also shown that the model is capable of explaining the abnormal beta-rebound seen in patients with schizophrenia.

By merging this new dynamical model of neural tissue with connectome data describing the anatomical structure of individual brains, we gain a perspective on whole brain dynamics that naturally complements many of the neuroimaging studies ongoing within the Beacon. A current use of the model is for in-silico experiments to help improve the design of transcranial magnetic stimulation protocols for the treatment of mental health conditions.

Future mathematical extensions will tap into powerful ideas from data-assimilation and uncertainty quantification that are commonly used in meteorology, and will ultimately allow for large scale brain state forecasting.