Throughout my research career I have been working in Mathematical Neuroscience with a particular interest in the use of nonlinear dynamics to understand aspects of the human central nervous system. These long-term goals have required me to build up, and help develop, a wide-ranging set of mathematical tools (for describing single cells through to networks). This has led to the development of Evans function techniques for the analysis of wave stability in neural fields as well as pioneered the use of nonsmooth dynamics for understanding spiking network states. In parallel with these theoretical developments, I have set out on the journey to learn the language of the experimentalists and now work in an environment that allows extensive collaboration with experimental groups within the University of Nottingham, including the Sir Peter Mansfield Imaging Centre, the MRC Institute of hearing research, Psychology, Electrophysiology, and the Nottingham Beacon in Precision Imaging. Such interaction has showcased the important contributions that mathematics can make to neuroscience. I am actively championing this new field of mathematics at the national and international level, co-ordinating a UK network on Mathematical Neuroscience and co-creating the new Journal of Mathematical Neuroscience.
I have made substantial contributions to neural field models of cortical and thalamic neural tissue. These models typically take the form of integro-differential equations. Their non-local nature has led to the development of a set of analytical and numerical tools for the study of waves, bumps and patterns, based around extensions of those used for local differential equation models. I am actively working to develop the mathematics relevant to heterogeneity, noise, delays, and plasticity in neural systems, and provide a focal point for community activity by organising conferences in these areas, most recently in France 2015 and 2016 (as co-founder of the International Conference on Mathematical Neuroscience series). My current mathematical work impacts on the analysis of brain rhythms seen in electro- and magneto-encephalography recordings, mode-locked spike trains seen in auditory neurons, models of branched dendritic tissue with active spines, neural network models of fear memory, and models of reinforcement learning in neural field models for spatial navigation.
I am developing new dynamical models of neural tissue with connectome data describing the anatomical structure of individual brains, to gain a perspective on whole brain dynamics. 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.