School of Mathematical Sciences

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Stephen Coombes

Professor of Applied Mathematics, Faculty of Science


Research Summary

My research interests lie in the area of mathematical biology and in particular the application of principles from nonlinear dynamics and statistical physics to the study of neural systems. Neural… read more

Selected Publications

Current Research

My research interests lie in the area of mathematical biology and in particular the application of principles from nonlinear dynamics and statistical physics to the study of neural systems. Neural field theories: Experimental studies have revealed the propagation of travelling bursts of activity in cortical and thalamic brain slices. Experiments in primates have also revealed that prefrontal cortical networks can support spatially localized areas of high activity, thought to be important for the functioning of working memory. Such waves and bumps of firing rate activity are a consequence of non-local synaptic interactions and the intrinsic behavior of local neuronal circuitry. The mathematical description of synaptically coupled neural tissue typically involves the use of integral equations. Apart from a spatial integral mixing the network connectivity function with space-dependent delays, arising from non-instantaneous axonal communication, these integral models can also include a temporal integration over some appropriately identified distributed delay kernel. These distributed delay kernels are biologically motivated and represent the response of biological synapses to spiking inputs. I am interested in using techniques from dynamical systems theory to study waves and patterns in such integral neural field models. Moreover, network studies that include a description of the slow T-type calcium current are known to support non-smooth waves of a type that can lurch through a medium. This work is highly relevant to modelling thalamic tissue and ralates to the emergence of EEG sleep rhythms in thalamo-cortical networks. Dendritic trees with active spines: Dendritic spines form the dominant component of many types of dendritic trees. They are small mushroom like appendages and may be found in their hundreds of thousands on the dendritic tree of a single cortical pyramidal cell. These extensions of the dendritic tree provide junction points for the axons of other neurons, and thus serve as loci for receiving inputs. Direct observations confirm earlier speculations that spine heads possess excitable channels capable of generating action potentials (spikes). Together with Dr Gabriel Lord (Maths, Heriot-Watt) I am developing a minimal model of a branched dendrite with active spines that may be used to i) explore issues of wave propagation and nonlinear signal processing in dendrites, ii) investigate biophysical learning rules for spike-time-dependent plasticity at the level of spines. Neurobiology of sensory gating (with Rob Mason, Electrophysiology Laboratory): Gating of sensory (e.g. auditory) information has been demonstrated as a reduction in the auditory-evoked potential responses recorded in the brain of both normal animals and human subjects. Auditory gating is perturbed in schizophrenic patients and pharmacologically by drugs such as amphetamine, PCP or ketamine, which precipitate schizophrenic-like symptoms in normal subjects. The neurobiological basis underlying this sensory gating is being investigated using i) multiple microelectrode array recordings, and ii) by developing mathematical (spiking network) models of the known underlying hippocampal circuitry. Dynamics of Calcium release and wave propagation in single cells: The existence of spatial and temporal signalling by calcium is one of the most significant findings of the last decade in the field of intracellular signalling. Calcium is stored intracellularly in the endoplasmic or sarcoplasmic reticulum at 2-3 orders of magnitude greater than its concentration in the cytosol and is released by a nonlinear feedback process referred to as calcium-induced calcium release (CICR). This mechanism for generating oscillations (or puffs) in the density of cytosolic free calcium is believed to underly the waves that propagate as intra and intercellular waves over distance as large as 1mm. I am interested in both the numerical solution of biophysical models (such as the de-Young/Keizer model) using techniques from dynamical systems theory and the exact solution of fire-diffues-fire (FDF) type models. The analysis of wave initiation and propagation in models with spatial inhomogeneities and nonlinear transport mechanisms is relevant to the understanding of recent experimental work by Martin Bootman and colleagues.

Future Research

EEG & functional MRI studies of brain microstates associated with anaesthesia-induced loss of consciousness (with Dorothee Auer, Academic Radiology) Anaesthesia-induced loss of consciousness (LOC) is a particularly revealing model for the study of electrical phenomena that differ between the conscious and unconscious brain. The objective of our work is to characterise spatio-temporal patterns of coherent and incoherent brain activation in altered levels of consciousness in humans using a combined fMRI and EEG approach. In parallel with this experimental work, defining specific regional brain activity and synchronisation levels between the brainstem, hypothalamus, thalamus and the neocortex as well as cortico-cortical interactions, we are developing coarse-grained neural field models of thalamo-cortical loops that will help us unravel the different modes of neural dynamics underlying consciousness, sedation and LOC.

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