Kyle C. A. Wedgwood

Phase-Amplitude Descriptions of Neural Oscillator Models

Phase oscillators are a common starting point for the reduced description of many single neuron models that exhibit a strongly attracting limit cycle. The framework for analysing such models in response to weak perturbations is now particularly well advanced, and has allowed for the development of a theory of weakly connected neural networks. However, the strong-attraction assumption may well not be the natural one for many neural oscillator models. For example, the popular conductance based Morris–Lecar model is known to respond to periodic pulsatile stimulation in a chaotic fashion that cannot be adequately described with a phase reduction. In this paper, we generalise the phase description that allows one to track the evolution of distance from the cycle as well as phase on cycle. We use a classical technique from the theory of ordinary differential equations that makes use of a moving coordinate system to analyse periodic orbits. The subsequent phase-amplitude description is shown to be very well suited to understanding the response of the oscillator to external stimuli (which are not necessarily weak). We consider a number of examples of neural oscillator models, ranging from planar through to high dimensional models, to illustrate the effectiveness of this approach in providing an improvement over the standard phase-reduction technique. As an explicit application of this phase-amplitude framework, we consider in some detail the response of a generic planar model where the strong-attraction assumption does not hold, and examine the response of the system to periodic pulsatile forcing. In addition, we explore how the presence of dynamical shear can lead to a chaotic response.

Spatiotemporal dynamics of insulitis in human Type 1 diabetes

Type 1 diabetes (T1D) is an auto-immune disease characterized by the selective destruction of the insulin secreting beta cells in the pancreas during an inflammatory phase known as insulitis. Patients with T1D are typically dependent on the administration of externally provided insulin in order to manage blood glucose levels. Whilst technological developments have significantly improved both the life expectancy and quality of life of these patients, an understanding of the mechanisms of the disease remains elusive. Animal models, such as the NOD mouse model, have been widely used to probe the process of insulitis, but there exist very few data from humans studied at disease onset. In this manuscript, we employ data from human pancreases collected close to the onset of T1D and propose a spatio-temporal computational model for the progression of insulitis in human T1D, with particular focus on the mechanisms underlying the development of insulitis in pancreatic islets. This framework allows us to investigate how the time-course of insulitis progression is affected by altering key parameters, such as the number of the CD20+ B cells present in the inflammatory infiltrate, which has recently been proposed to influence the aggressiveness of the disease. Through the analysis of repeated simulations of our stochastic model, which track the number of beta cells within an islet, we find that increased numbers of B cells in the peri-islet space lead to faster destruction of the beta cells. We also find that the balance between the degradation and repair of the basement membrane surrounding the islet is a critical component in governing the overall destruction rate of the beta cells and their remaining number. Our model provides a framework for continued and improved spatio-temporal modeling of human T1D.

Control of clustered action potential firing in a mathematical model of entorhinal cortex stellate cells.

Highlights • An SDE model of entorhinal cortex (EC) stellate cells is proposed.• Experimentally observed action potential clustering is investigated in the model.• Clusters are generated by subcritical-Hopf/homoclinic type bursting.• Potential mechanisms underlying changes in EC dynamics in dementia are presented.

Modelling of the cardiopulmonary responses to maximal aerobic exercise in patients with cystic fibrosis

Cystic fibrosis (CF) is a debilitating chronic condition, which requires complex and expensive disease management. Exercise has now been recognised as a critical factor in improving health and quality of life in patients with CF. Hence, cardiopulmonary exercise testing (CPET) is used to determine aerobic fitness of young patients as part of the clinical management of CF. However, at present there is a lack of conclusive evidence for one limiting system of aerobic fitness for CF patients at an individual patient level. Here, we perform detailed data analysis that allows us to identify important systems-level factors that affect aerobic fitness. We use patients’ data and principal component analysis to confirm the dependence of CPET performance on variables associated with ventilation and metabolic rates of oxygen consumption. We find that the time at which participants cross the anaerobic threshold (AT) is well correlated with their overall performance. Furthermore, we propose a predictive modelling framework that captures the relationship between ventilatory dynamics, lung capacity and function and performance in CPET within a group of children and adolescents with CF. Specifically, we show that using Gaussian processes (GP) we can predict AT at the individual patient level with reasonable accuracy given the small sample size of the available group of patients. We conclude by presenting future perspectives for improving and extending the proposed framework. Our modelling and analysis have the potential to pave the way to designing personalised exercise programmes that are tailored to specific individual needs relative to patient’s treatment therapies.

Networks of phase-amplitude neural oscillators.

In mathematical descriptions of oscillating neural cells, phase reduction techniques can be used to simplify the model to a one-dimensional system [1]. This reduction allows for deeper mathematical analysis of the system, and for simulation of larger networks, since the resulting model is computationally cheaper. However, if a limit cycle is not strongly attracting then this reduction may poorly characterise behaviour of the original system when under forcing, for example, synaptic input. Here we consider a coordinate transformation to a phaseamplitude framework [2] that allows one to track the evolution of distance from the cycle as well as phase on cycle. A number of common models in computational neuroscience (including FitzHugh-Nagumo and MorrisLecar) are revisited in this framework and their response to pulsatile current forcing is investigated. We highlight the differences between phase and phase-amplitude descriptions, and show that the former can miss some substantial features of neuronal response. Finally, we discuss extensions of this work that will allow for the description of networks of limit-cycle oscillators and improve upon the standard weakly coupled phase oscillator approach. In particular, we highlight the merits of piece-wise linear modelling for the development of a theory of strongly interacting systems.