Publications by year
Wedgwood K, Avitabile D, Davis J (In Press). Bump attractors and waves in networks of leaky integrate-and-fire neurons. SIAM Review
Zavala E, Gil-Gómez CA, Wedgwood KCA, Burgess R, Tsaneva-Atanasova K, Herrera-Valdez MA
(In Press). Dynamic modulation of glucose utilisation by glucocorticoid rhythms in health and disease.
Dynamic modulation of glucose utilisation by glucocorticoid rhythms in health and disease
AbstractA systems level coordination of physiological rhythms is essential to sustain healthy states, especially in the face of stimuli that may disrupt such rhythms. The timing of meals, medication and chronic stress can profoundly influence metabolism, which depends on the dynamic interactions between glucose, insulin and cortisol. Although the metabolic and stress endocrine axes are simultaneously disrupted in many diseases, a theoretical framework to understand how chronodisruption leads to disease is lacking. By developing a mathematical model of glucose utilisation that accounts for the antagonism between insulin and cortisol, we investigate the dynamic effects of glucose boluses under normal and disrupted cortisol rhythms, including the effects of cortisol agonists and antagonists. We also predict how cortisol rhythms modulate circadian responses to oral glucose diagnostic tests, and analyse the disruptions caused by hypercortisolism. Finally, we predict the mechanisms leading to type 2 diabetes in patients with normal and excess cortisol. Abstract
Wedgwood K, Creaser J, Diekman C (In Press). Entrainment dynamics organised by global manifolds in a circadian pacemaker model. Frontiers in Applied Mathematics and Statistics
Wedgwood K, Ashwin P
(In Press). Morphogen-directed cell fate boundaries: slow passage through bifurcation and the role of folded saddles. Journal of Theoretical Biology
Morphogen-directed cell fate boundaries: slow passage through bifurcation and the role of folded saddles
One of the fundamental mechanisms in embryogenesis is the process by which cells differentiate and create tissues and structures important for functioning as a multicellular organism. Morphogenesis involves diffusive process of chemical signalling involving morphogens that pre-pattern the tissue. These morphogens influence cell fate through a highly nonlinear process of transcriptional signalling. In this paper, we consider this multiscale process in an idealised model for a growing domain. We focus on intracellular processes that lead to robust differentiation into two cell lineages through interaction of a single morphogen species with a cell fate variable that undergoes a bifurcation from monostability to bistability. In particular, we investigate conditions that result in successful and robust pattern formation into two well-separated domains, as well as conditions where this fails and produces a pinned boundary wave where only one part of the domain grows. We show that successful and unsuccessful patterning scenarios can be characterised in terms of presence or absence of a folded saddle singularity for a system with two slow variables and one fast variable; this models the interaction of slow morphogen diffusion, slow parameter drift through bifurcation and fast transcription dynamics. We illustrate how this approach can successfully model acquisition of three cell fates to produce three-domain “French flag” patterning, as well as for a more realistic model of the cell fate dynamics in terms of two mutually inhibiting transcription factors. Abstract
Wedgwood KCA, Satin L (In Press). Six Degrees of Depolarisation: a Comment on Network Science of Biological Systems at Different Scales: a Review by Marko Gosak et al. Physics of Life Reviews
Wengert E, Miralles R, Wedgwood K, Wagley P, Strohm S, Panchal P, Idrissi AM, Wenker I, Thompson J, Gaykema R, et al (In Press). Somatostatin-positive Interneurons Contribute to Seizures in SCN8A Epileptic Encephalopathy. The Journal of Neuroscience
Galvis D, Hodson DJ, Wedgwood KCA
(2023). Spatial distribution of heterogeneity as a modulator of collective dynamics in pancreatic beta-cell networks and beyond. Frontiers in Network Physiology
Spatial distribution of heterogeneity as a modulator of collective dynamics in pancreatic beta-cell networks and beyond
We study the impact of spatial distribution of heterogeneity on collective dynamics in gap-junction coupled beta-cell networks comprised on cells from two populations that differ in their intrinsic excitability. Initially, these populations are uniformly and randomly distributed throughout the networks. We develop and apply an iterative algorithm for perturbing the arrangement of the network such that cells from the same population are increasingly likely to be adjacent to one another. We find that the global input strength, or network drive, necessary to transition the network from a state of quiescence to a state of synchronised and oscillatory activity decreases as network sortedness increases. Moreover, for weak coupling, we find that regimes of partial synchronisation and wave propagation arise, which depend both on network drive and network sortedness. We then demonstrate the utility of this algorithm for studying the distribution of heterogeneity in general networks, for which we use Watts–Strogatz networks as a case study. This work highlights the importance of heterogeneity in node dynamics in establishing collective rhythms in complex, excitable networks and has implications for a wide range of real-world systems that exhibit such heterogeneity. Abstract
Galvis D, Hodson DJ, Wedgwood KCA
(2022). The influence of spatial configuration in collective transitions: the. importance of being sorted.
The influence of spatial configuration in collective transitions: the. importance of being sorted
We studied the effects of spatial configuration on collective dynamics in a Abstract
nearest-neighbour and diffusively coupled lattice of heterogeneous nodes. The
networks contained nodes from two populations, which differed in their
intrinsic excitability. Initially, these populations were uniformly and
randomly distributed throughout the lattice. We then developed an iterative
algorithm for perturbing the arrangement of the network such that nodes from
the same population were increasingly likely to be adjacent to one another. We
found that the global input strength, or network drive, necessary to transition
the network from a state of quiescence to a state of synchronised and
oscillatory activity was decreased as network sortedness was increased.
Moreover, for weak coupling, we found that regimes of partial synchronisation
exist (i.e. 2:1 resonance in the activity of the two populations), which were
dependent both on network drive (sometimes in a non-monotonic fashion) and
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Chaffey JR, Young J, Leslie KA, Partridge K, Akhbari P, Dhayal S, Hill JL, Wedgwood KCA, Burnett E, Russell MA, et al
(2021). Investigation of the utility of the 1.1B4 cell as a model human beta cell line for study of persistent enteroviral infection. Sci Rep
Investigation of the utility of the 1.1B4 cell as a model human beta cell line for study of persistent enteroviral infection.
The generation of a human pancreatic beta cell line which reproduces the responses seen in primary beta cells, but is amenable to propagation in culture, has long been an important goal in diabetes research. This is particularly true for studies focussing on the role of enteroviral infection as a potential cause of beta-cell autoimmunity in type 1 diabetes. In the present work we made use of a clonal beta cell line (1.1B4) available from the European Collection of Authenticated Cell Cultures, which had been generated by the fusion of primary human beta-cells with a pancreatic ductal carcinoma cell, PANC-1. Our goal was to study the factors allowing the development and persistence of a chronic enteroviral infection in human beta-cells. Since PANC-1 cells have been reported to support persistent enteroviral infection, the hybrid 1.1B4 cells appeared to offer an ideal vehicle for our studies. In support of this, infection of the cells with a Coxsackie virus isolated originally from the pancreas of a child with type 1 diabetes, CVB4.E2, at a low multiplicity of infection, resulted in the development of a state of persistent infection. Investigation of the molecular mechanisms suggested that this response was facilitated by a number of unexpected outcomes including an apparent failure of the cells to up-regulate certain anti-viral response gene products in response to interferons. However, more detailed exploration revealed that this lack of response was restricted to molecular targets that were either activated by, or detected with, human-selective reagents. By contrast, and to our surprise, the cells were much more responsive to rodent-selective reagents. Using multiple approaches, we then established that populations of 1.1B4 cells are not homogeneous but that they contain a mixture of rodent and human cells. This was true both of our own cell stocks and those held by the European Collection of Authenticated Cell Cultures. In view of this unexpected finding, we developed a strategy to harvest, isolate and expand single cell clones from the heterogeneous population, which allowed us to establish colonies of 1.1B4 cells that were uniquely human (h1.1.B4). However, extensive analysis of the gene expression profiles, immunoreactive insulin content, regulated secretory pathways and the electrophysiological properties of these cells demonstrated that they did not retain the principal characteristics expected of human beta cells. Our data suggest that stocks of 1.1B4 cells should be evaluated carefully prior to their use as a model human beta-cell since they may not retain the phenotype expected of human beta-cells. Abstract
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Wedgwood K, Slowinski P, Manson J, Tsaneva-Atanasova K, Krauskopf B (2021). Robust spike timing in an excitable cell with delayed feedback. Journal of the Royal Society Interface, 18
Brunt L, Greicius G, Rogers S, Evans BD, Virshup DM, Wedgwood KCA, Scholpp S
(2021). Vangl2 promotes the formation of long cytonemes to enable distant Wnt/β-catenin signaling. Nat Commun
Vangl2 promotes the formation of long cytonemes to enable distant Wnt/β-catenin signaling.
Wnt signaling regulates cell proliferation and cell differentiation as well as migration and polarity during development. However, it is still unclear how the Wnt ligand distribution is precisely controlled to fulfil these functions. Here, we show that the planar cell polarity protein Vangl2 regulates the distribution of Wnt by cytonemes. In zebrafish epiblast cells, mouse intestinal telocytes and human gastric cancer cells, Vangl2 activation generates extremely long cytonemes, which branch and deliver Wnt protein to multiple cells. The Vangl2-activated cytonemes increase Wnt/β-catenin signaling in the surrounding cells. Concordantly, Vangl2 inhibition causes fewer and shorter cytonemes to be formed and reduces paracrine Wnt/β-catenin signaling. A mathematical model simulating these Vangl2 functions on cytonemes in zebrafish gastrulation predicts a shift of the signaling gradient, altered tissue patterning, and a loss of tissue domain sharpness. We confirmed these predictions during anteroposterior patterning in the zebrafish neural plate. In summary, we demonstrate that Vangl2 is fundamental to paracrine Wnt/β-catenin signaling by controlling cytoneme behaviour. Abstract
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Avitabile D, Davis JL, Wedgwood KCA
(2020). Bump attractors and waves in networks of leaky integrate-and-fire. neurons.
Bump attractors and waves in networks of leaky integrate-and-fire. neurons
Bump attractors are wandering localised patterns observed in in vivo Abstract
experiments of spatially-extended neurobiological networks. They are important
for the brain's navigational system and specific memory tasks. A bump attractor
is characterised by a core in which neurons fire frequently, while those away
from the core do not fire. We uncover a relationship between bump attractors
and travelling waves in a classical network of excitable, leaky
integrate-and-fire neurons. This relationship bears strong similarities to the
one between complex spatiotemporal patterns and waves at the onset of pipe
turbulence. Waves in the spiking network are determined by a firing set, that
is, the collection of times at which neurons reach a threshold and fire as the
wave propagates. We define and study analytical properties of the voltage
mapping, an operator transforming a solution's firing set into its
spatiotemporal profile. This operator allows us to construct localised
travelling waves with an arbitrary number of spikes at the core, and to study
their linear stability. A homogeneous "laminar" state exists in the network,
and it is linearly stable for all values of the principal control parameter.
Sufficiently wide disturbances to the homogeneous state elicit the bump
attractor. We show that one can construct waves with a seemingly arbitrary
number of spikes at the core; the higher the number of spikes, the slower the
wave, and the more its profile resembles a stationary bump. As in the
fluid-dynamical analogy, such waves coexist with the homogeneous state, are
unstable, and the solution branches to which they belong are disconnected from
the laminar state; we provide evidence that the dynamics of the bump attractor
displays echoes of the unstable waves, which form its building blocks.
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Rosenbauer J, Zhang C, Mattes B, Reinartz I, Wedgwood K, Schindler S, Sinner C, Scholpp S, Schug A (2020). Modeling of Wnt-mediated tissue patterning in vertebrate embryogenesis. PLOS Computational Biology, 16(6), e1007417-e1007417.
Williams CAP, Wedgwood K, Mohammadi H, Prouse K, Tomlinson O, Tsaneva K
(2019). Cardiopulmonary Responses to Maximal Aerobic Exercise in Patients with Cystic Fibrosis (data set).
Cardiopulmonary Responses to Maximal Aerobic Exercise in Patients with Cystic Fibrosis (data set)
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 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 gas exchange threshold (GET) 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 GET at the individual patient level with reasonable accuracy given the small sample size of the available group of patients. We conclude by presenting an example and future perspectives for improving and extending the proposed framework. Abstract
The 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.
Williams CA, Wedgwood KCA, Mohammadi H, Prouse K, Tomlinson OW, Tsaneva-Atanasova K
(2019). Cardiopulmonary responses to maximal aerobic exercise in patients with cystic fibrosis. PLoS ONE
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 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 gas exchange threshold (GET) 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 GET at the individual patient level with reasonable accuracy given the small sample size of the available group of patients. We conclude by presenting an example and future perspectives for improving and extending the proposed framework. The 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. Abstract
Zavala E, Wedgwood KCA, Voliotis M, Tabak J, Spiga F, Lightman SL, Tsaneva-Atanasova K
(2019). Mathematical Modelling of Endocrine Systems.
Mathematical Modelling of Endocrine Systems
Hormone rhythms are ubiquitous and essential to sustain normal physiological functions. Combined mathematical modelling and experimental approaches have shown that these rhythms result from regulatory processes occurring at multiple levels of organisation and require continuous dynamic equilibration, particularly in response to stimuli. We review how such an interdisciplinary approach has been successfully applied to unravel complex regulatory mechanisms in the metabolic, stress, and reproductive axes. We discuss how this strategy is likely to be instrumental for making progress in emerging areas such as chronobiology and network physiology. Ultimately, we envisage that the insight provided by mathematical models could lead to novel experimental tools able to continuously adapt parameters to gradual physiological changes and the design of clinical interventions to restore normal endocrine function. Abstract
Le Page G, Gunnarsson L, Trznadel M, Wedgwood KCA, Baudrot V, Snape J, Tyler CR
(2019). Variability in cyanobacteria sensitivity to antibiotics and implications for environmental risk assessment. Science of the Total Environment
Variability in cyanobacteria sensitivity to antibiotics and implications for environmental risk assessment
Once released into the environment antibiotics can kill or inhibit the growth of bacteria, and in turn potentially have effects on bacterial community structure and ecosystem function. Environmental risk assessment (ERA) seeks to establish protection limits to minimise chemical impacts on the environment, but recent evidence suggests that the current regulatory approaches for ERA for antibiotics may not be adequate for protecting bacteria that have fundamental roles in ecosystem function. In this study we assess the differences in interspecies sensitivity of eight species of cyanobacteria to seven antibiotics (cefazolin, cefotaxime, ampicillin, sufamethazine, sulfadiazine, azithromycin and erythromycin) with three different modes of action. We found that variability in the sensitivity to these antibiotics between species was dependent on the mode of action and varied by up to 70 times for β-lactams. Probabilistic analysis using species sensitivity distributions suggest that the current predicted no effect concentration PNEC for the antibiotics may be either over or under protective of cyanobacteria dependent on the species on which it is based and the mode of action of the antibiotic; the PNECs derived for the macrolide antibiotics were over protective but PNECs for β-lactams were generally under protective. For some geographical locations we identify a significant risk to cyanobacteria populations based upon measured environmental concentrations of selected antibiotics. We conclude that protection limits, as determined according to current regulatory guidance, may not always be protective and might be better derived using SSDs and that including toxicity data for a wider range of (cyano-) bacteria would improve confidence for the ERA of antibiotics. Abstract
Tait L, Wedgwood K, Tsaneva-Atanasova K, Brown JT, Goodfellow M
(2018). Control of clustered action potential firing in a mathematical model of entorhinal cortex stellate cells. J Theor Biol
Control of clustered action potential firing in a mathematical model of entorhinal cortex stellate cells.
The entorhinal cortex is a crucial component of our memory and spatial navigation systems and is one of the first areas to be affected in dementias featuring tau pathology, such as Alzheimer's disease and frontotemporal dementia. Electrophysiological recordings from principle cells of medial entorhinal cortex (layer II stellate cells, mEC-SCs) demonstrate a number of key identifying properties including subthreshold oscillations in the theta (4-12 Hz) range and clustered action potential firing. These single cell properties are correlated with network activity such as grid firing and coupling between theta and gamma rhythms, suggesting they are important for spatial memory. As such, experimental models of dementia have revealed disruption of organised dorsoventral gradients in clustered action potential firing. To better understand the mechanisms underpinning these different dynamics, we study a conductance based model of mEC-SCs. We demonstrate that the model, driven by extrinsic noise, can capture quantitative differences in clustered action potential firing patterns recorded from experimental models of tau pathology and healthy animals. The differential equation formulation of our model allows us to perform numerical bifurcation analyses in order to uncover the dynamic mechanisms underlying these patterns. We show that clustered dynamics can be understood as subcritical Hopf/homoclinic bursting in a fast-slow system where the slow sub-system is governed by activation of the persistent sodium current and inactivation of the slow A-type potassium current. In the full system, we demonstrate that clustered firing arises via flip bifurcations as conductance parameters are varied. Our model analyses confirm the experimentally suggested hypothesis that the breakdown of clustered dynamics in disease occurs via increases in AHP conductance. Abstract
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(2018). EPSRC blog.
I wrote a general audience blog on my fellowship project for EPSRC and my career and decisions leading up to it. Abstract
Tamagnini F, Cotton M, Goodall O, Harrison G, Jeynes C, Palombo F, Tomkow J, Tomkow T, Wedgwood K, Welsman J, et al
(2018). ’Of Mice and Dementia’: a filmed conversation on the use of animals in dementia research. Dementia (London, England)
’Of Mice and Dementia’: a filmed conversation on the use of animals in dementia research.
Preclinical science research focuses on the study of physiological systems regulating body functions, and how they are dysregulated in disease, in a non-human setting. For example, cells in a dish, computer simulations or animals. Scientific procedures traditionally involve a specialist scientist developing a hypothesis and subsequently testing it using an experimental set-up. The results are then disseminated to the wider scientific community, following peer review and only at the last stage the news will reach the general, lay public. In the last few years, some research funding institutions have promoted a different model, with the direct involvement of members of the public in the research co-creation, from the hypothesis development, to the grant revision, project monitoring and results communication. We personally experienced this model and brought it to a further level by producing a movie. Animal research is a very controversial topic as, while still being necessary for the investigation of body functions, it brings about issues related to the ethics, the regulation and the practical execution of experimental procedures on animals. Here we discuss the different stages of the ideation, production and outcomes of the movie ’Of Mice and Dementia’, a filmed conversation on animal experimentation in dementia research. The conversation was between scientists and lay people with a direct experience of dementia. Abstract
Bonilla-Quintana M, Wedgwood KCA, O’Dea RD, Coombes S (2017). An Analysis of Waves Underlying Grid Cell Firing in the Medial Enthorinal Cortex. The Journal of Mathematical Neuroscience, 7, n/a-n/a.
Avitable D, Wedgwood KCA
(2017). Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis. Journal of Mathematical Biology
Macroscopic coherent structures in a stochastic neural network: from interface dynamics to coarse-grained bifurcation analysis
We study coarse pattern formation in a cellular automaton modelling a spatially-extended stochastic neural network. The model, originally proposed by Gong and Robinson (Phys Rev E 85(5):055,101(R), 2012), is known to support stationary and travelling bumps of localised activity. We pose the model on a ring and study the existence and stability of these patterns in various limits using a combination of analytical and numerical techniques. In a purely deterministic version of the model, posed on a continuum, we construct bumps and travelling waves analytically using standard interface methods from neural field theory. In a stochastic version with Heaviside firing rate, we construct approximate analytical probability mass functions associated with bumps and travelling waves. In the full stochastic model posed on a discrete lattice, where a coarse analytic description is unavailable, we compute patterns and their linear stability using equation-free methods. The lifting procedure used in the coarse time-stepper is informed by the analysis in the deterministic and stochastic limits. In all settings, we identify the synaptic profile as a mesoscopic variable, and the width of the corresponding activity set as a macroscopic variable. Stationary and travelling bumps have similar meso- and macroscopic profiles, but different microscopic structure, hence we propose lifting operators which use microscopic motifs to disambiguate them. We provide numerical evidence that waves are supported by a combination of high synaptic gain and long refractory times, while meandering bumps are elicited by short refractory times. Abstract
Williams CA, Wedgwood KCA, Mohammadi H, Tomlinson OW, Tsaneva-Atanasova K
(2017). Modelling of the Cardiopulmonary Responses to Maximal Aerobic Exercise in Patients with Cystic Fibrosis.
, Cold Spring Harbor Laboratory. 0 pages.
Modelling of the Cardiopulmonary Responses to Maximal Aerobic Exercise in Patients with Cystic Fibrosis
Gommer F, Brown LP, Wedgwood KCA
(2016). Analytical method using gamma functions for determining areas of power elliptical shapes for use in geometrical textile models. COMPOSITES PART A-APPLIED SCIENCE AND MANUFACTURING
, 222-224. Author URL
Wedgwood KCA, Richardson SJ, Morgan NG, tsaneva-atanasova K
(2016). Spatiotemporal dynamics of insulitis in human type 1 diabetes. Frontiers in Physiology
Spatiotemporal dynamics of insulitis in human type 1 diabetes
Type 1 diabetes (T1D) is an auto-immune disease characterised by the selective destruction ofthe 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 insulinin order to manage blood glucose levels. Whilst technological developments have significantlyimproved both the life expectancy and quality of life of these patients, an understanding of themechanisms 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 humansstudied at disease onset. In this manuscript, we employ data from human pancreases collectedclose to the onset of type 1 diabetes and propose a spatio-temporal computational model forthe progression of insulitis in human T1D, with particular focus on the mechanisms underlyingthe development of insulitis in pancreatic islets. This framework allows us to investigate how thetime-course of insulitis progression is affected by altering key parameters, such as the number ofthe CD20+ B cells present in the inflammatory infiltrate, which has recently been proposed toinfluence the aggressiveness of the disease. Through the analysis of repeated simulations ofour stochastic model which track the number of beta cells within an islet, we find that increasednumbers of B cells in the peri-islet space lead to faster destruction of the beta cells. We also findthat the balance between the degradation and repair of the basement membrane surrounding theislet is a critical component in governing the overall destruction rate of the beta cells and theirremaining number. Our model provides a framework for continued and improved spatio-temporalmodelling of human T1D Abstract
Gommer F, Wedgwood KCA, Brown LP
(2015). Stochastic reconstruction of filament paths in fibre bundles based on two-dimensional input data. COMPOSITES PART A-APPLIED SCIENCE AND MANUFACTURING
, 262-271. Author URL
Wedgwood KCA, Lin KK, Thul R, Coombes S
(2013). Phase-amplitude descriptions of neural oscillator models. The Journal of Mathematical Neuroscience 2013, 3:2
Phase-amplitude descriptions of neural oscillator models
Phase oscillators are a common starting point for the reduced description of Abstract
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.
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Coombes S, Thul R, Wedgwood KCA
(2012). Nonsmooth dynamics in spiking neuron models. PHYSICA D-NONLINEAR PHENOMENA
(22), 2042-2057. Author URL
Lin KK, Wedgwood KCA, Coombes S, Young L-S
(2011). Limitations of perturbative techniques in the analysis of rhythms and. oscillations.
Limitations of perturbative techniques in the analysis of rhythms and. oscillations
Perturbation theory is an important tool in the analysis of oscillators and Abstract
their response to external stimuli. It is predicated on the assumption that the
perturbations in question are "sufficiently weak", an assumption that is not
always valid when perturbative methods are applied. In this paper, we identify
a number of concrete dynamical scenarios in which a standard perturbative
technique, based on the infinitesimal phase response curve (PRC), is shown to
give different predictions than the full model. Shear-induced chaos, i.e.
chaotic behavior that results from the amplification of small perturbations by
underlying shear, is missed entirely by the PRC. We show also that the presence
of "sticky" phase-space structures tend to cause perturbative techniques to
overestimate the frequencies and regularity of the oscillations. The phenomena
we describe can all be observed in a simple 2D neuron model, which we choose
for illustration as the PRC is widely used in mathematical neuroscience.
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