Monday 16 December
11:30 Arrival and coffee in Mathematical Sciences Atrium
12:00 Lukas Eigentler (University of Warwick)
Title: Delayed loss of stability of periodic travelling waves affects wavelength changes of patterned ecosystems
Abstract: Many patterned ecosystems, such as dryland vegetation patterns and intertidal mussel beds can be described by PDEs admitting periodic travelling waves (PTWs). Under a changing environment that increases stress, such systems undergo a cascade of wavelength changes before an extinction event occurs. Classically, wavelength changes have been predicted by identifying the intersection of a PTW’s wavelength contour with a stability boundary in the system’s Busse balloon. In this talk, I highlight that this information is often insufficient because of a delayed loss of stability phenomenon. I show that PTWs can persist as transients for ecologically significant times after the crossing of a stability boundary in the Busse balloon. I present a method that can predict the order of magnitude of the time delay between the crossing of a stability boundary and the occurrence of a wavelength change by linking the delay to features of the essential spectra of the PTWs.
12:40 Lunch buffet and posters in Mathematical Sciences Atrium
13:40 Stefan Ruschel (University of Leeds)
Title: Master stability curves for traveling waves on Zn-equivariant networks
Abstract: We present a framework for determining effectively the spectrum and stability of traveling waves in discrete systems with symmetries, such as rings and lattices, by computing master stability curves (MSCs). Unlike traditional methods, MSCs are independent of system size and, therefore, present an efficient pathway into studying the spectrum and stability of large networks. As a case study, we compute and analyze master stability curves of traveling waves (that take the form of pulse trains) in diffusively coupled rings of FitzHugh-Nagumo oscillators.
14:20 Maia Angelova (Aston University)
Title: Delay differential equations model for glucose-insulin regulation
Abstract: Diabetes Mellitus is a chronic metabolic illness that impairs the regulation of glucose and insulin blood levels. It affects millions of people around the world. In England, 7% of adults have evidence of Type 2 Diabetes Mellitus (T2DM) and 30% of those were underdiagnosed. Early diagnosis and prediction of T2DM is essential for effective treatment and positive outcomes. The diagnosis of T2DM is a medical procedure that requires multiple in hospital interventions to assess the glucose and insulin production.
In this talk, I will present our dynamic glucose-insulin regulation model with two delays [1-3]. The role of the delays will be discussed. Solutions, both analytical and numerical, will be considered and a range for the parameters will be discussed [3-4]. The model is designed to monitor and predict T2DM. It can be used embedded in a digital twin for monitoring and adjusting glucose levels without changing the lifestyle.
References
1. Huard, B., Easton, J., & Angelova, M. (2015). Investigation of stability in a two-delay model of the ultradian oscillations in glucose–insulin regulation. Communications in Nonlinear Science and Numerical Simulation, 26(1-3), 211-222.
2. Huard, B., Bridgewater, A., & Angelova, M. (2017). Mathematical investigation of diabetically impaired ultradian oscillations in the glucose–insulin regulation. Journal of theoretical biology, 418, 66-76.
3. Bridgewater, A., Huard, B., & Angelova, M. (2020). Amplitude and frequency variation in nonlinear glucose dynamics with multiple delays via periodic perturbation. Journal of Nonlinear Science, 30(3), 737-766.
4. Angelova, M., Beliakov, G., Ivanov, A., & Shelyag, S. (2021). Global stability and periodicity in a glucose-insulin regulation model with a single delay. Communications in Nonlinear Science and Numerical Simulation, 95, 105659.
15:00 Coffee break
15:20 Hil Meijer (University of Twente)
Title: Synchrony across the brain; a harmonic balance approach to delay-coupled oscillators
Abstract: Delays are a natural component of computational models of large-scale brain dynamics. Delays combined with local synaptic activity typically lead to oscillations, but the question is whether synchrony or some out-of-phase solution is stable. Here we present a machinery using harmonic balance and accounting for symmetries to look for instabilities of the synchronous solution.
We first analyse a simple model on a ring where ``travelling waves'' with activity jumping to the nearest or next-nearest neighbour appear. Employing numerical continuation, we also track which pattern exists as we change the delay. For stability of the asynchronous solutions, we rely on simulations. We then move on to the Wilson-Cowan model with similar results, and highlight some of the additional numerical challenges.
16:00 Benoit Huard (Northumbria University)
Title: Entrainment and amplitude variation in delayed models of glucose-insulin regulation
Abstract: The glycemic response to a glucose stimulus is an essential tool for detecting deficiencies in humans such as diabetes. In the presence of constant and periodic glucose infusions in healthy individuals, it is known that this control leads to slow oscillations as a result of feedback mechanisms at the organ and tissue level. These ultradian oscillations are typically modelled using systems of nonlinear equations with two discrete delays and here we give a particular attention to its periodic solutions. These arise from a Hopf bifurcation which is induced by an external glucose stimulus and the joint contributions of delays in pancreatic insulin release and hepatic glycogenesis. The effect of each physiological subsystem on the amplitude and period of the oscillations is exhibited by performing a perturbative analysis of its periodic solutions. It is shown that assuming the commensurateness of delays enables the Hopf bifurcation curve to be characterised by studying roots of linear combinations of Chebyshev polynomials. The impact of periodic (sinusoidal) infusions is characterised through numerical bifurcation analysis. The resulting expressions provide an invaluable tool for studying the interplay between physiological functions and delays in producing an oscillatory regime, as well as relevant information for glycemic control strategies.
Joint work with Maia Angelova, Gemma Kirkham and Stefan Ruschel.
17:00 & Dinner & Food provided in Mathematical Science Atrium followed by outing for drinks
Tuesday 17 December
10:00 Yuliya Kyrychko (University of Sussex)
Title: Imitation dynamics of vaccination with distributed delay risk perception
Abstract: In this talk I will discuss the dynamics of paediatric vaccination when modelled as a game, where increase in the rate of vaccination is taken to be proportional to the perceived payoff. Similarly to earlier models, this payoff is considered to be a difference between the perceived risk of disease, as represented by its momentary incidence, whereas for the perceived risk of vaccine side effects I will use an integral of the proportion being vaccinated with some delay kernel. This delay distribution can model two realistic effects: the fact that vaccine side effects take some time to develop after a person has been vaccinated, and that even after side effects have appeared, awareness of them will continue to impact vaccination choices for some period of time. I will discuss conditions of feasibility and stability of the disease-free and endemic steady states of the model for the general delay distribution, and for some specific delay distributions that include discrete delay, Gamma distribution (weak and strong cases), and the acquisition-fading kernel.
By computing bifurcation diagram of the endemic equilibrium we are able to establish parameter regions, where some steady level of infection is maintained, as well as regions where periodic solutions around the endemic steady state are observed. I will present a comparison of stability regions for endemic steady state, highlighting differences between distributions that are observed for the same values of parameters. This will demonstrate that not just the mean time delay, but also the details of the distribution are important when analysing the dynamics. To make the model more realistic, I will also consider the impact of a public health campaign on vaccination dynamics and contrast it to the case where vaccination choices are only dictated by information exchange between vaccinating and non-vaccinating people.
10:40 Jonathan Crofts (Nottingham Trent University)
Title: Network structure and time delays shape synchronisation patterns in brain network models
Abstract: In this talk, we investigate synchronisation patterns and coherence in a network of delayed Wilson-Cowan nodes. To model information processing across various brain regions, we incorporate two distinct time delays: an intra-nodal delay representing the time required for signals to traverse local circuitry within a cortical area, and an inter-nodal delay accounting for the longer communication times associated with white matter connections between brain regions. To investigate the role of network topology, we consider a range of toy network structures as well as the known cortical structure of the Macaque monkey. Our investigation identifies extensive regions of parameter space where synchronised states can exhibit transverse instabilities, resulting in diverse dynamic behaviours dependent on network structure, coupling dynamics, and delays. While complex partially synchronised states were observed across all network topologies, the cortical network exhibited unique time-dependent dynamics, including phase cluster formation, that were not seen in simpler toy networks, and which could be vital for its capacity to coordinate intricate cognitive processes.
11:20 Coffee Break
11:40 Francesca Scarabel (University of Leeds)
Title: Numerical stability and bifurcation analysis of equations with infinite delay
Abstract: Delay differential and renewal equations with infinite delay are widely used in mathematical biology, and in particular in ecology and epidemiology, to describe physiologically structured population models, in which the individual rates are assumed to depend on a structuring variable (e.g., age, size, time since infection) which evolves as a function of the individual’s age.
We consider nonlinear delay differential and renewal equations with infinite delay and derive a finite-dimensional ODE approximating the equivalent abstract differential equation. We discuss the one-to-one correspondence of equilibria between the original and the approximating system, and the convergence of the characteristic roots as the dimension of the approximation increases, when the collocation nodes are chosen as the zeros or the extrema of the Laguerre polynomials suitably scaled. We show examples of applications from population dynamics, including new analyses on periodic outbreaks emerging from time-since-infection models with waning immunity.
This work is in collaboration with Rossana Vermiglio (University of Udine) and based on the publication: F. Scarabel, R. Vermiglio, Equations with infinite delay: pseudospectral discretization for numerical stability and bifurcation in an abstract framework, SINUM (2024).
12:20 Robert Allen (University of Nottingham)
Title: Phase-Isostable Reduction of Coupled Oscillator Networks with Delays in the Node Dynamics and in the Coupling
Abstract: Phase-isostable reductions have for ODEs with limit cycle oscillations have been shown to produce much-improved descriptions of network behaviours as compared to a phase-only approach. [1] As such, there is interest in the development of analogous approaches for systems involving delays. This talk will be split into sections. In the first part, we develop a framework for the construction of phase and isostable response curves for systems where the oscillations are induced by DDEs, utilising the method of harmonic balance to aide calculations. [2]
In the second part, we will consider how the phase-isostable interaction functions are affected when the coupling between ODE oscillators is delayed. This brings us towards a more realistic model of synaptic coupling. We apply this approach to a very simple 2-node network to investigate how the interplay between the coupling strength and the coupling delay leads to different observations of network behaviour. In the final part, time-permitting, we will consider how to combine these two scenarios and apply this to simple Wilson-Cowan networks as a model of large-scale brain activity.
References
[1] R. Nicks, R. Allen, and S. Coombes. Insights into oscillator network dynamics using a phase-isostable framework. Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(1):013141, 01 2024.
[2] R. Nicks, R. Allen, and S. Coombes. Phase and amplitude responses for delay equations using harmonic balance. Phys. Rev. E, 110:L012202, 2024.
13:00 Lunch buffet in Mathematical Sciences Atrium
14:00 Jérémie Lefebvre (University of Ottawa)
Title: How white matter plasticity shapes brain dynamics, function and disease
Abstract: The white matter is composed of bundles of axons serving as cables connecting different brain regions and regulating neural traffic, a dense network occupying roughly half of the volume of the human brain. Conduction along these cables (i.e. how fast neural impulses go) is influenced by a substance called myelin, produced by glial cells. Myelin is crucial for memory function, and compromised white matter integrity is associated with severe conditions like multiple sclerosis and epilepsy. Traditionally thought to be static after development, new results show that white matter rewires itself, complementing synaptic plasticity to support learning. This activity-dependent myelination impacts brain dynamics and function across spatial (microns to meters) and temporal (milliseconds to years) scales, making it incredibly difficult to study using experiments alone. Luckily, this problem is amenable to mathematical modeling. We investigate activity dependent myelination and its role in maintaining the flexibility and stability of neural circuits function, using a combination of modelling and experimental data at cortical microcircuits and white matter scales. We analyze a variety of neural network models with adaptive, activity-dependent axonal conduction delays, to examine the consequences of neuron-glia feedback on the dynamics of these networks. We study axonal myelination patterns, how they change, and how axonal conduction influence neural activity to support neural communication and synchronization using simulations informed by human neuroimaging data. Our results show that adaptive myelination implements a gain control mechanism that enhances neural firing rates, correlations and synchrony through the temporal coordination of neural signaling. Taken together, these mathematical investigations provide new insights as to the potential role of myelin disruption in pathologies such as multiple sclerosis and epilepsy.
15:00 Coffee
15:20 Catherine Drysdale (University of Birmingham)
Title: A Novel Use of Pseudospectra in Mathematical Biology: Understanding HPA Axis Sensitivity
Abstract: The Hypothalamic-Pituitary-Adrenal (HPA) axis is a major neuroendocrine system, and its dysregulation is implicated in various diseases. This system also presents interesting mathematical challenges for modelling. We consider a nonlinear delay differential equation model and calculate pseudospectra of three different linearizations: a time-dependent Jacobian, linearization around the limit cycle, and dynamic mode decomposition (DMD) analysis of Koopman operators (global linearization). The time-dependent Jacobian provided insight into experimental phenomena, explaining why rats respond differently to perturbations during corticosterone secretion's upward versus downward slopes. We developed new mathematical techniques for the other two linearisations to calculate pseudospectra on Banach spaces and apply DMD to delay differential equations, respectively. These methods helped establish local and global limit cycle stability and study transients. Additionally, we discuss using pseudospectra to substantiate the model in experimental contexts and establish bio-variability via data-driven methods. This work is the first to utilize pseudospectra to explore the HPA axis.
16:00 Close