Abstracts
Asim Alawfi (University of Exeter)
Tracking symmetry-breaking in a model for hearing with delay (Poster)
Ferrario and Rankin (2021) introduced a model for auditory (hearing) perception, which couples two identical neural populations of excitatory and inhibitory neurons. The cross-inhibition coupling has a delay such that the model is a four-dimensional system of periodically forced delay differential equations with reflection symmetry. The symmetry breaking coincides with the switching between different auditory perceptions. Bifurcation analysis for this model finds the precise boundaries between the different regimes discussed by Ferrario and Rankin (2021) and predicts further phenomena, which may guide future experiments. An essential tool for our bifurcation analysis are extended systems for tracking symmetry-breaking bifurcations of periodic solutions in delay differential equations, newly implemented in DDE-Biftool.
Ferrario, A. and Rankin, J., 2021. Auditory streaming emerges from fast excitation and slow delayed inhibition. The Journal of Mathematical Neuroscience, 11(1), pp.1-32.
Tyler Cassidy (University of Leeds)
Numerics and approximations for gamma distributed delay differential equations
Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic gamma distributed DDEs have not previously been implemented. Modellers have therefore resorted to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations (ODEs). Unfortunately, this Erlang approximation imposes an artificial relationship between the mean and variance of the delayed process. Accordingly, we develop a numerical method to numerically integrate the gamma distributed DDE without relying on the Erlang approximation. Alternatively, we derive an ODE approximation of the gamma distributed DDE that is a more accurate than the common Erlang approximation. Using our numerical method to provide reference solutions, we show that the Erlang approximation may produce qualitatively different solutions that the underlying gamma distributed DDE while our proposed approximation does not.
Marianna Cerasuolo (University of Portsmouth)
Mathematical insights into the dynamics of human prostate cancer with neuroendocrine differentiation and distributed delay
Prostate cancer is the fifth most common cause of death from cancer, and the second most common diagnosed cancer in men. In the last few years many mathematical models have been proposed to describe the dynamics of prostate cancer under treatment. So far one of the major challenges has been the development of mathematical models that could represent experiments in vivo conditions (experiments on individuals) and therefore be suitable for clinical applications, while being mathematically tractable.
In this talk, taking a step in this direction, I am going to propose a nonlinear distributed-delay dynamical system that explores neuroendocrine transdifferentiation in human prostate cancer in vivo. Sufficient conditions for the existence and the stability of a tumour-present equilibrium will be given, and the occurrence of a Hopf bifurcation will be proven for a uniform delay distribution. Numerical simulations will be showed to explore differences in behaviour for uniform and exponential delay distributions. The study of the dynamical system will show how the choice of the delay distribution is key in defining the dynamics of the system and in determining the conditions for the onset of oscillations following a switch in the stability of the tumour-present equilibrium.
Rod Halburd (University College London)
Painlevé-type delay differential equations
The Painlevé equations are six classically know differential equations that arise in many applications and are closely connected with the theory of integrable systems and special functions. Many integrable difference equations are also considered to be of Painlevé type as they share several important integrability properties with the original differential equations and some of them arise as symmetries of the differential equations. In this talk I will discuss delay-differential equations that appear to fit with this picture.
Natalia Janson (Loughborough University)
Delay-induced bifurcations and optimisation
Finding the best solution, called optimisation, mathematically amounts to finding the global minimum of a cost function usually having several wells. Here, we conceive a novel principle aiding optimisation, based on delay-induced bifurcations in nonlinear systems with time delay [1].
To prevent confinement to a local minimum and to search for the global one, mechanisms are needed to overcome barriers between the minima. Simulated annealing digitally emulates pushing the particle over a barrier by random forces, whereas quantum computers utilise quantum tunnelling through barriers. The former can be very slow, whereas the latters are challenging and expensive to make.
We propose to enable the system to explore all available minima spontaneously, i.e. without the digital decision-making at every step, by replacing random forces with dynamical chaos. To generate global chaos, we utilise the time delay. Normally, it is hardly possible to predict how a delay would affect a non-linear system. However, we hypothesize and verify that for a certain broad class of non-linear systems, an increase of the delay predictably leads to a sequence of global bifurcations and then to global chaos. Such systems can be obtained by taking a standard gradient dynamical system and replacing the argument of its right-hand side with a delayed state variable.
The barriers between the minima disappear thanks to homoclinic or heteroclinic bifurcations, which reconfigure the manifolds in such a way, that they cease to serve boundaries of attractor basins. Our study verifies this principle for one-dimensional systems with delay [1,2]. The proposed principle could potentially be implemented in analogue non-quantum devices and lead to optimising machines faster than digital computers, and cheaper than quantum ones.
[1] Natalia B. Janson and Christopher J. Marsden, "Optimization with delay-induced bifurcations", Chaos 31, 113126 (2021) https://doi.org/10.1063/5.0058087
[2] N.B. Janson and C.J. Marsden, Delay-induced homoclinic bifurcations in modified gradient bistable systems and their relevance to optimization, Chaos 31, 093120 (2021) https://doi.org/10.1063/5.0035959
Andrew Keane (Cork University College)
A Delayed-Feedback Mechanism for Stable Oscillations in Large-Scale Ocean Circulation
The global ocean circulation plays a pivotal role in the regulation of the Earth's climate, as well as carbon and nutrient cycles and the habitability of the oceans for marine life. Transitions in circulation patterns are known to have occurred in the past and, considering their global importance, it is crucial to understand the nature and drivers of such transitions. Here we present stable oscillations observed in the ocean circulation of an Earth System Model of intermediate complexity. The presence of the oscillations depends on a circumpolar current. We adapt a simple ocean box model to include a delayed feedback to represent a circumpolar current and investigate stable oscillatory solutions by bifurcation analysis. Our results provide insight into oscillations observed in simulations based on land mass configurations typifying the geological past and also highlight the potential influence of changing circumpolar current speed on the stability of the ocean's meridional overturning circulation.
This is joint work with A. Pohl, H. Dijkstra and A. Ridgwell.
Andrew Krause (Durham University)
Gene Expression Time Delays in Reaction-Diffusion Systems
Reaction-diffusion systems are commonly used to model molecular mechanisms underlying pattern formation in developmental biology. Despite increasingly sophisticated variants of such systems, it is still unclear how common such hypothetical mechanisms are in nature. After reviewing these models from a wide perspective, we focus on the impact of gene-expression time delays on the robustness of reaction-diffusion systems as a model of biological pattern formation. We consider a large class of fixed and distributed delays, and explore these questions of robustness through numerical simulations and linear stability analyses of different reaction kinetics, as well as considering variations in initial and boundary conditions. We find that such delays can change the parameter regimes wherein patterns exist, but also importantly the time taken for patterns to form in presumptive models is always substantially longer in the presence of delay, independent of the details of the model. This leads to important questions about the role of cell-level mechanisms alongside tissue-level patterning. We discuss these results in terms of larger aspects of the philosophy of modelling multiscale phenomena in biology.
Yuliya Kyrychko (University of Sussex)
Imitation dynamics of vaccination with distributed delay risk perception
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.
Sarah Loos (University of Cambridge)
Time-delayed stochastic processes in nonequilibrium statistical physics
Many of the approaches in statistical physics and stochastic thermodynamics are fundamentally based on the Markov assumption, although real-world complex systems typically exhibit non-negligible memory effects. This is particularly true for systems that operate far from thermal equilibrium, such as molecular machines, flocks of bird, robotic systems, or systems subject to some sort of external control.
In this talk, I will discuss the origins and implications of time delays that arise in nonequilibrium systems from the perspective of statistical physics and stochastic thermodynamics [1-5]. As particular examples, I will consider time delays appearing in feedback control on mesoscale objects [2,3] and in the collective motion of swarms of insects [5].
[1] Loos, Klapp: Fokker–Planck Equations for Time-Delayed Systems via Markovian Embedding, J. Stat. Phys. 177, 95–118 (2019)
[2] Loos, Hermann, Klapp: Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay, Entropy 23, 696 (2021)
[3] Loos, Klapp: Heat flow due to time-delayed feedback, Sci. Rep. 9, 1-11 (2019)
[4] Holubec, Ryabov, Loos, Kroy: Equilibrium stochastic delay processes, New Journal of Physics 24, 023021, (2022)
[5] Holubec, Geiss, Loos, Kroy, Cichos: Finite-size scaling at the edge of disorder in a time-delay Vicsek model, Phys. Rev. Lett. 127, 258001 (2021)
Xuerong Mao (University of Strathclyde)
Advances in Nonlinear Hybrid SDDEs: Existence, Boundedness and Stability
This talk is concerned with a class of highly nonlinear hybrid stochastic differential delay equations (SDDEs). Different from the most existing papers, the time delay functions in the SDDEs are no longer required to be differentiable, not to mention their derivatives are less than 1. The generalized Hasminskii-type theorems are established for the existence and uniqueness of the global solutions. Comparing with the existing results, we show our new theorems are much more general and can be applied to a much wider class of highly nonlinear SDDEs. Further sufficient conditions are also obtained for the asymptotic boundedness and stability. This is a joint work with J. Hu and W. Mao.
Francesca Scarabel (University of Leeds)
Numerical bifurcation analysis of delay equations via reduction to ODEs
Via pseudospectral discretisation, a delay equation can be approximated with a system of ODEs, whose bifurcation properties can be studied with existing software for ODEs like MatCont for MATLAB. I will present the method and show examples, with special attention to a recent more efficient implementation for integral renewal equations with finite and infinite delay.
Kyle Wedgwood (University of Exeter)
Closed-Loop Interrogation of the Dynamics of Neuroendocrine Cells
This talk will discuss how mathematical modelling can be embedded within experiment protocols to study electrical behaviour in neurons and neuroendocrine cells in which delays play an important role. We discuss three examples, the first of which explores the capability of a neuron that is synaptically coupled to itself, to store and repeat patterns of precisely timed spikes, which we regard as single cell 'memories'. Drawing on analogies from semiconductor lasers, we append a delayed self-coupling term to the oft studied Morris-Lecar model of neuronal excitability and use bifurcation analysis to predict the number and type of memories the neuron can store. These results highlight the delay period as an important period parameter controlling the storage capacity of the cell. We then use the dynamic clamp protocol to introduce self-coupling to a mammalian cell and confirm the existence of the spiking patterns predicted by the model analysis. The second example covers preliminary work of investigating the origin of pulsatile secretion in corticotrophs in the pituitary gland. Such pulsatility has previously been conjectured to be strongly coupled to the delay period between secretion from the corticotrophs and feedback from the adrenal glands. Here, we combine Ca2+ imaging, mathematical modelling and dynamic perfusion to explore how delays influence behaviour of this combined system. The final example will explore how techniques combining control theory and bifurcation analysis with dynamic clamp can be used to probe single cell electrical excitability.
Serhiy Yanchuk (Potsdam Institute of Climate Impact Research)
Deep neural networks using a single neuron: folded-in-time architecture using feedback-modulated delay loops
Deep neural networks are among the most widely applied machine learning tools showing outstanding performance in a broad range of tasks. We present a method for folding a deep neural network of arbitrary size into a single neuron with multiple time-delayed feedback loops. This single-neuron deep neural network comprises only a single nonlinearity and appropriately adjusted modulations of the feedback signals. The network states emerge in time as a temporal unfolding of the neuron’s dynamics. By adjusting the feedback-modulation within the loops, we adapt the network’s connection weights. These connection weights are determined via a back-propagation algorithm, where both the delay-induced and local network connections must be taken into account. Our approach can fully represent standard Deep Neural Networks (DNN), encompasses sparse DNNs, and extends the DNN concept toward dynamical systems implementations. The new method, which we call Folded-in-time DNN (Fit-DNN), exhibits promising performance in a set of benchmark tasks.
Stelzer, F., Röhm, A., Vicente, R., Fischer, I., & Yanchuk, S. (2021). Deep neural networks using a single neuron: folded-in-time architecture using feedback-modulated delay loops. Nature Communications, 12(1), 5164. https://doi.org/10.1038/s41467-021-25427-4