Dynamical Systems Seminar

Abstract: 

Chaotic behavior can have severe consequences in biological systems such as the heart where it is linked to cardiac arrhythmias. There is a need to terminate this chaos in order to prevent such adverse effects. Understanding cardiac chaos therefore forms a societal challenge related to a biological problem with strong mathematical foundations. To gain deeper insight into these underlying processes, a truly interdisciplinary approach is needed. It is at this crossroads of fields (computer science, physics, mathematics, biology, medicine, engineering) that my research is located.

Cardiac dynamics takes the mathematical form of a reaction-diffusion equation, where the reaction term describes single-cell ionic flows using non-linear partial differential equations, and the diffusion makes wave propagation possible. Mathematical models vary from simplistic (Fitzhugh-Nagumo: 2 variables, 2 parameters) to including full biological detail (O’Hara-Rudy: 46 variables, >100 parameters). Most fundamental research is focused on the initiation of arrhythmias, where small changes to these models are made to see whether they influence arrhythmia onset (De Coster et al., Sci Rep 2018; De Coster et al., Cardiovasc Res 2023).

However, my current research is more translationally oriented and focusses on how we can stop arrhythmias (Majumder, De Coster, Kudryashova et al., eLife 2020). While arrhythmias initiate due to local effects, termination needs to rely on global cooperation of all the cells. This is why we defibrillate the heart with a large electrical shock to reset all the cells at the same time. However, this is very painful, posing the question whether we can obtain the same effect without these high-power shocks.

Peculiarly, the heart sometimes reaches emergent cooperative behaviour on its own. When we are young, our arrhythmias stop themselves naturally despite our same heart failing at older age. Hence, I believe that there are intrinsic mechanisms in the heart present that fail over time. By investigating which local ionic behaviour generates global cooperation in the prevention of arrhythmias, we might be able to strengthen these processes to prevent arrhythmias altogether. For this reason it is important to understand the nonlinearities in cardiac wave propagation in order to manipulate them, a project for which a KIEM grant was recently obtained to start and strengthen collaboration between Leiden departments related to cardiac chaos.

If you are interested in my research and want to know more, you can have a look at www.timdecoster.com

Abstract:

We will consider the following two component system of reaction diffusion equations.

The system is singularly perturbed with respect to the parameter epsilon. The case where $\delta=0$ is well understood, so the novelty lies in adding the function f(x) which depends on space. Our aim is to show that the travelling front solutions known for delta=0 and epsilon small will persist (in some suitable sense) when delta >0.

 On first sight the introduction of this term might appear harmless, but things get surprisingly complicated. The usual approach of introducing a comoving frame does not work. This is due to the fact that neither the speed of the wave nor the profile of the front are constant in time, moreover the front is asymptotic in space to background states which are not contant either (see the numeric simulation in the attacement to get a sense of how the front looks). It turns our that the position of the front satisfies a delay equation which we can study to get an understand of how the wave travels.

Abstract:

In biology, self-organization and pattern formation happen on many different scales. For example, in embryonic development, a homogeneous clump of cells evolves into a complex organism. We study so-called mechanochemical models. These models describe the deformation of a surface (for example a cell- or a tissue surface) caused by a diffusing chemical that is locally inducing curvature. This model captures a possible mechanism behind different shapes found in nature. We use tools such as numerical simulation and Geometric Singular Perturbation Theory to analyze and understand what type of patterns can occur.

Abstract: 

The Lévy Flight Foraging Hypothesis is a widely accepted dogma which asserts that animals using search strategies allowing for long jumps, also known as Lévy flights, have an evolutionary advantage over those animals using a foraging strategy based on continuous random walks modelled by Brownian motion. However, recent discoveries suggest that this popular belief may not be true in some geometric settings. In this talk we will explore some of the recent progress in this direction which combines Riemannian geometry with stochastic analysis to create a new paradigm for diffusion processes.

Abstract: 

We consider reversible vector fields in ℝ2n such that the set of fixed points of the involutory reversing symmetry is n-dimensional. Let such system have a smooth one-parameter family of symmetric periodic orbits which is of saddle type in normal directions. We establish that topological entropy is positive when the stable and unstable manifolds of this family of periodic orbits have a strongly-transverse intersection. This is joint work with Jeroen Lamb and Dmitry Turaev.

Abstract: We study the trapping of light in microscopically small optical Fabry-Perot cavities with mirror sizes and mirror distances of only a few optical wavelengths. It proofs to be surprisingly difficult to calculate the optical resonances of such cavities by ‘simply’ solving Maxwell equations with the boundary conditions imposed by the mirrors. As physicists we have tried to find solutions by applying perturbation theory, inspired by quantum mechanics, to the approximate paraxial solutions known for larger cavities. The perturbative analysis works reasonably well, has a beautiful symmetry, and provides lots of insight. But there seems to be more mathematics, like algebras of operators, propagators and generators, hidden under the surface. That’s why this talk has two purposes: (i) discuss the physics of resonant optical modes, including a perturbative analysis and experimental proofs of some non-paraxial effects, (ii) discuss possibilities to extend the current analysis and improve its mathematical basis. 

Abstract: 

The Janzen-Connell (J-C) hypothesis plays a central role in understanding biodiversity in ecological systems describing vegetation. Although the J-C hypothesis is a few decades old, and accepted to hold by the majority of ecologists, it has been proven very difficult to verify, due to the highly complex nature of vegetation dynamics, and the general class of vegetation systems to which the J-C hypothesis applies.

I have been part of a collaboration with ecologists to investigate the J-C hypothesis through modelling and analysis. I will describe the model we have developed, and will tell the story of our analytical investigations -- what we encountered, what we could and could not do, what I struggled with, and how useful mathematics can be to understand and (partially) verify the J-C hypothesis.

Abstract: 

Traveling pulses, and more generally waves, are an incredibly rich subset of feasible patterns in reaction-diffusion equations. Plentiful works have investigated their existence and stability, but what happens if the deterministic dynamics is affected by random occurrences? What kind of effect does noise have on the dynamics and thus on the previously observed behavior?

We will investigate reaction-diffusion systems, with spatial dimension two or higher, which are stochastically forced. Particularly, we add a small multiplicative noise term to the equation that is white in time, colored in space, and invariant under translations.  This translational invariance property of the noise is a natural assumption to pose, inspired by applications.  The multidimensional stability of planar traveling waves for the deterministic setting has been studied by many, see for instance  T. Kapitula (1997), Multidimensional Stability of Planar Traveling Waves. Existence and stability of wave profiles in one spatial dimension also has been established, by means of a phase tracking method, in C.H.S. Hamster, H.J. Hupkes (2020), Traveling Waves for Reaction-Diffusion Equations Forced by Translation Invariant Noise. Inspired by these  works, we set up a phase tracking framework for the higher dimensional setting.

Abstract: 

Pattern formation is often nucleated when a localized disturbance grows and spreads, generating the pattern in the wake of this spreading. The marginal stability conjecture postulates that invasion speeds and selected patterns in the wake may be predicted from spectral properties of the invaded ground state and an associated invasion front. That is, only those patterns and speeds are selected for which the associated invasion front is marginally spectrally stable. So far, the marginal stability conjecture has not been established in any pattern-forming system, mainly because this would require a sharp stability theory for localized perturbations to selected pattern-forming fronts, which is not yet available. In this talk, I will present such a theory for the FitzHugh-Nagumo system, which was originally proposed as a simplification of the Hodgkin Huxley model, describing the propagation of impulses in nerve fibers, and has since then attracted much interest as a phenomenological model of pattern formation. I will argue that the FitzHugh-Nagumo system admits families of marginally spectrally stable pattern-forming fronts and explain how their linear and nonlinear stability against suitably localized perturbations can be obtained. This is joint work with Montie Avery, Paul Carter and Arnd Scheel.

Abstract:

In this talk I’ll give an overview of the mathematical research on patterns in inhomogeneous systems I’ve been involved in. The set-up will be along (personal) historical lines: from the problem of ‘pinned fluxons’ in superconducting Josephson junctions coming from Twente, via inspiring and challenging simulations of a FitzHugh-Nagumo type ‘paradigm model’ in Japan, to – eventually – topographical effects in ecosystem models.  As is the nature of much mathematical research, the focus has initially been on the persistence of known structures – such as pulses and/or fronts – under perturbation by the inhomogeneous effects. I’ll steer the talk towards a discussion of a perhaps more interesting question: “What kind of new structures, i.e. structures that cannot exist in the (typically well-studied and well-understood) homogeneous case, may be created by inhomogeneous effects?”. Naturally, we will only consider this (way too general) question in the setting of specific (and as simple as possible) conceptual models.  

Abstract:

Machine learning techniques have become immensely popular for image recognition, playing games, and much more. To extend their use to physical systems described by differential equations with uncertainties, such as climate and energy applications, we propose to embed "structure" in machine learning methods. We call this "scientific machine learning". For example, by embedding physical conservations laws into neural networks architectures, we are able to "learn more with less": networks that generalize better, are more robust, and need less data.

Abstract: 

Pattern formation and modulation is an active branch of mathematics, not only from the perspective of fundamental theory but also for its huge applications in many fields. In this talk, the focus is given to pattern formation in two-compartments hyperbolic reaction-transport models, with particular emphasis on the role played by the inertial times on the dynamics of non-stationary spatially-periodic patterns. 

In the first part of the talk, the study of oscillatory periodic patterns arising in reaction-transport hyperbolic models is addressed in a framework where both species undergo diffusion but only one is subject to advection. Linear stability analysis is firstly addressed to deduce the conditions under which wave (or Turing-Hopf) instability takes place whereas multiple-scale weakly nonlinear analysis is applied to determine the equation which rules the spatio-temporal evolution of pattern amplitude close to criticality. Some intriguing consequences due to hyperbolicity are emphasized. 

Then, the focus is moved on hyperbolic models where both species undergo advection but only one is subject to diffusion. In this framework, linear stability analysis is performed to deduce the instability threshold and travelling wave solutions are inspected to extract more information on the wave speed.

To show in detail the richness of the abovementioned predictions, illustrative examples are provided through hyperbolic generalizations of the extended Klausmeier model where vegetation patterns arise along the hillside of semiarid terrains.

Abstract: 

For the sake of computational efficiency, in mathematical modelling, the Dirac Delta distributions are often utilized as a replacement for ​cells or vesicles, since the size of cells or vesicles is much smaller than the size of the surrounding tissue. In the current manuscript, we consider the scenario that the cell or the vesicle releases diffusive compounds to the immediate environment, which is modelled by the diffusion equation. Typically, one separates the intracellular and extracellular environment and uses homogeneous Neumann boundary condition for the cell boundary, while the Dirac delta approach neglects the intracellular environment. Hence, extra conditions are needed such that the solutions to the two approaches are consistent. The necessary and essential condition of consistency ​in the above setting has been derived and proved analytically. Suggested by the numerical results, we conclude that an initial condition in the form of a Gaussian kernel in the Dirac delta approach compensates for the discrepancy between the solutions to the two approaches in the numerical solutions, and various approaches determining the amplitude and variance of the Gaussian distribution have been discussed.

Abstract: 

Joint work with M. Eppinga, M. Baudena, F. Veerman, M. Rietkerk, and F. Giannino

A widely accepted explanation for the maintenance of tree species biodiversity in tropical (rain)forests is the Janzen-Connell hypothesis, which states that growth of seedlings is suppressed in the proximity of conspecific adult trees. As a result, optimal seedling growth occurs farther from conspecific adult trees than expected from seed dispersal patterns. Mounting evidence suggests that the accumulation of host-specific soil-borne pathogens, and the resulting negative plant-soil feedback, constitutes a mechanism driving spatial distributions of seedlings and parent trees in tropical (rain)forests. However, negative plant-soil feedback may involve additional mechanisms, including litter decomposition processes releasing host-specific harmful chemical compounds (i.e. autotoxicity). While soil-borne pathogens and autotoxicity may impact tree seedlings during different life stages, it is currently unclear whether this differential timing of plant-soil feedback affects the transient dynamics of tropical forest tree distributions. In our work, we theoretically explore how both mechanisms contribute to negative plant-soil feedback, and to what extent their presence is reflected in transient and stable distributions of tropical forest trees and seedlings.

By considering soil-borne pathogens and autotoxicity both separately and in combination, we can understand the influence of both factors on transient dynamics and emerging Janzen-Connell distributions. We also identify parameter regimes associated with different long-term behaviours. Moreover, we compare how the strength of negative plant-soil feedback is mediated by germination and growth strategies, using a combination of analytical approaches and numerical simulations.

Abstract: 

The Shigesada-Kawasaki-Teramoto model (SKT) was proposed to account for stable inhomogeneous steady states exhibiting spatial segregation, which describes a situation of

coexistence of two competing species. Even though the reaction part does not present the activator-inhibitor structure, the cross-diffusion terms are the key ingredient for the appearance

of spatial patterns. We provide a deeper understanding of the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearised and weakly non-linear analysis with advanced numerical bifurcation methods via the continuation software pde2path. We study the role of the additional cross-diffusion term in pattern formation, focusing on multistability regions and on the presence of time-periodic spatial patterns appearing via Hopf bifurcation points.

[1] C. Soresina, Hopf bifurcation in the full SKT model and where to find them, DCDS-S, 15(9):2673-2693, 2022.

Abstract: 

The hormone auxin plays a critical role in stimulating the growth of plants. We study a model of auxin propagation through an infinite, one-dimensional chain of neighboring cells and impose a traveling wave structure on the concentrations of auxin and related hormones. We solve the resulting nonlocal system in a long wave limit inspired by Friesecke and Pego’s analysis of monatomic Fermi-Pasta-Ulam-Tsingou lattice traveling waves. We find that auxin propagates as a solitary wave through the cell chain, while certain auxiliary proteins spread out as fronts. Our results uncover close connections between the biological parameters of the hormone system, the amplitude-wavelength scalings, and the leading order behavior of the traveling wave profiles. We provide ample numerical evidence that motivates and justifies our particular long wave scalings and that indicates the stability of our traveling waves.  This is joint work with Bente Hilde Bakker, Hermen Jan Hupkes, Roeland Merks, and Jelle van der Voort.