Group Meeting
Abstract
Tmoslav Plesa: Chemical Reaction Systems with a Homoclinic Bifurcation: An Inverse Problem, 25+5 min;
John Ockendon: Wave Homogenisation, 10 min + questions;
Hilary Ockendon: Sloshing, 10 min + questions
Forthcoming events in this series
Tmoslav Plesa: Chemical Reaction Systems with a Homoclinic Bifurcation: An Inverse Problem, 25+5 min;
John Ockendon: Wave Homogenisation, 10 min + questions;
Hilary Ockendon: Sloshing, 10 min + questions
We study the motion of a eukaryotic cell on a substrate and investigate the dependence of this motion on key physical parameters such as strength of protrusion by actin filaments and adhesion. This motion is modeled by a system of two PDEs consisting of the Allen-Cahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion perturbed by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters. In the subcritical regime, the well-posedness of the problem is proved (M. Mizuhara et al., 2015). Our main focus is the supercritical regime where we established surprising features of the motion of the interface such as discontinuities of velocities and hysteresis in the 1D model, and instability of the circular shape and rise of asymmetry in the 2D model. Because of properties (i)-(ii), classical comparison principle techniques do not apply to this system. Furthermore, the system can not be written in a form of gradient flow, which is why Γ-convergence techniques also can not be used. This is joint work with V. Rybalko and M. Potomkin.
Much of the recent interest in complex networks has been driven by the prospect that network optimization will help us understand the workings of evolutionary pressure in natural systems and the design of efficient engineered systems. In this talk, I will reflect on unanticipated attributes and artifacts in three classes of network optimization problems. First, I will discuss implications of optimization for the metabolic activity of living cells and its role in giving rise to the recently discovered phenomenon of synthetic rescues. Then I will comment on the problem of controlling network dynamics and show that theoretical results on optimizing the number of driver nodes/variables often only offer a conservative lower bound to the number actually needed in practice. Finally, I will discuss the sensitive dependence of network dynamics on network structure that emerges in the optimization of network topology for dynamical processes governed by eigenvalue spectra, such as synchronization and consensus processes. Optimization is a double-edged sword for which desired and adverse effects can be exacerbated in complex network systems due to the high dimensionality of their dynamics.
Networks form the backbones of a wide variety of complex systems,
ranging from food webs, gene regulation and social networks to
transportation networks and the internet. Due to the sheer size and
complexity of many of theses systems, it remains an open challenge to
formulate general descriptions of their large-scale structures.
Although many methods have been proposed to achieve this, many of them
yield diverging descriptions of the same network, making both the
comparison and understanding of their results very
difficult. Furthermore, very few methods attempt to gauge the
statistical significance of the uncovered structures, and hence the
majority cannot reliably separate actual structure from stochastic
fluctuations. In this talk, I will show how these issues can be tackled
in a principled fashion by formulating appropriate generative models of
network structure that can have their parameters inferred from data. I
will also consider the comparison between a variety of generative
models, including different structural features such as degree
correction, where nodes with arbitrary degrees can belong to the same
group, and community overlap, where nodes are allowed to belong to more
than one group. Because such model variants possess an increased number
of parameters, they become prone to overfitting. We demonstrate how
model selection based on the minimum description length criterion and
posterior odds ratios can fully account for the increased degrees of
freedom of the larger models, and selects the most appropriate trade-off
between model complexity and quality of fit based on the statistical
evidence present in the data.
Throughout the talk I will illustrate the application of the methods
with many empirical networks such as the internet at the autonomous
systems level, the global airport network, the network of actors and
films, social networks, citations among websites, co-occurrence of
disease-causing genes and many others.
Noise limits are one of the major constraints when designing
aircraft engines. Acoustic liners are fitted in almost all civilian
turbofan engine intakes, and are being considered for use elsewhere in a
bid to further reduce noise. Despite this, models for acoustic liners
in flow have been rather poor until recently, with discrepancies of 10dB
or more. This talk will show why, and what is being done to model them
better. In the process, as well as mathematical modelling using
asymptotics, we will show that state of the art Computational
AeroAcoustics simulations leave a lot to be desired, particularly when
using optimized finite difference stencils.
Michael Gomez:
Title: The role of ghosts in elastic snap-through
Abstract: Elastic `snap-through' buckling is a striking instability of many elastic systems with natural curvature and bistable states. The conditions under which bistability exists have been reasonably well studied, not least because a number of engineering applications make use of the rapid transitions between states. However, the dynamics of the transition itself remains much less well understood. Several examples have been studied that show slower dynamics than would be expected based on purely elastic timescales of motion, with the natural conclusion drawn that some other effect, such as viscoelasticity, must play a role. I will present analysis (and hopefully experiments) of a purely elastic system that shows similar `anomalous dynamics'; however, we show that here this dynamics is a consequence of the ‘ghost’ of the snap-through bifurcation.
Andrew Krause:
Title: Fluid-Growth Interactions in Bioactive Porous Media
Abstract: Recent models in Tissue Engineering have considered pore blocking by cells in a porous tissue scaffold, as well as fluid shear effects on cell growth. We implement a suite of models to better understand these interactions between cell growth and fluid flow in an active porous medium. We modify some existing models in the literature that are spatially continuous (e.g. Darcy's law with a cell density dependent porosity). However, this type of model is based on assumptions that we argue are not good at describing geometric and topological properties of a heterogeneous pore network, and show how such a network can emerge in this system. Therefore we propose a different modelling paradigm to directly describe the mesoscopic pore networks of a tissue scaffold. We investigate a deterministic network model that can reproduce behaviour of the continuum models found in the literature, but can also exhibit finite-scale effects of the pore network. We also consider simpler stochastic models which compare well with near-critical Percolation behaviour, and show how this kind of behaviour can arise from our deterministic network model.
Feedforward layers are integral step in processing and transmitting sensory information across different regions the brain. Yet experiments reveal the difficulty of stable propagation through layers without causing neurons to synchronize their activity. We study the limits of stable propagation in a discrete feedforward model of binary neurons. By analyzing the spectral properties of a mean-field Markov chain model, we show when such information transmission persists. Addition of inhibitory neurons and synaptic noise increases the robustness of asynchronous rate transmission. We close with an example of feedforward processing in the input layer to cerebellum.
We consider the impact of spatial heterogeneities on the dynamics of
localized patterns in systems of partial differential equations (in one
spatial dimension). We will mostly focus on the most simple possible
heterogeneity: a small jump-like defect that appears in models in which
some parameters change in value as the spatial variable x crosses
through a critical value -- which can be due to natural inhomogeneities,
as is typically the case in ecological models, or can be imposed on the
model for engineering purposes, as in Josephson junctions. Even such a
small, simplified heterogeneity may have a crucial impact on the
dynamics of the PDE. We will especially consider the effect of the
heterogeneity on the existence of defect solutions, which boils down to
finding heteroclinic (or homoclinic) orbits in an n-dimensional
dynamical system in `time' x, for which the vector field for x > 0
differs slightly from that for x < 0 (under the assumption that there is
such an orbit in the homogeneous problem). Both the dimension of the
problem and the nature of the linearized system near the limit points
have a remarkably rich impact on the defect solutions. We complement the
general approach by considering two explicit examples: a heterogeneous
extended Fisher–Kolmogorov equation (n = 4) and a heterogeneous
generalized FitzHugh–Nagumo system (n = 6).
Although not all complex networks are embedded into physical spaces, it is possible to find an abstract Euclidean space in which they are embedded. This Euclidean space naturally arises from the use of the concept of network communicability. In this talk I will introduce the basic concepts of communicability, communicability distance and communicability angles. Both, analytic and computational evidences will be provided that shows that the average communicability angle represents a measure of the spatial efficiency of a network. We will see how this abstract spatial efficiency is related to the real-world efficiency with which networks uses the available physical space for classes of networks embedded into physical spaces. More interesting, we will show how this abstract concept give important insights about properties of networks not embedded in physical spaces.
I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random data particle simulations, flocks and mills, and their qualitative behavior.
Despite many years of intensive research, the modeling of contact lines moving by spreading and/or evaporation still remains a subject of debate nowadays, even for the simplest case of a pure liquid on a smooth and homogeneous horizontal substrate. In addition to the inherent complexity of the topic (singularities, micro-macro matching, intricate coupling of many physical effects, …), this also stems from the relatively limited number of studies directly comparing theoretical and experimental results, with as few fitting parameters as possible. In this presentation, I will address various related questions, focusing on the physics invoked to regularize singularities at the microscale, and discussing the impact this has at the macroscale. Two opposite “minimalist” theories will be detailed: i) a classical paradigm, based on the disjoining pressure in combination with the spreading coefficient; ii) a new approach, invoking evaporation/condensation in combination with the Kelvin effect (dependence of saturation conditions upon interfacial curvature). Most notably, the latter effect enables resolving both viscous and thermal singularities altogether, without needing any other regularizing effects such as disjoining pressure, precursor films or slip length. Experimental results are also presented about evaporation-induced contact angles, to partly validate the first approach, although it is argued that reality might often lie in between these two extreme cases.
This talk covers two topics: (1) Phenotype change, where we consider the steady-fitness states, in a model developed by Korobeinikov and Dempsey (2014), in which the phenotype is modelled on a continuous scale providing a structured variable to quantify the phenotype state. This enables thresholds for survival/extinction to be established in terms of fitness.
Topic (2) looks at the steady-size distribution of an evolving cohort of cells, such as tumour cells in vitro, and therein establishes thresholds for growth or decay of the cohort. This is established using a new class of non-local (but linear) singular eigenvalue problems which have point spectra, like the traditional Sturm-Liouville problems. The first eigenvalue gives the threshold required. But these problems are first order unless dispersion is added to incorporate random perturbations. But the same idea will apply here also. Current work involves binary asymmetrical division of cells, simultaneous with growth. It has implications to cancer biology, helping biologists to conceptualise non-local effects and the part they may play in cancer. This is developed in Zaidi et al (2015).
Acknowledgement. The support of Gravida (NCGD) is gratefully acknowledged.
References
Korobeinikov A & Dempsey C. A continuous phenotype space model of RNA virus evolution within a host. Mathematical Biosciences and Engineering 11, (2014), 919-927.
Zaidi AA, van-Brunt B, & Wake GC. A model for asymmetrical cell division Mathematical Biosciences and Engineering (June 2015).
It is well known that low-Reynolds-number flows ($R_e\ll1$) have unique solutions, but this statement may not be true if complex solutions are permitted.
We begin by considering Stokes series, where a general steady velocity field is expanded as a power series in the Reynolds number. At each order, a linear problem determines the coefficient functions, providing an exact closed form representation of the solution for all Reynolds numbers. However, typically the convergence of this series is limited by singularities in the complex $R_e$ plane.
We employ a generalised Pade approximant technique to continue analytically the solution outside the circle of convergence of the series. This identifies other solutions branches, some of them complex. These new solution branches can be followed as they boldly go where no flow has gone before. Sometimes these complex solution branches coalesce giving rise to real solution branches. It is shown that often, an unforced, nonlinear complex "eigensolution" exists, which implies a formal nonuniqueness, even for small and positive $R_e$.
Extensive reference will be made to Dean flow in a slowly curved pipe, but also to flows between concentric, differentially rotating spheres, and to convection in a slot. In addition, certain fundamental exact solutions are shown to possess extra complex solutions.
by Jonathan Mestel and Florencia Boshier
In today's interconnected world, the dissemination of an idea, a trend, a rumor through social networks, as well as the propagation of information or cyber-viruses through digital networks are all common phenomena. They are conceptually similar to the spread of infectious diseases among hosts, as common to all these phenomena is the dissemination of a spreading agent on a networked system. A large body of research has been produced in recent years to characterize the spread of epidemics on static connectivity patterns in a wide range of biological and socio-technical systems. In particular, understanding the mechanisms and conditions for widespread dissemination represents a crucial step for its prevention and control (e.g. in the case of diseases) or for its enhancement (e.g. in the case of viral marketing). This task is however further hindered by the temporal nature characterizing the activation of the connections shaping the networked system, for which data has recently become available. As an example, in networks of proximity contacts among individuals, connections represent sequences of contacts that are active for given periods of time. The time variation of contacts in a networked system may fundamentally alter the properties of spreading processes occurring on it, with respect to static networks, and affect the condition at which epidemics become possible. In this talk I will present a novel theoretical framework adopting a multi-layer perspective for the analytical understanding of the interplay between temporal networks and spreading dynamics. The framework is tested on a set of time-varying network models and empirical networks.
Abstract: Motivated loosely by the problem of carbon sequestration in underground aquifers, I will describe computations and analysis of one-sided two-dimensional convection of a solute in a fluid-saturated porous medium, focusing on the case in which the solute decays via a chemical reaction. Scaling properties of the flow at high Rayleigh number are established and rationalized through an asymptotic model, that addresses the transient stability of a near-surface boundary layer and the structure of slender plumes that form beneath. The boundary layer is shown to restrict the rate of solute transport to deep domains. Knowledge of the plume structure enables slow erosion of the substrate of the reaction to be described in terms of a simplified free boundary problem.
Co-authors: KA Cliffe, H Power, DS Riley, TJ Ward
Particle-based stochastic reaction diffusion methods have become a
popular approach for studying the behavior of cellular processes in
which both spatial transport and noise in the chemical reaction process
can be important. While the corresponding deterministic, mean-field
models given by reaction-diffusion PDEs are well-established, there are
a plethora of different stochastic models that have been used to study
biological systems, along with a wide variety of proposed numerical
solution methods.
In this talk I will motivate our interest in such methods by first
summarizing several applications we have studied, focusing on how the
complicated ultrastructure within cells, as reconstructed from X-ray CT
images, might influence the dynamics of cellular processes. I will then
introduce our attempt to rectify the major drawback to one of the most
popular particle-based stochastic reaction-diffusion models, the lattice
reaction-diffusion master equation (RDME). We propose a modified version
of the RDME that converges in the continuum limit that the lattice
spacing approaches zero to an appropriate spatially-continuous model.
Time-permitting, I will discuss several questions related to calibrating
parameters in the underlying spatially-continuous model.
In order:
1. Michael Dallaston, "Modelling channelization under ice shelves"
2. Jeevanjyoti Chakraborty, "Growth, elasticity, and diffusion in
lithium-ion batteries"
3. Roberta Minussi, "Lattice Boltzmann modelling of the generation and
propagation of action potential in neurons"