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

21 January 2016

16:00

Tmoslav Plesa, John Ockendon, Hilary Ockendon

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

3 December 2015

16:00

Leonid v Berlyand

Abstract

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.

26 November 2015

16:00

Adilson E Motter

Abstract

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.

19 November 2015

16:00

Robert Style, Samuel Crew and Phil Trinh

Abstract

Samuel Crew (Lincoln College) and Philippe Trinh

In 1880, Stokes famously demonstrated that the singularity that occurs at the crest of the steepest possible water wave in infinite depth must correspond to a corner of 120°. Here, the complex velocity scales like the one-third power of the complex potential. Later in 1973, Grant showed that for any wave away from the steepest configuration, the singularity moves into the complex plane, and is instead of order one-half. Grant conjectured that as the highest wave is approached, other singularities must coalesce at the crest so as to cancel the square-root behaviour. Even today, it is not well understood how this process occurs, nor is it known what other singularities may exist.

In this talk, we shall explain how we have been able to construct the Riemann surface that represents the extension of the water wave into the complex plane. We shall also demonstrate the existence of a countably infinite number of singularities, never before noted, which coalesce as Stokes' highest wave is approached. Our results demonstrate that the singularity structure of a finite amplitude wave is much more complicated than previously anticipated,

12 November 2015

16:00

Tiago Peixoto

Abstract

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.

5 November 2015

16:00

Ed Brambley

Abstract

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.

29 October 2015

16:00

Michael Gomez, Jake Taylor-King, Andrew Krause, Zach Wilmott

Abstract

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.

Jake Taylor-King

Title:A Kinetic Approach to Evolving Spatial Networks, with an Application to Osteocyte Network Formation

Abstract:We study an evolving network where the nodes are considered as represent particles with a corresponding state vector. Edges between nodes are created and destroyed as a Poisson process, and new nodes enter the system. We define the concept of a “local state degree distribution” (LSDD) as a degree distribution that is local to a particular point in phase space. We then derive a differential equation that is satisfied approximately by the LSDD under a mean field assumption; this allows us to calculate the degree distribution. We examine the validity of our derived differential equation using numerical simulations, and we find a close match in LSDD when comparing theory and simulation. Using the differential equation derived, we also propose a continuum model for osteocyte network formation within bone. The structure of this network has implications regarding bone quality. Furthermore, osteocyte network structure can be disrupted within cancerous microenvironments. Evidence suggests that cancerous osteocyte networks either have dendritic overgrowth or underdeveloped dendrites. This model allows us to probe the density and degree distribution of the dendritic network. We consider a traveling wave solution of the osteocyte LSDD profile which is of relevance to osteoblastic bone cancer (which induces net bone formation). We then hypothesise that increased rates of differentiation would lead to higher densities of osteocytes but with a lower quantity of dendrites.

Abstract:We study an evolving network where the nodes are considered as represent particles with a corresponding state vector. Edges between nodes are created and destroyed as a Poisson process, and new nodes enter the system. We define the concept of a “local state degree distribution” (LSDD) as a degree distribution that is local to a particular point in phase space. We then derive a differential equation that is satisfied approximately by the LSDD under a mean field assumption; this allows us to calculate the degree distribution. We examine the validity of our derived differential equation using numerical simulations, and we find a close match in LSDD when comparing theory and simulation. Using the differential equation derived, we also propose a continuum model for osteocyte network formation within bone. The structure of this network has implications regarding bone quality. Furthermore, osteocyte network structure can be disrupted within cancerous microenvironments. Evidence suggests that cancerous osteocyte networks either have dendritic overgrowth or underdeveloped dendrites. This model allows us to probe the density and degree distribution of the dendritic network. We consider a traveling wave solution of the osteocyte LSDD profile which is of relevance to osteoblastic bone cancer (which induces net bone formation). We then hypothesise that increased rates of differentiation would lead to higher densities of osteocytes but with a lower quantity of dendrites.

22 October 2015

16:00

Alex Cayco Gajic

Abstract

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.

15 October 2015

16:00

Arjen Doelman

Abstract

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).

18 June 2015

16:00

Prof. Ernesto Estrada

Abstract

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.