# Workshop Programme

## for period 14 - 16 March 2012

### Pattern Formation: The inspiration of Alan Turing

14 - 16 March 2012

Timetable

Wednesday 14 March | ||||

12:30-13:15 | Lunch in St John's College Dining Hall | |||

13:10-13:40 | Registration | |||

13:40-13:50 | Opening & Welcome | |||

13:50-14:40 | Garfinkel, A (University of California, Los Angeles) |
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Far from Turing? Turing’s Paradigm in Physiology | ||||

Turing’s original model was a linear instability in a 2 variable PDE, describing imaginary “morphogens” reacting and diffusing in a 2D domain. Since the discovery of physiological morphogens in the past few decades, even this simple model has had successful applications. The growing maturity of the applications has now led modelers to more complex scenarios. Developments have included the extension of the original model to include cell density variables, the inclusion of mechanical factors, the extension to 3D spatial domains, and the study of patterns, such as isolated spots, that occur far from the linear instability first studied by Turing. We will review examples of these new developments in the field of physiology and pathophysiology. |
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14:40-14:50 | Break | |||

14:50-15:40 | Painter, K (Heriot-Watt University) |
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Integrating experiment and theory to elucidate the chemical basis of hair and feather morphogenesis | ||||

In his seminal 1952 paper, ‘The Chemical Basis of Morphogenesis’, Alan Turing lays down a milestone in the application of theoretical approaches to understand complex biological processes. The molecular revolution that has taken place during the six decades following this landmark publication has now placed this generation of theoreticians and biologists in an excellent position to rigorously test the theory and, encouragingly, a number of systems have emerged that appear to conform to some of Turing’s fundamental ideas. In this talk I will describe how the integration between experiment and theory has been used to enhance understanding in a model system of embryonic patterning: the emergence of feathers and hair in the skins of birds and mammals. |
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15:40-15:50 | Break | |||

15:50-16:10 | Grace, M; Hütt, M-T (Jacobs University Bremen) |
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Spiral-wave prediction in a lattice of FitzHugh-Nagumo oscillators | ||||

In many biological systems, variability of the components can be expected to outrank statistical fluctuations in the shaping of self-organized patterns. The distribution of single-element properties should thus allow the prediction of features of such patterns. In a series of previous studies on established computational models of Dictyostelium discoideum pattern formation we demonstrated that the initial properties of potentially very few cells have a driving influence on the resulting asymptotic collective state of the colony [1,2]. One plausible biological mechanism for the generation of variability in cell properties and of spiral wave patterns is the concept of a ‘developmental path’, where cells gradually move on a trajectory through parameter space. Here we review the current state of knowledge of spiral-wave prediction in excitable systems and present a new one-dimensional developmental path based on the FitzHugh-Nagumo model, incorporating parameter drift and concomitant variability in the distribution of cells embarking on this path, which gives rise to stable spiral waves. Such a generic model of spiral wave predictability allows new insights into the relationship between biological variability and features of the resulting spatiotemporal pattern. [1] Geberth, D. and Hütt, M.-Th. (2008) Predicting spiral wave patterns from cell properties in a model of biological self-organization. Phys. Rev. E 78, 031917. [2] Geberth, D. and Hütt, M.-Th. (2009) Predicting the distribution of spiral waves from cell properties in a developmental-path model of Dictyostelium pattern formation. PLoS Comput. Biol 5, e1000422. |
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16:10-16:30 | Heaton, L; Jones, N S; Lopez, E; Maini, P; Fricker, M (University of Oxford & Imperial College London) |
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Transport and development in fungal networks | ||||

Multi-cellular organisms have evolved sophisticated systems to supply individual cells with the resources necessary for survival. Plants circulate nutrients through the xylem and phloem, mammals have cardio-vascular systems, but how do fungi translocate materials? Cord-forming fungi form extensive networks that continuously adapt to their surroundings, but what is the developmental logic of such fungal networks, and how does fungal morphology enable efficient transport? In this talk I shall address these fundamental questions, and present the concept of growth-induced mass flows. The key idea is that aqueous fluids are incompressible, so as the fluid filled vessel expand, there must be movement of fluid from the sites of water uptake to the sites of growth. We have developed a model of delivery in growing fungal networks, and found good empirical agreement between our model and experimental data gathered using radio-labeled tracers. Our results lead us to suggest that in fora ging fungi, growth-induced mass flow is sufficient to account for long distance transport, if the system is well insulated. We conclude that active transport mechanisms may only be required at the very end of the transport pathway, near the growing tips. |
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16:30-16:50 | Gaffney, E; Lee, S S (University of Oxford & RIKEN, Japan) |
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The Sensitivity of Turing's Pattern Formation Mechanism | ||||

The prospect of long range signalling by diffusible morphogens initiating large scale pattern formation has been contemplated since the initial work of Turing in the 1950s and has been explored theoretically and experimentally in numerous developmental settings. However, Turing’s pattern formation mechanism is notorious for its sensitivity to the details of the initial conditions suggesting that, in isolation, it cannot robustly generate pattern within noisy biological environments. Aspects of developmental self-organisation, in particular a growing domain, have been suggested as a mechanism for robustly inducing a sequential cascade of self-organisation, thus circumventing the difficulties of sensitivity. However the sensitivity question emerges once more for generalisations of Turing's model which include further biological aspects, for example, the inclusion of gene expression dynamics: this will be explored in detail. |
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16:50-17:10 | Break | |||

17:10-18:00 | Mimura, M (Meiji University, Kanazawa, Japan) |
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Turing's instability versus cross-diffusion-driven instability | ||||

Turing' diffusion-driven instability has been observed in diverse complex and regularized spatio and/or temporal patterns in not only scientific but also engineering fields. Another instability on pattern formation, such ascross-diffusion-driven instability or chemotactic instability is discussed in ecological and biological systems. In this talk, I am concerned with the relation between two types of instabilities above. As an application, I discuss self-organized aggregation of biological individuals with aggregation pheromones. Keywords: cross-diffusion, active aggregation, infinite dimensional relaxation oscillation. |
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18:00-18:10 | Break | |||

18:10-19:00 | Sherratt, J (Heriot-Watt University) |
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Patterns of Sources and Sinks in Oscillatory Systems | ||||

In oscillatory systems, a fundamental spatiotemporal pattern is wavetrains, which are spatially periodic solutions moving with constant speed (also known as periodic travelling waves). In many numerical simulations, one observes finite bands of wavetrains, separated by sharp interfaces known as "sources" and "sinks". This talk is concerned with patterns of sources and sinks in the complex Ginzburg-Landau equation with zero linear dispersion; in this case the CGLE is a reaction-diffusion system. I will show that patterns with large source-sink separations occur in a discrete family, due to a constraint of phase-locking type on the distance between a source and its neighbouring sinks. I will then consider the changes in source-sink patterns in response to very slow increases in the coefficient of nonlinear dispersion. I will present numerical results showing a cascade of splittings of sources into sink-source-sink triplets, culminating in spatiotemporal chao s at a parameter value that matches the change in absolute stability of the underlying periodic travelling wave. In this case the gradual change in pattern form represents an ordered, structured transition from a periodic solution to spatiotemporal chaos. The work that I will present was done in collaboration with Matthew Smith (Microsoft Research. Cambridge) and Jens Rademacher (CWI, Amsterdam). |

Thursday 15 March | ||||

09:30-10:20 | Epstein, I (Brandeis University) |
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Experimental and Modeling Studies of Turing Patterns in Chemical Systems | ||||

I will review three decades of work on Turing patterns in our laboratory, from the discovery of pattern formation in the CIMA reaction to recent experiments on three-dimensional patterns using tomographic techniques. I will discuss modeling efforts (including the Lengyel-Epstein model), approaches to designing systems that display Turing patterns, the effects of external perturbation, and the role of “structured media” (gels, microemulsions, surfaces). |
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10:20-10:40 | Break | |||

10:40-11:00 | Holzer, M; Scheel, A (University of Minnesota) |
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A slow pushed front in a Lotka-Volterra competition model | ||||

We study the existence and stability of a traveling front in the Lotka-Volterra competition model when the rate of diffusion of one species is small. This front represents the invasion of an unstable homogeneous state by a stable one. It is noteworthy in two respects. First, we show that this front is the selected, or critical, front for this system. We utilize techniques from geometric singular perturbation theory and geometric desingularization. Second, we show that this front appears to be a pushed front in all ways except for the fact that it propagates slower than the linear spreading speed. We show that this is a result of the linear spreading speed arising as a simple pole of the resolvent instead of a branch pole. Using the pointwise Green's function, we show that this pole poses no a priori obstacle to stability of the nonlinear traveling front. |
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11:00-11:20 | Venkataraman, C (University of Warwick) |
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Turing patterns on growing surfaces | ||||

We investigate models for biological pattern formation via reaction-diﬀusion systems posed on continuously evolving surfaces. The nonlinear reaction kinetics inherent in the models and the evolution of the spatial domain mean that analytical solutions are generally unavailable and numerical simulations are necessary. In the ﬁrst part of the talk, we examine the feasibility of reaction-diﬀusion systems to model the process of parr mark pattern formation on the skin surface of the Amago trout. By simulating a reaction-diﬀusion system on growing surfaces of diﬀering mean curvature, we show that the geometry of the surface, speciﬁcally the surface curvature, plays a central role in the patterns generated by a reaction-diﬀusion mechanism. We conclude that the curvilinear geometry that characterises ﬁsh skin should be taken into account in future modelling endeavours. In the second part of the talk, we investigate a model for cell motility and chemotaxis. Our model consists of a surface reaction-diﬀusion system that models cell polarisation coupled to a geometric evolution equation for the position of the cell membrane. We derive a numerical method based on surface ﬁnite elements for the approximation of the model and we present numerical results for the migration of two and three dimensional cells. |
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11:20-11:40 | Fleck, C; Greese, B; Huelskamp, M (ZBSA, University of Freiburg & Botanical Institute, University of Cologne) |
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Influence of cell-to-cell variability on spatial pattern formation | ||||

Many spatial patterns in biology arise through differentiation of selected cells within a tissue, which is regulated by a genetic network. This is specified by its structure, its parameterisation, and the noise on its components and reactions. The latter, in particular, is not well examined because it is rather difficult to trace. Using experimental data on trichomes, i.e., epidermal plant hairs, as an example, we examine the variability in pattern formation that is due to small differences among the cells involved in the patterning process. We employ suitable local mathematical measures based on the Voronoi diagram of the trichome positions to determine the noise level in of the pattern. Although trichome initiation is a highly regulated process we show that the experimentally observed trichome pattern is substantially disturbed by cell-to-cell variations. Using computer simulations we find that the rates concerning the availability of the protein complex which triggers trichome formation plays a significant role in noise-induced variations of the pattern. The focus on the effects of cell-to-cell variability yields further insights into pattern formation of trichomes. We expect that similar strategies can contribute to the understanding of other differentiation processes by elucidating the role of naturally occurring fluctuations in the concentration of cellular components or their properties. |
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11:40-12:00 | Break | |||

12:00-12:50 | Nishiura, Y (Tohoku University) |
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Collision dynamics in dissipative systems | ||||

Spatially localized dissipative structures are ubiquitous such as vortex, chemical blob, discharge patterns, granular patterns, and binary convective motion. When they are moving, it is unavoidable to observe various types of collisions. One of the main questions for the collision dynamics is that how we can describe the large deformation of each localized object at collision and predict its output. The strong collision usually causes topological changes such as merging into one body or splitting into several parts as well as annihilation. It is in general quite difficult to trace the details of the deformation unless it is a very weak interaction. We need a change in our way of thinking to solve this issue. So far we may stick too much to the deformation of each localized pattern and become shrouded in mystery. We try to characterize the hidden mechanism behind the deformation process instead. It may be instructive to think about the following metaphor: the droplet falling down the landscape with many valleys and ridges. The motion of droplets on such a rugged landscape is rather complicated; two droplets merge or split at the saddle points and they may sin k into the underground, i.e., annihilation. On the other hand, the profile of the landscape remains unchanged and in fact it controls the behaviors of droplets. It may be worth to describe the landscape itself rather than complex deformation, namely to find where is a ridge or a valley, and how they are combined to form a whole landscape. Such a change of viewpoint has been proposed recently claiming that the network of unstable patterns relevant to the collision process constitutes the backbone structure of the deformation process, namely the deformation is guided by the connecting orbits among the nodes of the network. Each node is typically an unstable ordered pattern such as steady state or time-periodic solution. This view point is quite useful not only for the problems mentioned above but also for more generalized collision problems, especially, the dynamics in heterogeneous media is one of the interesting applications, since the encounter with heterogeneity can be regarded as a collision. Similarly questions of adaptability to external environments in biological systems fall in the above framework when they are reformulated in an appropriate way. In summary a highly unstable and transient state arising in collision problems is an organizing center which produces various outputs in order to adjust the emerging environments. |
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12:50-14:00 | Lunch in St John's College Dining Hall | |||

14:00-14:50 | Dahlem, M (Technische Universität Berlin) |
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Localized travelling pulses in mirgaine | ||||

Cortical spreading depression (SD) is a massive but transient perturbation in the brain's ionic homoeostasis. It is the underlying cause of neurological symptoms during migraine. I present a mechanism by which localized SD pulse segments are formed as long-lasting but transient patterns in a subexcitable medium, in which the homogeneous steady state is a global attractor. Initial perturbed states can develop into distinct transient pulses caused by a ghost of a saddle-node bifurcation that leads to a slow passage through a bottle-neck. The location of the bottle-neck in phase space is associated with a characteristic form (shape, size) of the pulse segment that depends on the curvature of the medium, i.e., the human cortex. Similar patterns have been observed with fMRI and in patient's symptom reports. The emerging transient patterns and their classification according to size and duration offers a model-based analysis of phase-depended stimulation protocols for non-i nvasive neuromodulation devices, e.g. utilizing transcranial magnetic stimulation, to intelligently target migraine. |
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14:50-15:10 | Break | |||

15:10-15:30 | Vaeth, M (Free Univ. of Berlin) |
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Spatial Patterns for Reaction-Diffusion Systems with Unilateral Conditions | ||||

It is well-known that Turing's effect can lead to stationary spatial pattern in reaction-diffusion systems of activator-inhibitor or substrate-depletion type. However, it is also known that this effect can occur only under certain assumptions about the diffusion speed. The aim of the talk is to discuss that conditions of unilateral type (obstacles) of various kind can lead to bifurcation of stationary spatial patterns even in a regime where classically no such phenomenon would occur. |
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15:30-15:50 | Pellicer, M (Universitat de Girona) |
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A PDE approach for the dynamics of the inflammatory stage in diabetic wound healing | ||||

Wound healing is an extremely complicated process and still not fully understood, moreover when diabetis mellitus is present. The inflammatory phase, the first one of this process, is where there exists a major difference between diabetic and nondiabetic wound healing. Here, we present a work in progress related with the modeling and analysis of the dynamics of some of the main agents involved in this first phase. We propose a reaction-diffusion system as a model for these dynamics. This model aims at generalizing the previous existing approach of J.Sherratt and H.Waugh, where an ODE system (only taking into account the time variable) is proposed as a simplified model for this situation. After obtaining this PDE approach, the well-posedness of the problem will be stated (both in a mathematical and a biological sense) and we will present some results related with the equilibria of the system. Finally, we will show some numerical simulations to illustrate the previous results. This is a joint work with Neus Consul (Universitat Politecnica de Catalunya) and Sergio M. Oliva (Universidade de Sao Paulo). |
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15:50-16:10 | Lee, S; Shibata, T (RIKEN, CDB) |
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Self-organisation of Cell Asymmetry : Turing’s inspiration is alive in a single cell | ||||

The development of multicellular organisms starts from a single fertilized egg but it finally involves the specification of diverse cell types. Such diversity is created via an asymmetric cell division, which is crucial to determine a distinct fate of the daughter cells and the most fundamental body plan for constructing a complex body system. The experimental investigation of the molecular levels proposes that differentially segregated protein or RNA determinants in the inside of a cell play a key role in the asymmetric cell divisions processes. The localization of specific proteins during an asymmetric cell division process is commonly observed in many model organisms, though related specific proteins or each step is observed in a slightly different way. Nonetheless, the schematic mechanism of the cell asymmetry division is still remained elusive. Moreover, the mechanism by which related proteins or RNA determinants can be localised to a specific region in a micro scale of small single cell is fully remained as a mystery. Thus to understand the mechanism by integrating the individual information of the molecular level is highly required. In this presentation, we present a mathematical model describing the asymmetric division process of C. elegans embryo cell. In particular, we explore the cortical flow effect on the localisation of membrane posterior PAR proteins and discuss the robustness of patterning length and timing arising in the establishment phase of the cell division. Finally, we show that how Turing’s spirit is alive in a single cell. |
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16:10-16:30 | Woolley, T; Baker, R E; Gaffney, E A; Maini, P K (University of Oxford) |
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Stochastic reaction and diffusion on growing domains: understanding the breakdown of robust pattern formation | ||||

All biological patterns, from population densities to animal coat markings, can be thought of as heterogeneous spatio-temporal distributions of mobile agents. Many mathematical models have been proposed to account for the emergence of this complexity, but, in general, they have consisted of deterministic systems of differential equations, which do not take into account the stochastic nature of population interactions. One particular, pertinent, criticism of these deterministic systems, is that the exhibited patterns can often be highly sensitive to changes in initial conditions, domain geometry, parameter values, etc. Due to this sensitivity, we seek to understand the effects of stochasticity and growth on paradigm biological patterning models. We extend spatial Fourier analysis and growing domain mapping techniques to encompass stochastic Turing systems. Through this we find that the stochastic systems are able to realise much richer dynamics than their deterministic counter parts, in that patterns are able to exist outside the standard Turing parameter range. Further, it is seen that the inherent stochasticity in the reactions appears to be more important than the noise generated by growth, when considering which wave modes are excited. Finally, although growth is able to generate robust pattern sequences in the deterministic case, we see that stochastic effects destroy this mechanism of pattern doubling. However, through Fourier analysis we are able to suggest a reason behind this lack of robustness and identify possible mechanisms by which to reclaim it. |
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16:30-16:50 | Break | |||

16:50-17:40 | Kondo, S (Nagoya University) |
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Turing pattern formation without diffusion | ||||

The reaction-diffusion mechanism, presented by AM Turing more than 60 years ago, is currently the most popular theoretical model explaining the biological pattern formation including the skin pattern. This theory suggested an unexpected possibility that the skin pattern is a kind of stationary wave (Turing pattern or reaction-diffusion pattern) made by the combination of reaction and diffusion. At first, biologists were quite skeptical to this unusual idea. However, the accumulated simulation studies have proved that this mechanism can not only produce various 2D skin patterns very similar to the real ones, but also predict dynamic pattern change of skin pattern on the growing fish. Now the Turing’s theory is accepted as a hopeful hypothesis, and experimental verification of it is awaited. Using the pigmentation pattern of zebrafish as the experimental system, our group in Osaka University has been studying the molecular basis of Turing pattern formation. We have identified the genes related to the pigmentation, and visualized the interactions among the pigment cells. With these experimental data, it is possible to answer the crucial question, “How is the Turing pattern formed in the real organism?” The pigmentation pattern of zebrafish is mainly made by the mutual interactions between the two types of pigment cells, melanophores and xanthophores. All of the interactions are transferred at the tip of the dendrites of pigment cells. In spite of the expectation of many theoretical biologists, there is no diffusion of the chemicals involved. However, we also found that the lengths of the dendrites are different among the interactions, which makes it possible to generate the conditions of Turing pattern formation, “local positive feedback and long range negative feedback”. Therefore, we think it is appropriate to call the identified mechanism as a Turing mechanism although it does not contain any diffusion. |
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17:50-18:40 | Jülicher, F (Max Planck Institute for the Physics of Complex Systems) |
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Pattern formation in active fluids | ||||

Biological matter is inherently dynamic and exhibits active properties. A key example is the force generation by molecular motors in the cell cytoskeleton. Such active processes give rise to the generation of active mechanical stresses and spontaneous flows in gel-like cytoskeletal networks. Active material behaviors play a key role for the dynamics of cellular processes such as cell locomotion and cell division. We will discuss intracellular flow patterns that are created by active processes in the cell cortex. By combining theory with quantitative experiments we show that observed flow patterns result from profiles of active stress generation in the system. We will also consider the situation where active stress is regulated by a diffusing molecular species. In this case, spatial concentration patterns are generated by the interplay of stress regulation and self-generated flow fields. |
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19:00-21:00 | Conference Dinner at St John's College, Oxford |

Friday 16 March | ||||

09:30-10:20 | Stevens, A (Westfalische Wilhelms-Universitat Munster) |
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Turing- and Non-Turing Type Pattern in Interacting Cell Systems | ||||

Examples for pattern formation in interacting cell sytems will be discussed, which result from direct cell-cell interaction and cellular motion. The analysis of the respective mathematical models - systems of partial differential equations of hyperbolic type and integro-differential equations - is partily done on the linearized level and partly done for suitable approximations. The long-time behavior of these models is discussed and the resulting patterns are set into context with experimental findings. |
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10:20-10:40 | Break | |||

10:40-11:00 | Walsh, J; Angstmann, C; Curmi, P (UNSW) |
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Linear Stability Analysis of Turing Patterns on Arbitrary Manifolds | ||||

Alan Turing’s mathematical model for pattern formation, based on linear instabilities in reaction-diffusion systems, has been widely applied in chemistry, biology and physics. Most of the modelling applications have been implemented on flat two dimensional domains, even though many of the patterns under investigation, including the celebrated application to animal coat patterns, occur on non-Euclidean two dimensional manifolds. In this work we have described an extension of Turing instability analysis to arbitrary manifolds. Our approach is simple to implement and it readily enables exploration of the effect of the geometry on Turing pattern formation in reaction-diffusion systems on arbitrarily shaped and sized domains. |
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11:00-11:20 | Madzvamuse, A (University of Sussex) |
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Pattern formation during growth development: models, numerics and applications | ||||

Mathematical modelling, numerical analysis and simulations of spatial patterning during growth development in developmental biology and biomedicine is an emerging young research area with significant potential of elucidating mechanisms for pattern formation on real biological evolving skin surfaces. Since the seminal work of Turing in 1952 which showed that a system of reacting and diffusing chemical morphogens could evolve from an initially uniform spatial distribution to concentration profiles that vary spatially - a spatial pattern - many models have been proposed on stationary domains exploiting the generalised patterning principle of short-range activation, long-range inhibition elucidated by Meinhardt of which the Turing model is an example. Turing's hypothesis was that one or more of the morphogens played the role of a signaling chemical, such that cell fate is determined by levels of morphogen concentration. However, our recent results show that in the presence o f domain growth, short-range inhibition, long-range activation as well as activator-activator mechanisms have the potential of giving rise to the formation of patterns only during growth development of the organism. These results offer us a unique opportunity to study non-standard mechanisms, either experimentally or hypothetically, for pattern formation on evolving surfaces, a largely unchartered research area. In this talk I will discuss modelling, numerical analysis, computations and applications of the models for pattern formation during growth. |
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11:20-11:40 | Kucera, M; Jaros, F; Vejchodsky, T (Institute of Mathematics of the Academy of Sciences of the Czech Republik) |
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The influence of non-standard boundary conditions on the generation of spatial patterns | ||||

The influence of certain unilateral boundary or interior conditions to spatial Turing's patterns described by reaction-diffusion systems will be discussed. The conditions considered can model sources reflecting concentration in their neighbourhood in the following way. If the concentration exceeds a given threshold then the source is inactive, when the concentration is about to decrease below the threshold then the source either prevents it or at least decelerates the decrease by producing a morphogen (or ligand) and supplementing it into an extracellular space. Some interesting consequences follow. For instance, spatial patterns can arise in general for an arbitrary ratio of diffusion speeds, e.g. for fast diffusion of activator and slow diffusion of inhibitor (the opposite situation than in Turing's original idea), and can arise also for arbitrarily small domains. Simple numerical simulations using a model proposed for a description of pigmentation in animals (in particular felids) promise to describe patterns with spots and, moreover, with a darker strip along the spine, which are observed among some felids. The unilateral conditions can be described mathematically by variational inequalities or inclusions. A more detailed explanation of the theory should be a subject of the talk of Martin Vaeth. |
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11:40-12:00 | Break | |||

12:00-12:50 | Bär, M (Physikalisch-Technische Bundesanstalt) |
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Pattern formation in multiscale systems: Homogenization and beyond | ||||

Various relevant pattern formation applications are observed in intrinsically heterogeneous reaction-diffusion systems. We derive a simple homogenization scheme and demonstrate that the resulting effective equations are sufficient to qualitatively reproduce the rich pattern dynamics of wave and Turing structures in the BZ-AOT microemulsion system. Furthermore, we validate this effective medium theory by simulations of wave propagation in discrete heterogeneous bistable and excitable media. We find that the approach fails if the heterogeneous medium is near a percolation threshold. For a simple discrete heterogeneous model of cardiac tissue, complex fractionated dynamics and reentrant dynamics appears in such a situation. Work done with Sergio Alonso, Raymond Kapral and Karin John. |
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12:50-14:00 | Lunch in St John's College Dining Hall | |||

14:00-14:50 | Scheel, A (University of Minnesota) |
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Wavenumber selection in closed reaction-diffusion systems | ||||

Motivated by the plethora of patterns observed in precipitation experiments starting with Liesegang's 1896 study, we investigate pattern formation in the wake of fronts in closed reaction-diffusion systems. We will briefly describe some models and the relation to phase separation models such as the Cahn-Hilliard equation and the Phase-Field System. We will then present results that characterize patterns formed in the wake of fronts. |
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14:50-15:10 | Break | |||

15:10-15:30 | Baier, G; Goodfellow, M; Taylor, P; Wang, Y; Garry, D (University of Manchester) |
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Epileptic Seizure Dynamics as a Selforganised Spatio-temporal Pattern | ||||

Since their original conception as manifestations of electric brain activity by Hughlings Jackson, epileptic seizures have been considered an example of a pathology that is best described by a complex spatio-temporal pattern. Nevertheless, their understanding in terms of nonlinear dynamics is still surprisingly limited. In particular, the transition into and out of seizure dynamics is widely regarded to be due to specific parameter changes into and out of a region of periodic solutions, although these changes have never been pinned down to actual physiological observables. Here we present a modelling framework for spatio-temporal epileptic dynamics in humans which builds on the notion of neural mass excitability of a generic cortical circuit. We justify the components of the model by comparison to experimental (animal) and clinical (human) data and study potential mechanisms underlying generalised and partial seizures. We find that, in addition to the dynamics provided by periodic attractors, spatio-temporal epileptic rhythms could also be explained by intermittency (spontaneous switching), and complex rhythmic transients following perturbations. We discuss these concepts using different clinical seizure types. Finally, we use the model framework to propose practical stimulation protocols to test for the presence of regions of abnormality ("epileptic foci") in the human brain. M. Goodfellow, K. Schindler, G. Baier, NeuroImage 55, 920-932 (2011); M. Goodfellow, K. Schindler, G. Baier, NeuroImage (2011), doi:10.1016/j.neuroimage.2011.08.060. P. Taylor, G. Baier, J. Comput. Neurosci. (2011), in print. Online: DOI 10.1007/s10827-011-0332-1. |
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15:30-15:50 | Khare, S; Singh Baghel, R; Dhar, J; Jain, R (Hindustan College of Sciences & Technolgy) |
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Spatial Pattern Formation in Phytoplankton Dynamics in 1-D and 2-D System | ||||

In this paper, we propose a mathematical model of infected phytoplankton dynamics with spatial movement. The reaction diffusion models in both one and two dimension space coordinates are studied. The proposed model is an extension of temporal model available [6], in spatiotemporal domain. It is observed that the reaction diffusion system exhibits spatiotemporal chaos in phytoplankton dynamics. The importantance of the spatially extension are established in this paper, as they display a wide spectrum of ecologically relevant behavior, including chaos. The stability of the system is studied with respect to disease contact rate and the growth fraction of infected phytoplankton indirectly rejoins the susceptible phytoplankton population. The results of numerical experiments in one dimension and two dimensions in space as well as time series in temporal models are presented using MATLAB simulation. Moreover, the stability of the corresponding temporal model is studied analytically . Finally, the comparisons of the three types of numerical experimentation are discussed in conclusion. Keywords: Reaction-diffusion equation, phytoplankton dynamics, Spatiotemporal pattern formation, Chaos, Local Stability. |
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15:50-16:10 | Weise, LD (Universiteit Utrecht) |
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Spiral wave initiation in the Reaction-Diffusion-Mechanics models | ||||

We introduce a discrete reaction-diffusion-mechanics (dRDM) model to study the effects of deformation on reaction-diffusion (RD) processes. The dRDM framework employs a FitzHugh-Nagumo type RD model coupled to a mass-lattice model, that undergoes finite deformations. The dRDM model describes a material whose elastic properties are described by a generalized Hooke's law for finite deformations (Seth material). Numerically, the dRDM approach combines a finite difference approach for the RD equations with a Verlet integration scheme for the equations of the mass-lattice system. Using this framework we study and find new mechanisms of spiral wave initiation in the contracting excitable medium in homogeneous and heterogeneous cases. In particular, we show that deformation alters the "classical," and forms a new vulnerable zone at longer coupling intervals. This mechanically caused vulnerable zone results in a new mechanism of spiral wave initiation, where unidirectional conduction block and rotation directions of the consequently initiated spiral waves are opposite compared to the mechanism of spiral wave initiation due to the "classical vulnerable zone." We also study and classify mechanisms of spiral wave initiation in excitable tissue with heterogeneity in passive and in active mechanical properties. |
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16:10-16:30 | Vasiev, B; Harrison, N; Diez del Corral, R; Weijer, C (Liverpool University) |
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Chemotaxis and morphogen dynamics in biological tissues | ||||

In developmental biology patterns formed by morphogens are often affected by movement of cells producing the morphogens. The mutual effects of cell movement and dynamics of concentration patterns are enhanced when the movement is due to chemotactic response to the morphogens. Here we present a set of cell movement patterns with associated patterns in concentration fields of chemotactic agents obtained analytically in continuous model and numerically in individual-cell based model. We have found that group of cells can push itself to move, provided that it produces a chemical which acts as a chemorepellent to its constituent cells. Also the group of cells can be pulled to move by the chemoattractor produced by the surrounding cells in a tissue. Many other chemotactic scenarios are in play when the group of cells is inhomogeneous, i.e. when only part of cells is reacting chemotactically to the morphogen produced by the other part or in the surrounding tissue. We demonstrate the se scenarios on the models of primitive streak extension and regression in the chick embryo. |
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16:30-16:50 | Break | |||

16:50-17:40 | De Kepper, P; Szalai, I; Cuiñas, D; Horváth, J (Centre national de la recherche scientifique) |
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The design of Turing patterns in solution chemistry | ||||

Twenty-two years ago, the first experimental observation of the stationary symmetry breaking reaction-diffusion patterns, predicted by Alan Turing, was made. It boosted theoretical and experimental studies in this field. Though a considerable variety of patterns had been found after that first observation, the number of isothermal reactions producing such patterns was limited to only two for fifteen years. Recently, we proposed an effective method for producing stationary and non-stationary symmetry breaking patterns in open spatial reactors. In the last three years, stationary reaction-diffusion patterns have been found in four new reactions. Among these, two are for the first time not based on oxihalogen chemistry. We shall briefly present this design method and give an overview of experimental contributions in the field. |
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17:40-17:50 | Break | |||

17:50-18:40 | Othmer, H (University of Minnesota) |
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The Effect of the Signaling Scheme on the Robustness of Pattern Formation in Development | ||||

Pattern formation in development is a complex process that involves spatially-distributed signals called morphogens that influence gene expression and thus the phenotypic identity of cells. Usually different cell types are spatially segregated, and the boundary between them may be determined by a threshold value of some state variable. The question arises as to how sensitive the location of such a boundary is to variations in properties, such as parameter values, that characterize the system. In this talk we discuss recent work on both deterministic and stochastic reaction-diffusion models of pattern formation with a view toward understanding how the signaling scheme used for patterning affects the variability of boundary determination between cell types in a developing tissue. |