Nonlocal reaction–diffusion equations describe various biological and biomedical applications. Their mathematical properties are essentially different in comparison with the local equations, and this difference can lead to important biological implications. This review will present the state of the art in the investigation of nonlocal reaction–diffusion models in biomedical applications. We will consider various models arising in mathematical immunology, neuroscience, cancer modelling, and we will discuss their mathematical properties, nonlinear dynamics, resulting spatiotemporal patterns and biological significance.

Reaction-diffusion systems which have reaction term satisfying f(-q) = -f(q) tend strongly to form striped patterns. Haken’s slaving principle is used to derive differential equations for unstable mode amplitudes close to the Turing instability. This connects a dynamical symmetry to pattern selection, with possible relevance to biological and chemical pattern-forming phenomena.

Possible effects of interaction (cross-talk) between signaling pathways is studied in a system of Reaction–Diffusion (RD) equations. Furthermore, the relevance of spontaneous neurite symmetry breaking and Turing instability has been examined through numerical simulations. The interaction between Retinoic Acid (RA) and Notch signaling pathways is considered as a perturbation to RD system of axon-forming potential for N2a neuroblastoma cells. The present work suggests that large increases to the level of RA–Notch interaction can possibly have substantial impacts on neurite (...) outgrowth and on the process of axon formation. This can be observed by the numerical study of the homogeneous system showing that in the absence of RA–Notch interaction the unperturbed homogeneous system may exhibit different saddle-node bifurcations that are robust under small perturbations by low levels of RA–Notch interactions, while large increases in the level of RA–Notch interaction result in a number of transitions of saddle-node bifurcations into Hopf bifurcations. It is speculated that near a Hopf bifurcation, the regulations between the positive and negative feedbacks change in such a way that spontaneous symmetry breaking takes place only when transport of activated Notch protein takes place at a faster rate. (shrink)

Two general features of metameric patterning in Drosophilaare considered: (1) maintenance of a constant number of metameres (segments or parasegments) in the face of variation in length of the embryo; (2) expression of pattern by on‐off switchings of particular genes, with only three or four rows of cells to each element of pattern. For each of these features, the general strategic question is raised: could reaction‐diffusion theory account for this? In both cases, it is answered affirmatively. For the (...) second feature, this review contains some hitherto unpublished computer simulations by one of us (K. Y. T.), illustrating that a reaction‐diffusion mechanism can be transformed into a patterned switching mechanism by nothing more than compartmenting of the diffusion region. For the scale of three compartments to one pattern repeat unit (representing three rows of cells to a segment) the switching pattern predicted by computation is two‐off to one‐on. This resembles the pattern of expression of the engrailed gene, posteriorly localized in each segment. (shrink)

This study addresses the problem of output quasisynchronization for coupled complex-valued memristive reaction-diffusion complex networks via the distributed event-triggered control scheme. First, by using the separate method, set value mapping, and intermediate value theorem, the complex-valued memristive reaction-diffusion complex networks can be transferred into two semi-uncertain real-valued reaction-diffusion complex networks. Second, a distributed output piecewise event-triggered control scheme with spatial sampled-data is first proposed including a spatial sampling event-triggered generator and spatiotemporal sampling state feedback (...) controller. Furthermore, this scheme can effectively save the measurement resources and lower the update rate of controllers in spatial and time domain. Third, the synchronization analysis is considered by utilizing an appropriate Lyapunov function, the Halanay inequality, and the improved Wirtinger inequality. Subsequently, several output event-triggered quasisynchronization criteria are derived. The relations among event trigger conditions, spatial sampling interval, convergence rate, and control gain are given by rigorous mathematical derivation. Finally, multiple simulations are compared to substantiate the validation of the OPETC scheme. (shrink)

It will be settled out for the open problem of designing an r-order finite-time state observer for reaction-diffusion genetic regulatory networks with time-varying delays. By assuming the Dirichlet boundary conditions, aiming to estimate the mRNA and protein concentrations via available network measurements. Firstly, sufficient F-T stability conditions for the filtering error system have been investigated via constructing an appropriate Lyapunov–Krasovskii functional and using several integral inequalities and convex technique simultaneously. These conditions are delay-dependent and reaction-diffusion-dependent and (...) can be checked by MATLAB toolbox. Furthermore, a method is proposed to design an r-order F-T state observer, and the explicit expressions of observer gains are given. Finally, a numerical example is presented to illustrate the effectiveness of the proposed method. (shrink)

This paper investigates the stochastically exponential stability of reaction-diffusion impulsive stochastic cellular neural networks. The reaction-diffusion pulse stochastic system model characterizes the complexity of practical engineering and brings about mathematical difficulties, too. However, the difficulties have been overcome by constructing a new contraction mapping and an appropriate distance on a product space which is guaranteed to be a complete space. This is the first time to employ the fixed point theorem to derive the stability criterion of (...) reaction-diffusion impulsive stochastic CNN with distributed time delays. Finally, an example is provided to illustrate the effectiveness of the proposed methods. (shrink)

This article is devoted to the characterization of the basin of attraction of pattern solutions for some slow–fast reaction–diffusion systems with a symmetric property and an underlying oscillatory reaction part. We characterize some subsets of initial conditions that prevent the dynamical system to evolve asymptotically toward solutions which are homogeneous in space. We also perform numerical simulations that illustrate theoretical results and give rise to symmetric and non-symmetric pattern solutions. We obtain these last solutions by choosing particular (...) random initial conditions. (shrink)

Social media offers a platform for diffused stakeholders to interact with firms—alternatively praising, questioning, and chastising businesses for their CSR performance and seeking to engage in two-way dialogue. In 2014, 163,402 public messages were sent to Fortune 200 firms’ CSR-focused Twitter accounts, each of which was either shared, replied to, “liked,” or ignored by the targeted firm. This paper examines firm reactions to these messages, building a model of firm response to stakeholders that combines the notions of CSR communication and (...) stakeholder salience. Our findings show that firm response to a stakeholder on social media is positively and most significantly associated with what we refer to as the stakeholder’s connective power but negatively associated with the firm’s own connective power. To a lesser extent, firm response is positively associated with the stakeholder’s normative power but negatively associated with the firm’s own normative power. Firm response is also shown to be positively associated with stakeholder urgency in terms of both the originality of a stakeholder message and the expression of positive sentiment. (shrink)

This article addresses the physical chemical processes underlying biological self-organisation by which a homogenous solution of reacting chemicals spontaneously self-organises. Theoreticians have predicted that self-organisation can arise from a coupling of reactive processes with molecular diffusion. In addition, the presence of an external field, such as gravity, at a critical moment early in the process may determine the morphology that subsequently develops. The formation, in-vitro, of microtubules, a constituent of the cellular skeleton, shows this type of behaviour. The preparations (...) spontaneously self-organise by reaction-diffusion and the morphology that develops depends upon the presence of gravity at a critical bifurcation time early in the process. Here, we present numerical simulations of a population of microtubules that reproduce this behaviour. Microtubules can grow from one end whilst shrinking from the other. The shrinking end leaves behind a chemical trail of high tubulin concentration. Neighbouring microtubules preferentially grow into these regions, whilst avoiding regions of low tubulin concentration. The chemical trails produced by individual microtubules thus activate and inhibit the formation of neighbouring microtubules and this progressively leads to self-organisation. Gravity acts by way of its directional interaction with the macroscopic density fluctuations present in the solution arising from microtubule disassembly. (shrink)

In this paper, a multigroup SVIR epidemic model with reaction-diffusion and nonlinear incidence is investigated. We first establish the well-posedness of the model. Then, the basic reproduction number ℜ 0 is established and shown as a threshold: the disease-free steady state is globally asymptotically stable if ℜ 0 1. Moreover, applying the classical method of Lyapunov and a recently developed graph-theoretic approach, we established the global stability of the endemic equilibria for a special case.

Based on the heterogeneous multiscale method, this paper presents a finite volume method to solve multiscale convection-diffusion-reaction problem. The paper constructs an algorithm of the optimal order convergence rate in H1-norm under periodic medias.

Mathematical models of tumour invasion appear as interesting tools for connecting the information extracted from medical imaging techniques and the large amount of data collected at the cellular and molecular levels. Most of the recent studies have used stochastic models of cell translocation for the comparison of computer simulations with histological solid tumour sections in order to discriminate and characterise expansive growth and active cell movements during host tissue invasion. This paper describes how a deterministic approach based on reaction- (...) class='Hi'>diffusion models and their generalisation in the mechano-chemical framework developed in the study of biological morphogenesis can be an alternative for analysing tumour morphological patterns. We support these considerations by reviewing two studies. In the first example, successful comparison of simulated brain tumour growth with a time sequence of computerised tomography (CT) scans leads to a quantification of the clinical parameters describing the invasion process and the therapy. The second example considers minimal hypotheses relating cell motility and cell traction forces. Using this model, we can simulate the bifurcation from an homogeneous distribution of cells at the tumour surface toward a nonhomogeneous density pattern which could characterise a pre-invasive stage at the tumour-host tissue interface. (shrink)

In this paper we try to answer the question whether diffusion is a possible mechanism to explain mesoderm induction in Amphibians. First the embryological data are discussed and a hypothesis for mesoderm formation is set forth. The blastula being essentially a hollow sphere, we assume that the induction mechanism in an embryo at the blastula stage can be simulated by diffusion-reaction processes on spherical surfaces. A model is constructed for the simple case when the source is held (...) constant with respect to time, the decay proportional to the concentration and the diffusion coefficient a constant, From simulation we find a (best) value for the decay constant to be 6 × 10–5/sec and for the diffusion constant to be 0.24 × 10– 6 cm2/sec. The relation between the parameters is derived from an analytic solution for the diffusion process on a spherical surface with a continuously producing point source and the concentration proportional to the decay. The form and regulative properties of the steady concentration gradient are discussed. (shrink)

This paper discusses the stability and Hopf bifurcation analysis of the diffusive Kaldor–Kalecki model with a delay included in both gross product and capital stock functions. The reaction-diffusion domain is considered, and the Galerkin analytical method is used to derive the system of ordinary differential equations. The methodology used to determine the Hopf bifurcation points is discussed in detail. Furthermore, full diagrams of the Hopf bifurcation regions considered in the stability analysis are shown, and some numerical simulations of (...) the limit cycle are used to confirm the theoretical outcomes. The delay investment parameter and diffusion coefficient can have great impacts on the Hopf bifurcations and stability of the business cycle model. The investment parameters for the gross product and capital stock as well as the adjustment coefficient of the production market are also studied. These parameters can cause instability in, and the stabilization of, the business cycle model. In addition, we point out that, as the delay investment parameter increases, the Hopf bifurcation points for the diffusion coefficient values decrease considerably. When the delay investment parameter has a very small value, the solution of the business cycle model tends to become steady. (shrink)

In this essay, the first statistical and stochastic treatments of chemical dynamics are analyzed and discussed, in particular the diffusive description of chemical reactions. The first part of the paper introduces the historical and methodological basis of the theories about stochastic processes and diffusion as well as their lesser-known applications in chemical kinetics, which were advanced by Jens Anton Christiansen (1888–1969). In the second, part we will focus our attention on the early works of Oskar Benjamin Klein (1894–1977) and (...) Hendrik Anton Kramers (1894–1952) on electrolytes and the latter’s more mature work, which completes and gives a firm theoretical background to Christiansen’s description. (shrink)

Nitric oxide (NO) is involved in synaptic long-term potentiation (LTP) by multiple signaling pathways. Here, we show that LTP of synaptic transmission can be explained as a feature of signal transduction—bistable behavior in a chain of biochemical reactions with positive feedback, formed by diffusion of NO to the presynaptic site and facilitating the release of glutamate (Glu). The dynamics of Glu, calcium (Ca2+) and NO is described by a system of nonlinear reaction–diffusion equations with modified Michaelis–Menten (MM) (...) kinetics. Numerical investigation reveals that the chain of biochemical reactions analyzed can exhibit a bistable behavior under physiological conditions when production of Glu is described by MM kinetics and decay of NO is modeled by means of two enzymatic pathways with different kinetic properties. Our finding extends understanding of the role of NO in LTP: a short high-intensity stimulus is “memorized” as a long-lasting elevation of NO concentration. The conclusions obtained by analysis of the chain of biochemical reactions describing LTP can be generalized to other chains of interactions or for creating the logical elements for biological computers. (shrink)

This paper investigates the diffusive predator-prey system with nonmonotonic functional response and fear effect. Firstly, we discussed the stability of the equilibrium solution for a corresponding ODE system. Secondly, we established a priori positive upper and lower bounds for the positive solutions of the PDE system. Thirdly, sufficient conditions for the local asymptotical stability of two positive equilibrium solutions of the system are given by using the method of eigenvalue spectrum analysis of linearization operator. Finally, the existence and nonexistence of (...) nonconstant positive steady states of this reaction-diffusion system are established by the Leray–Schauder degree theory and Poincaré inequality. (shrink)

Diffusion of responsibility is a well-known effect widely studied in a real-life setting. It can occur in a situation in which the more people observe a crisis event, the less likely it is that someone will react and provide real assistance. These days of a galloping digital revolution a question is to be raised as to whether the same effect can be observed in the online space of communication. In order to investigate this phenomenon we designed a study aimed (...) at testing whether people exposed to a situation of cyberbullying will decide to take action against it depending on how many other Internet users are also aware of that crisis. Results obtained by us confirmed the existence of the diffusion of responsibility in the Internet similar to that observed in our daily lives. We also confirmed that a well-known influence technique “Even a penny will help” (in our study “every reaction will help”) can be effectively used to model behaviour online. In our times of digital revolution, those outcomes can be a step both toward understanding human behaviour in the online setting, showing us that it is not that different from the one presented in real live face-to-face communication and toward helping deal with antisocial behaviour people face online on a daily basis. (shrink)

In fear differential conditioning, stimuli that resemble the conditioned stimulus (CS+) are more likely to trigger fear responses. Excessive fear responses on stimuli not like CS + are often associated with anxiety. However, the threat judgments process and how this process manifests itself differently in subgroups with different generalisation rule applications, is unclear. This study examines whether anxiety biases the threat decision process in fear generalisation paradigm and whether subgroups characterised by different generalisation gradients was interpreted differently by drift-diffusion (...) model. We gathered behavioural data through a binary fear generalisation judgment task and clustered participants based on their responses. Reaction time distributions and individual scale scores were analyzed using the hierarchical drift-diffusion model. The model results suggested that similarity and state anxiety facilitated evidence-gathering processes that favoured “threat” judgments, but at the same time, state anxiety weakened the effect of stimulus similarity as evidence. Further cluster analyses revealed that this effect of anxiety on threat judgments only held true for specific subgroups of participants. This pioneering computational modelling effort in fear generalisation underscores the significant role of strategy preference and its complex interaction with anxiety in shaping stimulus processing. (shrink)

The global spread of corporate social responsibility (CSR) practices is widely explained in institutional-isomorphic terms: Corporations worldwide adopt CSR in reaction to isomorphic pressures exerted on them by a pro-CSR global environment, including normative calls for CSR, activist targeting, civil regulation frameworks, and educational activities. By contrast, this article considers the proactive agency of corporations in CSR diffusion, which is informed by nonmarket strategies that seek to instrumentally reshape the political and social environment of corporations. Applying a “channels-of- (...) class='Hi'>diffusion” perspective, we show that in the initial phase of CSR’s transnational diffusion—as exemplified by the cases of Venezuela (1962-1967) and Britain (1977-1981)—CSR traveled through learning exchanges between business elite “exporters” and “importers” whose engagement in diffusion addressed crisis-enhanced political threats and opportunities in the receiving country. The focal agents established national CSR business associations, which disseminated among local corporations CSR practices adapted to confront the challenges at hand. We identify the features of such “business-led cross-national diffusions of CSR”; formulate propositions regarding their conditions, dynamics, and effects; and suggest that further research of this mode of diffusion would advance a more nuanced and balanced understanding of CSR’s globalization. (shrink)

Travelling waves crossing the nervous networks at mesoscopic/macroscopic scales have been correlated with different brain functions, from long-term memory to visual stimuli. Here we investigate a feasible relationship between wave generation/propagation in recurrent nervous networks and a physical/chemical model, namely the Belousov–Zhabotinsky reaction. Since BZ’s nonlinear, chaotic chemical process generates concentric/intersecting waves that closely resemble the diffusive nonlinear/chaotic oscillatory patterns crossing the nervous tissue, we aimed to investigate whether wave propagation of brain oscillations could be described in terms of (...) BZ features. We compared experimentally detected oscillations during the spontaneous activity of the brain with BZ-like concentric waves simulated by a recently introduced artificial network. The observed overlap and agreement between simulated and measured oscillatory patterns suggests that changes in cortical areas’ neural activity might be described in terms of a recognizable diffusion pattern. We describe biological plausibility, benefits and limits of our approach and discuss the relationship among BZ-like networks, Pandemonium-like architectures and the spontaneous activity of the brain. ER -. (shrink)

Professor Zenkovsky's work is a history in the grand manner --and a grand history it is. We are much indebted to Dr. George Kline for his lucid, readable translation. Our historian takes the basic theme of Russian thought to have been the relations of Christianity and secularism. This does not mean that there has been a dearth of studies in logic, philosophy of science, and so on; or that Zenkovsky neglects these or treats them unfairly or unintelligently. But his belief (...) is that "the genuinely vital creative problems of philosophic reflection derive from the Christian gospel and hence cannot be resolved apart from it." In the West, however, it has, he thinks, been difficult to see this because freedom of investigation has there become identified with opposition to the Church, while in Russia it has been possible to defend such freedom "from the religious point of view, taking the ecclesiastical tradition as point of departure, and giving oneself up fully to the 'free spontaneous force of theological investigation,' in Karsavin's familiar expression." The Russian view of the Church, as set forth by Khomyakov and Kireyevski, a hundred years ago, is not that of an "authority" wielded by officials or even councils, but of a community of members forming a spiritual organism in which they reach a level of life involving integral experience of the nature of reality. Ecclesiastical coercion, as in the West, is viewed as a heresy. Faith calls us to a life of freedom, rather than of "obedience." Zenkovsky regards as "romantic" the notion of a generally diffused spiritual insight which would make all important truths perfectly evident, but he seems to agree with the rejection of coercion. (shrink)

In this paper we describe a test for Nijhout's hypothesis that the eyespot patterns on butterfly wings are the result of a threshold reaction of the epidermal cells to a concentration gradient of a diffusing degradable morphogen produced by focal cells at the centre of the future eyespot. The wings of the nymphalid butterfly, Bicyclus anynana, have a series of eyespots, each composed of a white pupil, a black disc and a gold outer ring. In earlier extirpation and transplantation (...) experiments it has been established that these eyespots are indeed organised around groups of signalling cells active during the first hours of pupal development. If these cells were to supply the positional information for eyespot formation in accordance with Nijhout's diffusion-degradation gradient model, then, when two foci are close together, the signals should sum, and this effect should be apparent in the detailed shape of the resulting pigment pattern. We give an equation for the form of the contours that would be obtained in this manner. We use this to test the morphogen gradient hypothesis on measurements of the outlines of fused eyespots obtained either by grafting focal cells close together, or by using a mutation that produces adjacent fused eyespots. The contours of the fused patterns were found to satisfy our equation, thus corroborating Nijhout's hypothesis to the extent possible with this particular type of experiment. (shrink)

To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a 0+a 1tan ξ+a 2tan 2 ξ, which indicates that (...) some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method. (shrink)

Research on employees’ responses to corporate social responsibility has recently accelerated and begun appearing in top-tier academic journals. However, existing findings are still largely fragmented, and this stream of research lacks theoretical consolidation. This article integrates the diffuse and multi-disciplinary literature on CSR micro-level influences in a theoretically driven conceptual framework that contributes to explain and predict when, why, and how employees might react to CSR activity in a way that influences organizations’ economic and social performance. Drawing on social identity (...) theory and social exchange theory, we delineate the different but interdependent psychological mechanisms that explain how CSR can strengthen the employee–organization relationship and subsequently foster employee-related, micro-level outcomes. Contributions of our framework to extant literature and potential extensions for future research are then discussed. (shrink)

Research on employees’ responses to corporate social responsibility has recently accelerated and begun appearing in top-tier academic journals. However, existing findings are still largely fragmented, and this stream of research lacks theoretical consolidation. This article integrates the diffuse and multi-disciplinary literature on CSR micro-level influences in a theoretically driven conceptual framework that contributes to explain and predict when, why, and how employees might react to CSR activity in a way that influences organizations’ economic and social performance. Drawing on social identity (...) theory and social exchange theory, we delineate the different but interdependent psychological mechanisms that explain how CSR can strengthen the employee–organization relationship and subsequently foster employee-related, micro-level outcomes. Contributions of our framework to extant literature and potential extensions for future research are then discussed. (shrink)

A reaction diffusion system is used to study the interaction between species in a population dynamic system. It is not only used in a population dynamic system with the diffusion phenomenon but also used in physical chemistry, medicine, and animal and plant protection. It has been studied by more and more scholars in recent years. The FitzHugh–Nagumo model is one of the most famous reaction-diffusion models. This article takes a deeper look at a FitzHugh–Nagumo model (...) in a network with time delay. Firstly, we studied the linear stability of the equilibrium, then the existence of Hopf bifurcation is given, and finally, the stability of the Hopf bifurcation is introduced. (shrink)

A nonlinear two-variable reaction-diffusion model of bone mineral metabolism, built from an overall self-oscillatory compartmental model of calcium metabolism in vivo, has been studied for its ability to generate spatial and spatio-temporal self-organizations in a two-dimensional space. Analytical and numerical results confirm the theoretical properties previously described for this kind of model. In particular, it is shown that, for a given set of reactional parameter values and certain values of the ratio of the two diffusion coefficients, there (...) exists a set of unstable wavenumbers leading spontaneously to the development, from the homogeneous steady state, of either different types of stationary spatial patterns (hexagonal, striped and re-entrant hexagonal patterns) or more or less complex spatio-temporal expressions. We discuss the relevance of analogies established between some spatial or spatio-temporal structures predicted by the model and some peculiar features of the primary bone trabecular architecture which appear during embryonic ossification. (shrink)

Traditional frameworks for evaluating scientific models have tended to downplay their exploratory function; instead they emphasize how models are inherently intended for specific phenomena and are to be judged by their ability to predict, reproduce, or explain empirical observations. By contrast, this paper argues that exploration should stand alongside explanation, prediction, and representation as a core function of scientific models. Thus, models often serve as starting points for future inquiry, as proofs of principle, as sources of potential explanations, and as (...) a tool for reassessing the suitability of the target system. This is illustrated by a case study of the varied career of reaction-diffusion models in the study of biological pattern formation, which was initiated by Alan Turing in a classic 1952 paper. Initially regarded as mathematically elegant, but biologically irrelevant, demonstrations of how, in principle, spontaneous pattern formation could occur in an organism, such Turing models have only recently rebounded, thanks to advances in experimental techniques and computational methods. The long-delayed vindication of Turing’s initial model, it is argued, is best explained by recognizing it as an exploratory tool. (shrink)

New results have been obtained by dynamical systems theory which allow an ammendment of the narrow vision of mechanicism. Forms of behavior richer than the ones considered by classicalmechanicism can be exhibited by the new structures studied by system theory (feedback, reaction-diffusion, hierarchical structures, multiple timescales...). In this paper a critical presentation of these ideas is developed.

This paper is devoted to develop a nonlocal and time-delayed reaction-diffusion model for HIV infection within host cell-to-cell viral transmissions. In a bounded spatial domain, we study threshold dynamics in terms of basic reproduction numberR0for the heterogeneous model. Our results show that ifR0 1, virus will persist in the host environment.

The contribution of this present paper is to propose a method that combines a chemical Brusselator reaction-diffusion system with a biological cell system via gap junction for controlling and visualizing the frequency and magnitude of chaotic intracellular calcium oscillations in two cell types, including nonexcitable cells and the glial cells. This produces a wide variety of oscillatory behaviors similar to those reported in numerous biological experiments. We particularly show that in the majority of chaos cases, the reactor to (...) cell coupling can induce the generation of regular calcium oscillations as the coupling strength is varied. Together with the proposed method of coupled models, the regularity of these chaotic oscillations enables us to gain better understanding and extensive insights into the overall coupling dynamics. (shrink)

This personal, yet scientific, letter to Alan Turing, reflects on Turing's personality in order to better understand his scientific quest. It then focuses on the impact of his work today. By joining human attitude and particular scientific method, Turing is able to “immerse himself” into the phenomena on which he works. This peculiar blend justifies the epistolary style. Turing makes himself a “human computer”, he lives the dramatic quest for an undetectable imitation of a man, a woman, a machine. He (...) makes us see the continuous deformations of a material action/reaction/diffusion dynamics of hardware with no software. Each of these investigations opens the way to new scientific paths with major consequences for contemporary live and for knowledge. The uses and the effects of these investigations will be discussed: the passage from classical AI to today's neural nets, the relevance of non-linearity in biological dynamics, but also their abuses, such as the myth of a computational world, from a Turing-machine like universe to an encoded homunculus in the DNA. It is shown that these latter ideas, which are sometimes even made in Turing's name, contradict his views. (shrink)

The calcium signals regulate the production and secretion of many signaling molecules like inositol trisphosphate ( $$IP_{3}$$ ) and adenosine triphosphate (ATP) in various cells including pancreatic $$\beta$$ -cells. The calcium signaling mechanisms regulating $$IP_{3}$$, ATP and insulin responsible for various functions of $$\beta$$ -cells are still not well understood. Any disturbance in these mechanisms can alter the functions of $$\beta$$ -cells leading to diabetes and metabolic disorders. Therefore, a mathematical model is proposed by incorporating the reaction-diffusion equation (...) for calcium dynamics and a system of first-order differential equations for $$IP_{3}$$, ATP-production and insulin secretion with initial and boundary conditions. The model incorporates the temporal dependence of $$IP_{3}$$ -production and degradation, ATP production and insulin secretion on calcium dynamics in a $$\beta$$ -cell. The piecewise linear finite element method has been used for the spatial dimension and the Crank-Nicolson scheme for the temporal dimension to obtain numerical results. The effect of changes in source influxes and buffers on calcium dynamics and production of $$IP_{3}$$, ATP and insulin levels in a $$\beta$$ -cell has been analyzed. It is concluded that the dysfunction of source influx and buffers can cause significant variations in calcium levels and dysregulation of $$IP_{3}$$, ATP and insulin production, which can lead to various metabolic disorders, diabetes, obesity, etc. The proposed model provides crucial information about the changes in mechanisms of calcium dynamics causing proportionate disturbances in $$IP_3$$, ATP and insulin levels in pancreatic cells, which can be helpful for devising protocols for diagnosis and treatment of various metabolic diseases. (shrink)

Patterns resulting from the sole interplay between reaction and diffusion are probably involved in certain stages of morphogenesis in biological systems, as initially proposed by Alan Turing. Self-organization phenomena of this type can only develop in nonlinear systems (i.e. involving positive and negative feedback loops) maintained far from equilibrium. We present Turing patterns experimentally observed in a chemical system. An oscillating chemical reaction, the CIMA reaction, is operated in an open spatial reactor designed in order to (...) obtain a pure reaction-diffusion system. The two types of Turing patterns observed, hexagonal arrays of spots and parallel stripes, are characterized by an intrinsic wavelength. We identify the origin of the necessary difference of diffusivity between activator and inhibitor. We also describe a pattern growth mechanism by spot splitting that recalls cell division. (shrink)

Examples are given of applications by Pauling, Mulliken, Marcus and G.E.Kimball of the three Pythagorian means to formulate the scales of electronegativity of the elements, to the calculations of rate constants of electron transfer cross-reactions, to the calculation of the observed rate constant as function of activation and diffusion rate constants in the case of mixed reaction-diffusion rates and to the calculation of the effective diffusion coefficient in solution of a salt AB as a whole from (...) the diffusion coefficients of the ions in which it dissociates. (shrink)

The fate of root epidermal cells is controlled by a complex interplay of transcriptional regulators, generating a genetically determined, position‐biased arrangement of root hair cells. This pattern is altered during postembryonic development and in response to environmental signals to confer developmental plasticity that acclimates the plant to the prevailing conditions. Based on the hypothesis that events downstream of this initial mechanism can modulate the pattern installed during embryogenesis, we have developed a reaction diffusion model that reproduces the root (...) hair patterning previously observed experimentally. Under all growth conditions, an almost equal spacing between root hair forming cells was observed both in vitro and in silico, indicating that long‐range intercellular communication is crucial for the trichoblasts' decision to form a root hair. We assume that a hair growth promoter (HGP) is upregulated in root‐hair‐forming cells by a trichoblast‐specific component. Once established, HGP production is self‐enhancing. The autocatalytic regulation of HGP is antagonized by an HGP‐produced hair growth inhibitor (HGI). HGI is exported from trichoblasts and diffuses to neighboring cells, where it inhibits further HGP production and promotes the non‐hair cell fate. Under conditions of phosphate deficiency, we hypothesise that HGP production is increased and HGI diffusion rate is reduced, leading to a position‐independent formation of extra root hairs. BioEssays 30:75–81, 2008. © 2007 Wiley Periodicals, Inc. (shrink)

We present a technique for the analysis of pattern formation by a class of models for the formation of ocular dominance stripes in the striate cortex of some mammals. The method, which employs the adiabatic approximation to derive a set of ordinary differential equations for patterning modes, has been successfully applied to reaction-diffusion models for striped patterns [1]. Models of ocular dominance stripes have been studied [2,3] by computation, or by linearization of the model equations. These techniques do (...) not provide a rationale for the origin of the stripes. We show here that stripe formation is a non-linear property of the models. Our analysis indicates that stripe selection is closely linked to a property in the dynamics of the models which arises from a symmetry between ipsilateral and contralateral synapses to the visual cortex of a given hemisphere. (shrink)

In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a diffusible substance known as tumour angiogenesis factor (TAF) into the surrounding tissue. Endothelial cells which form the lining of neighbouring blood vessels respond to this chemotactic stimulus in a well-ordered sequence of events comprising, at minimum, of a degradation of their basement membrane, migration and proliferation. Capillary sprouts are formed which migrate towards the tumour eventually penetrating it and permitting vascular growth to take place. It (...) is during this stage of growth that the insidious process of invasion of surrounding tissues can and does take place. A model mechanism for angiogenesis is presented which includes the diffusion of the TAF into the surrounding host tissue and the response of the endothelial cells to the chemotactic stimulus. Numerical simulations of the model are shown to compare very well with experimental observations. The subsequent vascular growth of the tumour is discussed with regard to a classical reaction-diffusion pre-pattern model. (shrink)