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Primary and Secondary Waves in Developmental Biology
... To illustrate this, I shall discuss two early examples in detail, both depending on catastrophe theory. In the first, Zeeman (1974) predicted that there could be a wave associated with the formation of a boundary in a previously undifferentiated tissue. In the second, Bazin and Saunders (1978) predicted that an amoeba should modify a certain chemoattractant. ...
... The application that became the centre of the controversy about catastrophe theory was Zeeman's (1974) paper on travelling wave fronts. Zeeman claimed that when a frontier forms in a region that was previously undifferentiated, i.e. either homogeneous or with at most a gradient in properties along it, then this frontier does not first appear in its final position. ...
Biologists and social scientists often carry out their research in ways that are quite different from those used by physical scientists. Their results are often different as well; the nature of their subjects makes it less likely that they can produce the firm and generally quantitative predictions that are standard in physics. As mathematics is being applied more and more outside the physical sciences, a new methodology is appearing that better reflects the nature of these other subjects. This is happening only slowly, chiefly because the relevant work is generally seen only in its context of trying to solve a particular problem rather than as a contribution to applied mathematics as such. The aim of this paper is to encourage progress by describing some results that have already been obtained, and by discussing explicitly some of the issues that arise.
... This theory focuses on solving the problem that how and why a dynamic system becomes unstable under the continuous changes with the influence of one or more parameters (Thom, 1972;Wang et al., 2017). Catastrophe Theory can find out the inherent instability in a nonlinear dynamical system (Diks and Wang, 2016), and it has been proven to be a successful tool to investigate the qualitative properties in many fields, e.g., in physics (Berry, 1976;Holmes and Rand, 1976;Poston and Stewart, 1978), in biology (Zeeman, 1974;Bazin and Saunders, 1979), in social sciences (Zeeman, 1976a;Zeeman, 1976b), in engineering and technology (Henley, 1976;John and Brain, 1976), in railway system safety (Wang et al., 2017;Huang et al., 2019b), in traffic flow analysis (Acha-Daza and Hall, 1994) and so on. ...
In this paper, the cusp catastrophe model is applied to analyze the railway dangerous goods transportation system risk state changes. Firstly, a Risk-Accident Catastrophe Tree is proposed to formulate the process that how the risk factors cause the accident. Secondly, the whole risk factors are classified into five categories including failure of human behavior, machine failure, transported materials, environment problems and management problems. Next, the cusp catastrophe model of railway dangerous goods transportation system is established, the Split Coefficient is defined to evaluate bifurcation set. The results of the case study show that the risk state changes of railway dangerous goods transportation system satisfy bimodality, inaccessibility, sudden transitions and divergence, but not satisfy hysteresis. The larger the Split Coefficient is, the easier the trajectory of system control point crosses with bifurcation sets. The reason of the system state changes from a safe state to a risk state is: as long as the trajectory of system control point crosses with bifurcation sets, there must be a cusp catastrophe in
the system, and the system control point will cross the fold surface, which makes the risk energy increase sharply, the structure, information and energy of the system will also be destroyed.
... We illustrate the two alternatives by a couple of examples. EXAMPLE 1. AGGRESSION [22]. According to Konrad Lorenz [7] fear and rage are conflicting drives influencing aggression. ...
Catastrophe theory is a method discovered by Thorn [14] of using singularities of smooth maps to model nature. In such models there are often several levels of structure, just as in a geometry problem there can be several levels of structure, for instance the topological, dif-ferential, algebraic, and affine, etc. And, just as in geometry the topological level is generally the deepest and may impose limitations upon the higher levels, so in applied mathematics, if there is a catastrophe level, then it is generally the deepest and likely to impose limitations upon any higher levels, such as the differential equations involved, the asymptotic behaviour, etc. Again, in geometry the com-plexity of the higher levels may render them inaccessible, so that they can only be handled implicitly rather than explicitly, while at the same time the underlying topological invariants may even be computable. Similarly in applied mathematics the complexity of the differential equations may sometimes render them inacces-sible (even to computers), so that they can only be handled irnplicitly rather than explicitly, while the underlying catastrophe can be modeled, possibly even to the extent of providing quantitative prediction. Therefore catastrophe theory pifers two attractions: On the one hand it some-times provides the deepest level of insight and lends a simplicity of understanding. On the other hand, in very complex systems such as occur in biology and the social sciences, it can sQmetimes provide a model where none was previously thought possible. In this paper we discuss various levels of structure that can be superim-posed upon an underlying catastrophe and illustrate them with an assortment of examples. For convenience we shall mostly use the familiar cusp catastrophe (see [5], [13], [14], [24]). <P 1975, Canadian Mathematical Congress 533 534 E. C. ZEEMAN Level 1. Singularities. Level 2. Fast dynamic (homeostasis). Level 3. Slow dynamic (development). Level 4. Feedback. Level 5. Noise. Level 6. Diffusion. Thorn's classification of elementary catastrophes belongs to Level 1. Levels 2,3,4 refer to ordinary differential equations, and Level 6 refers to partial differential equations. Level 1. Singulairties. We begin by recalling the main classification theorem. Let C 9 X be manifolds with dim C ^ 5, and let/e C°°{C x X). Suppose that/is generic in the sense that the related map C -» C°°{X) is transverse to the orbits of the group Diff {X) x Diff {R) acting on C°°{X). (Genericity is open-dense in the Whitney C°°-topology.) Let M c C x X be given by V x f = 0, and let i\M- • C be in-duced by projection C x X -> C.
... In [6] and [63], Cooke and Zeeman propose a clock and wavefront model to explain somite formation. They postulate the existence of a longitudinal positional information gradient down the AP axis of the embryo which determines regional development by setting the time in each cell at which it will undergo a catastrophe. ...
... In many cases, skin organ patterns are actually laid down sequentially, and it is believed that travelling waves of determination (Zeeman (1974)) often initiate morphogenetic processes. For example, stripe pigment patterns on the alligator develop sequentially. ...
001 Abstract. A tissue interaction model for skin organ pattern formation is presented. Possi- ble spatially patterned solutions on rectangular domains are investigated. Linear stability analysis suggests that the model can exhibit pattern formation. A weakly nonlinear two-dimensional pertur- bation analysis is then carried out. This demonstrates that when bifurcation occurs via a simple eigenvalue, patterns such as rolls, squares, and rhombi can be supported by the model equations. Our nonlinear analysis shows that more complex patterns are also possible if bifurcation occurs via a double eigenvalue. Surprisingly, hexagonal patterns could not develop from a primary bifurcation.
... Such waves play an organizing role in the general processes of structure formation in biological systems. Very interesting discussions of similar phenomena can be also found in [12,13]. I n the former, for instance, Zeeman discusses the wave propagation of a frontier in a homogeneous tissue, to its final position, a process which can be described by means of state variables (of the cells) in a n-dimensional space (Rn) and of specific control parameters. ...
A general review is presented of some of the approaches to the general problem of morphogenesis which originate either directly or indirectly in the work of Turing. The main points are the stability analysis of the Gierer-Meinhardt equations and the discussion on the applicability to biological systems of the concepts and techniques of field theory.
... Although the applied theories that have given rise to the MPSTW such as catastrophe theory [20], Hjelmslevian and Greimasian semiotics [19], graph theory and crystalographic groups in its beginnings orbited around the static conception of relation while product, later they have approached the dynamic conception of relation while production which is in tune with an ever emerging Quality. Lourenci has always been aware that if it were possible to show that architectural design could be considered a language, necessarily it should resemble the nature of a Chinese language [3] which is essentially emergent, relational and dynamic. ...
The tight coupling quantity/quality in nature reflects an efficient use of exergy in constructing biological material leading to a very effective storage of information per unit of mass maintaining homeostatic conditions. It seems the entity called incipient Emergy applied to the modeling of holarchic man-made eco-mimetic systems enables us to evince a sustainable basis for the design of our cities due to its explicitation of the relationship of the quantity/quality. Resembling autocatalytic feedback loops, the balanced trade-off between the knowledge of the knowing subject and the known (phenomenological) object is introduced in the eco-design model and underlying geometric and computational models precisely because its nature tunes in to the premises of incipient Emergy.
... We proved that the wave of determination is not stopped by a cut across its path. Such a 'kinematic' wave (Zeeman, 1974) does not depend upon the propagation of a signal across the tissue, but results from the fact that the cells are timers laid out in the order in which they are preset to change -anterior cells are everywhere in advance of posterior cells. ...
Following neurulation, the frog segments c.40 somites and concurrently undergoes a striking elongation along the anteroposterior axis. This elongation (excluding the head) is largely the result of a presegmental extension of posterior tissue with a lesser contribution from the extension of segmented tissue. Presegmental extension is entirely the result of activity within a narrow zone of extension that occupies the central region in the tail bud. Within the zone of extension, a minimum of six prospective somites undergo an eight- to ten-fold extension along the axis. The zone passes posteriorly across the tissue of the tail tip. The anterior of the tail bud contains three extended prospective somites in the course of segmentation. The anterior boundary of the zone of extension coincides in space exactly with the anterior boundary of the zone of abnormal segmentation that results from temperature shock. This means that extension ceases immediately before the sudden tissue change associated with segmentation.
... The development of the pattern is, therefore, a strikingly dynamic process. A morphogenetic 'wave' (Zeeman, 1974) sweeps across the skin: ahead of the wave, the tissue lacks any obvious pattern; behind it, primordia have formed in an orderly triangular array. ...
A regular array of feather primordia covers chick dorsal skin in vivo. The pattern develops over a period of 2 days as a morphogenetic wave sweeps across either side of the back of the chick, forming successive anteroposterior rows of primordia. This paper describes a new method for the culture of chick skin which allows the development of large areas of the feather pattern to be investigated experimentally. Skin is cultured on a substratum of hydrated collagen; since the collagen is transparent, feather primordium development can be observed in detail. The new method has been used to investigate the problem of when the positions of feathers are determined. I show that during the time when the first few rows of primordia are forming, skin taken from just lateral to the most recently formed row can be caused to form an increased number of primordia per row by stretching it anteroposteriorly. This result indicates that the positions of feathers are determined sequentially along an invisible wave which moves just ahead of the visible wave of primordium morphogenesis.
... With the recent experimental findings concerning the existence of a segmentation clock and a wavefront of FGF8 along the AP of vertebrate embryos, the Clock and Wavefront model has received a large amount of support. First proposed by Cooke and Zeeman (Cooke, 1975(Cooke, , 1998Cooke and Zeeman, 1976;Zeeman, 1974), the model assumed the existence of a longitudinal positional information gradient along the AP axis which interacts with a smooth cellular oscillator (the clock) to set the times at which cells undergo a catastrophe. In this context, Cooke and Zeeman were referring to the changes in adhesive and migratory behavior of cells as they form somites. ...
Somitogenesis is the process of division of the anterior-posterior vertebrate embryonic axis into similar morphological units known as somites. These segments generate the prepattern which guides formation of the vertebrae, ribs and other associated features of the body trunk. In this work, we review and discuss a series of mathematical models which account for different stages of somite formation. We begin by presenting current experimental information and mechanisms explaining somite formation, highlighting features which will be included in the models. For each model we outline the mathematical basis, show results of numerical simulations, discuss their successes and shortcomings and avenues for future exploration. We conclude with a brief discussion of the state of modeling in the field and current challenges which need to be overcome in order to further our understanding in this area.
Catastrophe theory can describe a continuous process that is undergoing abrupt changes. A dynamic process can be considered a cusp catastrophe if it has the following five qualities: bimodality, sudden transitions, hysteresis, inaccessibility, and divergence. In this paper, the cusp catastrophe model is applied to describe the dynamic changing process of railway system safety. This dynamic process is also proved to have the five qualities of a cusp catastrophe. With the use of the cusp catastrophe model, a framework for the system risk of railway systems is constructed, and some ideas on the risk field of railway system safety are presented. On the basis of the framework and ideas, the dynamic changes in the system risk in a railway accident evolution process can be described in a novel approach. The concept of system risk in railway system safety is also discussed in three aspects. We hope that the theoretical description of and discussion on system risk could facilitate a profound analysis of railway system safety.
Experimental investigations and theoretical analyses of embryo-genesis have attempted to discover and describe the processes whereby a newly fertilized egg cell is transformed into an adult organism. These investigations have centered on the possible organismic, multicellular, cellular, cytoplasmic and genetic organization responsible for this transformation. Theoretical models, reflecting to some extent contemporary activity in the physical sciences and mathematics, have attempted to provide more or less precise mechanisms for these processes. The present paper reviews some of these studies and their contribution to an understanding of the nature of embryogenesis and the emergence of adult organisms.
Catastrophe Theory, proposed by Thom (1969), describes the behavior of a dynamical system in terms of the maxima and minima of the associated potential energy function. The minima represent stationary or (quasi-) equilibrium conditions for the energy function and serve as states of attraction for the dynamical system, while the maxima act as repellor states. The potential energy function V is parameterized by a manifold C, the control space, on a manifold X, the behavior space. The set
$$M = \left\{ {\left( {c,x \in } \right)CxX\left| {{\nabla _x}\nabla \left( {c,x} \right) = 0} \right.} \right\}
$$
defines the catastrophe manifold when V is differentiable everywhere (Woodcock and Poston, 1974a). A catastrophe is a singularity of the map
$$X:M \to C$$
$$\left( {c,x} \right) \to c
The electrical activity of a single neurone varies stochastically with time, and for neurones which generate action potentials the output from the neurone can be treated as a stochastic point process. Such a description is incomplete, in that it ignores the spatial and temporal changes of voltage with distance and time within the neurone, but this incomplete description appears to be a useful way of characterizing the output from a neurone. The action potentials from a neurone will disperse spatially and temporally down the axonal branches, and will form the inputs to synapses on other neurones and effectors. Thus it might appear reasonable to describe the electrical activity of neurones forming the nervous system as a multi-variate point process: however, there are too many neurones for such an approach to be feasible.
While self-organization has been studied for some time and is now a familiar concept in science, there is as yet no general theory nor even a generally accepted definition. Much of what has been done has been within the context of dissipative structures, so much so that for many people the two ideas have become totally conflated. But the concept of self-organization is much broader; it encompasses systems described by catastrophe theory, for example. Here we draw on the experience of catastrophe theory to suggest features that a theory of self-organization should have, and we illustrate this by the problem of segmentation. We point out that self-organization provides an alternative to natural selection as an explanation of order and organization in biological systems.
The problem of discontinuity in the behavior of culture systems-the last stronghold of the anti-processualists-is discussed. Abrupt change in behavior can now be described in terms of smooth changes in the underlying causative factors by means of Rene Thom's Theory of Elementary Catastrophes. The theory suggests insights not only into discontinuities with respect to time ("sudden" changes) but into the differentiation of forms as the result of bifurcations (morphogenesis). Although existing applications of the Theory in the social sciences lack quantitative precision, they offer a deeper understanding of crucial mechanisms of social evolution and, it is suggested, go far toward solving the discontinuity problem in archaeology.
In view of the long and fruitful partnership between mathematics and physics, it was only natural that the first applications of mathematics in biology should take theoretical physics as a model. Indeed, Lotka (1924) entitled his pioneering work Elements of Physical (not, as in the 1956 reprint, Mathematical) Biology, claiming as his intention the ‘application of physical principles and methods in the contemplation of biological systems’.
Embryogenesis depends on a series of processes which generate specific patterns at each stage of development. For example, gastrulation, chondrogenesis, formation of scale, feather and hair primordia all involve major symmetry breaking. These ubiquitous spatial pattern formation requirements depend on specific pattern generation mechanisms which are still unknown. They are the subject of much research both theoretical and experimental. In the case of integumental patterns, for example, we do not in general even know when in development the pattern is actually formed. This was the key question studied by Murray et al. (1990) in a recent theoretical and experimental paper on alligator (Alligator missippiensis) stripes.
A tissue interaction model for skin organ pattern formation is presented. Possible spatially patterned solutions on rectangular domains are investigated. Linear stability analysis suggests that the model can exhibit pattern formation. A weakly nonlinear two-dimensional perturbation analysis is then carried out. This demonstrates that when bifurcation occurs via a simple eigenvalue, patterns such as rolls, squares, and rhombi can be supported by the model equations. Our nonlinear analysis shows that more complex patterns are also possible if bifurcation occurs via a double eigenvalue. Surprisingly, hexagonal patterns could not develop from a primary bifurcation.
The geometrical nature of the Elementary catastrophes (Thom, 1969) is reviewed. Histories of the movement of catastrophe manifolds and bifurcation sets are presented for some of the space-equivalent unfoldings described by Wassermann (1975). These unfoldings provide descriptions of the variation with time of the stability of stationary states of associated potential energy functions. Identification of these stationary energy states with stationary states of a system therefore provides a description of its behavior with time. Qualitative descriptions of this type are particularly useful when the complexity of a system prevents a detailed quantitative description. Histories of bifurcation set movements suggest different types of system behavior at different space-like coordinates. This type of theory may be a useful model for the processes leading to differentiation of cells and to emergence of adult forms of a biological organism.
The general theory of elastic stability is extended to include the
imperfection-sensitivity of twofold compound branching points with
symmetry of the potential function in one of the critical modes
(semi-symmetric points of bifurcation). Three very different forms of
imperfection-sensitivity can result, so a subclassification into
monoclinal, anticlinal and homeoclinal semi-symmetric branching is
introduced. Relating this bifurcation theory to Rene Thom's catastrophe
theory, it is found that the anticlinal point of bifurcation generates
an elliptic umbilic catastrophe, while the monoclinal and homeoclinal
points of bifurcation lead to differing forms of the hyperbolic umbilic
catastrophe. Practical structural systems which can exhibit this form of
branching include an optimum stiffened plate with free edges loaded
longitudinally, and an analysis of this problem is presented leading to
a complete description of the imperfection-sensitivity. The paper
concludes with some general remarks concerning the nature of the
optimization process in design as a generator of symmetries,
instabilities and possible compound bifurcations.
The differentiation of cells in developmental biology the emergence of order in chemical and biochemical reactions gravitational collapse in astrophysics and the instability of an atomic lattice are all considered in the present article as examples of an equilibrium path of a system which suddenly becomes unstable. The theory of elastic stability offers one set of explanations. Another comes from the more recent catastrophe theory associated with the name of René Thorn. In the following contribution these two approaches are taken a aspects of a unified non-linear bifurcation theory, able to reduce an original multi-dimensional problem to one which can be described in term of a small number of active internal coordinates and a corresponding number of external control parameters. This allow three-dimensional picture to be drawn for many of the most relevant singularities.
This chapter discusses survey of research in theoretical biology that addresses issues in embryology and the physiology of activity, and then states opinions on the status of Bernstein's philosophical position. By contrast, as Bernstein implies, the Russians developed a separate tradition of the study of the physiology of regulation, which may be traced back to Sechenov's study of “Reflexes of the Brain.” In addressing the neuropsychological problem of predicting lesion sites in the brain from symptom-complexes and vice versa, Luria explicitly bases his idea of “functional system” on the work of Anokhin, Bernstein and Vygotsky, asserting that the fundamental task of neuropsychology is to ascertain “which groups of concertedly working zones are responsible for the performance of complex mental activity (and) what contribution is made by each of these zones to the complex functional system.” Two approaches are discussed to the “units of control” employed by the brain in controlling movement. Both are called synergies. The first, due to Sherrington, posits a reflex unit above that of the motor unit; while the second, due to Bernstein, suggests that a restricted number of programs may underlie most of behavior.
Elementary catastrophe theory provides a method for the qualitative description of systems with associated potential energy functions. It can serve at least as an illustrative metaphor for the local description of biological development. It replaces earlier theories of development that utilized paths on an ad hoc epigenetic landscape, for example, with one that generates trajectories on a well-defined hypersurface, the catastrophe manifold. While the elementary theory does provide a much richer mathematical language than the earlier geometric theories, it suffers from several drawbacks, outlined in this paper, which prevent it from being a conclusive theory for biological development.
It is shown that a large class of dynamics can give rise to primary and secondary waves, with the important property that the secondary wave does not have to follow in the path of the primary wave; it can be at a considerable angle to it or even in the opposite direction. To illustrate how this result can be applied in developmental biology, it is used to provide support for a two-wave model for prepattern formation and determination in Drosophila.
The geometrical nature of the Elementary catastrophes (Thom, 1969) is reviewed. Histories of the movement of catastrophe manifolds
and bifurcation sets are presented for some of the space-equivalent unfoldings described by Wassermann (1975). These unfoldings
provide descriptions of the variation with time of the stability of stationary states of associated potential energy functions.
Identification of these stationary energy states with stationary states of a system therefore provides a description of its
behavior with time. Qualitative descriptions of this type are particularly useful when the complexity of a system prevents
a detailed quantitative description. Histories of bifurcation set movements suggest different types of system behavior at
different space-like coordinates. This type of theory may be a useful model for the processes leading to differentiation of
cells and to emergence of adult forms of a biological organism.
Since Darwin's theory of evolution by natural selection, the idea of descent with modification came to dominate systematics, and so the study of morphology became subgugated to the reconstruction of phylogenies. Reinstating the organism in the theory of evolution (Ho & Saunders, 1979; Webster & Goodwin, 1982) leads to a project inrational taxonomy (Ho, 1986, 1988a), which attempts to classify biological forms on the basis of transformations on a given dynamical structure.Does rational taxonomy correspond to thenatural system that Linnaeus and his contemporaries as well as all pre-Darwinian morphologists had in mind? Here, we examine how rational taxonomy and the natural system can coincide in the dynamics of processes generating forms during development, which conferexclusivity, genericity androbustness to the forms that do exist. We use the example of segmentation, especially inDrosophila, as an illustration to explore the implications of rational taxonomy for evolution and systematics, and the relationship between ontogeny and phylogeny.
Examen critique approfondi de la theorie des catastrophes appliquee, principalement au domaine biologique (E. C. Zeeman). Que ce soit dans ses applications neurophysiologiques ou ethologiques, etc., cette theorie, qui n'est pas discutee ici sur le plan mathematique pur, se revele peu feconde, et surtout peu informative.
The role of stretching-generated tensile stresses upon the organization of axial rudiments have been studied. Pieces of the dorsal wall ofXenopus laevis andRana temporaria embryos at the late gastrula stage were rotated through 90, transplanted into the field of neurulation tensions of another embryo and replaced by ventral tissues already insensitive to inductive influences. The axial rudiments which developed from rotated and transplanted dorsal tissues oped from rotated and transplanted dorsal tissues almost completely reorientated according to the tensile patterns in adjacent host tissues. Some of the donor cells changed their presumptive fates in accordance with their new positions in the host tensile field. Transplanted ventral tissues were involved in the morphogenetic movements specific for the dorsal regions and imitated some typical dorsal structures. In the regions without pronounced tensions the structure of transplanted axial rudiments was chaotic. It is suggested that the organization of the axial structures is established and maintained by tensile fields created by uniformly polarized cells. Cell polarization can be transmitted by contact from host to donor tissues. The specificity of this propagating process and its morphogenetical role is discussed.
Catastrophe theory is a mathematical theory which, allied with a new and controversial methodology, has claimed wide application,
particularly in the biological and the social sciences. These claims have recently been heatedly opposed. This article describes
the debate and assesses the merits of the different arguments advanced.
The world I grew up in believed that change and development in life are part of a continuous process of cause and effect, minutely and patiently sustained throughout the millenniums. With the exception of the initial act of creation ..., the evolution of life on earth was considered to be a slow, steady and ultimately demonstrable process. No sooner did I begin to read history, however, than I began to have my doubts. Human society and living beings, it seemed to me, ought to be excluded from so calm and rational a view. The whole of human development, far from having been a product of steady evolution, seemed subject to only partially explicable and almost invariably violent mutations. Entire cultures and groups of individuals appeared imprisoned for centuries in a static shape which they endured with long-suffering indifference, and then suddenly, for no demonstrable cause, became susceptible to drastic changes and wild surges of development. It was as if the movement of life throughout the ages was not a Darwinian caterpillar but a startled kangaroo, going out towards the future in a series of unpredictable hops, stops, skips and bounds. Indeed, when I came to study physics I had a feeling that the modern concept of energy could perhaps throw more light on the process than any of the more conventional approaches to the subject. It seemed that species, society and individuals behaved more like thunder-clouds than scrubbed, neatly clothed and well-behaved children of reason. Throughout the ages life appeared to build up great invisible charges, like clouds and earth of electricity, until suddenly in a sultry hour the spirit moved, the wind rose, a drop of rain fell acid in the dust, fire flared in the nerve, and drums rolled to produce what we call thunder and lightening in the heavens and chance and change in human society and personality.
LAURENS VAN DER POST, The Lost World of the Kalahari
The Volterra equations which represent competitions between two species are utilized to examine the phenomenon of boundary
formation between two species of plants. The set of stable stationary points for these equations is determined and is illustrated
in a product space of parameters and dynamical variables. The stages of boundary appearance and succession are visualized
by considering slow changes of the parameters as functions of time and space.
This paper addresses itself to long-term conflicting issues in environmental policy analysis. After a brief discussion of the ‘new scarcity’ and its subsequent potential perturbations of our economic system in the long run, attention will be paid to some elements of catastrophe theory which may be helpful in gaining insight into the future impacts of inertia and conflicts in environmental management. Next, a fairly simple model for integrating natural resources and production will be constructed so as to illustrate the usefulness of catastrophe theory in economic-energy-environmental systems. The paper will be concluded with a brief judgement of using catastrophe notions in the social sciences.
Kinetic instabilities and self-organization phenomena in chemical and biological systems are reviewed and discussed from the point of view of the unified qualitative theory of non-equilibrium processes.
We examine sequential spatial pattern formation in a tissue interaction model for skin organ morphogenesis. Pattern formation occurs as a front sweeps across the domain leaving in its wake a steady state spatial pattern. Extensive numerical simulations show that these fronts travel with constant wave speed. By considering the envelope of the solution profile we present a novel method of calculating its wave speed.
Disruption of normal vertebral development results from abnormal formation and segmentation of the vertebral precursors, called somites. Somitogenesis, the sequential formation of a periodic pattern along the antero-posterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. We review the most popular models of somite formation: Cooke and Zeeman's clock and wavefront model, Meinhardt's reaction-diffusion model and the cell cycle model of Stern and co-workers, and discuss the consistency of each in the light of recent experimental findings concerning FGF-8 signalling in the presomitic mesoderm (PSM). We present an extension of the cell cycle model to take account of this new experimental evidence, which shows the existence of a determination front whose position in the PSM is controlled by FGF-8 signalling, and which controls the ability of cells to become competent to segment. We conclude that it is, at this stage, perhaps erroneous to favour one of these models over the others.
Catastrophe theory can be used to illustrate the results of some of the events that occur during embryogenesis. Trajectories over the surface of catastrophe manifolds provide an extension of Waddington's concept of a description of embryogenesis based on chreodic paths on an epigenetic landscape. The development of Elementary Catastrophe Theory to describe time- and space-equivalent events provides a rich qualitative “language” that seems well suited for the description of cellular behavior. Space-equivalent catastrophes have been used to illustrate the processes of cellular determination, differentiation, and transdetermination with initial success. This use should be tested further, for example, by experimental manipulation of embryonic tissue at well defined intervals of time during its development. The relationship of this approach to some others and to possible future theoretical developments has been discussed.
Morphological evidence is presented that definitive mesoderm formation in Xenopus is best understood as extending to the end of the neurula phase of development. A process of recruitment of cells from the deep neurectoderm layers into mesodermal position and behaviour, strictly comparable with that already agreed to occur around the internal blastoporal 'lip' during gastrula stage 20 (earliest tail bud). Spatial patterns of incidence of mitosis are described for the fifteen hours of development between the late gastrula and stage 20--22. These are related to the onset of new cell behaviours and overt cyto-differentiations characterizing the dorsal axial pattern, which occur in cranio-caudal and then medio-lateral spatial sequence as development proceeds. A relatively abrupt cessation of mitosis, among hitherto asynchronously cycling cells, precedes the other changes at each level in the presumptive axial pattern. The widespread incidence of cells still in DNA synthesis, anterior to the last mitoses in the posterior-to-anterior developmental sequence of axial tissue, strongly suggests that cells of notochord and somites in their prolonged, non-cycling phase are G2-arrested, and thus tetraploid. This is discussed in relation to what is known of cell-cycle control in other situations. Best estimates for cell-cycle time in the still-dividing, posterior mesoderm of the neurula lie between 10 and 15 h. The supposition of continuing recruitment from neurectoderm can resolve an apparent discrepancy whereby total mesodermal cell number nevertheless contrives to double over a period of approximately 12 h during neurulation when most of the cells are leaving the cycle. Because of pre-existing evidence that cells maintain their relative positions (despite distortion) during the movements that form the mesodermal mantle, the patterns presented in this paper can be understood in two ways: as a temporal sequence of developmental events undergone by individual, posteriorly recruited cells as they achieve their final positions in the body pattern, or alternatively as a succession of wavefronts with respect to changes of cell state, passing obliquely across the presumptive body pattern in antero-posterior direction. These concepts are discussed briefly in relation to recent ideas about pattern formation in growing systems.
Multistable figures show that the stimulus-percept relation is not a single valued function. We therefore propose a tentative nonlinear model on the hypothesis that the graph of this relation is the equilibrium set of a dynamic system. For simplicity and to obtain testable predictions, we consider a system whose bifurcations are gradient-like and thus generically described by the elementary catastrophes. We motivate this general model, and then show how, in conjunction with the principle of minimal singularity, it implies cusp catastrophe geometry in a specific perceptual example. Indeed, we argue for canonical cusp geometry in this case. The model incorporates naturally certain observed features of multistable perception, such as hysteresis and bias effects. Despite being a continuum model it is naturally compatible with the subjective dichotomy of bistable perception. The model makes testable predictions which may easily be extended to other specific examples of multistable perception.
Catastrophe Theory was developed in an attempt to provide a form of Mathematics particularly apt for applications in the biological sciences. It was claimed that while it could be applied in the more conventional “physical” way, it could also be applied in a new “metaphysical” way, derived from the Structuralism of Saussure in Linguistics and Lévi-Strauss in Anthropology.
Since those early beginnings there have been many attempts to apply Catastrophe Theory to Biology, but these hopes cannot be said to have been fully realised.
This paper will document and classify the work that has been done. It will be argued that, like other applied Mathematics, applied Catastrophe Theory works best where the underlying laws are securely known and precisely quantified, requiring those same guarantees as does any other branch of Mathematics when it confronts a real-life situation.
It is argued that all chronic gastroduodenal peptic ulcers result from localised increase in mucosal susceptibility to acid attack at the interface between a segment of gastroduodenitis and gastric fundus or duodenal mucosa. The site is predetermined by the background mucosal pattern. Changes can occur in the differentiated gastroduodenal mucosa that closely resemble cell population transformations described in embryology and regeneration biology. A second pathological process, gastroduodenitis, may develop that does not of itself predispose to ulceration, but the combination of factors can produce a zone of increased acid susceptibility. These complex changes could be generated by immunologically activated gastroduodenitis. Destructive or stimulatory immune reactions, analogous to those seen in the thyroid gland, could affect the gastrin-secreting G cells and other paracrine cells. The resulting tropic and inflammatory reactions would provide the background for peptic ulceration.
How do animals wiggle and bend? Throughout the animal kingdom may be found a persistent body organization of repeated metameric segments. In practically all metazoans, this meristic body pattern provides an efficient arrangement for body movement and locomotion. This is particularly striking in the annelids, in which this meristic pattern is seen in body segments and organs within the segments. Naturalists as far back as Aristotle (1910, translated by Thompson) have noted the common feature of segmentation in animals as diverse as worms, maggots, and snakes and in chicken embryos. Because of the ready availability of chicken eggs and the accessibility of the developing embryo, the segmentation of the vertebrate embryo was noted as early as the sixteenth century. This segmentation was alluded to as the “division of the embryo,” which could seldom be seen in the early stages of development without aids for magnification. Although lenses for magnification were noted as early as Aristophanes (fifth century BC), Roger Bacon (thirteenth century) is thought to have been the first scientist—naturalist to use lenses for investigative purposes (Needham, 1959). Embryologists were slow to use microscopes in their observations, although occasional use of the “perspicilli” was reported, and undoubtedly this instrument helped Harvey in his pioneer observations (1651) on early chick development (Meyer, 1936). Others in the seventeenth century also represented somites in their drawings of chicken embryos, although they did not remark upon them.
One of the most visually striking patterns in the early developing embryo is somite segmentation. Somites form as repeated, periodic structures in pairs along nearly the entire caudal vertebrate axis. The morphological process involves short- and long-range signals that drive cell rearrangements and cell shaping to create discrete, epithelialized segments. Key to developing novel strategies to prevent somite birth defects that involve axial bone and skeletal muscle development is understanding how the molecular choreography is coordinated across multiple spatial scales and in a repeating temporal manner. Mathematical models have emerged as useful tools to integrate spatiotemporal data and simulate model mechanisms to provide unique insights into somite pattern formation. In this short review, we present two quantitative frameworks that address the morphogenesis from segment to somite and discuss recent data of segmentation and epithelialization.
This tutorial paper explains some of the basic mathematical ideas involved in René Thom's Catastrophe Theory, in the simplest and most accessible case: the Elementary Catastrophes. Topics discussed include the classification of local behavior of smooth functions, determinacy (how much of a Taylor series is adequate to capture the function's behavior), unfoldings (what are the possible perturbations?), and structural stability (robustness). These and other concepts are exemplified using simple models from science: the buckling of an arch, the response of patients suffering from hyperthyroidism to therapy, and cellular differentiation. These models have been selected to act as illustrative examples, and no attempt is made here to survey the applications of the theory since this has already been done in Stewart (1981,1982).
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