Purpose of this review

Patterns in nature have attracted attention for centuries because of their complexity and regularity. A great leap forward in the theoretical understanding of their development from simple interacting constituents was made by Alan Turing in his classic work on morphogenesis [1]. The static "Turing" patterns that develop may take the forms of, among others, spots or stripes, famously reminiscent of those found on animal coats. A simple reaction-diffusion model that can display Turing patterns is the Schnakenberg model (Fig 1A) [2,3]. An analogous mechanism has recently been identified in the form of a cellular interaction network in zebrafish, responsible for its skin patterns [4], and another involved in the mammalian palate [5].

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Fig 1. Introduction to pattern types and spiral wave formation.

A: (Left to right) spatial and spatiotemporal pattern examples. Spot and stripe Turing patterns both in coupled Schnakenberg elements; target wave formation from a central pacemaker and established spiral wave, both in coupled FitzHugh-Nagumo oscillators. B: Snapshots of spiral wave patterns from diverse biological systems: (left to right) cAMP signaling in a Dictyostelium discoideum colony, local contraction in neonatal rat cardiac monolayer cultures, MinD protein density in a lipid bilayer and simulated cytokine levels in a two-dimensional grid of cells. See Acknowledgments for image sources. C (upper row): The update rules of the minimal three-state cellular automaton model lead to spiral wave formation, when applied to an open wave front (consisting of a layer of excited cells, depicted in black, and an adjacent layer of refractory cells, depicted in gray). Lower row: a similar numerical experiment for the model from [6,7].

https://doi.org/10.1371/journal.pcbi.1004367.g001

Turing patterns represent one of the major categories of patterns that may be observed. Another important and widespread type is the travelling wave; such patterns are periodic and vary dynamically with time. These wave patterns manifest in biology in situations such as the circular "target" wavefronts produced by the sinoatrial node in the heart, or the open-ended, spiral waves of signaling found in the development of the frog Xenopus laevis [8]. Examples of both these wave types are shown in Fig 1A. While target waves require an oscillating "pacemaker" cell in their center, spiral waves have a self-sustaining spiral core and typically originate from the breaking of the wavefront of target waves. The deep link between excitability and wave propagation allows us to look at the underlying mechanisms in an algorithmic fashion: A highly stylized, minimal model of excitable dynamics can reproduce both the schematics of wave propagation and the basic principle of the transition from an open wave front to a self-sustained spiral (see Fig 1C). In this review, we are particularly concerned with understanding aspects of such transitions: in particular, how they are regulated by features of the nonlinear relationships governing the system at hand.

Spatiotemporal patterns have been a highly successful perspective in the study of complex systems for two main reasons: (1) Patterns often fall into a small number of generic categories (e.g., aggregation patterns or propagating waves [9]). They provide fundamental links between local interaction rules and large-scale collective behaviors [10]. The types of interactions and the nonlinearities of the reactions establishing such large-scale patterns are ubiquitous, with reaction-diffusion systems providing the most prominent example; see, e.g., [11]. The prediction derived from the observation of patterns of biochemical agents and corresponding biochemical mechanisms implementing a reaction-diffusion system (e.g., in hydra development and head-region regeneration [12]) constitutes another example of the enormous power that lies in the observation of patterns in real systems and the subsequent dissection (or deconstruction) of these patterns towards an identification of local interaction rules guided by our understanding of self-organization. (2) Patterns are abstract, generic objects that can establish (or make visible) unexpected parallels between diverse biological, technological, and social systems. The ubiquity of spiral wave patterns across many systems and many scales, from molecular-level interactions to ecosystem-wide responses [13], is a good example of this unifying capacity of a pattern perspective. Fig 1B contains some examples of spiral waves in real biological systems.

As high-throughput analysis techniques provide ever more quantitative and often spatially resolved biological data (see, e.g., [14–16] as examples of such technological advances), we now have the means to explore the merits of a biology-inspired view, in which the (statistical) details of the patterns matter. The essential features of a biologically oriented perspective on pattern formation are (firstly) that in the majority of pattern-forming biological systems, the patterns often, though not exclusively (see, e.g., [17] for a critique of an indiscriminately adaptionist perspective on biology), serve a purpose and therefore can be considered to be evolutionarily optimized. A variety of examples exist: the size of spiral waves regulates the size of multicellular aggregates in subsequent stages of aggregation in the slime mold Dictyostelium discoideum. Bacteria stabilize themselves via aggregation against water flow. Another slime mold, Physarum plasmodium, employs pattern formation to sample space for food sources. Even circadian rhythms in plants can have a complex underlying spatiotemporal organization, leading to propagating waves along leaves, coordinating photosynthetic activity [18]. The stomatal cells of plants respond with complex patterns (termed "stomatal patchiness") to changes in the environment, which effectively implement a density classification algorithm [19].

Noise has always been at the forefront of interest in the analysis of nonlinear systems. For biological systems, over the last few years, intrinsic and extrinsic noise in gene expression [20–22] and in cell fate decisions [23,24] has been investigated. While the variability we refer to here represents fixed differences in cell-to-cell properties, varying, if at all, on a much larger time scale than the resulting patterns, the "extrinsic noise" in these works contains both variability and the collective noise (dynamically) acting upon all cells simultaneously. This distinction is important, as the distribution of cell properties can thus be tuned to achieve certain preferential collective states. Furthermore, some interesting theoretical work has emerged from studying the interplay of noise and variability on pattern formation (see "Background"). Similar to these influences of noise, variability can also shape patterns. As these differences between the constituents are "hard-wired" into the system, we expect a tight evolutionary control on the shaping of patterns by variability.

Secondly, it is plausible, and supported by diverse evidence [25–29], that in biology, the exact spatial layout of the patterns (among the diverse pattern arrangements possible in this systemic configuration) is selected by the distribution of cell properties (i.e., by biological variability), rather than solely—like in physics or chemistry—by spontaneous fluctuations [30]. In this way, spatiotemporal patterns are becoming ever more relevant for computational biology and systems biology because the intracellular "implementation" of such optimized collective states can be investigated. The patterns can serve as a "microscope" for the underlying single-cell mechanisms of regulation.

Understanding how variability (i.e., the magnitude of cell-to-cell differences) shapes collective states (i.e., the emerging patterns) is instrumental to a biology-oriented view of self-organization. We here review the role of variability in the self-organization of biological systems. Our key example is the establishment of spiral wave patterns in excitable media, which is used to model such phenomena as, e.g., the self-organization of Dictyostelium discoideum [15,31–33], cardiac arrhythmia [34] and even the epidemic spreading of infectious diseases (see, e.g., [35,36]). We thus explore the relationship between variability in cell properties and features of spiral wave patterns for a variety of mathematical models.

Basic principles of spiral wave formation and the influence of biological variability

In biology, variability in the form of cell-to-cell differences (in some biologically relevant parameter) can be expected to contribute more significantly than stochastic noise to the heterogeneity in a biological system (see, e.g., [29] for experimental evidence and [37–39] for theoretical arguments). The potential importance of variability in biological pattern formation is one of the most striking differences between biological systems and those in physics or chemistry, where system components are essentially identical and random fluctuations are the only factor determining the details of the self-organization process and the resulting patterns. This has far-reaching implications for our understanding of biological systems, in which patterns are linked to function. The patterns frequently constitute a (precursor of a) collective systemic mode and are thus subject to evolutionary selection. Variability may be seen as a mechanism by which individual elements shape these collective modes.

The authors in [33] have convincingly made this case for Dictyostelium, relating the strength of a regulatory feedback loop to the spatial density of spiral-wave patterns. By studying mutants in key components of this loop, the authors systematically varied this intrinsic parameter and observed the corresponding changes in the patterns. Their main finding is that wild-type feedback strength is optimized to yield maximal aggregation territory size. This allows for optimally sized basins of attraction for the consecutive aggregation process of the Dictyostelium cells in the formation of the multicellular stage capable of spore generation, which completes the developmental cycle. It is also mechanistically plausible that variability in cell properties could be an important driving mechanism behind spatiotemporal patterns in biological systems, as the degree of such differences determines the capacity of the cells to locally synchronize and, consequently, form patterns on a larger spatial scale. Variability, the existing pattern of cell-to-cell differences, is constant in time. It gives rise to heterogeneity in pattern-forming substances (e.g., in the distribution of a chemotactic substance). This heterogeneity, in turn, leads to the establishment of self-organized patterns. In Dictyostelium, the concept of a genetically encoded "developmental path," a starvation-induced transition of the individual cells through different dynamical regimes of behavior, initially the excitable rest state, then oscillatory, before once more becoming excitable, provides a framework through which the effects of heterogeneity can be implemented in a realistic theoretical setup (see [40] and "Models").

We start out with an example to illustrate how the distribution of cell properties can influence self-organized spatiotemporal patterns. Spiral waves can be generated in a variety of different ways: one of the most accessible basic mechanisms can be shown with an extremely simple model of excitable dynamics. Fig 1 introduces the main mechanistic idea of spiral wave pattern formation, illustrates their ubiquity in biological systems and provides a first example of how they are influenced by biological variability. The model used is a three-state cellular automaton on a two-dimensional lattice. Each lattice site is either susceptible S, excited E, or refractory R. An excitable (susceptible) element S acquires the state E, if there is an excitation E in its four-element neighborhood; a refractory element R returns to the susceptible state S after one time step (see "Models" for details). In Fig 1C, upper row, an open-ended wave front (first snapshot) curls up to form a spiral wave. In spite of the geometrical artifacts due to the simple four-element neighborhood and the simplistic state space, the general principle is clearly visible: the initial conditions of a wavefront segment (black elements) directly followed by a layer of refractory lattice sites (gray elements) trigger two forms of patterns, a propagating wave front (oriented upwards in the figure) and the open end of the wave front giving rise to first an additional excited element and then to a curved wave segment, which subsequently propagates outwards in a spiral shape.

A more sophisticated model (by [7]), which we will refer to as the "Levine model," brings the whole scenario much closer to biological reality. It was originally formulated to account for the basic mechanisms behind spiral wave formation in Dictyostelium discoideum. Technically, it is an ordinary differential equation (ODE)—cellular automaton hybrid. It uses effective degrees of freedom (excitability, refractory phase), along with biologically plausible variables (the local concentration of the intercellular signaling substance, cAMP) and parameters (e.g., the feedback strength, describing how an external cAMP signal affects cAMP production) to account for the most important properties of the actual biological system. This model has also been used in [37] and [33]. The lower row in Fig 1C shows a similar open-wavefront simulation using the Levine hybrid model, indicating that even with this more complex model, the behavior of spiral wave formation is qualitatively the same.

As these two examples demonstrate, spirals, and more generally any pattern in excitable media, may arise through different mechanisms. In general, a propagating circular wavefront must be interrupted. This may occur when such a wavefront encounters an obstacle in the form of refractory or less excitable elements (S1 Fig), when a target wave is generated inside another wave and can only partially propagate, or through special initial conditions which mimic this process, as in our first examples. Though their origins are still not always fully understood, spiral waves are ubiquitous in biological systems. Fig 1B gives examples of spatial snapshots from diverse biological situations. The first is cAMP concentrations, spreading through a colony of Dictyostelium cells prior to the aggregation phase of the amoeba’s life cycle (image from the database described in [44]; see also [33]). The second snapshot shows a spiral wave of electric activity forming in cardiac tissue: here, in neonatal rat cell cultures, imaged using a dye-free modality that tracks local contraction (see [41]). Such spiral waves are pathological, and their breakup appears to lead to cardiac arrhythmia. The third image shows spiral waves of protein density at a sub-cellular scale: the proteins MinD and MinE, which have a key function in Escherichia coli cell division, are here reconstituted in vitro on a supported lipid bilayer in the presence of ATP as the energy source (see [42]). The last snapshot in Fig 1B shows a simulation of spiral waves occurring in the context of signaling in mammalian tissue. The cytokine production of cells distributed on a regular 2-D lattice is shown. Suitable amounts of noise change the wave structure to the point where spiral waves appear in the tissue (see [43]).

In Dictyostelium, such spiral waves represent a chemotactic signal guiding each cell towards an aggregation site, at which the formation of a multicellular structure is initiated. The distribution of spiral waves thus directly affects the fate of each cell. The spiral wave density (qualitatively speaking) translates into the cell counts of the multicellular mounds. With spiral waves, a pathological dynamical state of cardiac tissue, identifying the spatial distributions of cell properties that lead to reduced spiral wave probability would be advantageous.

The growing accessibility of Dictyostelium mutant data and their resulting pattern-level phenotypes, through studies such as [44], also enhance the prospects of gaining insight into pattern formation in this model organism via mathematical models that incorporate detailed and realistic biological knowledge. Table 1 contains some examples of biological properties in Dictyostelium, with their corresponding methods of implementation in a computer model and the potential effects on patterns that result.

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Table 1. Sources of variability affecting details of macroscale patterns in Dictyostelium and their representations in computer-based mathematical models.

https://doi.org/10.1371/journal.pcbi.1004367.t001

These quantities are, of course, interdependent. Cell density [6] and excitability [33], for example, vary with the current stage in the cell cycle [25,40]. That responses of Dictyostelium to chemotactic stimuli are cell-specific has been shown in [29]. One method of incorporating variability into Dictyostelium models is the use of agent-based models, with applications to processes such as cell sorting [45]. The role of coupling strength was investigated in [46] and [47]; effects such as target-spiral competition were shown in [48].

Different cell properties will influence the exact layout of the emerging patterns in different ways. Similarly, different forms of cellular interactions and signal processing will modulate the effects of this biological variability in different ways. When such interdependences are well understood, the statistical properties of the patterns can thus serve as a "microscope" for the underlying principles of regulation.

Theoretical investigations of biological variability

Over several decades, a range of theoretical investigations has shown that variability can play an essential role in achieving a qualitative understanding of the processes at hand. The corresponding theoretical approaches range from variability as a driving force in evolutionary processes [77] to the generation of endogenous circadian oscillations from fast dynamics [78]. The vast field of research on synchronization in networks of nonlinear oscillators under the influence of variability and stochastic contributions has been described, e.g., from the point of view of synchronization [79] and from the network point of view [80].

The effect of variability on spatiotemporal chaos has been studied in [81–83]. Additionally, various numerical investigations have explored variability-induced pattern formation, compared to and in cooperation with noise using arrays of coupled oscillators [28,58,84]. The fact that, similarly to noise, variability can induce patterns or trigger a transition from one pattern to another has been a focus of research in the 1990s and early 2000s. In the more theoretically oriented studies, our biologically motivated term "variability" often appears as “diversity,” “disorder,” or “quenched noise.” For coupled FHN oscillators, it has been shown that variability can induce both a phase transition from oscillatory to excitable behavior [28] and, in a subexcitable system, pattern formation in the form of spiral waves of excitation [58]. Pattern complexity has been shown to be highest at intermediate variability [84]. Such studies culminated in the phenomenon of diversity-induced resonance [57,58], where the response of an excitable or bistable system to a subthreshold stimulus is optimal at intermediate levels of diversity. In [85] such a diversity-induced resonance was found in simulations of calcium dynamics, showing that calcium waves propagate optimally at intermediate cell-to-cell variability.

In recent work on excitable dynamics and spiral wave patterns, there is a strong trend to take into account realistic structural geometries in order to better understand the observed spatiotemporal patterns (see, e.g., [86]). On the methodological side, target and spiral wave identification techniques have been developed and implemented [49,87,88] and simulation methods have been advanced [89]. The authors of [90] also discuss how spiral tip identification in the FHN model influences the reconstructed patterns of meandering spiral waves. The use of generalized recurrence plots for reconstructing the phase diagram of nonlinear spatiotemporal systems from a limited set of observations has been described by [91]. In spite of its historical roots in the 1960s and 1970s, the explanation of spiral wave patterns still receives an enormous amount of scientific attention. The reason is 2-fold: (1) Our understanding of pattern formation processes still has severe gaps on the theoretical side; in particular, the routes from homogeneous patterns to fully established spiral waves are surprisingly dependent on the details of the system at hand. (2) It is currently being noticed that (precisely due to the dependence of patterns on the regulatory details within the system) a deep and detailed analysis of spiral wave patterns can help access the underlying principles of regulation. Detailed theoretical studies have provided evidence that the correlations between cell property distributions and patterns strongly depend on the regulatory mechanisms at the level of individual cells (e.g., comparing the results from [37] and [25]).

Returning to the purely theoretical description of properties of spatiotemporal patterns, it would of course be a major step forward if some aspects of the predictability of patterns from cell properties could also be understood analytically. Two approaches are particularly promising for tackling this question. Firstly, in [92] a mathematical framework has been developed for describing diffusion and annihilation of spiral wave tips within a simple kinematical model. Secondly, in [93] a response theory of (spiral wave) patterns under small perturbations has been formulated, in particular, the sensitivity of the spiral’s drift velocity.

Even in studies in which medium heterogeneity in the system is taken explicitly into account (e.g., [94]), the correlation between heterogeneity and the distribution properties of spiral waves is not discussed. We believe that this link could convey important insights into the prediction of spiral wave events, and on a wider level, the relationship between heterogeneity and system behavior has many additional practical applications in fields ranging from engineering to disease outbreak prevention. The strong practical interest in understanding spiral wave patterns in cardiac tissue has also led to a range of theoretical studies on the pinning of spiral waves by properties of the medium [95–98].

Dictyostelium pattern formation

A model system for which regulation by variability could be a principal mechanism (see, e.g., [7,40]) is the slime mold Dictyostelium discoideum. In this paradigmatic example of biological pattern formation, individual amoeba cells aggregate under the influence of the chemotactic signal cAMP and form a multicellular organism [99]. This process is initiated by nutrient deprivation; this causes single cells to emit cAMP into their environment. These molecules are detected by neighboring cells via highly specific surface receptors [100], initiating the intracellular autocatalytic synthesis of additional cAMP by the enzyme adenylyl cyclase (ACA) and its subsequent secretion into the environment. Time-delayed receptor desensitization and halting of ACA activity are involved in the following refractory period. Extracellular cAMP is degraded by membrane-bound and secreted phosphodiesterase, which is, on the other hand, regulated by its inhibitor. The coupling of the underlying reaction kinetics with diffusion results in wave propagation. As long as the local cAMP concentration increases with time, the cells react with positive chemotaxis, resulting in periodic movement perpendicular to the wave front, i.e., towards the origin of the chemical signal.

As more details become available concerning the molecular network responsible for cAMP oscillations in Dictyostelium (e.g., [101,102]), modelling efforts increase in their corresponding complexity (see, e.g., [103,104], which are typical of this transition towards a more "systems biology" approach). In terms of the local Dictyostelium response to the cAMP gradient, progress is still being made in the understanding of the small-scale decisions underlying pattern formation [105]. Another evolving avenue of interest is Dictyostelium’s contribution to understanding the origins of multicellularity [106,107]; such efforts are spurred on by the sequencing of Dictyostelium discoideum’s genome [108]. These recent examples illustrate the continually increasing, vast amount of research that has been done over more than four decades into many aspects of the Dictyostelium life cycle. In particular, several findings of the last few years have added new ideas to the view that in the case of Dictyostelium, biological variability is responsible for certain stages of symmetry breaking in the usual course of the developmental cycle (local pattern initiation, spatial distribution of cell streams, and distribution and proportions of differentiated cell types).

In the work by the authors of [33], the strength of a regulatory feedback loop is related to the spatial density of spiral wave patterns in cell colonies of Dictyostelium. By studying mutants in key components of the regulatory feedback loop, the authors systematically varied this intrinsic parameter and observed how the spatiotemporal patterns changed accordingly. In order to display the systematics of their finding more clearly, the authors of [33] simulate patterns using a simple cellular-automaton based model, in which the feedback strength (i.e., the cAMP pulse-dependent increase in excitability) appears as an explicit parameter. Their observable is the spatial frequency of spiral waves, which they obtain by counting the phase singularities in their spatiotemporal patterns. Remarkably, the intermediate feedback strength found in wild-type cells turns out to produce an optimal (i.e., minimal) number of phase singularities compared to higher and lower feedback strength mutants, respectively. In the extreme case, in which feedback is constantly absent, no stable spiral wave pattern evolves. This observation has two implications. Firstly, aggregation territory size is optimized by the pulse-dependent development of feedback strength. For the wild type, this allows for optimally sized basins of attraction for the consecutive aggregation process of the cells leading to the multicellular organism capable of spore generation and thus completing the developmental cycle of Dictyostelium. Secondly, wave geometry is determined by feedback strength. Spiral waves seem to be favored by the system compared to target waves, although both wave geometries result in fruiting bodies for the experimental system. The biological advantage of spiral-based signaling is that spirals are self-sustaining continuous structures, which preserve their stability, to a certain extent outside the excitable regime. This allows for maintenance of the aggregation process under developmentally and environmentally conditioned changes. In contrast, target waves require the periodic activity of oscillatory regions (i.e. pacemakers).

Strong support for the hypothesis that cell properties can indeed affect the collective patterns in Dictyostelium has come from the recent observation that the direction and magnitude of a cell’s response to a signal pulse is an individual cell property, which remains constant in time [29]. In that work the behavior of single cells under periodic cAMP signals was analyzed and it is observed that the characteristics of the gradient sensing response of an individual cell at a certain time point strongly correlate with those of the same cell at a later time point. A recent paper [15] provides further support for our hypotheses, demonstrating that Dictyostelium cells’ internal cAMP concentrations oscillate at a frequency determined by intracellular machinery. The authors found that experimentally observed rhythmic cAMP synthesis could only be replicated successfully in their mathematical model when the stochastic pulsing of individual cells in response to subthreshold cAMP levels was included in the model. The implication is that organized group dynamics in a Dictyostelium population depend on the random behavior of individual cells (see also [109]). Another study finds that isogenic Dictyostelium cells have diverse sensitivities to cAMP, and that this may facilitate collective behavior [110]. Similarly, in [72] the roles of noise and variability in Dictyostelium cAMP signaling were explored in a phenomenological FitzHugh-Nagumo model. Here, variability was incorporated in the form of individual cells’ thresholds of response to extracellular cAMP. The results suggested that stochastic noise can account for population-level cAMP oscillations, while variability in this cell property alone cannot. These recent works, in addition to greatly enhancing the experimental accessibility of the system (via the direct measurement of cAMP concentrations), emphasize the role of individual cells and the debate whether it is the activity of dedicated pacemaker cells or the spontaneous firing of random cells which constitutes the main driving mechanisms behind the initiation of patterns during this early phase (see also [111]). While the work from [15] emphasizes the latter aspect, the results of [29] suggest an important role for the former, and it is likely that both noise and variability interact in the shaping of spatiotemporal patterns.

These diverse sets of questions, all of which are related to biological variability and their influence on pattern formation, illustrate the importance and timeliness of establishing a biology-oriented perspective on spatiotemporal patterns.

In the following, we will focus on two additional mathematical models. The first is a highly detailed model of pattern formation in Dictyostelium, which was initially formulated in [112]; we also refer to this model as the "Goldbeter" model. This model incorporates relevant biological details, such as the relay of suprathreshold cAMP pulses and autonomous cAMP oscillations, and the phosphorylation-dependent modification of the cAMP receptor, in the same way that current modeling attempts in systems biology would (see, e.g., [104,113]). Standard model reduction techniques (such as time scale separation) yielded a three-dimensional model that has also been elegantly used to analyze spatiotemporal patterns [40]. The dynamical variables are the fraction of active cAMP receptors and the concentrations of external and intracellular cAMP, respectively (see "Models" for details).

As in many such modeling situations, one now has the choice to either explore the biological details of these spatiotemporal patterns or, alternatively, aim at understanding the generic features. The second model to be discussed in detail in the following is therefore an array of coupled FitzHugh-Nagumo oscillators as a generic model for spatiotemporal patterns arising from excitable dynamics [53,54].

A novel ingredient organizing the interplay between biological variability and the resulting patterns in a system is the concept of a "developmental path." This is a specific parameter drift with time which couples small cell-to-cell property differences to the cell’s dynamical behavior, thus turning them into "organizers" of the spatiotemporal patterns. The "developmental path" concept was proposed by Lauzeral and coworkers [40], in the context of the model of Dictyostelium discoideum pattern formation developed by Martiel and Goldbeter [112] (see also [114]). It was created as a hypothetical mechanism of the development of heterogeneity from homogeneous conditions for the synthesis of spiral waves of cAMP in Dictyostelium. According to this concept, the cells in a population of Dictyostelium undergo time-dependent changes in cell properties, and intercellular variation in these properties places the cells at differing points along this developmental path. The path in [40] results from sigmoidal variations in the maximum activity of adenylate cyclase and the rate of extracellular cAMP degradation. The variation from cell to cell in the cell cycle when starvation begins is a possible origin of the different time offsets of the cells on the path. Such a path would be expected to occur in a higher-dimensional space in reality, but even in this simple form can provide an adequate representation of the actual biological dynamics. We consider the developmental path form of the Goldbeter model as an example of predicting patterns arising from a biologically plausible source of variability. Furthermore, we discuss the extension of the concept to the FHN model, showing that the principle is versatile and can be applied to diverse biological models as a method of generating heterogeneity.

Models

Here we describe the models used throughout this review. The symbol ∇² represents the discretized Laplacian for 2-D diffusion, and we use the five-point Laplacian except in the case of the Levine model, where the eight-point Laplacian is used. D is the value of the respective diffusion coefficient.

Schnakenberg model.

This simple model [2] appears in Fig 1A, as an example of a driver of Turing-type patterns. Here, c₁ = 0.05, c_-1 = 1.0, c₃ = 1.0, D_u = 1.0, D_v = 20.0, and γ = 2.0. The value of the parameter c₂ determines whether the system gives rise to spots (c₂ = 1.00) or stripes (c₂ = 1.57). The initial values of u and v are randomly varied between -0.01 and 0.01.

(1)