Qualitative and Quantitative Cellular Automata from Differential Equations

Narbel, Philippe

doi:10.1007/11861201_16

Cited by 3 publications

(7 citation statements)

References 21 publications

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“…In contrast to implicit numerical schemes, they suffer from stability restrictions, which can cause higher computational costs [35]. On the other hand, explicit numerical schemes are closely related to CA [24] and, thus, easily adapt to discrete phenomena in space such as excluded volumes. CA represent a versatile and flexible approach that enables the combination of representations at different scales [17].…”

Section: An Explicit Numerical Scheme To Approximate the Diffusion Eqmentioning

confidence: 98%

“…Approximating Equation (1) by using a second-order difference scheme as presented in [24,33] yields directly the rule for the state transition (4)): (f) biological scenario, which includes excluded volumes. (g) The molecular mass diffuses in consideration of excluded volumes, (h) resulting in a distribution of molecular mass on all grid cells assigned to accessible space.…”

Section: An Explicit Numerical Scheme To Approximate the Diffusion Eqmentioning

confidence: 99%

“…This method is based on the replacement of the differential operators by finite differences [24]. The difference operators involve only points of the discretized source space spanned by vectors of the norms in time St and space δ, δ [24], where δ 2 denotes the size of a grid cell in the CA approach.…”

Section: An Explicit Numerical Scheme To Approximate the Diffusion Eqmentioning

confidence: 99%

“…The difference operators involve only points of the discretized source space spanned by vectors of the norms in time St and space δ, δ [24], where δ 2 denotes the size of a grid cell in the CA approach. This method has often been used by CA to model diffusion at the macroscopic scale [24] by applying a cell-based view on molecular diffusion: the state of each grid cell depends on the mass gradients to its specific local neighbourhood. We assume that the gradients are sufficiently…”

Section: An Explicit Numerical Scheme To Approximate the Diffusion Eqmentioning

confidence: 99%

“…As a consequence, spatial structures are simplified and excluded volumes are frequently neglected in practice [13]. Explicit numerical schemes are closely related to cellular automata (CA) [24] and, thus, adapt better to discrete phenomena such as a heterogeneous spatial environment that includes dynamic excluded volumes. CA represent a simple, versatile, and flexible approach that has often been applied to simulate biophysical systems [17,25].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An explicit numerical scheme to efficiently simulate molecular diffusion in environments with dynamically changing barriers

Kossow

Rybacki

Millat

et al. 2015

Mathematical and Computer Modelling of Dynamical Systems

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Despite temporal changes in the quantities of molecules, the functioning of cells also depends on their distribution within cells and in their extracellular environment. The dynamics of molecules are often governed by diffusion in heterogeneous environments consisting of dynamically changing impenetrable barriers (excluded volumes). This provides a challenge for efficient simulations of cellular processes with large numbers of molecules. To model the diffusion of molecular mass in consideration of excluded volumes, we here present an explicit numerical scheme that approximates the diffusion equation by using cellular automata. Because this approach represents molecular diffusion at the macroscopic scale, it is more amenable for efficient simulations than comparable microscopic approaches that treat diffusing molecules individually. In contrast to implicit numerical schemes (macroscopic approach), our approach is capable of accounting for excluded volumes, even if those exhibit a dynamic of their own, without increasing computational costs. The presented approach can easily be integrated into certain types of spatio-temporal multiscale models, as demonstrated by an existing model investigating cancer progression. Thereby, it allows to take the spatial effects of a heterogeneous environment on diffusing molecules into account.

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Section: An Explicit Numerical Scheme To Approximate the Diffusion Eqmentioning

confidence: 98%

Section: An Explicit Numerical Scheme To Approximate the Diffusion Eqmentioning

confidence: 99%

Section: An Explicit Numerical Scheme To Approximate the Diffusion Eqmentioning

confidence: 99%

Section: An Explicit Numerical Scheme To Approximate the Diffusion Eqmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An explicit numerical scheme to efficiently simulate molecular diffusion in environments with dynamically changing barriers

Kossow

Rybacki

Millat

et al. 2015

Mathematical and Computer Modelling of Dynamical Systems

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A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address broad spatial scales in spite of scarce data

Yatat

Tchuinte²,

Dumont³

et al. 2018

BIOMATH

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The savanna biome encompasses a variety of vegetation physiognomies that traduce complex dynamical responses of plants to the rainfall gradients leading from tropical forests to hot deserts. Such responses are shaped by interactions between woody and grassy plants that can be either direct, disturbance-mediated or both. There has been increasing evidence that several vegetation physiognomies, sometimes highly contrasted, may durably coexist under similar rainfall conditions suggesting multi-stability or at least not abrupt transitions. These fascinating questions have triggered burgeoning modelling efforts which have, however, not yet delivered an integrated picture liable to furnish sensible predictions of potential vegetation at broad scales. In this paper, we will recall the key ecological processes and resulting vegetation dynamics that models should take into account. We will also present the main modelling options present in the literature and advocate the use of minimalistic models, capturing only the essential processes while retaining sufficient mathematical tractability and restricting themselves to a minimal set of parameters assessable from the overall literature.

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Evolutionary Motifs for the Automated Discovery of Self-Organizing Dimer Automata

Arendt

Cao

2012

Advs. Complex Syst.

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It is usually difficult to reverse engineer a simple rule that exhibits some desirable and interesting behavior. We approach this problem by searching for dimer automaton rules exhibiting a broadly defined behavior, self-organization. We expected the simple and asynchronous nature of dimer automata to hinder self-organization, but an exhaustive search quickly yielded three rules that do, in fact, exhibit properties of self-organization. Two of these rules are applicable to actual physical phenomena, motivating searching for additional, more complex rules. However, exhaustive searches scale poorly here because of the rarity of interesting rules combined with the fast growth rate of the search space. To address these challenges we developed the evolutionary motifs algorithm. This algorithm finds the building blocks of the previously found dimer automaton rules, and combines them to form new rules in an evolutionary manner. Our evolutionary algorithm was more effective than an exhaustive search, producing a diverse population of rules exhibiting self-organization.

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Qualitative and Quantitative Cellular Automata from Differential Equations

Cited by 3 publications

References 21 publications

An explicit numerical scheme to efficiently simulate molecular diffusion in environments with dynamically changing barriers

An explicit numerical scheme to efficiently simulate molecular diffusion in environments with dynamically changing barriers

A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address broad spatial scales in spite of scarce data

Evolutionary Motifs for the Automated Discovery of Self-Organizing Dimer Automata

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