Predictable topological sensitivity of Turing patterns on graphs

Hütt, Marc‐Thorsten; Armbruster, Dieter; Lesne, Annick

doi:10.1103/physreve.105.014304

Cited by 9 publications

(2 citation statements)

References 45 publications

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“…Small systems may behave differently from macroscopic ones due to system subdivision and quantum interactions, leading occasionally to the violation of extensivity and the Gibbs–Duhem relation. , Diffusion Maps can identify thermodynamic variables through the eigendecomposition of the particles affinity matrix . Macroscopic transport phenomena, at the boundary of chemistry and physics, involve gradual information spread within the component network, while the synchronization of CRN or particle concentration field harmonizes dynamics. Coupled diffusion-reaction phenomena may drive self-organizing spatiotemporal patterns found in combustion, biology, or catalysis, such as “chemical turbulence” .…”

Section: Continuum-scale Chemical Mixturesmentioning

confidence: 99%

Mixtures Recomposition by Neural Nets: A Multidisciplinary Overview

Nicolle,

Deng,

Ihme

et al. 2024

J. Chem. Inf. Model.

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Artificial Neural Networks (ANNs) are transforming how we understand chemical mixtures, providing an expressive view of the chemical space and multiscale processes. Their hybridization with physical knowledge can bridge the gap between predictivity and understanding of the underlying processes. This overview explores recent progress in ANNs, particularly their potential in the ’recomposition’ of chemical mixtures. Graph-based representations reveal patterns among mixture components, and deep learning models excel in capturing complexity and symmetries when compared to traditional Quantitative Structure–Property Relationship models. Key components, such as Hamiltonian networks and convolution operations, play a central role in representing multiscale mixtures. The integration of ANNs with Chemical Reaction Networks and Physics-Informed Neural Networks for inverse chemical kinetic problems is also examined. The combination of sensors with ANNs shows promise in optical and biomimetic applications. A common ground is identified in the context of statistical physics, where ANN-based methods iteratively adapt their models by blending their initial states with training data. The concept of mixture recomposition unveils a reciprocal inspiration between ANNs and reactive mixtures, highlighting learning behaviors influenced by the training environment.

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Section: Continuum-scale Chemical Mixturesmentioning

confidence: 99%

Mixtures Recomposition by Neural Nets: A Multidisciplinary Overview

Nicolle,

Deng,

Ihme

et al. 2024

J. Chem. Inf. Model.

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“…However, their analyses were restricted to regular lattices [ 8 , 9 ] and small networks [ 10 , 11 ]. Recent theoretical work has elucidated the relationship between network architecture and diversity of Turing patterns [ 37 ].…”

Section: Introductionmentioning

confidence: 99%

Reaction-diffusion models in weighted and directed connectomes

Schmitt

Nitzsche²,

Eipert³

et al. 2022

PLoS Comput Biol

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Connectomes represent comprehensive descriptions of neural connections in a nervous system to better understand and model central brain function and peripheral processing of afferent and efferent neural signals. Connectomes can be considered as a distinctive and necessary structural component alongside glial, vascular, neurochemical, and metabolic networks of the nervous systems of higher organisms that are required for the control of body functions and interaction with the environment. They are carriers of functional epiphenomena such as planning behavior and cognition, which are based on the processing of highly dynamic neural signaling patterns. In this study, we examine more detailed connectomes with edge weighting and orientation properties, in which reciprocal neuronal connections are also considered. Diffusion processes are a further necessary condition for generating dynamic bioelectric patterns in connectomes. Based on our high-precision connectome data, we investigate different diffusion-reaction models to study the propagation of dynamic concentration patterns in control and lesioned connectomes. Therefore, differential equations for modeling diffusion were combined with well-known reaction terms to allow the use of connection weights, connectivity orientation and spatial distances. Three reaction-diffusion systems Gray-Scott, Gierer-Meinhardt and Mimura-Murray were investigated. For this purpose, implicit solvers were implemented in a numerically stable reaction-diffusion system within the framework of neuroVIISAS. The implemented reaction-diffusion systems were applied to a subconnectome which shapes the mechanosensitive pathway that is strongly affected in the multiple sclerosis demyelination disease. It was found that demyelination modeling by connectivity weight modulation changes the oscillations of the target region, i.e. the primary somatosensory cortex, of the mechanosensitive pathway. In conclusion, a new application of reaction-diffusion systems to weighted and directed connectomes has been realized. Because the implementation were performed in the neuroVIISAS framework many possibilities for the study of dynamic reaction-diffusion processes in empirical connectomes as well as specific randomized network models are available now.

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Turing Bifurcation in the Swift–Hohenberg Equation on Deterministic and Random Graphs

Medvedev,

Pelinovsky

2024

J Nonlinear Sci

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The Swift–Hohenberg equation (SHE) is a partial differential equation that explains how patterns emerge from a spatially homogeneous state. It has been widely used in the theory of pattern formation. Following a recent study by Bramburger and Holzer (SIAM J Math Anal 55(3):2150–2185, 2023), we consider discrete SHE on deterministic and random graphs. The two families of the discrete models share the same continuum limit in the form of a nonlocal SHE on a circle. The analysis of the continuous system, parallel to the analysis of the classical SHE, shows bifurcations of spatially periodic solutions at critical values of the control parameters. However, the proximity of the discrete models to the continuum limit does not guarantee that the same bifurcations take place in the discrete setting in general, because some of the symmetries of the continuous model do not survive discretization. We use the center manifold reduction and normal forms to obtain precise information about the number and stability of solutions bifurcating from the homogeneous state in the discrete models on deterministic and sparse random graphs. Moreover, we present detailed numerical results for the discrete SHE on the nearest-neighbor and small-world graphs.

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Predictable topological sensitivity of Turing patterns on graphs

Cited by 9 publications

References 45 publications

Mixtures Recomposition by Neural Nets: A Multidisciplinary Overview

Mixtures Recomposition by Neural Nets: A Multidisciplinary Overview

Reaction-diffusion models in weighted and directed connectomes

Turing Bifurcation in the Swift–Hohenberg Equation on Deterministic and Random Graphs

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