Stochastic Turing patterns on a network

Asslani, Malbor; Patti, Francesca Di; Fanelli, Duccio

doi:10.1103/physreve.86.046105

Cited by 39 publications

(44 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is anticipated that, in this case, the effective equation for the evolution of the unstable mode is of the Ginzbourg-Landau type. Moreover, we aim at considering the problem of pattern formation on a stochastic perspective [8,9] and consequently revisit the formal derivation here discussed to assess the role played by endogenous noise.…”

Section: Discussionmentioning

confidence: 99%

“…In general, when space reduces to a regular lattice or a symmetric graph, the dynamics is uniquely responsible for the onset of the instability which eventually materializes in the observed macroscopic and collective patterns. These are, for instance, the celebrated Turing patterns that, in recent years, have received much attention also in light of their applicability on networks [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Multiple-scale theory of topology-driven patterns on directed networks

Contemori¹,

Patti²,

Fanelli³

et al. 2016

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

Dynamical processes on networks are currently being considered in different domains of cross-disciplinary interest. Reaction-diffusion systems hosted on directed graphs are in particular relevant for their widespread applications, from computer networks to traffic systems. Due to the peculiar spectrum of the discrete Laplacian operator, homogeneous fixed points can turn unstable, on a directed support, because of the topology of the network, a phenomenon which cannot be induced on undirected graphs. A linear analysis can be performed to single out the conditions that underly the instability. The complete characterization of the patterns, which are eventually attained beyond the linear regime of exponential growth, calls instead for a full nonlinear treatment. By performing a multiple time scale perturbative calculation, we here derive an effective equation for the nonlinear evolution of the amplitude of the most unstable mode, close to the threshold of criticality. This is a Stuart-Landau equation the complex coefficients of which appear to depend on the topological features of the embedding directed graph. The theory proves adequate versus simulations, as confirmed by operating with a paradigmatic reaction-diffusion model.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multiple-scale theory of topology-driven patterns on directed networks

Contemori¹,

Patti²,

Fanelli³

et al. 2016

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

show abstract

“…In a spatial continuum, pattern formation arising from stochastic transport with reactions or forcing has been widely studied [5]. However, the study of pattern formation for stochastic transport on a network with reactions or forcing is a recent field of study [6][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Continuous-time random walks on networks with vertex- and time-dependent forcing

et al. 2013

View full text Add to dashboard Cite

We have investigated the transport of particles moving as random walks on the vertices of a network, subject to vertex-and time-dependent forcing. We have derived the generalized master equations for this transport using continuous time random walks, characterized by jump and waiting time densities, as the underlying stochastic process. The forcing is incorporated through a vertex-and time-dependent bias in the jump densities governing the random walking particles. As a particular case, we consider particle forcing proportional to the concentration of particles on adjacent vertices, analogous to self-chemotactic attraction in a spatial continuum. Our algebraic and numerical studies of this system reveal an interesting pair-aggregation pattern formation in which the steady state is composed of a high concentration of particles on a small number of isolated pairs of adjacent vertices. The steady states do not exhibit this pair aggregation if the transport is random on the vertices, i.e., without forcing. The manifestation of pair aggregation on a transport network may thus be a signature of self-chemotactic-like forcing.

show abstract

“…The stochastic component ultimately results from the inherent discreteness of the system, and can significantly modify the idealized mean-field predictions. Endogenous fluctuations induced by the finiteness of the system can, for instance, seed the emergence of regular oscillations, when parameters are set so to drive deterministic convergence towards a trivial equilibrium [24][25][26][27][28][29][30][31].In this Letter, we put forward a minimal model for discrete collections of excitatory and inhibitory agents in mutual interaction with excitatory and inhibitory loops, bearing universality traits in light of its inherent simplicity. Endogenous-noise induces quasi-cyclic dynamics that display unusual long range correlations, persisting over arbitrary large network structures.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Intertangled stochastic motifs in networks of excitatory-inhibitory units

et al. 2017

Self Cite

View full text Add to dashboard Cite

A stochastic model of excitatory and inhibitory interactions which bears universality traits is introduced and studied. The endogenous component of noise, stemming from finite size corrections, drives robust inter-nodes correlations, that persist at large large distances. Anti-phase synchrony at small frequencies is resolved on adjacent nodes and found to promote the spontaneous generation of long-ranged stochastic patterns, that invade the network as a whole. These patterns are lacking under the idealized deterministic scenario, and could provide novel hints on how living systems implement and handle a large gallery of delicate computational tasks.PACS numbers: 02.50. Ey, 87.18.Sn, 87.18.Tt Living systems execute an extraordinary plethora of complex functions, that result from the intertwined interactions among key microscopic actors [1]. Positive and negative feedbacks appear to orchestrate the necessary degree of macroscopic coordination [2], by propagating information to distant sites while supporting the processing steps that underly categorization and decisionmaking. Excitatory and inhibitory circuits play, in this respect, a role of paramount importance. As an example, networks of excitatory and inhibitory neurons constitute the primary computational units in the brain cortex [3][4][5]. They can flexibly adjust to different computational modalities, as triggered by distinct external stimuli [6,7]. Genetic regulation is also relying on sophisticated inhibitory and excitatory loops [8][9][10][11]. Specific genes are customarily assigned to the nodes of a given constellation, which can be abstractly pictured as a complex network [12][13][14]. Weighted edges between adjacent nodes encode for the characteristics of the interaction. Simple deterministic models can be put forward to reproduce at the mesoscopic, coarse grained level, the prototypical evolution of excitatory and inhibitory units, organized in two mutually competing populations. Continuous variables are customarily introduced to quantify the activity of each selected population. Non linearities prove crucial to adequately represent the threshold mechanism of activation that modulates, from neurons to genes, the system response [15,16]. One can then assemble large networks with designated topology, by replicating the aforementioned module on each node of the collection and incorporating the specific nature of the coupling [17]. Stationary patterns [18] of asynchronous activity for the scrutinized species can eventually emerge, following symmetry breaking instabilities that necessitate a fine tuning of the parameters involved. These patterns could define the basic architectural units for natural systems to perform efficient computations [19][20][21].As opposed to the deterministic formulation, an individual-based description -hence intrinsically stochastic for any finite population -can be invoked [22,23]. This amounts to characterizing the microscopic dynamics via transition rates, that govern the interactions among individuals and with the surro...

show abstract

Stochastic Turing patterns on a network

Cited by 39 publications

References 29 publications

Multiple-scale theory of topology-driven patterns on directed networks

Multiple-scale theory of topology-driven patterns on directed networks

Continuous-time random walks on networks with vertex- and time-dependent forcing

Intertangled stochastic motifs in networks of excitatory-inhibitory units

Contact Info

Product

Resources

About