Stochastic Turing patterns in the Brusselator model

Abstract

A stochastic version of the Brusselator model is proposed and studied via the system size expansion. The mean-field equations are derived and shown to yield to organized Turing patterns within a specific parameters region. When determining the Turing condition for instability, we pay particular attention to the role of cross-diffusive terms, often neglected in the heuristic derivation of reaction-diffusion schemes. Stochastic fluctuations are shown to give rise to spatially ordered solutions, sharing the same quantitative characteristic of the mean-field based Turing scenario, in term of excited wavelengths. Interestingly, the region of parameter yielding to the stochastic self-organization is wider than that determined via the conventional Turing approach, suggesting that the condition for spatial order to appear can be less stringent than customarily believed.

Linear noise approximation for stochastic oscillations of intracellular calcium

“…Such stochastic contributions, endogenous to the system, can amplify via a resonant mechanism and so yield macroscopic oscillations in the discrete concentration in a region of the parameters for which a stable fixed point is predicted, as follows the deterministic linear stability analysis [8,9]. Similar conclusions apply to spatially extended systems [12][13][14][15]. As concerns calcium dynamics, stochasticity has been mainly associated to external disturbances [10,11].…”

Intertangled stochastic motifs in networks of excitatory-inhibitory units

“…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.…”

“…The factor V −1/2 reflects the Gaussian nature of the approximation. The linearized fluctuations can be shown to obey a Langevin equations [27] in the formξ i = j J ij ξ + η i , where η i (t) is a Gaussian noise term with zero mean and with correlator…”

“…The factor V −1/2 reflects the Gaussian nature of the approximation. The linearized fluctuations can be shown to obey a Langevin equations [27] in the formξ i = j J ij ξ + η i , where η i (t) is a Gaussian noise term with zero mean and with correlatorEndogenous oscillations can be analyzed by computing the power spectral density matrix (PSDM):3) The diagonal entries of the PSDM are real and coincide with the power spectra for the fluctuations, associated to each species. The (generally complex) offdiagonal elements of the PSDM can be properly normalized so to yield the Complex Coherence Function…”

Preface. Bifurcations and Pattern Formation in Biological Applications

“…Chemical Master Equations [2], Stochastic ODEs and Stochastic PDEs [9] are used to describe the evolution of biological systems with external noise or intrinsic stochasticity. The influence of noise on the dynamics of an ecosystem is considered in the work by D. Valenti, A. Giuffrida, G. Denaro, N. Pizzolato, L. Curcio, B. Spagnolo, S. Mazzola, G. Basilone and A. Bonanno, presented in this issue in [20].…”