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A mechanism of synchronization in interacting multi-cell genetic systems. (English) Zbl 1093.34042

The authors investigate the following coupled noisy system with time delays \[ \frac{dx_j}{dt} = f(x_j(t),x_j(t-\tau),\eta_j(t)) + \sum_{k=1}^{m} D_k^{j} x_k(t-\tau_E) +\xi_j(t), \] where \(x=(x_1,\dots,x_m)^{T} \in \mathbb{R}^{m}\), \(\eta(t)= (\eta_1(t),\dots, \eta_m(t))\) and \(\xi(t)=(\xi_1(t),\dots,\xi_m(t))\) are independent Gaussian noises with zero means. This model is quite general and can describe, for example, a cell to cell communication network. In this case, \(x\) represents the concentration of species in living cells, \(D_k^{j}\) are coupling coefficients, \(\tau>0\) is the intracellular time delay, and \(\tau_E>0\) is the extracellular time-delay due to diffusion or transport process.
The main conclusion of the paper is that appropriate noise intensities and coupling strengths are capable of driving the system to be phase-locked, i.e., the differences between the appropriately introduced phases of the oscillators stay bounded. The authors provide an analytical treatment of the locking mechanism.

MSC:

34K25 Asymptotic theory of functional-differential equations
34K50 Stochastic functional-differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
92D10 Genetics and epigenetics
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