Universal Scaling of Wave Propagation Failure in Arrays of Coupled Nonlinear Cells

Abstract

We study the onset of the propagation failure of wave fronts in systems of coupled cells. We introduce a new method to analyze the scaling of the critical external field at which fronts cease to propagate, as a function of intercellular coupling. We find the universal scaling of the field throughout the range of couplings, and show that the field becomes exponentially small for large couplings. Our method is generic and applicable to a wide class of cellular dynamics in chemical, biological, and engineering systems. We confirm our results by direct numerical simulations. PACS: 87.18.Pj, 82.40.Bj, 87.19.Hh, The impact of discreteness on the propagation of phase fronts in biophysical, chemical, and engineering systems has been intensively studied during the last decade. Among the diverse examples are calcium release waves in living cells [1], reaction fronts in chains of coupled chemical reactors [2,3], arrays of coupled diode resonators [4,5], and discontinuous propagation of action potential in cardiac tissue [6][7][8]. All these disparate systems share a common phenomenon of wave front propagation failure, independently of specific details of each system. Recently this effect has drawn considerable attention (see, e.g., Refs. [2,9]). Numerous experimental evidences show that the propagation failure occurs at finite values of the coupling strength (a critical coupling) [2,4,6]. This is contrary to continuous systems, where wave fronts propagate for arbitrary couplings [10]. A challenging problem is to establish the universal properties of the critical coupling; this is crucial for making predictions of qualitatively different regimes of system dynamics.In this Letter we consider the universal behavior of phase separation fronts in one-dimensional nonlinear discrete systems in an external field. We study the propagation failure transition for a class of simple dynamical models describing experimental observations in arrays of coupled nonlinear cells, such as chains of bistable chemical reactors [2,3], systems of cardiac cells [8], etc. A new analytical method is presented to study generic properties of the critical external field of the transition. We find, using this method, how the critical field scales with the intrachain coupling. This method is applicable for a wide range of the couplings. We confirm our analytical predictions by direct numerical simulations of the full system. Our model in general is given by the following set of coupled nonlinear equations:Here u n is the order parameter at the n-th site, γ is the damping coefficient, C is the coupling constant, and G(u n , E) is the onsite potential, where E is the applied field. The potential has at least two minima separated by a barrier u B . The external field E is responsible for the energy difference between the minima. This provides for one globally stable and one metastable minimum, u = u + and u = u − , respectively. Phase fronts connect the two minima and tend to propagate, to increase the size of the energetically more favorable phase, u + .

Continuous description of lattice discreteness effects in front propagation

“…The model (2.1) has been used to describe propagation phenomena in various physical contexts such as nonlinear electrical lattices [34,35], individual cells in the cardiac tissue, which are resistively coupled through gap junctions (e.g. Keener [36] and references therein), in arrays of coupled nonlinear cells [37], cellular differentiation [38] and coupled chemical reactors [20]. The system (2.1) exhibits bistability in the h-interval [−2(3/3) 3/2 , 2(3/3) 3/2 ].…”

Waves and propagation failure in discrete space models with nonlinear coupling and feedback

“…Travelling wave solutions are calculated in a novel way by noting that the advanced and retarded terms in the travelling wave equation may be dealt with by considering a vector of variables v j defined on [0, 1], where u(ξ ±1) = v j±1 , and boundary conditions ensure continuity. This technique is also applicable to the discrete bistable reaction diffusion equation which has been extensively studied previously [20][21][22][23]. In Section 5 I demonstrate the behaviours outlined for the simple relay model in a more detailed model which includes ligand-receptor binding.…”

“…For diffusive discrete bistable systems, the potential formulation has been used to predict depinning by perturbing analytical solutions by a shift, and determining the loss of energy minima (and therefore stationary fronts) as a function of that shift [23]. However, this method relies upon an analytical expression for solutions which is not currently available for models of the form (2).…”

“…Conditions for stationary fronts in discrete diffusive bistable systems have been derived by considering a limited number of "active sites", where only those sites feel nonlinearity, and all other sites are close enough to the homogeneous steady states to be in the linear regime [23,25], or to be considered as at those states [20]. With Hill functions (10) there is no linear approach of u j to zero, rather, all sites experience nonlinearity unless they are at zero.…”

“…However, this method relies upon an analytical expression for solutions which is not currently available for models of the form (2). The analytical solution used by Kladko et al [23] is constructed as a perturbation of the solution to the continuum reaction diffusion system, but here there is no equivalent continuous model. The continuation of previously known steady solutions (in the zero coupling limit) has also been proved using the implicit function theorem [24], but the juxtacrine relay model cannot be considered as a network of weakly coupled bistable units.…”