March 9, 2007 10:18 Int J. Bifurcation and Chaos/INSTRUCTION FILE Int J. Bifurcation and Chaos Submission Style EFFECT OF NOISE AND STRUCTURAL INHOMOGENEITIES IN REACTION DIFFUSION MEDIA S. Morfu Laboratoire d Electronique, Informatique et Image (LE2i) UMR C.N.R.S 5158, Aile des Sciences de l Ingénieur, BP 47870,21078 Dijon Cedex, France, smorfu@u-bourgogne.fr http://<www.le2i.com> P. Marquié Laboratoire d Electronique, Informatique et Image (LE2i) UMR C.N.R.S 5158, Aile des Sciences de l Ingénieur, BP 47870,21078 Dijon Cedex, marquie@u-bourgogne.fr Received (Day Month Year) Revised (Day Month Year) We investigate the pinning and de-pinning condition of a kink in both homogeneous and inhomogeneous reaction-diffusion media. We show that structural inhomogeneities hinder information propagation whereas noise enhances propagation. Keywords: Reaction-diffusion media; inhomogeneous medium; propagation failure. 1. Introduction Reaction-diffusion equations arise in many fields of physics, chemistry and biology [Murray (1989)]. In recent years, a considerable attention has been devoted to these systems in the interdisciplinary field of neuro-science since reaction diffusion equations can model the behavior of nerve fibers and cardiac tissues. In this paper, we focus our study on the Nagumo equation expressed here in its discrete form [Nagumo et al. (1962)]: du n = D(u n+1 + u n 1 2u n )+ f(u n ), (1) where D represents the intercellular coupling and f(u n ) = u n (u n a)(u n 1) represents a cubic on-site nonlinearity. For weak couplings below a critical value D, propagation fails which may induce fatal consequences in the field of cardiophysiology [Keener (1987)]. Various studies have Laboratoire d Electronique, Informatique et Image (LE2i) UMR C.N.R.S 5158, Aile des Sciences de l Ingénieur, BP 47870,21078 Dijon Cedex, France. France smorfu@u-bourgogne.fr. 1
= March 9, 2007 10:18 Int J. Bifurcation and Chaos/INSTRUCTION FILE 2 S. Morfu and P. Marquié attempt to determine the analytical value of this critical coupling D which depends on the nonlinearity threshold a [Keener (1987); Laplante and Erneux (1992); Mitkov (1999); Kladko et al. (2000); Comte et al. (2001)]. Figure 1 summarizes the analytical results reported in [Keener (1987); Laplante and Erneux (1992); Comte et al. (2001)] (curves (a), (b) and (c) respectively) and the numerical ones obtained with a direct simulation of eq. (1) using a fourth order Runge-Kutta algorithm ( signs). As represented & $ ", = & $ =? " # # #!! # " " # # > Fig. 1. Critical coupling inducing propagation failure versus the threshold nonlinearity a. The numerical results ( signs) are compared to the theoretical laws D = [2a 2 a+2 2(a+1)] a 2 3a+1/25 for curve (a), D = a 2 /4 for curve (b) and D = (a 2)[a 1+ 1 2a]/4 for curve (c). in figure 1, propagation failure is now well understood in non disturbed media. However, cardiac tissues modeled by these reaction diffusion equations are rather inhomogeneous than homogenous and often submitted to perturbation wether random or not, which leads to new conditions for propagation [Kulka et al. (1995); Keener (2000); Carpio and Bonilla (2001)]. For instance, propagation failure may also occur in a discrete medium with intercellular coupling D 1 owing to a coupling inhomogeneity D 2 greater than D 1 (that is not necessary lower than D ) [Mornev O.A. et al. (1991); Morfu et al. (2002); Morfu et al. (2002)]. By contrast, it has been shown that noise acts against propagation failure in an electrical lattice of modified Chua circuits [Báscones et al. (2002)] or in an
March 9, 2007 10:18 Int J. Bifurcation and Chaos/INSTRUCTION FILE Effect of noise and structural inhomogeneities in reaction diffusion media 3 homogeneous Nagumo chain [Morfu (2003)]. In this paper, we propose to restrict our study to the Nagumo model considering both structural inhomogeneities and noise contribution. In the second section, we first propose to investigate the effect of a Heaviside type distribution of coupling on the propagation conditions. Then, in the third section, we consider the joint effect of noise and inhomogeneities. 2. Inhomogeneous medium In this section, we consider the junction of two homogeneous Nagumo media M 1 and M 2 with respective coupling D 1 and D 2, the junction being located at the interface site m. This configuration of inhomogeneities can be identified to the works of Wang and Rudy who have investigated action potential propagation in cardiac tissues presenting an abrupt change of the intercellular coupling [Wang and Rudy (2000)]. They have experimentally shown that propagation fails if the coupling D 2 in the second medium is lower than a critical value D in f but also if the coupling D 2 exceeds a second critical value D sup. We propose here to investigate numerically this phenomenon using the Nagumo model. The inhomogeneous Nagumo medium is described by eq. (1) with D = D 1 before the interface (namely for n < m), and with D = D 2 after the interface (namely for n > m), whereas the dynamics of the interface site m is given by du m = D 1 (u m 1 u m )+D 2 (u m+1 u m )+ f(u m ). (2) Note that in eq. (2), the term D 1 (u m 1 u m ) expresses the coupling between the cell of the interface and its neighborhood in the first homogeneous medium, whereas the other term D 2 (u m+1 u m ) represents the coupling between the interface cell and its neighborhood cells in the second homogeneous medium. For a given coupling D 1 in the first medium, we analyze numerically versus the coupling D 2 in the second medium wether a kink propagating in the first medium overcomes the interface site m or not. The length of the medium is set to 50 and the interface site to m = 25. We find numerically the same behaviour reported by Wang and Rudy, that is the existence of two critical values D in f and D sup for the coupling D 2 defining the propagation condition. We perform several numerical simulations of the system using a fourth order Runge-Kutta algorithm with integrating time step = 0.01, changing each time the coupling D 2 to estimate by dichotomy the value of the critical value D sup beyond which a kink is pinned at the interface site m. Using the same methodology, we determine numerically the second critical value D in f below which the kink cannot overcome the interface. The two critical values are plotted in dotted line figure 2 and provide the range of parameters allowing propagation in an inhomogeneous Nagumo chain with a Heaviside-type distribution of coupling. We have also investigated the propagation failure effect induced by a single inhomogeneity of coupling D 2 in a Nagumo chain with intercellular coupling D 1. The main feature is observed, that is the existence of two critical couplings values D in f and D sup defining the propagation conditions (continuous line of figure 2).
, March 9, 2007 10:18 Int J. Bifurcation and Chaos/INSTRUCTION FILE 4 S. Morfu and P. Marquié ' 2 H F = C = J E, I K F & %, I K F $ # " 2 H F = C = J E, E B! 2 H F = C = J E &!!! "! $! & " " " ",, E B Fig. 2. Propagation failure induced by two distributions of coupling. Critical values D in f and D sup of the coupling D 2 defining the propagation condition for a Heaviside distribution of coupling (dotted line) and for a Dirac distribution of coupling (solid line). Nonlinearity threshold a = 0.3. 3. Noisy inhomogeneous Medium In this section, we numerically investigate the behavior of the previous inhomogeneous Nagumo chain initially in propagation failure condition owing to the interface site m and submitted to a spatiotemporal noise. The equations describing the evolution of such a noisy inhomogeneous Nagumo medium are given by: du n = D 1 (u n+1 + u n 1 2u n )+ f(u n )+η n (t) f or n < m du m du n = D 1 (u m 1 u m )+D 2 (u m+1 u m )+ f(u m )+η m (t) f or n = m = D 2 (u n+1 + u n 1 2u n )+ f(u n )+η n (t) f or n > m. where η n (t) is a spatially independent uniform white noise over [ 3σ; 3σ]. We set the coupling D 1 in the first medium to 0.03. Furthermore, the coupling D 2 in the second medium exceeds the critical value D sup = 0.036 so that without noise the kink propagating in the first medium cannot overcome the interface site m. We investigate versus (3)
I, March 9, 2007 10:18 Int J. Bifurcation and Chaos/INSTRUCTION FILE Effect of noise and structural inhomogeneities in reaction diffusion media 5 the Root Mean Square (R.M.S.) amplitude σ of the spatio-temporal noise η n (t) wether the kink overcomes the interface site m or not. Our simulation results reveals that there exists a minimum amount of noise σ that allows the kink to overcome the interface site m. This value, represented figure 3 versus D 2, is estimated by dichotomy for each value of D 2 simulating the system with an integrating time step = 10 2 for a sufficiently long time (namely t = 30000 in our case, that is 3 10 6 integrating time steps, which corresponds to 265 wave propagation time scales). "! #! # F H F = C = J E I # F H F = C = J E # # # # Fig. 3. Minimum R.M.S noise amplitude σ allowing to overcome the interface site versus the coupling D 2. D 1 = 0.03, a = 0.3. 4. Conclusion Considering the discrete Nagumo model, we have numerically shown that the presence of structural inhomogeneities can hinder information propagation whereas noise can reduce the fatal consequences of these inhomogeneities. We think that this result could be extended to other distributions of inhomogeneities wether periodic or not and could concern also other parameters of the medium presenting inhomogeneities like the threshold of the nonlinearity a. In fact, this work extends to inhomogeneous media the property of noise enhanced propagation usually observed in an homogeneous medium [Lindner et al. (1998);
March 9, 2007 10:18 Int J. Bifurcation and Chaos/INSTRUCTION FILE 6 S. Morfu and P. Marquié Löcher et al. (1998); Morfu (2003)]. Endly, we think that this work could be useful in better understanding cardiac tissue behaviour and could also have potential applications to many clinical problems such has why atrial ablation fails. References Báscones R., Garcìa-Ojalvo J., Sancho J.M., (2002). Pulse propagation sustained by noise in arrays of bistable electronic circuits. Phys. Rev. E, 65: 061108. Carpio A., Bonilla L.L., (2001). Wave front depinning transition in discrete one-dimensional reactiondiffusion systems. Phys. Rev. Lett, 86: 6034 6037. Comte J.C., Morfu S. and Marquié P. (2001). Propagation failure in discrete bistable reactiondiffusion systems: theory and experiments. Phys. Rev. E, 64: 027102. Keener J.P. (1987). Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math., 47: 556 572. Keener J.P. (1991), The effects of discret gap junction coupling and propagation in myocardium J. Theor. Biol. 148:49 82. Keener J.P. (2000). Homogenization and propagation in the bistable equation. physica D, 136: 1 17. Kladko K, Mitkov. I., Bishop A.R. (2000). Universal scaling of wave propagation failure in arrays of coupled nonlinear cells Phys. Rev. Lett., 84: 4505 4508. Kulka A., Bode M., Purwins H.-G, (1995).On the influence of inhomogeneities in a reaction-diffusion system. Phys. Lett. A, 203: 33 39. Laplante J.P. and Erneux T. (1992). Propagation failure and multiple steady states in an array of diffusion coupled flow reactor. Physica A, 188 : 89 98. Lindner J.F., Chandramouli S., Bulsara S.R., Löcher M., and Ditto W.L. (1998). Noise enhanced propagation. Phys. Rev. Lett., 23: 5048 5051. Löcher M., Cigna D., and Hunt E.R., (1998). Noise sustained propagation of a signal in coupled bistable electronic element. Phys. Rev. Lett., 80: 5212 5215. Mitkov I. (1999). One-and two-dimensional wave fronts in diffusive systems with discrete sets of nonlinear sources. Physica D, 133: 398 403. Morfu S., Comte J.C., Marquié P. and Bilbault J.M. (2002). Propagation failure induced by coupling inhomogeneities in a nonlinear diffusive medium. Phys. Lett. A, 294: 304-307. Morfu S., Nekorkin V.I., Bilbault J.M. and Marquié P. (2002). The wave front propagation failure in an inhomogeneous discrete Nagumo chain: theory and experiments. Phys. Rev. E, 66: 046127 1/8. Morfu S. (2003). Propagation failure reduction in a Nagumo chain Phys. Lett. A, 317, 73-79. Mornev O.A. (1991). Elements of the Optics of Autowaves in Nonlinear Wave Processes in Excitable Media, ed. A.V. Holden, M. Markus and H.G. Othners, Plenum Press, N.Y.: 111 118. Murray J.D. (1989). Mathematical biology, ed. Springer-Verlag, Berlin. Nagumo J., Arimoto S. and Yoshisawa S., (1962). An active pulse transmission line simulating nerve axon. Proc. IRE, 50: 2061. Wang Y. and Rudy Y. (2000). Action potential propagation in inhomogneous cardiac tissue: safety factor consideration and ionic mechanism. Am. J. Physiol. Heart Circ. Physiol., 278: H1019.