Transient Localized Patterns in Noise-Driven Reaction-Diffusion Systems

Noise can induce excitable systems to make time-limited transitions between quiescent and active states. Here we investigate the possibility that these transitions occur locally in a spatially extended medium, leading to the occurrence of spatiotemporal patches of activation. We show that this can in fact occur in a parameter range such that there exist (in general unstable) localized solutions of the governing deterministic reaction-diffusion equations. Our work is motivated by a recent biological example showing transiently excited cell membrane regions.

Figure 1

(a) Cell at different times after stimulation with $1\mu \mathrm{M}$ cAMP. (b) PHCrac-GFP at the boundary, presented as the difference of fluorescence intensity between the boundary and the cytosol, color coded as shown below in the panel. The ordinate refers to the position along the cell with 0 indicated by the arrow in the cell image. This figure is taken with permission from [14].

Figure 2

(a) Randomly created localized patches, obtained by adding temporally correlated and uniformly distributed noise to a small number of the points (as described in the text). The patches have a typical size and lifetime that depend on their density. In this system $\alpha =0.5$, $\beta =0.7$, $ϵ=0.01$, $D_u=1$, $D_v=5$ and noise frequency is 0.002%. (b)–(c) Patch width (b) and lifetime (c) distribution [system parameters as in (a)].

Figure 3

(a) Stationary pulse solution for the system of Eqs. (1). $u(x)$ (solid blue line) and $v(x)$ (dashed red line). System parameters as in Fig. 2. (b) Deterministic simulation (no noise). The stationary pulse may be stable or unstable, as seen here for $\alpha =0.5$, $\beta =0.6$, $ϵ=0.01$, $D_u=1$, $D_v=6$. This stationary pulse eventually vanishes via an oscillatory instability.

Figure 4

(a) Phase space for the parameters $\beta$ and $ϵ$, obtained by deterministic simulations. The simulations were typically run ${10}^8$ iterations before stability was asserted. $D_u=0.2$, $D_v=0.5$, $a=0.5$. $dx$ and $dt$ were chosen as described in the text. (b) The lifetime of the randomly created patches is correlated with the decay time of the unstable pulse solution.