An integro-differential equation for dynamical systems with diffusion-mediated coupling

Abstract

Many systems of biological interest can be modeled as pointlike oscillators whose coupling is mediated by the diffusion of some substance. This coupling occurs because the dynamics of each oscillator is influenced by the local concentration of a substance which diffuses through the spatial medium. The diffusion equation, on its hand, has a source term which depends on the oscillator dynamics. We derive a mathematical model for such a system and obtain an integro-differential equation. Its solution can be obtained by an approximation scheme for which the unperturbed solution is used to obtain a first-order solution to the coupled oscillators and so on. We present numerical results for the special case of a one-dimensional bounded domain in which the oscillators are randomly placed. Our results show the influence of the coupling parameters on some aspects of the dynamics of the coupled oscillators, like phase and frequency synchronization.

Appendix: Fast diffusion limit of the interaction kernels

It is instructive to verify that the Green functions we have obtained for the diffusion process reduce to the expressions already obtained by Kuramoto and coworkers in the case of very fast diffusion, for which the concentration of the chemical mediator achieves immediately its stationary state.

From the Green function of the specific geometry in which the system is defined, the corresponding interaction kernel is

$$\begin{aligned} \sigma (\mathbf{r}_j,\mathbf{r}_k;t) = \int _0^{t} \mathrm{d}t' \, G\left( \mathbf{r}_j,t;\mathbf{r}_k,t'\right) . \end{aligned}$$

(38)

Using the Green’s function of the $d=1$-dimensional case in free space given by (10), the interaction kernel reads

$$\begin{aligned} \sigma (x_j,x_k;t)\,&=\, {} \int _0^{t} \mathrm{d}t' \frac{\mathrm{e}^{-\eta (t-t')}}{\sqrt{4\pi D(t-t')}} \exp \nonumber \\&\quad\left[-\frac{{(x_j-x_k)}^2}{4D(t-t')}\right], \end{aligned}$$

(39)

which, after a change of variables, reads

$$\begin{aligned} \sigma (x_j,x_k;t) = \frac{x_j-x_k}{4D\sqrt{\pi }} \int _{u_1}^\infty \frac{\mathrm{d}u}{u^{3/2}} \, \exp \left( -u-\frac{a_1}{u}\right) , \end{aligned}$$

(40)

where

$$\begin{aligned} a_1\,=\, & {} \frac{\eta {(x_j-x_k)}^2}{4D} = {\left\{ \frac{\gamma (x_j-x_k)}{2}\right\} }^2, \nonumber \\ u_1\,=\, & {} \frac{{(x_j-x_k)}^2}{4Dt}, \end{aligned}$$

(41)

and we have defined a coupling length

$$\begin{aligned} \gamma = \sqrt{\frac{\eta }{D}}. \end{aligned}$$

(42)

In fast diffusion case is equivalent to take the $t\rightarrow \infty$ limit for the interaction kernel, for which $u_1\rightarrow 0$. Taking this limit we have

$$\begin{aligned} \sigma (x_j,x_k)\,=\, & {} \lim _{t\rightarrow \infty } \sigma (x_j,x_k;t)\nonumber \\\,=\, & {} \frac{\gamma }{2\eta } \, \exp \left\{ -\gamma (x_j-x_k)\right\} , \end{aligned}$$

(43)

which coincides with the earlier results of Kuramoto and coworkers [17, 19], in their analysis of the fast-relaxation case.

For the two-dimensional case ($d=2$) in free space, we use the Green function (11), and the interaction kernel becomes

$$\begin{aligned} \sigma ({\mathbf {r}}_j,{\mathbf {r}}_k;t) = \frac{1}{4\pi D} \int _{u_1}^\infty \frac{\mathrm{d}u}{u} \, \exp \left( -u-\frac{a_2}{u}\right) , \end{aligned}$$

(44)

where

$$\begin{aligned} a_2\,=\, & {} \frac{\eta [{(x_j-x_k)}^2+{(y_j-y_k)}^2]}{4D} = {\left| \frac{\gamma ({\mathbf {r}}_j-{\mathbf {r}}_k)}{2}\right| }^2, \nonumber \\ u_1\,=\, & {} \frac{{\vert {\mathbf {r}}_j-{\mathbf {r}}_k \vert }^2}{4Dt}, \end{aligned}$$

(45)

which, in the $t\rightarrow \infty$ limit, reduces to the result already found by Nakao [19]:

$$\begin{aligned} \sigma ({\mathbf {r}}_j,{\mathbf {r}}_k) = \lim _{t\rightarrow \infty } \sigma ({\mathbf {r}}_j,{\mathbf {r}}_k;t) = \frac{1}{2\pi D} \, K_0 \left( \gamma \vert {\mathbf {r}}-{\mathbf {r}}_k \vert \right) , \end{aligned}$$

(46)

where $K_0$ is the modified Bessel function of the second kind and zeroth order.

Finally, for the three-dimensional free-space case ($d=3$) we use the Green function (11) to obtain the corresponding the interaction kernel

$$\begin{aligned}\sigma ({\mathbf {r}}_j,{\mathbf {r}}_k;t) &= - \frac{1}{4D \pi ^{3/2}} \frac{1}{{|{\mathbf {r}}_j-{\mathbf {r}}_k|}} \nonumber \\&\quad \times\int _{u_1}^\infty \frac{\mathrm{d}u}{\sqrt{u}} \exp \left( -u-\frac{a_3}{u}\right) , \end{aligned}$$

(47)

where

$$\begin{aligned} a_3 = \frac{\eta }{4D} {\vert {\mathbf {r}}_j-{\mathbf {r}}_k\vert }^2, \qquad u_1 = \frac{{\vert {\mathbf {r}}_j-{\mathbf {r}}_k \vert }^2}{4Dt}, \end{aligned}$$

(48)

which, in the limit $t\rightarrow \infty$, becomes

$$\begin{aligned} \sigma ({\mathbf {r}}_j,{\mathbf {r}}_k) &=& {} \lim _{t\rightarrow \infty } \sigma ({\mathbf {r}}_j,{\mathbf {r}}_k;t) \nonumber \\ & = \frac{1}{4D\pi } \frac{1}{\vert {\mathbf {r}}_j-{\mathbf {r}}_k\vert } \exp \left( -\gamma \vert {\mathbf {r}}_j-{\mathbf {r}}_k\vert \right) , \end{aligned}$$

(49)

in accordance with the result previously found by Nakao [19].