Dynamical equivalence between resonant translocation of a polymer chain and diversity-induced resonance

TL;DR

The paper tackles how heterogeneity promotes collective oscillations in networks of nonlinear oscillators, linking diversity-induced resonance (DIR) to a mechanical analog of a dimer/polymer diffusing on a substrate. It introduces a dimer-diffusion resonance (DDR) mechanism and shows that, for networks with zero-mean bias distributions, the system behaves like a 1D polymer whose monomer ordering mirrors the bias values. An analytical condition emerges: resonance occurs when the effective rest length or bias-to-coupling ratio satisfies a critical relation, e.g., $\bar{a}/c \approx \lambda/2$, which for a bistable quartic potential with $\lambda=2$ yields $a^* \approx 0.5$ (for $N=100$, $c=1$). The findings extend from simple two-oscillator models to networks of quartic and FitzHugh–Nagumo oscillators, offering a general, intuitive framework for synchronization in heterogeneous systems with broad biological and technological relevance.

Abstract

Networks of heterogeneous oscillators are often seen to display collective synchronized oscillations, even when single elements of the network do not oscillate in isolation. It has been found that it is the diversity of the individual elements that drives the phenomenon, possibly leading to the appearance of a resonance in the response. Here we study the way in which heterogeneity acts in producing an oscillatory regime in a network and show that the resonance response is based on the same physics underlying the resonant translocation regime observed in models of polymer diffusion on a substrate potential. Such a mechanical analog provides an alternative viewpoint that is useful to interpret and understand the nature of collective oscillations in heterogeneous networks.

Figures (5)

• Figure 1: Example of bimodal bias distribution function $P(a)$ (black curve) for $\sigma_a/\bar{a} = 2/3$, resulting from the superposition of the partial bias distribution functions $\frac{1}{2} P_\pm(a)$ of the two different types of oscillators (orange and green dashed curves) --- these distributions are defined in Eq. \ref{['eq:bimodal']}. For comparison, we draw also: the limiting Gaussian distribution function $P_1(a)$ (blue dot-dashed curve), given by Eq. \ref{['eq:gaussian']}, obtained for $\bar{a} \to 0$ keeping the standard deviation $\sigma_a$ constant; and the two-value $\delta$-distribution function $P_2(a)$ (visualized as two red vertical lines), given by Eq. \ref{['eq:twovalue']}, obtained for $\sigma_a \to 0$ keeping $\bar{a}$ constant.
• Figure 2: Asymptotic global oscillatory activity $\sqrt{\langle\delta X^2\rangle}$ defined in Eqs. \ref{['eq:dX2']}--\ref{['eq:dX2_C']} in the $\bar{a}$-$\sigma_a$-plane of a heterogeneous network of (a) quartic oscillators subject to a periodic forcing and (b) FN oscillators. The blue curves represent the limit $\bar{a}\to 0$, thus reproducing the results of DIR Tessone-2006a, while the red curves correspond to the limit $\sigma_a \to 0$ for a network with a two-point bias distribution Cartwright-2000a. Notice that both these curves have a peak around the value $1/2$. The potential $V(x)$ defined in Eq. \ref{['eq:quartic']} is the same for the two types of oscillators, as is the total number of oscillators $N = 100$, the coupling constant $c = 1$, and the final simulation time $t = 2000$. The oscillating force acting on the quartic oscillators, defined in Eq. \ref{['eq:oscillatingforce']}, has period $\tau = 200$ and amplitude $b = 0.2$; the constants regulating the dynamics of the FN slow degrees of freedom $y_i$, Eq. \ref{['eq:oscillator4']}, are $\alpha = 0.02$, $\beta = 0.04$.
• Figure 3: (a) Regular network composed of alternating types of oscillators with bias $a = -\bar{a}$ (yellow nodes) and $a = +\bar{a}$ (green nodes). Each node is coupled to two nearest neighbors on both sides, so the degree is $k_0=4$. Blue links represent interactions between two oscillators of the same type, green links between oscillators of different types. (b) Polymer mechanical analog in $x$-space. The harmonic forces between particles of the same type tend to induce localized clusters, while harmonic interactions between particles of different type induce the formation of two clusters at a distance $\ell$. As a result, the system behaves similarly to two interacting monomers $A$ and $B$ that compose a dimer with equilibrium length $\ell$.
• Figure 4: Mechanical analog of a small network of four oscillators. Each oscillator interacts with all the other oscillators but the equilibrium distances of all the different interactions are consistent with each other according to Eq. \ref{['eq:length_i']} and produce a robust 1D chain structure.
• Figure 5: Snapshots of the translocation of a polymer composed of $N=20$ monomers, representing a dynamical analog of a network of quartic oscillators subject to a time-periodic forcing defined in Eq. \ref{['eq:oscillatingforce']}. The snapshots are taken at different times within a single time period $T = 2\pi/\omega$. The monomers are color-coded according to the respective values of the external biases $a_i$. One can notice that during a polymer translocation (a network oscillation), monomers move in a file maintaining an order based on the bias value, in which the leftmost (rightmost) monomer, with the smallest (largest) $x$-coordinate, has also the smallest (largest) value of the constant external bias. The other parameters are as in Fig. \ref{['fig:dX2']}.