Response to synchronization in distance-dependent coupled heterogeneous oscillators

Fireflies are often presented as a main example of a natural system that exhibits synchronization~[1]. However, there are species of fireflies in which males synchronize first and only after that phase females are capable of exhibiting such unison behavior; stated differently, female fireflies exhibit {\em response to synchronization}~[2]. In this work, we consider a dynamical model based on bursting (male) and nonbursting (female) oscillators~[3] which are closely related to the integrate-and-fire oscillators whose characteristics are shown in Fig~1(a)--(b).<br/><br/>The dynamical system composed of $N=N_F+N_M$ is given by Eq. (1),<br/>where $\theta_i=\pm 1$, according to the oscillator represents a male or a female, respectively, and $\beta_{ij}$ are the elements of a weighted adjacency matrix and determine the distance-dependent coupling strength, situation depicted in Fig.~1(c).<br/><br/>The transition between the states determined by $\varepsilon$ is described by the following relation: Eqs. 2(a)-(c).<br/>We first studied the case of a stationary weighted adjacency matrix and secondly we considered a time-varying matrix, the latter mimicking a relatively small motion of oscillators. In both cases, aspects such as transients, clustering, and response temporal patterns are outlined. As an example of the phenomenon, Fig.~1(d) illustrates the response to synchronization for the case of a population of 20 oscillators: 10 males and 10 females.<br/><br/>We conclude our work by discussing the potential and possible pitfalls of implementing these dynamic networks on a robotic platform.