Mathematical modelling and application of frog choruses as an autonomous distributed communication system

Abstract

Interactions using various sensory cues produce sophisticated behaviour in animal swarms, e.g. the foraging behaviour of ants and the flocking of birds and fish. Here, we investigate the behavioural mechanisms of frog choruses from the viewpoints of mathematical modelling and its application. Empirical data on male Japanese tree frogs demonstrate that (1) neighbouring male frogs avoid call overlaps with each other over a short time scale and (2) they collectively switch between the calling state and the silent state over a long time scale. To reproduce these features, we propose a mathematical model in which separate dynamical models spontaneously switch due to a stochastic process depending on the internal dynamics of respective frogs and also the interactions among the frogs. Next, the mathematical model is applied to the control of a wireless sensor network in which multiple sensor nodes send a data packet towards their neighbours so as to deliver the packet to a gateway node by multi-hop communication. Numerical simulation demonstrates that (1) neighbouring nodes can avoid a packet collision over a short time scale by alternating the timing of data transmission and (2) all the nodes collectively switch their states over a long time scale, establishing high network connectivity while reducing network power consumption. Consequently, this study highlights the unique dynamics of frog choruses over multiple time scales and also provides a novel bio-inspired technology that is applicable to the control of a wireless sensor network.

Interaction mechanisms quantified from dynamical features of frog choruses

“…Here we use four datasets of the audio data in which all the frogs stably called more than 1400 times in four hours, allowing us to precisely estimate a phase oscillator model by utilizing the large sample size of call timing. [20][21][22][23] including alternating chorus patterns of male Japanese tree frogs [13][14][15]19]. In this study, we first calculate a phase φ n (t i ) (n = 1, 2, 3) with discrete time t i for the nth frog from the separated audio data according to Eq.…”

“…To investigate interaction mechanisms inherent in the acoustic communication of actual animals, we analysed empirical data of male Japanese tree frogs that were obtained from our previous laboratory experiment and data analysis [19,26] (see §4.1 for details). In each trial of the experiment, we randomly captured three male frogs at a field site, and then placed them along a straight line at intervals of 50 cm between nearest neighbours.…”

“…A phase oscillator model is a well-known mathematical model that is theoretically derived from general classes of coupled oscillators [16] (see §4.2 for the details of a phase oscillator model). Experimental and theoretical studies have shown that the phase oscillator model with a simple interaction term can qualitatively reproduce the synchronization phenomena in various types of actual biological oscillators [27][28][29][30] including the chorus structures of male Japanese tree frogs [18][19][20]26]. In this study, we first calculated a phase ϕ n (t i ) (n = 1, 2, 3) with discrete time t i according to equation (4.1) using the call timing of actual frogs (see §4.1 for details).…”

“…The second condition (i.e., T n,k+1 − T n,k < 0.9 sec) is assumed because of the property of the frog chorus in which male Japanese tree frogs intermittently start and stop their periodic calling behavior over a long time scale. Namely, each frog periodically calls almost at the interval of 0.3 sec for several tens of seconds, stays silent for several minutes, and then repeats this cycle [19]. Because such an intermittency over a long time scale is out of the scope of the phase oscillator model, we assume the second condition and then calculate the phase φ n (t i ) only during the periodic calling behavior.…”

“…This paper is organized as follows: (i) a mathematical model (a phase oscillator model [16]) is identified for each pair of male Japanese tree frogs based on our empirical data using a Bayesian approach, (ii) the attention paid among the male frogs is quantified on the basis of the identified model, and (iii) the relationship between the quantified attention and behavioural parameters is statistically analysed. Note that the phase oscillator model is theoretically derived from general classes of periodic oscillators [16], and can describe synchronized features in coupled biological oscillator including calling behaviour of male frogs [18][19][20]26]. Thus, our main goal in this study is to identify the interaction mechanisms of frog choruses within a reliable mathematical framework, and then infer the behavioural strategies in male frogs.…”