# American Institute of Mathematical Sciences

February  2020, 14(1): 97-115. doi: 10.3934/ipi.2019065

## Electrocommunication for weakly electric fish

 Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland

* Corresponding author: Andrea Scapin

Received  March 2019 Revised  August 2019 Published  November 2019

Fund Project: The author is supported by SNF grant 200021-172483

This paper addresses the problem of the electro-communication for weakly electric fish. In particular we aim at sheding light on how the fish circumvent the jamming issue for both electro-communication and active electro-sensing. Our main result is a real-time tracking algorithm, which provides a new approach to the communication problem. It finds a natural application in robotics, where efficient communication strategies are needed to be implemented by bio-inspired underwater robots.

Citation: Andrea Scapin. Electrocommunication for weakly electric fish. Inverse Problems & Imaging, 2020, 14 (1) : 97-115. doi: 10.3934/ipi.2019065
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##### References:
Before the JAR (the EOD frequencies of the two fish are the same). Plot of $u(x) = u_1(x) + u_2(x)$, where $u(x,t) = u(x) e^{i\omega_0 t}$
After the JAR (the EOD frequencies $\omega_1$ and $\omega_2$ of the two fish are apart from each other)
Standard shape of the pulse wave $h(t)$
The setting. $\mathfrak{F}{1}$ is acquiring measurements at $N_s = 150$ different closely spaced positions
Estimate of the position and the orientation of $\mathfrak{F}{2}$, with noise level $\sigma_0 = 0.1$. The dashed red curve represents the estimated body of $\mathfrak{F}{2}$, whereas the green one represents the true body of $\mathfrak{F}{2}$. The white circle represents a small dielectric object placed between $\mathfrak{F}{1}$ and $\mathfrak{F}{2}$
Plot of the imaging functional $\mathcal{I}_2$ that the fish $\mathfrak{F}{1}$ uses to track $\mathfrak{F}{2}$
Plot of the linear trajectory tracking. $N_{\text{exp}} = 10$ trials have been considered
Plot of the trajectory tracking when the leading fish is swimming in circle, clockwisely. $N_{\text{exp}} = 10$ trials/realizations have been considered
Plot of the isopotential lines when $\mathfrak{F}{2}$ (on the left) is passive (electrically silent) and $\mathfrak{F}{1}$ (on the right) is active
Plot of the MUSIC imaging functional used in Algorithm 3 by using $N_r = 32$ receptors and $N_f = 100$ frequencies, with noise level $\sigma_0 = 0.1$. The square and the diamond indicate the approximation of the center and the true center of the target D, respectively. $\mathfrak{F}{1}$ (right) can image the target despite the presence of $\mathfrak{F}{2}$ (left), which is estimated by applying Algorithm 2
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