Minimal Model of Prey Localization through the Lateral-Line System

The clawed frog Xenopus is an aquatic predator catching prey at night by detecting water movements caused by its prey. We present a general method, a “minimal model” based on a minimum-variance estimator, to explain prey detection through the frog’s many lateral-line organs, even in case several of them are defunct. We show how waveform reconstruction allows Xenopus’ neuronal system to determine both the direction and the character of the prey and even to distinguish two simultaneous wave sources. The results can be applied to many aquatic amphibians, fish, or reptiles such as crocodilians.

Figure 1

The clawed frog Xenopus laevis laevis. Its lateral-line organs can be seen clearly as white “stitches.” In the model presented here they are arranged on a circle of diameter 4 cm, a convenient but irrelevant simplification.

Figure 2

Connections of a neuron, with preferred direction $\phi$, to the lateral-line organs $i$ (open circles). Lines denote axons, small filled circles denote synapses with strengths $J_{ik}$. Axonal delays ${\Delta }_{ik}$ and synaptic strengths are designed so that the membrane potential $V_{\phi }(t)$ of the neuron approximates the original waveform $x(t)$ at the source of the water wave. ${\Delta }_{ik}$ and $J_{ik}$ are provided by our model. Its localization and source-reconstruction ability needs no precise tuning of ${\Delta }_{ik}$ and $J_{ik}$. The inset shows a typical reverse transfer function $s_j$ provided by Eq. (5) and its approximation through delta functions.

Figure 3

A “map” of norms $\Vert V_{\phi }\Vert$ of membrane potentials of 72 model neurons, each representing a different direction $\phi$ (horizontal axis). In our model, Xenopus swims in the direction $\phi$ where $\Vert V_{\phi }\Vert$ has its maximum, which in this case is $\phi \approx 0$. The sinusoidal wave source with frequency 10 Hz is positioned 10 cm in front of Xenopus ($\phi =0$). Each neuron reconstructs the waveform of the stimulus by means of its membrane potential $V_{\phi }(t+T)$ (solid lines in the insets), “assuming” that the actual stimulus comes from direction $\phi$. If this assumption is correct (at $\phi =0$), Xenopus’ approximation resembles the original waveform $x(t)$ (dotted lines in the insets) quite well. For wrong directions (e.g., $\phi =180°$), Xenopus’ approximation is noise with a small amplitude. The present model therefore serves two purposes. First, neurons responding strongest tell Xenopus the direction of the wave source. Second, the membrane potential of these neurons gives Xenopus an approximation to the actual waveform and allows the animal to distinguish different kinds of prey.

Figure 4

Top: Xenopus’ experimental response angle [9] versus stimulus angle $\phi$ during 25 trials at angle $\phi =n\times 5°,-36<n\le 36$. Left: intact Xenopus. Right: lateral-line organs at the right-hand side have been deactivated. Bottom: Model response of neurons labeled by the $\phi$ of maximal $\Vert V_{\phi }\Vert$; cf. Fig. 3. The agreement with experiment is fair.

Figure 5

“Map” like that of Fig. 3 for two simultaneous wave sources of 10 and 15 Hz, positioned at $\phi =-45°$ and $45°$, 10 cm in front of Xenopus. The animal’s approximations are shown in the insets. In this way it could easily distinguish position and waveform of the sources, as in experiment [10].