How a Frog Can Learn What Is Where in the Dark

During the night 180 lateral-line organs allow the clawed frog Xenopus to localize prey by detecting water waves emanating from insects floundering on the water surface. Not only can the frog localize prey but it can also determine its character. This suggests waveform reconstruction, and a key question is how the frog can establish the appropriate neuronal hardware. Detecting time differences arising from the input on the skin is a key to neuronal information processing, and spike-timing-dependent synaptic plasticity (STDP) therefore seems to be the natural tool. We show how supervised STDP allows a frog to learn what is where in the dark. Learning can also be derived from a minimization principle.

Figure 1

Power spectrum $|S^{\mathbf{p}}(\omega )|$ of a theoretical reverse transfer function computed from (6) (solid line) and power spectra of learned reverse transfer functions according to (4) after different numbers of learning steps (see legend, $\eta ={10}^{-5}$). White noise (${\sigma }_x=1$) at random angles 10 cm away from the frog was used as learning stimulus. As the number of learning steps increases, the learned transfer functions converge to the theoretical result.

Figure 2

A model minimizing the error $E_W$ from (8) has been learned applying the method of Fig. 1 with the window of reference (${\sigma }_{\phi }=14°$) of (7) for 200 000 times. The figure shows a map of norms of reconstructions $\Vert {\widehat{x}}^{\mathbf{p}}\Vert$ (thick solid line) at different angles $\phi (\mathbf{p})$ around Xenopus. As expected, the norm is maximal where the actual test stimulus is, viz., at 0°, 10 cm right in front of Xenopus. Moreover, at 0° the model reconstructs (solid line in the inset on the right) the given test stimulus (sinusoidal with amplitude 1, dashed line in the inset) quite well. At angles far off ($-180°$ in the inset on the left), the reconstruction is basically noisy with a low amplitude. The Gaussian white noise added to the deflections $y_i$ of the lateral-line organs has a standard deviation of ${\sigma }_n=0.01$.

Figure 3

Circuit diagram of a neuron with membrane potential $V$ that is connected to lateral-line nerve fiber $j$ by excitatory (full circles) and inhibitory (open circles) synapses with synaptic strengths $J_{jk}$ and delays ${\Delta }_{jk}$. The neurons labeled by “$+$” and “$-$” signs accommodate the observation that a single fiber can have synapses of only one type, excitatory or inhibitory.

Figure 4

A map of 180 neurons learned to reconstruct the source at 180 positions arranged in a circle with radius 10 cm around Xenopus. After 50 000 learning steps, i.e., presentations of Gaussian white-noise stimuli during 2 s, performed according to (11), the map is able to accurately localize both of two simultaneous wave sources at $\phi =-45°$ (17 Hz) and 45° (18 Hz). In addition, the neurons indicating the respective angles reconstruct the wave source (solid lines in the insets). The delays ${\Delta }_{ik}^{\mathbf{p}}$, $k=1,\dots ,100$, of (9) are randomly distributed in an interval of 500 ms. In this way the animal could easily distinguish position and waveform of the sources. As ${\sigma }_{\phi }$ (here 14°) becomes larger, the two “hills” become broader and the discrimination ability of Xenopus decreases accordingly.

Figure 5

Xenopus’ simulated response angle versus stimulus angle $\phi$ during 25 trials at angle $\phi =n\times 5°$, $-36<n\le 36$. Xenopus was assumed to turn into the direction $\phi (\mathbf{p})$, such that the model response $\Vert V^{\mathbf{p}}\Vert$ of the neuron reconstructing the stimulus at position $\mathbf{p}$ is maximal. The neuronal system has learned as in Fig. 4. The frog’s performance is excellent.