The complexity of high-frequency electric fields degrades electrosensory inputs: implications for the jamming avoidance response in weakly electric fish

2.1. Modelling the electric field

We perform a detailed comparison of the spatio-temporal features of jamming signals between Eigenmannia and Apteronotus using computational models. We have previously developed a model for the time varying electric field of a 21 cm long Apteronotus leptorhynchus [26], and describe here a similar model for Eigenmannia. In brief, a two-dimensional cross section of the Eigenmannia body (26 cm in length) was divided into three compartments: skin, body and electric organ. The contour of the fish (not including fins) was an idealized approximation of the tracings provided by Assad et al. [27] and Rasnow & Bower [30] and was symmetrical about the rostro-caudal axis. The length of each model was given by the original tracings from the Assad and Rasnow recordings. The skin comprised a 100 µm layer and the electric organ was defined as a 0.01 × 22 cm rectangle centred about the rostro-caudal axis and beginning 3 cm from the tip of the nose. The electrical conductivities for each body compartment were taken from the Apteronotus model [26,31]. To replicate the experimental recording conditions, the fish model was placed in a 70 × 70 cm ‘tank’ with water conductivity of 200 µS cm⁻¹ [27,32], and a reference (ground) electrode in the corner of the tank nearest the head.

As in previous studies, we used a finite-element approach (COMSOL Inc.) to model the electric field [26,31,33,34]; we solved the forward quasi-electrostatic problem: Poisson's equation (equation (2.1)), in which the voltage (Φ) in space with conductivity (σ) is calculated from the current density (J). The lateral edges of the electric organ were defined as boundary current sources with the current following equation (2.2). Note that in Eigenmannia, the electric organ is enveloped by a resistive sheath [25], which would bias current flow parallel to the rostral–caudal axis in a manner that may not be consistent with these boundary conditions. Nonetheless, since our model accurately reproduced the spatio-temporal variations in the EOD and electric field (see following), it was sufficient to address the JAR-related questions in this study. Future work will be aimed at understanding the current generating properties of the electric organ itself.

2.1

Due to the relatively simple dipolar nature of the EOD in Eigenmannia [27,32], the current density on the organ was defined as the sum of two Gaussian curves, as was done previously for a single-phase model in Apteronotus [31,33]. Each Gaussian curve had three free parameters: amplitude (A), peak centre (c) and standard deviation (σ), with the current density function J(x) as given in equation (2.2).

2.2

The model was fitted to previously recorded EOD data, kindly provided by C. Assad and B. Rasnow [27,30]. The EOD was sampled at 25 timepoints over one EOD cycle, and spatially interpolated (linear, over triangular mesh) to 0.2 cm resolution within 2 cm lateral of the fish, and 0.6 cm resolution outside that region. This variable density implicitly weights data points closer to the body, allowing for better fits to the increased heterogeneities in this region [26]. The fitting procedure involved finding the parameters for the current density (equation (2.2)) that minimized the sum of squared error (SSE, equation (2.3)) between model potential (Φ^m) and recorded potential (Φ^r). Optimization was performed using the Neldler–Mead simplex algorithm [35] which was implemented in the Java Apache Math Library through COMSOL's JAVA API. In order to compare the error across phases, the SSE was normalized (normalized root mean squared error, NRMSE; equation (2.3)).

2.3

and

2.4

To bias continuity of the current density solutions between each EOD phase, the solution (optimized current) for one phase was used as the initial condition for the fitting algorithm at the next phase. Note that the recorded data for phase 2 were uncharacteristically discontinuous in comparison to the rest of the data, and were therefore excluded from the optimization. Once fitting was complete, the solutions for each phase were stored as a numerical lookup table, with a temporal resolution of 25 points per EOD cycle and a spatial resolution of 1000 points over the electric organ.

2.2. Amplitude and phase modulations in multiple sine waves

Electric fish generate a quasi-sinusoidal, oscillating electric field. The fields (EODs) of multiple fish with different frequencies sum to create an amplitude-modulated signal; as in previous studies [4,36], we illustrate this using sine waves that represent the EODs of two fish (equation (2.5)). The reference (or carrier) frequency is F₀ (i.e. receiving fish), and the modulating frequency is F₁ (neighbouring fish) with a difference frequency [8] (this is the standard convention in which a positive DF indicates that the neighbouring fish has the higher EOD frequency). When the DF is non-zero, an ‘apparent frequency’ appears (the beat frequency). In the time domain, this results in a beat (amplitude modulation) at a frequency |DF| (where |⋅| denotes absolute value). Eigenmannia is able to sense differences in EOD periods as low as 400 ns (i.e. DF ∼ 0.036 Hz in a 300 Hz fish) [37,38]. For a relatively small modulating signal, the amplitude modulation is given by equation (2.6) [7], where α, F and ϕ represent the amplitude, frequency and phase shift respectively. In general, the modulating signal will be smaller than the carrier as electric fields decay over space and the self-generated EOD is generally larger than that sensed from neighbouring fish [7]. It should be noted that for more complicated signals (more than two sine waves, or harmonics present) these analytic forms are not always applicable, so in the case of the simulated EODs, the AM, and later the envelope (when multiple DFs are present), is calculated using peak tracing (see later).

2.5

and

2.6

A second order effect of amplitude modulation is phase modulation; the modulating signal effectively pushes the phase of the carrier around in time. This phase modulation is given by equation (2.7) for pure sine waves, where ω₀ and ω₁ are the angular frequencies of fish 1 and 2 respectively and α₁ and α₂ are their respective amplitudes [7]. Again, in general for complex signals, an analytic form of the phase is not applicable, so the phase ϕ of a signal x is given by the angle of the Hilbert transform (equation (2.8)). The phase difference Δϕ between the modulated signal and a reference signal x(t) is given by equation (2.9). The phase modulation is shown in figure 2b.

2.7

2.8

2.9

In our analyses, we calculated the AM and envelope by tracing the local minima. The differential phase modulation was calculated by equation (2.9) and sampled at these minima. For the Lissajous plots shown in figures 4 and 6, the reference signal was taken at a single body location (50%); for the JAR model (see below), the unperturbed signal at each body location served as the reference signal for that location. The choice of the 50% point was arbitrary; however, the Lissajous plots are similar regardless of point chosen.

Given the modulations to both phase and amplitude, the sign of the DF can be unambiguously determined. The amplitude modulation as a function of phase forms an enclosed orbit, or Lissajous plot (figure 2b) with the direction of rotation denoting the sign of the DF (clockwise: DF < 0, and counterclockwise: DF > 0) [7,8]. While normally these are plotted in Cartesian coordinates, we used polar coordinates (the comparison is shown in figure 2b). Although small phase modulations are clear when portrayed in Cartesian representations, large modulations can lead to discontinuities in these plots. Polar representations are much easier to interpret in general, due to the continuous phase wrapping.

As noted earlier, interactions involving more than two fish involve additional considerations. For example, with three sine waves, not only are there DFs between each of the three signals, there are also the differences between each DF (DDFs) [7]. When one modulating sine wave produces a positive DF and the other produces a negative DF, the Lissajous plots will trace out modulations that oscillate between positive and negative DFs. In this case, the Lissajous plots will form rotating ‘hypotrochoid-like’ patterns with the net rotation given by the difference between the two DFs (the DDF) [7,39].

2.3. Calculating jamming avoidance response-related sensory inputs

To investigate the JAR in the laboratory, a pair of transverse electrodes (orthogonal to the body) is typically used to mimic another fish [7,8,19]. We replicate this experimental paradigm in our models, using two transverse electrodes to deliver a sine wave stimulus at frequencies of ±1% that of the model fish. Specifically, we used EOD frequencies of 300 Hz (Eigenmannia) and 600 Hz (Apteronotus) with DFs of ±3 Hz and ±6 Hz respectively (i.e. DF = ±1%). These DFs typically result in JARs with similar magnitudes in each species [6,13]. The stimulus was calibrated so that the potential across the electrodes was equal to the peak-to-peak amplitude of the fish's head–tail EOD; this resulted in a peak modulation depth (contrast) of the order of 60%. The electrodes were positioned at 14 cm lateral to the body (on both sides) and at a rostro-caudal location (nose at 0 cm) of 10 cm for Eigenmannia and 8 cm for Apteronotus; note that the body lengths for the Eigenmannia and Apteronotus model fish are 26 cm and 21 cm respectively. We also consider interactions among three fish [7,39]. In this case, the electrode stimulus is the sum of two sine waves with different frequencies (representing two different fish). In Apteronotus, the baseline DF between each sine wave and the receiving fish (which had EODf = 600 Hz) was 103 Hz and −97 Hz (DDF =+6 Hz) and −103 Hz and +97 Hz (DDF =−6 Hz). This was halved in Eigenmannia to match the 300 Hz EODf of the receiving fish, with baseline DFs of 51.5 Hz and −48.5 Hz (DDF = +3 Hz) and −51.5 Hz and +48.5 Hz (DDF =−3 Hz). To simplify further discussion, we will refer to this condition as DDF =±1%. In this three-fish case, the peak-to-peak stimulus amplitudes were scaled, as in the two-fish case, such that the maximal amplitude of the electrode signal was equal to the head–tail EOD amplitude.

To evaluate the sensory information underlying the JAR, we use a well-established model [39] in which EOD amplitude and phase modulations can be used to predict the relative magnitude and sign of the JAR. The model directly maps phase and amplitude modulations onto JAR magnitude (fig. 11 in [29], and fig. 9 in [39]). We calculated this mapping by using the previously described fits to behavioural data [39]. The phase and amplitude modulations are determined at different body locations and through this mapping, each provides an estimate of JAR magnitude. Despite this model being developed for Eigenmannia, it allows us to characterize the quality and reliability of sensory inputs in both species of fish, but does not imply that the neural algorithms underlying the JAR are the same. As a point of note, due to windowing effects of the Hilbert transform used to calculate the phase differences, the first and last 8 cycles of the AM cycle were excluded to eliminate errors.

3.1. Eigenmannia model

In previous work, we developed a finite-element model of the spatio-temporal electric field generated by Apteronotus [26]. To compare the electric signals available for sensing by Eigenmannia and Apteronotus, we developed a similar model of the Eigenmannia EOD (see Methods). The fitting error for the Eigenmannia model, as compared to the experimentally measured potentials, was 2% ± 0.15% NRMSE (mean ± s.d.). Overall, the EOD of Eigenmannia is relatively uniform and dipolar in nature (figure 1). The zero-crossing (white band where the sign of the EOD changes in space, i.e. voltage is zero) is relatively static in space over the EOD cycle. In contrast, the zero-crossing in Apteronotus oscillates in space near the tail. Furthermore, the fraction of time that the EOD has a multipolar topology is much greater in Apteronotus than in Eigenmannia (note that at phase 12, both fish have a multipolar EOD). Given the accurate models of the spatio-temporal EOD of these species, we were able to compare the sensory signals that would be available to each during a JAR experiment.

3.2. Jamming avoidance response-related sensory inputs for experiments involving two and three fish

In typical experiments involving the JAR, a pair of transverse electrodes are used to deliver a sine wave stimulus that mimics one or more fish. We considered similar stimuli, using a pair of transverse electrodes (see Methods) to deliver a signal with a DF equal to 1% (i.e. 6 Hz for Apteronotus and 3 Hz for Eigenmannia). In figure 3, we show the resulting AMs in each species at different rostral–caudal locations; these locations were also used as representative locations in the following analyses. While the AMs at each location were quite similar between the two species, there were also apparent differences, suggesting that the available sensory information differs between the two species of fish. In the following, we look more closely at the phase and amplitude modulations to examine the information content of these signals.

As described previously [4,5,39], to determine the sign of the DF and produce the appropriate JAR, the fish must determine the amplitude and relative phase of the modulated signal with respect to its own unperturbed EOD. In figure 4, we compare the phase and amplitude modulations at multiple body locations for a DF of 1%. In these Lissajous plots, the radial angle indicates the phase difference between the modulated signal and the reference signal (taken here as the unperturbed EOD at 50% body length, see Methods), and the radius represents the relative amplitude of the AM (note that each plot was scaled independently to its own maximum radius). While some of these Lissajous plots form simple loops, others show more complicated double-loops (see Eigenmannia at 0.25 body fraction for example); these doublets occur when the AM depth is large (α₀ is similar to α₁; equations (2.5)–(2.8)); the resulting phase precession can span the entire circle causing the double-loops to appear. In general, the results in figure 4 are consistent with the theory and previous experiments [4,8]: the sign of the DF unambiguously determines the direction of rotation in both species.

We next considered what happens with three fish. In a natural setting with more than two fish, the sensory signals underlying the JAR become more complicated. For example, when one fish is joined by two other fish, two DFs can arise with respect to the receiving fish. In addition, the DDF can also result in relevant modulations; this is referred to as the ‘social envelope’ [7]. When the DDF is small enough (similar to a DF that would produce a JAR), Eigenmannia will change EODf in a manner predicted by the JAR, producing a SER. It is not known if Apteronotus will respond similarly but in a natural setting they do not usually create social envelopes [7,15,16]. Nonetheless, we can use our models, as in the previous section, to explore the effects of social envelopes on the sensory information available for a SER.

The social envelopes are shown in figure 5 (the three-fish analogue of figure 3). The high-frequency modulation occurs at the DF (approximately ±15% of the EODf) and the slow envelope (smaller than that in the two-fish case) occurs at the DDF (approximately 1% of the EODf). To explore how a fish would perceive this DDF, we show the resulting Lissajous plots in figure 6 (the three-fish analogue of figure 4). The previously predicted hypotrochoid-like patterns (see Methods) were most apparent for large amplitude modulations (e.g. body fraction of 0.25 in Eigenmannia) [7,39]. Such patterns are due to the superposition of a positive and negative DF (counterclockwise and clockwise rotations respectively) and in both species, the net direction of rotation (given by the dominant DF) gives the appropriate sign of the DDF.

In general, the Lissajous plots for Eigenmannia in both two-fish and three-fish conditions (figures 4 and 6) encompass a larger area than those for Apteronotus, indicating deeper, and hence more robust modulations. This suggests there is a higher signal-to-noise ratio for JAR-related sensory inputs in Eigenmannia. In the next section, we considered how this might relate to the robustness of these sensory inputs.

3.3. Predicting the quality of jamming avoidance response-related sensory inputs

The Lissajous plots described in the previous section suggest that the quality of sensory information available for the JAR is higher in Eigenmannia. More reliable sensory inputs should lead to the production of a more robust JAR with larger magnitude. We tested this by adopting a previously described behaviour-based model of the JAR [39] (see Methods). The algorithm that fish use for the JAR is well accepted [4], and involves the phase and amplitude modulations represented in figures 4 and 6 as inputs. The JAR model maps these inputs onto the relative magnitude of a JAR; a larger JAR represents increased certainty in the decision process and thus we considered JAR magnitude as a proxy for the reliability of sensory inputs. Figure 7 shows the JAR magnitude at different locations along the body in both Apteronotus and Eigenmannia for the two-fish condition. The JAR prediction depends on the body location used for a reference, so the data are presented as ±1 s.d. (shaded) over all reference locations (figure 7). It is immediately apparent in Eigenmannia that the sensory input predicts the appropriate direction of the JAR (positive JAR for a negative DF and vice versa) for almost all body locations (figure 7). Importantly, there is negligible overlap between the +DF and −DF condition, indicating relatively unambiguous sensory inputs. Conversely, in Apteronotus there is an extensive overlap between the two DF conditions, implying more ambiguous, or less robust sensory inputs.

As expected, extending this analysis to three fish introduced complexity: since the modulation rotates in the ‘wrong’ direction at some points in time, the change in EODf produced by social envelopes should be reduced in magnitude [7]. We confirmed this using the same JAR model as the two-fish case [39]. In Eigenmannia (at least in the front half of the body) the sensory inputs result in a smaller but reliable SER in the correct direction (figure 8). On the other hand, Apteronotus show a much more variable response due to contradictory signals arising from different body locations. Overall, as for the two-fish case, this analysis suggests that the sensory signals underlying the SER are also more reliable for Eigenmannia, and predicts a small unreliable SER, if any, in Apteronotus. Thus, it appears that the complex EOD topology exhibited by Apteronotus degrades sensory information in both JAR and SER contexts, and is one of the causes of a less reliable response in this species.