SUMMARY
Recent studies have shown that central nervous system neurons in weakly electric fish respond to artificially constructed electrosensory envelopes, but the behavioral relevance of such stimuli is unclear. Here we investigate the possibility that social context creates envelopes that drive behavior. When Eigenmannia virescens are in groups of three or more, the interactions between their pseudosinusoidal electric fields can generate ‘social envelopes’. We developed a simple mathematical prediction for how fish might respond to such social envelopes. To test this prediction, we measured the responses of E. virescens to stimuli consisting of two sinusoids, each outside the range of the Jamming Avoidance Response (JAR), that when added to the fish's own electric field produced lowfrequency (below 10 Hz) social envelopes. Fish changed their electric organ discharge (EOD) frequency in response to these envelopes, which we have termed the Social Envelope Response (SER). In 99% of trials, the direction of the SER was consistent with the mathematical prediction. The SER was strongest in response to the lowest initial envelope frequency tested (2 Hz) and depended on stimulus amplitude. The SER generally resulted in an increase of the envelope frequency during the course of a trial, suggesting that this behavior may be a mechanism for avoiding lowfrequency social envelopes. Importantly, the direction of the SER was not predicted by the superposition of two JAR responses: the SER was insensitive to the amplitude ratio between the sinusoids used to generate the envelope, but was instead predicted by the sign of the difference of difference frequencies.
INTRODUCTION
Weakly electric fish generate an electric organ discharge (EOD) that results in an electric field that surrounds the fish's body. In Eigenmannia, the EOD is quasisinusoidal and when fish are in close proximity (~1 m or less) their EODs interact. In the case of two nearby conspecifics, the combined EOD signal has a modulation, termed the amplitude modulation (AM). If there are more than two nearby conspecifics or relative movements between conspecifics, the combined EOD signal contains modulations of the AM, which has been termed the electrosensory envelope (Middleton et al., 2006).
The interactions of two EODs have been well studied in relation to the Jamming Avoidance Response (JAR). When two nearby conspecifics have EOD signals S_{1} and S_{2} at frequencies of f_{1} and f_{2}, respectively, the combined signal, S_{1}+S_{2}, has an emergent AM. The AM frequency is at the frequency difference, df, where df=f_{2}−f_{1}. When two neighboring Eigenmannia have EODs of similar frequency (e.g. 500 and 505 Hz, with df=5 Hz) they perform the JAR, during which each fish will raise or lower their individual EOD frequency to increase df, and thus the AM frequency.
When there are three or more EOD signals it is possible that fish are responding not only to the AM but also to the emergent envelope. Here we define a ‘social envelope’ as the modulation of the AM that occurs when at least three EODs are added. For example, if there are three EOD signals, S_{1}, S_{2} and S_{3}, at frequencies f_{1}, f_{2} and f_{3}, respectively, the combined signal S_{1}+S_{2}+S_{3} can have an AM; the magnitude of this AM also fluctuates over time, referred to here as the envelope of the AM (Fig. 1A) (see Appendix, Definition of envelopes). Thus, it is possible that even with high df values there could be a lowfrequency envelope (as shown in Fig. 1).
Understanding the behavioral relevance and sensory processing of envelope information has proven challenging in part because the extraction of envelope information requires nonlinear processing (Fig. 1B; see Appendix, Methods for envelope extraction). However, recent neurophysiological studies have already identified enveloperelated neural activity at each level from the receptor afferents to the midbrain in weakly electric fish (Middleton et al., 2006; Middleton et al., 2007; Longtin et al., 2008; Savard et al., 2011; McGillivray et al., 2012), suggesting that not only can the fish extract envelope information but there might also be behavioral relevance of these signals for the animals.
Modelbased prediction of the Social Envelope Response
The beauty of the JAR is that the behavioral response can be predicted based on a simple algorithm (Heiligenberg, 1991). For the fish to shift its EOD frequency in the ‘correct’ direction (e.g. the direction that increases df), the fish must be able to compute the sign of the df. The fish does this without an efference copy of its own EOD (Bullock et al., 1972) using amplitude and phase modulation information measured across the body (e.g. multiple electroreceptors) (Metzner, 1999).
The JAR computation is diagrammatically represented as a Lissajous figure in the amplitudephase plane (Fig. 2A), which was pioneered by Heiligenberg and Bastian (Heiligenberg and Bastian, 1980) and has been verified through electrophysiological recordings (Bastian and Heiligenberg, 1980). The plot is the magnitude (xaxis) versus the phase (yaxis) of the complex representation of the combined signal (see Appendix, The amplitudephase Lissajous). The Lissajous trajectory will rotate clockwise for negative df and counterclockwise for positive df at a frequency of df. The direction of rotation of the Lissajous predicts the direction that the fish will shift its EOD during the JAR.
When three sinusoids (or EODs) interact, AMs emerge at each of the df values, and the Lissajous is more complicated (Fig. 2B). For example, if there are three EOD signals, S_{1}, S_{2} and S_{3}, at frequencies f_{1}, f_{2} and f_{3}, there will be AMs at the magnitudes of the following df values: df_{2}=f_{2}−f_{1}, df_{3}=f_{3}−f_{1} and df_{1}=f_{3}−f_{2}. However, the signal measured by each fish is typically dominated by its own EOD. So, for fish 1 the signal S_{1} dominates the others (S_{2} and S_{3}) and correspondingly, the AMs at df_{2} and df_{3} dominate the AM at df_{1}. In this case, the AM at df_{1} can be considered negligible, and the dominant envelope frequency emerges at ddf=df_{3}−df_{2} (see Eqn A18). Note that ddf is a signed quantity, which is important to the predictions stated below.
In this paper, we hypothesize that the JAR circuit can be extended to predict a behavioral response to signals outside the range of the JAR that nevertheless generate lowfrequency envelopes. In some cases, two conspecific signals (S_{2} and S_{3}) when added to S_{1} produce a lowfrequency envelope (see Appendix, Caveats of the analytic signal method). Two such cases are depicted in Fig. 2B – positive and negative ddf. At first glance, the ‘floral’ pattern of the Lissajous seems to lack a consistent rotation. However, each of the ‘petals’ precesses in a direction corresponding to the sign of the ddf, at frequency ddf (see Appendix, The amplitudephase Lissajous). Upon lowpass filtering of both the amplitude and phase signals, the petals are filtered out and the general, slow precession emerges (Fig. 2C).
Interestingly, as the amplitude ratio between the signals is inverted, the direction of rotation of individual petals flips, but the direction of the precession remains unchanged. Does the fish respond to the direction of the petals (df values and amplitude ratio) or the precession (ddf)? When the df values are within the JAR range the response of the fish follows the petals (Partridge and Heiligenberg, 1980). But what happens when the df values are outside the JAR range? We hypothesize that fish respond to the emergent envelope at the ddf, governed by the precession as revealed by the lowpass filtered model (see Appendix, The amplitudephase Lissajous). If, as our model predicts, the fish uses a downstream lowpass filter from the JAR circuit to extract envelope information, it could drive a behavioral Social Envelope Response (SER), much like the JAR to AM stimuli.
MATERIALS AND METHODS
Adult Eigenmannia virescens (Valenciennes 1836) (10–15 cm in length) were obtained through a commercial vendor and housed in aquarium tanks with a water temperature of ~27°C and conductivity in the range of 150–300 μS cm^{−1} (Hitschfeld et al., 2009). All fish used in these experiments were housed in social tanks that contained two to five individuals. These experiments were conducted at the Johns Hopkins University between 2009 and 2012. All experimental procedures were approved by the Johns Hopkins Animal Care and Use Committee and followed guidelines established by the National Research Council and the Society for Neuroscience.
Experimental procedure
Each individual fish (N=4) was transferred to the testing tank (27±1°C and 175±25 μS cm^{−1}) and allowed to acclimate for 2–12 h before experiments began. During the acclimation period a second fish was also in the testing tank, to provide recent social experience, but was removed before the start of the experiment. For testing, the experimental fish was restricted in a chirp chamber to prevent movement. Experiments were started when the EOD frequency did not change by more than ±1 Hz for at least 25 consecutive minutes, which typically took 1–3 h.
Trials were presented across multiple testing blocks that lasted 1–3 h and were completed on different days. Between testing sessions the fish was returned to its home tank to reduce changes in response due to motivation, fatigue or other unknown factors. If the EOD responses of the fish deteriorated over the course of testing the fish was placed back in the home tank for 1–5 days and then retested.
Experimental setup
The chirp chamber was positioned such that the fish was located in the middle of two electrodes (headtotail) separated by 25 cm (Fig. 3, red electrodes), used to record the EOD. All stimuli were applied into the tank via transverse electrodes separated by 25 cm with the fish located in the middle (Fig. 3, black electrodes). A 1 cm transverse dipole (Fig. 3, yellow electrodes) adjacent to the head of the fish measured the local electric field, which was used to estimate stimulus amplitude.
At the start of each trial the initial EOD frequency of the fish (f_{1i}) was extracted. All trials within a testing block were presented randomly for each fish. Each trial lasted 200 s with an intertrial interval of 200 s. For each trial, the fish was presented with a stimulus that was either a single sinusoid (control trials; S_{2}) or a sum of two sinusoids (envelope trials; S_{2}+S_{3}).
For the envelope trials, the frequencies of the individual sinusoids (f_{2} and f_{3}) were calculated by adding a specified initial frequency difference (df_{i}) to f_{1i}, i.e. f_{2}=f_{1i}+df_{2i} and f_{3}=f_{1i}+df_{3i}. For control trials, only f_{2} was calculated and applied. The frequencies f_{2} and f_{3} were held constant, i.e. not clamped to f_{1}, so changes in the fish's EOD frequency results in changes in the value of each df and the ddf.
Experimental stimuli
Control trials
All fish completed trials (N=20) with a single sinusoid stimulus (S_{2}) at a specified high df (>40 Hz). The initial df values used were ±52, 58, 72, 78, 92 and 98 Hz, which are outside the range of frequencies known to elicit the JAR. These df values were a subset of those used to create the envelope stimuli in other trials (see below). For all control trials the stimulus amplitude was 0.74 mV cm^{−1} and the stimulus amplitude ramp time was 20 s.
Amplitude trials
All fish completed trials (N=10), with a sum of two sinusoids (S_{2}+S_{3}) that produced a ddf_{i} of ±4 Hz. The initial df values used were ±48 and ±52, such that there were two trial types: df_{2i}=−48 and df_{3i}=+52 or df_{2i}=−52 and df_{3i}=+48, which resulted in a +4 Hz and a −4 Hz envelope, respectively. These trials were repeated at five different stimulus amplitudes (0.15, 0.45, 0.74, 1.05 and 1.34 mV cm^{−1}) with a ramp time of 20 s.
Envelope trials
All fish completed trials (N=48) with a sum of two sinusoids (S_{2}+S_{3}) that produced a specified ddf_{i}. Trials were completed with df_{2i}=±50, ±70 or ±90 with df_{3i} sweeping from −df_{2i}−8 to −df_{2i}+8 at intervals of 2 Hz. For example, for df_{2i}=50, df_{3i} was set at each of the following values for individual trials: −58, −56, −54 or −52 (resulting in initial ddf values of −8, −6, −4 and −2 Hz) or −48, −46, −44 and −42 (resulting in initial ddf values of 2, 4, 6 and 8 Hz). For trials where df_{2i}=−50, the df_{3i} values were the same as the above example, except with a positive sign. This was repeated for df_{2i}=±70 and ±90, resulting in trials with ddf_{i} values from −8 to +8 in increments of 2 Hz (excluding 0), produced by df values of varying frequencies. All trials were completed with combined stimulus amplitude of 0.74 mV cm^{−1} and a ramp time of 20 s.
Ramptime trials
One fish completed trials (N=30) with a sum of two sinusoids (S_{1}+S_{2}), where three amplitude ramp times were tested (1, 20 and 100 s). Each ramp time was repeated for two envelope frequencies (+4 Hz: df_{2i}=−48, df_{3i}=+52; and −4 Hz: df_{2i}=−52, df_{3i}=+48) and five stimulus amplitudes (0.15, 0.45, 0.74, 1.05 and 1.34 mV cm^{−1}).
Ratio trials
One fish completed trials (N=10) with a sum of two sinusoids (S_{2}+S_{3}), where the relative amplitudes of each individual component were varied at a ratio of 1:1, 1:3, 2:3, 3:2 and 3:1 for envelopes of +4 Hz (df_{2i}=−48, df_{3i}=+52) and −4 Hz (df_{2i}=−52, df_{3i}=+48).
Data analysis
For each trial the recorded EOD was used to compute the EOD frequency as a function of time, f_{1}(t). This was achieved via postprocessing with a custom script in MATLAB (MathWorks, Natick MA, USA) that computed the spectrogram of the recorded signal and determined f_{1}(t) as the frequency with the highest power near the fish's baseline EOD frequency. The baseline f_{1i} was measured at the start of each trial using a frequencytovoltage (F2V) converter (FV1400, OnoSokki, Yokohama, Japan). For 60 trials the output of the F2V converter was verified against postexperiment Fourier analysis. The error between the two measurements was negligible (mean ± s.d.=0.0008±0.054 Hz). f_{1} stabilized by the last 60 s of each trial; f_{1f} is the mean frequency measured over this period. The change in frequency was calculated as Δf_{1}=f_{1f}−f_{1i}.
For each trial, Δf_{1} was normalized to the individual fish's maximum response, Δf_{1max}, to allow responses to be compared across fish. Because fish could raise or lower their EOD frequency, some measures are normalized as Δf_{1}/Δf_{1max}. Dependent measures were analyzed using oneway repeatedmeasures ANOVA. For significant main effects, effect size (ηp^{2}) is given to allow comparison between measures. Additionally, post hoc Tukey's honestly significant difference (HSD) tests were run on each significant main effect. We indicate the critical value (Q_{crit}) for each test and provide the obtained values (Q_{obt}) only for those that were statistically significant (i.e. greater than the critical value).
RESULTS
EOD frequency changes were not elicited by high df values
To ensure that observed responses were not due to the individual df values, we conducted control trials where fish were presented with a single sinusoid stimulus that had a high df. We measured Δf_{1} during the last 60 s of the intertrial interval and found that the EOD frequency was stable without stimulation (mean ±s.e.m.=0.05±0.006 Hz). In addition, Δf_{1} across the first 10 s (0.23±0.03=Hz) and the last 60 s (0.52±0.04 Hz) of control stimulus presentation produced only nominal changes to the EOD frequency. Responses to all control trials by a single fish are shown in Fig. 4A. Thus, it is unlikely that the observed Δf_{1} to the sum of sinusoid stimuli (which has an emergent envelope) was due to a response to any individual component alone.
Fish exhibited an SER
The sum of two sinusoid stimuli (S_{2}+S_{3}) elicited changes in EOD frequency. Fig. 4 shows a characteristic SER of a single fish to two replicates of a +2 Hz envelope (df_{2}=−50, df_{3}=+52; blue) and two replicates of a −2 Hz envelope (df_{2}=−52, df_{3}=+50; red). The figure shows that the envelope response differs from the response observed to control stimuli (gray).
Across all fish, responses to control stimuli were minimal (range=0.05 to 0.74 Hz) compared with the SERs (range=1 to 4 Hz). Moreover, the time course of EOD change during control trials was much larger than the time course of the SER, which corresponded to the stimulus ramp time (20 s). In addition, responses to control trials were biased downward, while the SERs were bidirectional. The direction of the SER shows that the fish shifts its EOD frequency down when the envelope frequency (ddf) is positive and up when the envelope frequency is negative. The direction of the SER was typically opposite the sign of the ddf, resulting in the EOD shifting towards the closer df (although the final df values were 40 Hz or above).
SER was stronger for lowerfrequency envelopes
Fish changed f_{1} in response to sumofsinusoid stimuli that created initial envelopes in the frequency range of ddf_{i}=2 to 8 Hz as illustrated for a single fish in Fig. 5. The figure also illustrates that Δf_{1} is qualitatively similar across all df values used. However, the strength of the SER (the change in EOD frequency during a trial) is dependent upon on ddf_{i} (Fig. 6A). The effect of the initial absolute envelope frequency, ddf_{i}, on the normalized absolute EOD frequency change, Δf_{1}/Δf_{1max}, was significant (F_{3,9}=6.45, P=0.04, ηp^{2}=0.68). Normalized Δf_{1} is generally smaller as a function of larger initial ddf: mean ± s.e.m.=0.59±0.04 for 2 Hz, 0.52±0.03 for 4 Hz, 0.34±0.03 for 6 Hz and 0.39±0.04 for 8 Hz. The only significant pairwise differences (Tukey's HSD, Q_{crit}=4.41) were between the lowest envelope frequency (2 Hz) and those higher than 6 Hz (2 Hz versus 6 Hz: Q_{obt}=4.45; 2 Hz versus 8 Hz: Q_{obt}=5.51; Fig. 6A, asterisks). The rest of the pairwise comparisons were not significant (Q_{obt}<4.41).
SER increased the envelope frequency
Fish change f_{1} in response to initial envelope stimuli, which resulted in a change in the envelope frequency (Fig. 6B). In general, the final absolute envelope frequency settles in the range of 5–15 Hz (mean ± s.e.m.=8.87±0.20 Hz). The change in envelope frequency (Δddf=ddf_{f}−ddf_{i}) as a function of ddf_{i} is shown in Fig. 6C. We found a significant effect of the initial envelope frequency (ddf_{i}) on the Δddf (F_{3,9}=6.32, P=0.01, ηp^{2}=0.68) such that the change in envelope frequency, Δddf, was smaller as a function of larger ddf_{i}: mean ± s.e.m.=4.78±0.68 for 2 Hz, 4.57±0.51 for 4 Hz, 2.86±0.36 for 6 Hz and 3.29±0.59 for 8 Hz. The only significant pairwise differences (Tukey's HSD, Q_{crit}=4.41) were between 2 and 6 Hz (Q_{obt}=5.05) and between 4 and 6 Hz (Q_{obt}=4.50), where the change in envelope frequency was greater for the lower initial envelope frequency in each pair.
SER depended on stimulus amplitude and not the rate of amplitude change
The strength of the SER, measured by the magnitude Δf_{1}, increased as a function of stimulus amplitude (shown for one fish in Fig. 7A). The effect of stimulus amplitude on the normalized Δf_{1} was significant (F_{4,12}=7.16, P=0.02, ηp^{2}=0.71; Fig. 7B). The change in frequency, Δf_{1}, was generally larger for larger stimulus amplitudes: mean ± s.e.m.=0.32±0.07 for 0.15 mV cm^{−1}, 0.49±0.09 for 0.45 mV cm^{−1}, 0.68±0.11 for 0.74 mV cm^{−1}, 0.63±0.09 for 1.05 mV cm^{−1} and 0.83±0.06 for 1.34 mV cm^{−1}. There were significant pairwise differences (Tukey's HSD, Q_{crit}=4.21) between the lowest stimulus amplitude (0.15 mV cm^{−1}) and those higher than 0.74 mV cm^{−1} (0.15 versus 0.74: Q_{obt}=5.16; 0.15 versus 1.05: Q_{obt}=4.49 and 0.15 versus 1.34: Q_{obt}=7.20) and between the second lowest stimulus amplitude (0.45 mV cm^{−1}) and the highest stimulus amplitude (0.45 versus 1.34: Q_{obt}=4.73; Fig. 7B, asterisks). Other pairwise comparisons were not significant (Q_{obt}<4.20).
The effect of stimulus amplitude on final ddf_{f} was significant (F_{4,12}=7.99, P=0.04, ηp^{2}=0.73; Fig. 7C). There was a significant pairwise difference (Tukey's HSD, Q_{crit}=4.20) between the lowest stimulus amplitude (0.15 mV cm^{−1}) and those higher than 0.74 mV cm^{−1} (0.15 versus 0.74: Q_{obt}=5.25; 0.15 versus 1.05: Q_{obt}=4.71; and 0.15 versus 1.34: Q_{obt}=7.65) and between the second lowest stimulus amplitude (0.45 mV cm^{−1}) and the highest stimulus amplitude (0.45 versus 1.34: Q_{obt}=4.83). Other pairwise comparisons were not significant (Q_{obt}<4.20).
In data from one fish, differences in ramp time did not effect the strength of the SER, ゔf_{1} (Fig. 8). Thus, the SER strength depended on the amplitude of the stimulus, but not on the rate of change of amplitude.
SER did not switch direction with changes in amplitude ratio
For a given df_{2} and df_{3} pair, the relative amplitudes of S_{2} and S_{3} determine the rotation of the ‘petals’ of the Lissajous but not the general precession. As can be seen for ddf=−4Hz, the petals rotate counterclockwise for ratios 1:3 and 2:3, and clockwise for 1:1, 3:1 and 3:2, but the graph precesses clockwise in all cases (Fig. 9, top). Similarly, for ddf=+4 Hz the petals rotate clockwise for ratios 3:1 and 3:2 and counterclockwise for 1:1, 1:3 and 2:3, but the graph precesses counterclockwise in all cases (Fig. 9, bottom).
We examined the sign of SER, measured by the sign of Δf_{1} in response to different stimulus amplitude ratios S_{2}:S_{3} (1:1, 1:3, 2:3, 3:2 and 3:1) for ddf=±4 Hz (Fig. 9). We found that the direction of the SER depended only on the sign of ddf, not the amplitude ratio; i.e. f_{1} shifts up when ddf is negative, and f_{1} shifts down when ddf is positive (Fig. 9, middle). This supports our hypothesis that the SER is driven by the precession of the Lissajous rather than the local rotation of the petals when the df values are outside the JAR range.
DISCUSSION
Forty years of analysis of the JAR (Watanabe and Takeda, 1963; Bullock et al., 1972) have focused on mapping a welldefined computation (Heiligenberg, 1991) through all stages of neural processing, from sensory receptors to motor units (Metzner, 1999). This work was successful mainly due to a sharp focus on the specific parameters that were necessary and sufficient to drive the behavior, thereby putting aside potentially complex temporal features – such as social and movement envelopes – that are likely to be ubiquitous in a fish's electrosensory milieu (Tan et al., 2005; Stamper et al., 2010). Recent neurophysiological studies have identified neurons that respond to such electrosensory envelopes (Middleton et al., 2006; Middleton et al., 2007; Longtin et al., 2008; Savard et al., 2011; McGillivray et al., 2012), but the function of this brain activity was unknown.
Here we show the behavioral relevance of one category of electrosensory envelopes. We measured the EOD responses of E. virescens to envelope stimuli like those that would arise from the electrical interactions of three or more motionless conspecifics. We call this behavior the SER. We also proposed a simple extension of the algorithm for the JAR, a lowpass filter of the instantaneous amplitude and phase of the combined signal, which accurately predicts SER behavior.
In the SER, E. virescens raised or lowered their EOD frequency, which resulted in an increase in frequency of the envelope by approximately 2–6 Hz, with final envelope frequencies between 5 and 15 Hz. The strength of the SER depended on the initial envelope frequency and the stimulus amplitude: low initial frequencies and high stimulus amplitudes elicited the largest changes in EOD frequency. The SER direction was insensitive to the relative amplitude ratio between stimulus signals, indicating dependence on the slow precession of the Lissajous, as opposed to fast local rotations of the petals, as predicted by our model (see Fig. 9).
Mechanisms for the SER
We extended the widely known model for the control of the JAR with the addition of a lowpass filter that eliminates responses to the local rotations of the Lissajous while retaining its precession. The model does not predict where and how this computation may be implemented in the brain. Part of this computation could be implemented as a saturation nonlinearity of amplitudecoding Preceptors, which would cause them to encode envelopes (Savard et al., 2011). When combined with a rectification circuit in the electrosensory lateral line lobe (ELL) (Middleton et al., 2006; Middleton et al., 2007; Longtin et al., 2008), the amplitude axis of the Lissajous would oscillate at the envelope frequency (Eqn A27). In this case, the phase axis would be filtered independently in downstream circuits to yield the circular Lissajous that precesses at the ddf. Alternatively, amplitude and phase filtering may both occur in downstream circuits. In this case, the higher response thresholds (as compared with the JAR) may be necessary to overcome the attenuation caused by the filter.
Possible functional relevance of the SER
In their natural habitat, weakly electric fish are commonly found in groups of three or more conspecifics (Tan et al., 2005; Stamper et al., 2010), which is a necessary condition for the SER. We showed that fish exhibited SERs that increased the frequency of envelopes to higher frequencies (up to 15 Hz). The SER appears to be analogous to the JAR, in which fish also shift their EOD frequency, effecting an increase in the frequency of the AM (Heiligenberg, 1991). It has been shown that lowfrequency AMs impair aspects of electrolocation and that the JAR may allow fish to avoid this detrimental interference (Heiligenberg, 1973; Bastian, 1987). In addition to the behavioral impairment, it has also been shown that neural responses to moving objects are impaired by lowfrequency jamming (Ramcharitar et al., 2005). If the SER functions analogously to the JAR, one would predict that lowfrequency envelopes might also degrade electrosensory performance and impair the underlying neural responses to moving objects.
Movement envelopes
Fish are rarely completely motionless; therefore, we expect that movementrelated envelopes commonly emerge in groups of two or more fish. These envelopes can encode the relative velocity between fish and possibly provide reliable cues about distance (Yu et al., 2012). We suspect that fish may also exhibit a ‘movement envelope response’ that can be driven by modulations due to the relative movement between individuals. These movementbased envelopes indeed arise in a social context, but for clarity we distinguish them from ‘social envelopes’ as defined in this paper. This distinction is important because social envelopes constitute a special class of signals that arises solely due to the details of the interactions between electric fields of three or more wavetype weakly electric fish. Movementrelated envelopes, however, can arise in a variety of contexts, including from nonsocial sources such as the interaction of fish with objects in their environment.
In the natural habitat, a cacophony of stimuli contribute to modulations of the EOD in Eigenmannia, including simple moving objects (Carlson and Kawasaki, 2007), summations of multiple electric signals (as examined in this paper) and movements of nearby electrogenic animals (Metzen et al., 2012). In addition, amplitude and phase modulations influence each other, creating cross interactions, which also have behavioral implications (Carlson, 2008). These and related behaviors appear to be mediated by sets of simple computational rules that are instantiated in the ascending electrosensory pathways of Eigenmannia and other closely related species of weakly electric fishes. The behavioral results in this paper provide yet another platform for the analysis and reanalysis of a welldescribed neural circuit that is used in the control of multiple behaviors.
APPENDIX
This Appendix focuses on social envelopes arising from multiple interacting motionless wavetype weakly electric fish. We address the following questions: (1) what are amplitude modulations and envelopes in the context of interacting EODs; (2) how do AMs and envelopes emerge from sums of sinusoids; and (3) what are the constraints on biological mechanisms for the extraction of AMs and envelopes?
Definition of envelopes
Interactions of the electric fields of two motionless fish with different EOD frequencies give rise to a ‘beat’ pattern at the df of the two EOD signals. The sum of two sinusoids can be mathematically decomposed into an amplitude and phasemodulated signal: (A1) However, the structure of M(t) and ψ(t) is complicated: (A2) When a_{1}>>a_{2}>0, these signals can simplified to an intuitive expression: (A3) This holds for electric fish because the selfgenerated EOD for a fish generally dominates the EODs of conspecifics.
Mathematically, the ‘modulator’ M(t) above is referred to as the envelope of the signal. However, this quantity is termed an AM by the electric fish community, because envelope coding in electric fish is observed in the afferents of Ptype electroreceptors, which code EOD amplitude increases. Thus the source signal for envelope extraction is the AM of the EOD, not the underlying EOD signal itself. We will refer to the envelope of the EOD as the AM, and the envelope of the envelope of the EOD as, simply, the envelope.
For three interacting, motionless EODs (modeled as sinusoids), M(t) and ψ(t) are more complicated. The interactions can produce a combined signal that has higherorder features, such as ‘beats of beats’ (primarily at the ddf), which are termed social envelopes.
The AM M(t) of a signal s(t) is the signal such that, when multiplied by a carrier signal, cosψ(t), reproduces s(t), namely s(t)=M(t)cosψ(t). Under appropriate assumptions, M(t) is a smooth curve that approximately traces the local maxima of s(t), and −M(t) its minima, and these local extrema approximately correspond to the peaks and troughs of cosψ(t). Thus, for an AM to be ‘well defined’ (in a sense formalized below), the extrema should oscillate at slower frequencies than the carrier. In the case of two sinusoids, the expression for M(t) in Eqn A2 represents a pure AM only if it is in a lower frequency band than the carrier signal cosψ(t), i.e. the signals are spectrally separated. In fact the Hilbert transform can be used to decompose such a signal into a product of its amplitude and carrier if those signals are spectrally separated (Bedrosian, 1963; Rihaczek and Bedrosian, 1966). M(t) and cosψ(t) resulting from mixing of two sinusoids as in Eqn A2 generally results in infinite harmonics; however, when a_{1}>>a_{2}>0 and ω_{2}–ω_{1}<ω_{1}, the majority of the spectral content of the M(t) and cosψ(t) is bandseparated, so they form welldefined AMs (Lerner, 1960; Rihaczek and Bedrosian, 1966). These restrictions can also be explained in the context of a signal, initially constructed as . The AM and carrier extracted by the Hilbert transform, M(t) and cosψ(t), will generally not be equal to and unless and are themselves bandseparated. For three or more sinusoidal EODs, there are analogous constraints on amplitudes and frequencies of the individual EODs in order that their sum produces a welldefined envelope of the AM (see Fig. A1).
Consider a group of N weakly electric fish, assumed to be motionless, with approximated sinusoidal EODs. The EOD of fish k, where kϵ{1,2,…,N}, is perceived at an electroreceptor of fish 1 as a_{k}c_{k}, where a_{k} is the amplitude, and ω_{k}=2πf_{k} is in radians, where f_{k} is the frequency of fish k in Hz. a_{k} is a function of the relative distance and orientation between fish 1 and fish k for k≠1, and a_{1} would depend on body bending of fish 1. The total signal at the electroreceptor is: (A4) where cos(ω_{k}t+ϕ_{k})=cos(θ_{k})=c_{k}. Because s(t) is assumed to be the signal at a receptor of fish 1, the amplitude of fish 1 will generally be greatest, i.e. a_{1}>>a_{k}, k=2,3,…,N.
We have stated that amplitudes and envelopes contain frequencies at the df and ddf values, but it is clear from Eqn A4 that the signal at the electroreceptor has only components at ω_{k}. Thus, a nonlinear method is required to extract AMs and envelopes.
Methods for envelope extraction
Magnitude of the analytic signal
A common definition of AMs is as the magnitude of the analytic signal. For a given real input signal s(t), its complex analytic signal is defined as , where ŝ is obtained by the Hilbert transform: (A5) where P denotes the Cauchy principal value and * denotes the convolution operator.
There are two key properties of the Hilbert transform used extensively below. First, the Hilbert transform is linear: (A6) Second, the Hilbert transform of sinusoids is given by: (A7)
We can express the analytic signal in polar form as . The AM is and the phase function is . The nonlinearity in this form of AM extraction arises not from the Hilbert transform or analytic signal construction – which are both linear – but rather from the magnitude operation.
To apply this to the sum of sinusoids signal, let be the complex analytic signal corresponding to a_{k}c_{k}, namely . Then: (A8) is the analytic signal of s(t) in Eqn A4. Graphical representation of this decomposition is shown in Fig. A2, for the threesinusoid case. The real part of the analytic signal is the real signal s(t)=M(t)cosψ(t). The magnitude is calculated via: (A9) where , and the summation notation k≠j refers to the combinations of k,jϵ{1,…,N}, k≠j. Hence, we obain the following: (A10) The Taylor series expansion can be firstorder approximated as: (A11) when x is small. To apply this firstorder approximation to Eqn A10, note that: (A12) When a_{1} dominates, the upper bound is approximately 4(a_{2}+…+a_{N})/a_{1}, which for sufficiently large a_{1} is small, and the approximation in Eqn A11 is justified. Thus: (A13) Subtracting DC, we obtain a ‘Hilbert approximation’, M_{H}, of the AM: (A14) where b_{kj}=a_{k}a_{j}/α. The AM is approximately a sum of () cosines at the df values ω_{k}−ω_{j}.
The envelope of this AM is obtained as the magnitude of the analytic signal of the expression in Eqn A14. Repeating steps in Eqns A9, A10, A11, A12, A13 and A14, we obtain: (A15) where the summation is over the set of all [N(N^{2}−1)(N−2)]/8 combinations of {{k,j},{p,q}} such that k,j,p,qϵ{1,…,N}, k≠j, p≠q, {k,j}≠{p,q} and: (A16) The envelope can be approximated as the sum of sinusoids at ddf values of the frequencies contained in the original sum of sinusoids. In the context of this paper (mixing of three EODs), from Eqn A15: (A17) where Δθ_{12}=(ω_{1}−ω_{2})t+(ϕ_{1}−ϕ_{2})=2π(df_{1})t+(ϕ_{1}−ϕ_{2}). Similarly, Δθ_{13}=2π(df_{2})t+(ϕ_{1}−ϕ_{3}) and Δθ_{23}=2π(df_{3})t+(ϕ_{2}−ϕ_{3}). If we assume that a_{1}>>a_{2} and a_{2}>>a_{3}, we can further approximate this as: (A18) where ddf=df_{1}−df_{2} and Φ=(ϕ_{1}−ϕ_{2}−ϕ_{1}−ϕ_{3}). Thus the envelope is primarily composed of the ddf, namely the difference of difference frequencies between fish 1 and other conspecifics.
Caveats of the analytic signal method
Realtime envelope extraction using the analytic signal is not biologically plausible, as the Hilbert transform is a noncausal operator. The analytic envelope of a narrowband Gaussian white noise, as discussed in Middleton et al. (Middleton et al., 2006), is generally well defined. However, this breaks down as the number of sinusoidal components is reduced (unless one of the amplitudes is dominant), as shown in the case of three EODs in Fig. A1C. The approximation in Eqn A18 is not applicable for a wide range of parameters, e.g. any conspecific amplitude is nonnegligible relative to a_{1}, or insufficient band separation between f, df and ddf. These limitations do not apply to our experimental setup because: (1) the combined signal perceived by the fish is dominated by its own EOD and (2) the ddf and df values used were sufficiently separated.
Envelope extraction via rectification and lowpass filtering
A commonly used method of envelope extraction is rectification of the signal followed by filtering. A similar mechanism may be used in electric fish via rectification by having the firing threshold of a neuron close to the mean of the input, and then filtering through a slow synapse (Middleton et al., 2007; Longtin et al., 2008).
Fullwave rectification of a zeromean signal is the absolute value of the signal. For the sum of sines from Eqn A4: (A19) Using the firstorder Taylor approximation from Eqn A11, we obtain the following: (A20)
If the band of difference frequencies ω_{k}−ω_{j} is spectrally separate from that of the sum frequencies ω_{k}+ω_{j}, an appropriate filter can extract the df values only, which form the AM. Let L(·) be such a filter, which highpasses DC and lowpasses sum frequencies; applying this filter to the rectified signal yields: (A21) One more rectification and lowpass filtering step provides us with the envelope, at the ddf: (A22) Envelopes extracted by both methods thus have similar spectral content, and the difference frequency components are separated only by scale. Envelopes can also be extracted by halfwave rectification, with qualitatively similar results (see Fig. A3).
The amplitudephase Lissajous
Here, we explain why the Lissajous figure as described in the paper rotates according to the df values, and why the lowpass filtered version of the Lissajous precesses according to the ddf. Consider the summed signal Eqn A4. The phase function ψ(t) is: (A23) Using tan^{−1}z=(i/2){ln[(1−iz)/(1+iz)]} (where z is a complex number), we have: (A24) The analytic phase relative to that of fish 1 is: (A25) This is the phase of the sum of cosines at the df values, . The analytic signal magnitude (Eqn A14), when a_{1} dominates, can be further simplified to: (A26) The Lissajous (M_{L} versus P_{L}) consists primarily of df components, which is why the graph rotates according to the df values. However, M_{L} is a firstorder Taylor approximation. If, instead, we approximate to the second order, we obtain the following: (A27) The squared term, when expanded, will contain product terms of the form c_{k−j}c_{p−q}, which can be written as follows: (A28) This shows that the amplitude of the analytic signal contains ddf terms [(a_{k}a_{j}a_{p}a_{q})/4α^{3}]c_{k−j−p−q} (Fig. A3, top). Hence a lowpass filtered version of the Lissajous also serves as a means to extract out components at ddf, albeit with reduced magnitudes. Fig. A3 (middle, bottom) shows that a ddf component is also present in the fullwave and halfwave rectified signals. Interestingly, when we apply each extraction method twice in succession, it affects the ddf component differently. It turns out that the halfwave rectified signal contains a stronger component at the ddf than the halfwave rectified AM.
FOOTNOTES

↵† These authors contributed equally to this work

FUNDING
This material is based on work supported by the National Science Foundation (NSF) [grants IOS0817918 to E.S.F and N.J.C.; CMMI0941674 to N.J.C. and E.S.F.; and CISE0845749 to N.J.C.] and the Office of Naval Research [grant N000140910531 to N.J.C. and E.S.F.]. S.A.S. was supported by an NSF Graduate Research Fellowship.
LIST OF SYMBOLS AND ABBREVIATIONS
 a_{k}
 amplitude of the kth sinusoid
 AM
 amplitude modulation
 c_{k}, s_{k}
 cos(θ_{k}), sin(θ_{k})
 c_{k−j}, c_{k+j}
 cos(θ_{k}−θ_{j}), cos(θ_{k}+θ_{j})
 ddf
 difference of df values, e.g. df_{3}−df_{2}
 df
 frequency difference
 df_{1}, df_{2}, df_{3}
 f_{3}−f_{2}, f_{2}−f_{1}, f_{3}−f_{1}
 E(t)
 envelope of the combined signal
 EOD
 electric organ discharge
 f_{1}, f_{2}, f_{3}
 frequencies of S_{1}, S_{2} and S_{3}, respectively
 f_{1f}, df_{2f}, df_{3f}, ddf_{f}
 final values of f_{1}, df_{2}, df_{3} and ddf
 f_{1i}, df_{2i}, df_{3i}, ddf_{i}
 initial values of f_{1}, df_{2}, df_{3} and ddf
 JAR
 Jamming Avoidance Response
 M(t)
 AM of the combined signal
 S_{1}
 EOD signal (measured)
 S_{2}, S_{3}
 applied sinusoid signals
 SER
 Social Envelope Response
 t
 time
 Δf_{1}
 f_{1f}−f_{1i}
 θ_{k}
 angle (rad) of the kth sinusoid, i.e. (ω_{k}t+θ_{k})
 ϕ_{k}
 phase (rad) of the kth sinusoid
 ψ(t)
 phase modulation of the combined signal
 ω_{k}
 frequency (rad s^{−1}) of the kth sinusoid
 © 2012.