Beyond the Jamming Avoidance Response: weakly electric fish respond to the envelope of social electrosensory signals

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 quasi-sinusoidal 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 S₁ and S₂ at frequencies of f₁ and f₂, respectively, the combined signal, S₁+S₂, has an emergent AM. The AM frequency is at the frequency difference, |df|, where df=f₂−f₁. 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, S₁, S₂ and S₃, at frequencies f₁, f₂ and f₃, respectively, the combined signal S₁+S₂+S₃ 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 low-frequency 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 envelope-related 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.

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 amplitude-phase 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 (x-axis) versus the phase (y-axis) of the complex representation of the combined signal (see  Appendix, The amplitude-phase Lissajous). The Lissajous trajectory will rotate clockwise for negative df and counter-clockwise 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, S₁, S₂ and S₃, at frequencies f₁, f₂ and f₃, there will be AMs at the magnitudes of the following df values: df₂=f₂−f₁, df₃=f₃−f₁ and df₁=f₃−f₂. However, the signal measured by each fish is typically dominated by its own EOD. So, for fish 1 the signal S₁ dominates the others (S₂ and S₃) and correspondingly, the AMs at |df₂| and |df₃| dominate the AM at |df₁|. In this case, the AM at |df₁| can be considered negligible, and the dominant envelope frequency emerges at ddf=|df₃|−|df₂| (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 low-frequency envelopes. In some cases, two conspecific signals (S₂ and S₃) when added to S₁ produce a low-frequency 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 amplitude-phase Lissajous). Upon low-pass 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 low-pass filtered model (see  Appendix, The amplitude-phase Lissajous). If, as our model predicts, the fish uses a downstream low-pass 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.

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.

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 re-tested.

The chirp chamber was positioned such that the fish was located in the middle of two electrodes (head-to-tail) 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 inter-trial interval of 200 s. For each trial, the fish was presented with a stimulus that was either a single sinusoid (control trials; S₂) or a sum of two sinusoids (envelope trials; S₂+S₃).

For the envelope trials, the frequencies of the individual sinusoids (f₂ and f₃) were calculated by adding a specified initial frequency difference (df_i) to f_1i, i.e. f₂=f_1i+df_2i and f₃=f_1i+df_3i. For control trials, only f₂ was calculated and applied. The frequencies f₂ and f₃ were held constant, i.e. not clamped to f₁, so changes in the fish's EOD frequency results in changes in the value of each df and the ddf.

All fish completed trials (N=20) with a single sinusoid stimulus (S₂) 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.

All fish completed trials (N=10), with a sum of two sinusoids (S₂+S₃) 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.

All fish completed trials (N=48) with a sum of two sinusoids (S₂+S₃) 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.

One fish completed trials (N=30) with a sum of two sinusoids (S₁+S₂), 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).

One fish completed trials (N=10) with a sum of two sinusoids (S₂+S₃), 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).

For each trial the recorded EOD was used to compute the EOD frequency as a function of time, f₁(t). This was achieved via post-processing with a custom script in MATLAB (MathWorks, Natick MA, USA) that computed the spectrogram of the recorded signal and determined f₁(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 frequency-to-voltage (F2V) converter (FV-1400, Ono-Sokki, Yokohama, Japan). For 60 trials the output of the F2V converter was verified against post-experiment Fourier analysis. The error between the two measurements was negligible (mean ± s.d.=0.0008±0.054 Hz). f₁ 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 Δf₁=f_1f−f_1i.

For each trial, Δf₁ 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 |Δf₁|/|Δf_1max|. Dependent measures were analyzed using one-way repeated-measures ANOVA. For significant main effects, effect size (ηp²) 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).

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 Δf₁ during the last 60 s of the inter-trial interval and found that the EOD frequency was stable without stimulation (mean ±s.e.m.=0.05±0.006 Hz). In addition, Δf₁ 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 Δf₁ to the sum of sinusoid stimuli (which has an emergent envelope) was due to a response to any individual component alone.

The sum of two sinusoid stimuli (S₂+S₃) elicited changes in EOD frequency. Fig. 4 shows a characteristic SER of a single fish to two replicates of a +2 Hz envelope (df₂=−50, df₃=+52; blue) and two replicates of a −2 Hz envelope (df₂=−52, df₃=+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).

Fish changed f₁ in response to sum-of-sinusoid 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 Δf₁ 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, |Δf₁|/|Δf_1max|, was significant (F_3,9=6.45, P=0.04, ηp²=0.68). Normalized |Δf₁| 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).

Fish change f₁ 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, ηp²=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.

The strength of the SER, measured by the magnitude |Δf₁|, increased as a function of stimulus amplitude (shown for one fish in Fig. 7A). The effect of stimulus amplitude on the normalized |Δf₁| was significant (F_4,12=7.16, P=0.02, ηp²=0.71; Fig. 7B). The change in frequency, |Δf₁|, 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, ηp²=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f₁| (Fig. 8). Thus, the SER strength depended on the amplitude of the stimulus, but not on the rate of change of amplitude.

For a given df₂ and df₃ pair, the relative amplitudes of S₂ and S₃ determine the rotation of the ‘petals’ of the Lissajous but not the general precession. As can be seen for ddf=−4Hz, the petals rotate counter-clockwise 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 counter-clockwise for 1:1, 1:3 and 2:3, but the graph precesses counter-clockwise in all cases (Fig. 9, bottom).

We examined the sign of SER, measured by the sign of Δf₁ in response to different stimulus amplitude ratios S₂:S₃ (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. f₁ shifts up when ddf is negative, and f₁ 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.

Forty years of analysis of the JAR (Watanabe and Takeda, 1963; Bullock et al., 1972) have focused on mapping a well-defined 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 low-pass 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).

We extended the widely known model for the control of the JAR with the addition of a low-pass 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 amplitude-coding P-receptors, 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.

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 low-frequency 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 low-frequency jamming (Ramcharitar et al., 2005). If the SER functions analogously to the JAR, one would predict that low-frequency envelopes might also degrade electrosensory performance and impair the underlying neural responses to moving objects.

Fish are rarely completely motionless; therefore, we expect that movement-related 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 movement-based 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 wave-type weakly electric fish. Movement-related envelopes, however, can arise in a variety of contexts, including from non-social 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 re-analysis of a well-described neural circuit that is used in the control of multiple behaviors.

This Appendix focuses on social envelopes arising from multiple interacting motionless wave-type 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?

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 P-type 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 higher-order 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 a₁>>a₂>0 and |ω₂–ω₁|<ω₁, the majority of the spectral content of the M(t) and cosψ(t) is band-separated, so they form well-defined 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 band-separated. 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 well-defined envelope of the AM (see Fig. A1).

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.

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.

Real-time envelope extraction using the analytic signal is not biologically plausible, as the Hilbert transform is a noncausal operator. The analytic envelope of a narrow-band 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 non-negligible relative to a₁, 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.

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).

 
• 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. |df₃|−|df₂|

 
• df

  frequency difference

 
• df₁, df₂, df₃

  f₃−f₂, f₂−f₁, f₃−f₁

 
• E(t)

  envelope of the combined signal

 
• EOD

  electric organ discharge

 
• f₁, f₂, f₃

  frequencies of S₁, S₂ and S₃, respectively

 
• f_1f, df_2f, df_3f, ddf_f

  final values of f₁, df₂, df₃ and ddf

 
• f_1i, df_2i, df_3i, ddf_i

  initial values of f₁, df₂, df₃ and ddf

 
• JAR

  Jamming Avoidance Response

 
• M(t)

  AM of the combined signal

 
• S₁

  EOD signal (measured)

 
• S₂, S₃

  applied sinusoid signals

 
• SER

  Social Envelope Response

 
• t

  time

 
• Δf₁

  f_1f−f_1i

 
• θ_k

  angle (rad) of the kth sinusoid, i.e. (ω_kt+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