Limitations of a previous model

Since Eigenmannia electrocytes’ continual APs demand narrowly-maintained equilibrium potentials (E_K and E_Na), homeostatic processes evidently succeed in minimizing fluctuations of [Na⁺] and [K⁺] close to the AP-generating membrane. A previously developed excitability model for isolated electrocytes (termed here, “MKZ” for Markham, Kaczmarek, Zakon, 2013) [16] depicts a system that is quiescent until stimulated by intracellular current injection (I_clamp). This model does not include an explicit Na⁺/K⁺ pump so initially we intended to incorporate one and explore E_K and E_Na stability for APs at frequencies over the species range (200–600 Hz). Because pump function is electrogenic, a key system requirement is to stabilize membrane potential (V_m) at the resting value (V_rest) of the unstimulated excitable system. Stabilization is accomplished via Na⁺ and K⁺ leaks of appropriate magnitudes [18] This however renders MKZ inexcitable because the very small V_rest-E_K necessitates an enormous K⁺-specific leak to counter pump electrogenicity and this makes V_m ultra-stable near V_rest.

Fig 1C shows MKZ APs during a 200 Hz stimulus train (same parameters as in Fig 3A in Ref. [16]); an expanded AP (right panel) reveals that the regenerative ΔV_m covers ~35 mV (from ~-17 mV to +17 mV) and that V_peak depends strongly on I_stim (see inset) and so is not strictly all-or-none. Excitability that depends strongly on large amplitude stimuli in this way seems ill-suited for Eigenmannia’s robust high frequency EOD output.

The MKZ model assumes an isopotential electrocyte in which the membrane current (I_membrane) charges and discharges 50 nF of membrane capacitance. While space-clamp (isopotential) is necessary for electrocyte voltage clamp recordings, electrocytes are in general not isopotential in situ as Na⁺-entry occurs exclusively via Nav channels at the innervated posterior membrane, located ~1 mm distant from the anterior membrane, which is channel-rich but inexcitable [5] In Sternopygus [19] and presumably Eigenmannia current flow across the intracellular series resistance results in an anterior-posterior potential difference of several tens of millivolts.

An excitability model applicable to EODs should mimic AP features of healthy electrocytes under current clamp (I_clamp) where AP thresholds are close to -45 mV during tonic firing when evoked by a constant current stimulus (40 ms) [16]. Applying constant current stimuli to the MKZ model (rather than the much larger 0.2 ms current pulses used by Lewis et al. [6] however, elicits only one high threshold AP and after a damped oscillation, the V_m settles to a depolarized plateau. Inspection of Fig 1C shows that for modeled MKZ-APs elicited by a 0.2 ms pulse, most of the depolarization (~60 mV of the total AP voltage excursion) is accomplished not by I_Na but by the large injected current. The model developed in the next sections, Epm, addresses this and several other limitations of the MKZ model.

The new excitability model (Epm) considers only the posterior excitable membrane. Demand on the Na⁺/K⁺ pump resulting from APs is inferred from net Na⁺-entry across this membrane. No specific inference is made about where and how pump electrogenicity is managed in the electrocyte. This issue, plus anterior membrane contributions to high-frequency EODs and ion homeostasis will be addressed in later work.

APs are spikes in the transmembrane potential, V (or V_m), and are solutions of the equation: (1) where C is the capacitance of the electrocyte membrane, I_NaT (transient) and I_NaP (persistent) are the currents through Nav channels, I_K, the current through Kv channels, I_L, the generic leak current and I_stim, the stimulus current initiating the APs. We only show the currents included in the Epm model. The reader is referred to [16] and [6] for other currents in the MKZ model. (2) (3) (4) where the kinetic variables j = m, h, and n obey the equation: (5) while I_L is given by: (6) and I_stim can flow either through a current clamp electrode or through post-synaptic cation channels, as presented in the next section. Values for parameters in the models MKZ and Epm appear in Table 1.

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Table 1. Kinetic constants for the MKZ and Epm models.

MKZ parameters are from Ref [6]. Listed gNa_max values are for the 200 Hz case. For Epm, the gNa_max values used for other frequencies are discussed in the Section Frequency-dependent cost per AP.

https://doi.org/10.1371/journal.pone.0196508.t001

Further details on the above currents are discussed in the following sections starting with the cation channel version of I_stim.

A post-synaptic cation conductance for Epm

EOD frequency-dependent O₂ consumption [6] pertains to the in vivo situation, with APs triggered by post-synaptic current through acetylcholine receptor channels (AChRs). In Epm, an electro-diffusion description of ion permeation through this class of channel allows for separate tallying of Na⁺ and K⁺ fluxes. (7) with: (8) where X is either Na⁺ or K⁺, the two ions that are permeant through the AChR channels, F is the Faraday constant, R the gas constant, V the transmembrane potential and z the valence of the ions. The ratio of the permeabilities P_K/P_Na is 1.11 as given in Table 1 and the absolute value of P_Na was chosen to yield an appropriate-sized stimulus for the modeled posterior membrane. In the standard pulsatile stimulation used here the maximum value of syn_clamp(t) is 1; higher values will imply correspondingly higher densities of activatable AChRs. In what follows syn_clampW or syn_clamoW_0Hz will indicate a constant “cation permeability clamp” of magnitude W (for example syn_clamp0.5 means that syn_clamp(t) = 0.5).

For Sternopygus [20] and Electrophorus [21] excitatory postsynaptic potentials (epsp) and miniature post-synaptic currents at the “electroplax” (mepsc), respectively, have been recorded. Electrophorus miniatures are slower than those at frog endplates [21], nevertheless, during predatory strikes, the electric eel can produce brief EOD bursts at ~400 Hz [22] (mechanistic details are unknown). Eigenmannia EODs demand continuous APs at up to 600 Hz [23] presenting an even more extreme challenge, assuming the electrocytes fire 1:1 with the EOD frequency [14, 19]. Even a subset of exceptionally short-lived events recorded from Torpedo electroplax (an elasmobranch, not a teleost) [24] or the fast rising endplate currents reported for lizard muscle [25] would be too slow overall. We model high frequency synaptic AChR activation using a pulsatile syn_clamp stimulus that could operate effectively up to just beyond 600 Hz. Because such short-lived stimuli are strictly conjectural, we also consider steady-state (0 Hz) syn_clamp, and mixed stimuli (i.e. pulsatile atop some background AChR activation).

gNa(V) characteristics for Epm

The voltage dependent kinetic variables α_j and β_j in Epm (and MKZ) have the form or where the pre-factors k_αj and k_βj, and the exponential factors η_αj and η_βj are given in Table 1. The one exception is the kinetic variable β_h which describes the rate of inactivation. In Epm it has the functional form used by Hodgkin and Huxley for squid axon: (9)

For MKZ and Epm, Fig 2 plots the characteristics in Hodgkin-Huxley terms [26, 27] of the voltage-gated conductances. In MKZ the m³(V) midpoint is -13 mV (Fig 2). The G/G_max(V) reported for Eigenmannia electrocytes is, however, -37 mV (Fig 3 of Markham et al., 2013 [16]); for Epm the choice of prefactor k_αm set the midpoint of m³(V) at -28mV (see Fig 2). For the rate of recovery from inactivation (forward inactivation rate) α_h (V) (Eq (3)), the exponential factor η_αh has the value used by Cannon and colleagues (Table 1 in [28]; originally from Pappone [29] for mammalian skeletal muscle).

(10)

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Fig 2. Graphical comparison of the V-gated conductances of MKZ and Epm.

The plots are generated using parameters in Table 1. Left: equilibrium (i.e., t → ∞) values of the Hodgkin-Huxley parameters. As labeled, gNa and gK are proportional to m³ and n⁴ respectively, and, h is proportional to the availability (or inactivation status) of the Na conductance. Right: time constants for these processes (see text for more explanation). For tau_h (V) curves, the voltage regions dominated by the recovery and the inactivation processes are indicated.

https://doi.org/10.1371/journal.pone.0196508.g002

The prefactor k_αh was adjusted to yield a robust window conductance (see m³h for Epm in Fig 2). Our choice of the backward inactivation rate β_h(V)(Eq (9)) ensures the fast development of the APs and a very brief refractory period. We note that while in the Hodgkin-Huxley papers, τ_h (for the squid axon) converges to 1 ms for large depolarizations, the parameters here make τ_h go asymptotically to 0.3 ms as reported by Shenkel and Sigworth [30] (see their Fig 3B up to ~+20 mV and their Fig 8C) for Electrophorus electrocytes. Epm inactivation kinetics, unlike in MKZ, conform to a standard Hodgkin-Huxley form (see τ_h plots, Fig 2). In the electric fish Electrophorus, Sternopygus, and Eigenmannia, τ_h begins to increase at large depolarizations [16, 20, 30], but because the voltages in question are beyond the physiological range, this feature was not included in the model.

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Fig 3. Sustained step and ramp I_clamp stimulation of Epm.

Standard gNa_max (700 μS) is used unless otherwise specified, I_stim as labeled, other parameters as in Table 1. See text for further explanation.

https://doi.org/10.1371/journal.pone.0196508.g003

Epm retains a simple persistent gNa(V) with the same activation kinetics as for the transient gNa(V) and no inactivation (see Eq (3). It has a lower magnitude than MKZ, i.e., γ = 0.02 in Epm versus 0.1 in MKZ (Table 1)).

K conductances for Epm

It was formerly thought that Na⁺-activated I_K (I_KNa) and persistent I_Na recorded from Eigenmannia electrocytes form an interacting pair [16], but subsequent immunocytochemical evidence located K_Na and Nav channel proteins exclusively to anterior and posterior membranes, respectively [5]. While Epm retains a (diminished) persistent gNa, it does not include the gK_Na. Moreover, its description in MKZ as a “Na⁺-activated gK” notwithstanding, this conductance operates as a classic delayed rectifier under the modelled conditions. Its modulation by s, the non-dimensional parameter sensitive to [Na⁺]_{intracellular} (see Materials and Methods section in [16]), is inconsequential since the value of s converges within ~ 1 ms to unity. Additionally, ∆[Na⁺]_{intracellular} had no impact on E_Na which is held constant in that model. Thus with or without the MKZ “Na-modulation” (asterisks (*) in Table 1), MKZ APs (e.g. Fig 1C) are identical. Accordingly, additional, experimental and computational, efforts are needed to resolve the functional contributions of gK_Na in Eigenmannia electrocytes.

Eigenmannia electrocytes have an inwardly rectifying gK [16] whose normal operating range is sufficiently hyperpolarized that even if some were present at the posterior membrane, it would neither contribute to nor interfere with excitability. Skeletal muscle t-tubular membrane [31] has abundant Kir2 protein. Kir2 mRNA transcripts have been detected in Eigenmannia electrocytes [32], but these channels have not to date been localized in the electrocyte, whereas Kir6 protein (an ATP-sensitive gIR) is expressed only on the anterior electrocyte membrane [5]. Thus, MKZ included gIR, but Epm does not.

Although a delayed rectifier current is evident in Sternopygus [20], a Na⁺-independent delayed rectifier K⁺ current is not present in Eigenmannia electrocytes [16]. However, without a gK(V), high frequency excitability fails. It has been suggested for Electrophorus that “the membrane potential is repolarized by Na⁺ channel inactivation, and from the conductance of Cl^- and K⁺ through leak channels” (p. 230 in [33]), in other words, without a delayed rectifier. This seems untenable for Eigenmannia whose electrocytes must fire from 200 Hz up to 600 Hz. With no delayed rectifier, the 50 nF membrane would need an RC time constant of ~1 ms to repolarize by ~ -100 mV (+20 mV to -80 mV). This requires a leak (g_K and/or g_Cl) of 50 μS (for reference g_Leak = 0.76 μS in MKZ) that would render the membrane inexcitable unless gNa_max was massive. For Epm, we retained the C_m = 50 nF of MKZ, thus requiring the inclusion of a delayed rectifier K⁺ conductance in Epm. For this delayed rectifier we retained the MKZ gK_max of 2000 μS. With these constants, use of gNa_max = 700 μS gives a reasonable overshooting V_peak value for APs at the low end of the EOD range (200 Hz) (e.g. use of at gNa_max = 400 μS at 200 Hz gives V_peak = ~0 mV). Accordingly, for Epm, gNa_max = 700 μS was made the “standard” for APs at 200 Hz (Table 1).

Having settled on the parameters for gNa(V) and established the need for a delayed rectifier (gK(V)) within Epm, it was necessary to design the latter to allow for APs up to 600 Hz. We assumed that gK(V), when activating, should not interfere with the depolarizing I_Na, but instead should develop rapidly just after the AP has overshot 0 mV, and thus drive repolarization. An I_K(V) that activates in this manner would be consistent with an impedance analysis of Sternopygus electrocyte APs (see Fig 28 in [19] or Fig 41 in [34]). That analysis shows that after a conductance associated with rapid depolarization falls rapidly, another conductance grows rapidly. These characteristics were achieved via the Epm gK(V) parameters listed in Table 1, which result in a gK activation curve (n⁴(V) in Fig 2) whose midpoint is -2.6 mV and whose slope is 0.017 mV^-1 (equivalent to 14.7 mV per e-fold). Interestingly, these values are very similar to those reported for a skeletal muscle gK(V) (Kv3.4, Fig 2C of [35]).

Epm responses to constant current stimulation

For Epm, a sustained constant current stimulation (I_stim) (40 ms, as per Eigenmannia I-clamp experiments: Fig 1 of [16]) yields trains of overshooting, all-or-none, strongly regenerative APs that, depending on I_stim intensity, fire at frequencies from below to above the species EOD range (200–600 Hz) (Fig 3Ai-iv). Even after I_stim ends Epm can fire a full-amplitude AP before settling to V_rest (Fig 3Aii). Note that as I_stim is increased the frequencies near 600 Hz (Fig 3Aiv) can be attained but at diminished AP amplitude. Another indicator that Epm is robustly excitable is seen in Fig 3B, where an AP fires as the membrane depolarizes past ~-70 mV during a small I_stim (Fig Bi) or even after, provided the I_stim value is extremely close to threshold (Fig Bii). For larger values of gNa_max (grey in Fig 3Bii), minor quantitative differences in AP characteristics are evident. The model electrocyte would be even more prone to fire in the presence of the current noise typical during synaptic activation. The Epm system exhibits “Class 1 excitability” (pp 219–221 of [36]); APs of arbitrarily low frequency can be triggered, depending on the strength and duration of I_stim.

In Fig 3C, firing is elicited by an I_stim ramp of 40 ms duration going to a maximum of 2400 nA, resulting in frequencies that begin near ~320 Hz and increase to ~600 Hz (note the progressively diminishing AP amplitude for both gNa_max = 700 and 1126 μS). The initial frequency depends on ramp steepness (the shallower the slope the lower the initial frequency, not shown) and the maximal frequency, which is set by channel kinetics, depends only on I_stim magnitude.

Reduced V_peak values at high frequencies (e.g., see Fig 3Aiv and 3C) point to a question addressed in the next section and raised previously [6], namely, is there a need for higher Nav channel density to sustain higher frequency APs? For Eigenmannia individuals this would correspond to a positive correlation between EOD frequency and Nav channel density.

Epm responses to pulsatile synaptic current stimulation

We assume that each electrocyte AP is triggered by a brief pulsatile AChR-mediated post-synaptic current intense enough to yield full-amplitude APs at frequencies up to 600 Hz (see Fig 4). The shape of this standard stimulus (syn_clamp1) is based on synaptic currents in Torpedo [24] and lizard [25] see Fig 4 legend and Table 1.

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Fig 4.

A. The standard pulsatile stimulus shape syn_clamp (t) used in Epm has 3 segments: a linear rise for 0.05 ms, a plateau for 0.200 ms and an exponential decay to zero (specifically the segments are: syn_clamp1.0 (t) = t/0.05 (t<0.05); 1 (0.05<t<0.25); and exp[-(t-0.25)/0.1] (t>0.25), t in ms)). Bi. For comparison, a depiction of the fastest cholinergic post-synaptic event we found in the literature for electroplaques, shown at the same time scale as the standard pulsatile stimulus in A. The miniature electroplaque current (mepc) depicted exemplifies a fast subclass of of mepc recorded in relatively intact Torpedo electroplaques [24]. Bii depicts the average time course of mepcs measured from lizard intercostal muscle [25]. The pulsatile stimulus used in Epm is speedier in all respects than these mepcs. Whereas a mepc results from a quantal release event that would be insufficient to stimulate an entire electroplaque, the syn_clamp stimuli of Epm must provide sufficient “ACh-activated” I_cation at the Eigenmannia electroplaque to rapidly depolarize to threshold 50 nF of post-synaptic membrane. Thus, 1:1 triggering (1 electromotor neuron AP: 1 electrocyte AP) is assumed, but how it would be achieved at up to 600 Hz in vivo is not understood.

https://doi.org/10.1371/journal.pone.0196508.g004

Fig 5A plots the membrane potential and underlying variables for a single Epm AP in a 200 Hz train. During the stimulus (top) net Na⁺ influx (I_Na) is substantial while I_K is minimal (E_K is near V_rest). Because the stimulus ends before V_m(t) reaches the V_rev for AChRs (+2.2mV) both I_K and I_Na components of the syn_clamp remain unidirectional. As in I-clamped Eigenmannia electrocytes [16] the AP shows an apparent threshold region near -45 mV (Fig 5A). The AP peaks at ~13 mV (for computations V_peak = +12.86 mV), a value used in connection with syn_clamp1 as the “standard” AP-amplitude. Early I_Na(V,t) is unopposed but the rapid-onset of I_K(V,t) then drives AP repolarization in co-operation with gNa inactivation. Recovery from inactivation is almost complete ~1 ms after V_peak. As a result Epm APs at 200 Hz are indistinguishable from APs at lower frequencies.

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Fig 5. Epm-simulated APs stimulated by pulsatile synaptic current.

A. One AP from a 200 Hz AP train (stimulus: syn_clamp1.0_200Hz) with gNa_max = 700 μS. B. For gNa_max values as labeled, APs from trains with stimulus syn_clamp1.0_200Hz or syn_clamp1.0_600Hz. First column, gNa_max = 700 μS; second column, for syn_clamp1.0_600Hz, gNa_max is raised to 1126 μS to bring V_peak to 13 mV. See text for further explanation.

https://doi.org/10.1371/journal.pone.0196508.g005

To maintain a constant V_peak at higher frequencies, gNa_max values must increase (as with MKZ; [6]). Fig 5B compares model variables at 200 Hz and 600 Hz with gNa_max kept at 700 μS for both frequencies (left) or with gNa_max increased to 1126 μS for the 600 Hz APs (right). These values plus those needed for intervening frequencies are also indicated as plot labels in Fig 6B. The bottom panels of Fig 5B show that Na⁺ entry at 200 Hz vs 600 Hz is unaffected if the AP amplitude (top panels) is allowed to fall (left) but increases markedly if gNa_max is adjusted to attain the standard V_peak (top right). The middle panels reveal that the voltage-gated channels’ activation kinetics are not limiting here; the rise and fall of parameters m³ and n⁴ (which reflect open-gNa and open-gK) is rapid enough at both 200 and 600 Hz (middle panels).

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Fig 6. Cost of Epm APs at different AP frequencies based on Na⁺-entry.

A. Illustrated are APs firing at 200 Hz and 600 Hz and then failing at 700 Hz which, however, exceeds the biological range of Eigenmannia (700 Hz stimulation elicits APs with irregular amplitudes and timing). For all frequencies the pulsatile stimulus amplitude was syn_clamp1.0 and gNa_max was adjusted to yield V_peak = 13 mV. To calculate Na⁺-entry (= the time integral of the three sources of I_Na seen in Fig 5A) trains of at least 20 APs were used. B. Na⁺ entry plotted to assess the cost/AP at different frequencies, as labeled and as explained in the text. Starred region beyond 600 Hz signifies that although this is beyond the species range, Epm is still able to produce regular APs at 650 Hz (though not, as seen in B, at 700 Hz). Calculations were done by requiring that V_peak = 12.86 mV for each frequency but this is referred to throughout the paper as 13 mV. Larger fonts for gNa_max values at 200 Hz and 600 Hz emphasize that these represent the extremes of the biological range for Eigenmannia EODs.

https://doi.org/10.1371/journal.pone.0196508.g006

Because gK_max was held constant across frequencies while gNa_max was increased, the AP depolarization rate and hence the speed of gNa activation and inactivation onset increases with frequency. The limiting factor for maintaining full amplitude APs is recovery from inactivation. The middle panels of Fig 5B show that h (sodium channel availability or inactivation status) at 600 Hz has insufficient time to return to unity after each AP; at 600 Hz h barely reaches 0.8 before the next AP. Failure to fully recover from inactivation underlies the requirement for increased gNa_max at 600 Hz. Thus, while overall AP shapes differ only subtly between 200 and 600 Hz, the faster rise to threshold (and then to peak) at 600 Hz is not inconsequential. It is possible only because of the augmented Na⁺-entry (Fig 5B bottom panel, right).

Frequency-dependent cost per AP

Thus, in the Epm model, the increased gNa_max needed to sustain APs at full amplitude with increasing frequency predicts that the cost/AP (Hz) will rise. Fig 6A illustrates a few APs from trains of Epm APs elicited by pulsatile syn_clamp at 200 Hz, 600 Hz, and the unphysiologically rapid 700 Hz. Such APs are used to calculate Na⁺-entry (trains of at least 20 APs were generated), which is a proxy for energetic cost/AP. Given sufficient gNa_max, AP trains are attainable up to ~650 Hz in Epm, whereafter irregularities appear (e.g., see 700 Hz in Fig 6A).

The upper panel of Fig 6B combines Na⁺ entry through V-gated (Nav) and synaptic (AChR) channels. The cost/AP increases nonlinearly over this range of frequencies. However, separating the synaptic cost/AP(Hz) reveals that the cost of synaptically stimulating APs is essentially constant. Hence, the non-linearity in cost/Hz can be ascribed to Nav currents. Since our initial choice of gNa_max = 700 μS at 200 Hz may seem arbitrary, we also ran calculations with gNa_max = 900 μS. Briefly, this increases V_peak to ~18 mV, which means larger values of Na⁺ entry for the gNa, but nearly unchanged synaptic costs.

Is all depolarizing Na⁺-entry equivalent?

If synaptic stimuli with amplitudes different from syn_clamp1 are used, does this picture change? Fig 7 examines this question, with Fig 7A showing V_m(t), plus the key attendant currents for a single AP triggered by pulsatile (200 Hz) syn_clamp1 or syn_clamp5 (a value 5 times larger than syn_clamp1). As expected, the AP rise time is faster with the stronger synaptic input, but the I_Na and I_K profiles are similar for both inputs. Fig 7B provides a comparison over this range of stimulus amplitudes. The black dashed-line shows that syn_clamp1 triggers APs of V_peak = 13 mV and that V_peak changes little with larger stimuli. And strikingly, the larger stimuli hardly alter the total Na⁺-entry cost. Even though the increased stimulus (“more clamped-open AChRs”) is allowing more Na⁺-entry by this route (see the strong rise of the red line), total Na⁺-entry (green line) into the electrocyte increases only slightly. With much more of the depolarization accomplished early due to Na⁺ influx through AChR channels, the need for influx through Nav channels once they activate is diminished.

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Fig 7. Effect of pulsatile stimulus amplitude.

A. Single APs elicited by the standard pulsatile stimulus (V_m(t) units are arbitrary, with the syn_clamp1_200Hz AP as reference) calculated for syn_clamp1_200Hz or syn_clamp5_200Hz, and V-gated currents (INa_T, where T signifies the transient component; persistent INa not shown), IK (delayed rectifier) plus the INa component of current through the AChR channels; gNa_max = 700 μS, stimuli at 200 Hz. B. For such APs elicited by pulsatile synaptic stimulation at increasing intensities (from syn_clamp0.5_200Hz to syn_clamp5.0_200Hz), Na⁺-entry via the stimulus (AChR) channels increases dramatically (red) but total Na⁺-entry (green) is almost unaffected. The V_peak is relatively insensitive to stimulus intensity beyond syn_clamp = 1 (an “all-or-none” AP feature). C. Further testing of pulsatile syn_clamp intensity, for APs at 200 Hz and 600 Hz with the gNa_max values indicated. The dashed black line in C is equivalent to the dashed black line in B (where values are normalized).

https://doi.org/10.1371/journal.pone.0196508.g007

Would this outcome signify that a fish could be energetically “indifferent” to which class of channels carry the major load for depolarization? We suggest not. A cost analysis based solely on redressing Na⁺-entry into electrocytes ignores metabolic costs incurred pre-synaptically. Costs related to producing and packaging AChR molecules is not captured by the Na⁺-entry tally. Biologically, therefore, if EOs are driven by pulsatile synaptic stimuli, operating near our reference state (i.e. coordinates 1,1 of Fig 7B) would be optimal, ensuring signal integrity while avoiding high presynaptic costs.

Whereas Fig 7B deals only with APs at 200 Hz, Fig 7C looks also at 600 Hz. Both Fig 7B and 7C have dashed black lines for the 200Hz V_peak, but note that the Y-axis in Fig 7C is the absolute value of V_peak. Note too that for X-axis position “syn_clamp1” in Fig 7C, for each of the 3 plot lines, the relevant APs would be those from Fig 5B (top panels).

The question here: to produce a full sized V_peak, could larger pulsatile stimuli (up to 10X syn_clamp1) compensate for an otherwise too-low gNa_max? The consequences of larger stimuli are qualitatively the same across the frequency range, as shown in Fig 7C which varies syn_clamp for 200 Hz and 600 Hz, with appropriate gNa_max values (see mauve and black dashed lines; the blue plot simply demonstrates that gNa_max appropriate to 200 Hz makes full amplitude APs at 600 Hz impossible for any stimulus intensity). Near syn_clamp2, AP peaks are slightly more depolarized than for syn_clamp1 and all-or-none behavior is particularly robust, with stimulus intensity variations of ~±0.3 in yield essentially the same AP amplitude. Rightward in the plots, V_peak values fall; larger syn_clamp becomes counterproductive.

Would the somewhat more robust all-or-none behavior near syn_clamp2 make it preferable to syn_clamp1? Not necessarily, since synaptic transmitter costs would nearly double. Taken together, Epm predicts (based on the three curves of Fig 7C) that across Eigenmannias’s 200–600 Hz EOD range, adequately large APs would be attained via appropriately-sized gNa_max, not by augmenting stimulatory AChR currents. Similarly, variations in gK_max cannot maintain V_peak across these frequencies. A Nav channel density adequate for the low end of the Eigenmannia EOD frequency range could not be compensated for in high frequency fishes by adjusting gK_max (data not shown). This leads to the broad prediction that AChR density would be similar for fish operating at low and high frequencies, while Nav channel density would increase for higher frequency fish. With <2-fold difference in gNa_max predicted for 200 Hz vs 600 Hz fish, however, experimentally quantifying such moderate density differences may be challenging.

Simulating electrocyte action potentials during rapid changes in EOD frequency

In Eigenmannia and related electric fish, the close proximity of two fish with similar EOD frequencies produces beat frequencies that are detrimental to electrolocation [37–40]. Fish respond with a jamming avoidance response to increase beat frequency (i.e. increase their frequency differences), wherein the fish with the higher frequency increases its EOD frequency by ~5–10 Hz while the fish with the lower EOD frequency decreases its frequency by a similar magnitude. When fish increase EOD frequency during the JAR, how is the amplitude of the electrocyte AP maintained? To achieve higher AP frequencies for a JAR, Eigenmannia individuals could quickly augment the Nav channel density in their electrocytes’ posterior membranes as proposed in earlier work [6]. Based on the Epm however, this costly procedure might be unnecessary. Within this model, extant excitability machinery could produce a typical JAR (i.e. 10 Hz above baseline frequency) with minimal loss of AP integrity. Fig 8 shows that in a fish with baseline EOD 400 Hz (with gNa_max = 783 μS), V_peak would change by only ~-0.25 mV during a +10Hz JAR., at 410 Hz, a ΔV_peak of ~-0.25 mV. This slightly diminished V_peak it should be noted, would be available with no time lag and would reverse immediately when the need for the JAR ended. The difference would be “within the noise” and thus would not be expected to jeopardize EOD signalling integrity.

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Fig 8. Simulation of a JAR.

For APs at frequencies ± 10 Hz near 400 Hz, with gNa_max at 783 μS, V_peak differs minimally. This suggests that typical ± 10 Hz JARs in Eigenmannia might require no increased expression of Nav channels.

https://doi.org/10.1371/journal.pone.0196508.g008

If JARs occurred frequently, and if signal integrity needed to be even better than the ± 0.25 mV of Fig 8, then the issue of JAR dynamic range could be an important determinant of a fish’s “choice” of (central nervous system-determined) stimulus-frequency, given its electrocyte gNa_max. If a 400 Hz fish operated with a gNa_max somewhat greater than 783 μS, then its JAR plot (not shown) would be shallower than in Fig 8; likewise across the species frequency range.

Synaptic stimulation with a non-zero background

On timescales of many hours, Eigenmannia individuals rigorously maintain their particular EOD frequency. Thus far (Figs 5–7), the background [ACh] in the synaptic cleft (“[ACh]_synaptic”) is assumed to decay rapidly toward zero after each invariant pulsatile stimulus. But instead, could maintenance of [ACh]_synaptic be responsible for setting an invariant frequency? Here we briefly consider this other extreme, plus the more likely situation in which reliably large pulsatile ∆[ACh]_synaptic stimuli ride atop a non-zero background [ACh]_synaptic.

Fig 9Ai shows Epm APs for various constant or steady-state levels of AChR activation (i.e., 0 Hz stimulation) for gNa_max = 700 μS. Fig 9Aii shows that a threshold for AP generation occurs at ~syn_clamp0.0092_0Hz (initial point in the plot). As syn_clamp0Hz increases above this threshold, the AP frequency rises from 0 Hz until at syn_clamp0.05_0Hz APs (with V_peak = 9.0 mV) fire at ~200 Hz, the bottom of the species’ EOD range. With increasing syn_clamp0Hz intensity (as with increasing I_clamp intensity, see Figs 5 and 6) the AP frequency increases but V_peak falls (see syn_clamp0.40_0Hz in Fig 8Ai) unless, as shown in Fig 9Aiii, gNa_max is made larger. This shows that, in principle, APs at frequencies required to drive EODs up to 600 Hz could be elicited using a constant-amplitude AChR conductance. In practice, inherent sources of noisy Δ[ACh]_synaptic (e.g. as in Torpedo electroplax recordings by Giirod et al [24]) would make the EOD frequency unacceptably variable.

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Fig 9. Background synaptic stimulation.

Ai. Epm APs for three steady-state (i.e. 0 Hz) levels of AChR activation, syn_clamp0.03_0Hz, syn_clamp0.05_0Hz, and syn_clamp0.4_0Hz fire at 137 Hz, 202 Hz, and 541 Hz respectively (gNa_max = 700 μS throughout). Aii. Epm AP frequency as a function of stimulus intensity (conditions as in Ai); squares color-coded for the plots in Ai. Aiii. Red trace at 541 Hz firing (conditions as in Ai) with a second overlaid trace showing APs when gNa_max is increased to 1155 μS (this produces APs of V_peak = 9 mV, i.e., the same as for syn_clamp0.03_0Hz with gNa_max = 700 μS). Bi. APs elicited by a 200 Hz pulsatile stimulus in the presence of a subthreshold background stimulus (syn_clamp0.00736_0Hz). For syn_clamp0.1_200Hz (orange) the output AP frequency is 50 Hz whereas for syn_clamp0.4_200Hz (blue) the target frequency of 200 Hz is attained. Bii. This background stimulus (syn_clamp0.00736_0Hz) plus 200 Hz pulsatile stimuli at the amplitudes indicated elicits APs at different frequencies as shown here in a “devil’s staircase” plot.

https://doi.org/10.1371/journal.pone.0196508.g009

In Fig 9B APs are stimulated by an imposed Δ[ACh]_synaptic that combines pulsatile and background syn_clamp (the 0 Hz level was set at 80% of threshold, i.e., syn_clamp0.0074_0Hz). Fig 9Bi shows V_m(t) when either syn_clamp0.1_200Hz (orange) or syn_clamp0.4_200Hz (blue) are added atop this 0 Hz background. The former (orange) is insufficient to achieve the target AP frequency (200 Hz) whereas the latter (blue) is sufficient. Over 4 cycles, the syn_clamp0.1_200Hz stimuli summate to trigger 50 Hz APs. When the membrane depolarizes due to Na⁺ entry from pulsed cholinergic stimuli, both transient and persistent gNa changes contribute to maintaining the voltage gain between pulses because in that voltage range inactivation is inconsequential (h≈1). With a fixed subthreshold background (syn_clamp0.0074_0Hz) a plot of AP frequency versus 200 Hz stimuli of increasing amplitude generates a “devil’s staircase” plot [41] Fig 9Bii). Below the critical pulsatile stimulus amplitude required for any given AP frequency, it would be possible, given a noise-free system, to discern period doubling, tripling etc. In the case illustrated here (i.e., 200 Hz APs) syn_clamp0.32_200Hz is the relevant critical stimulus, and because the modeling is noise-free, period doubling and tripling is evident (plateaus are evident at 100 Hz and 66.6 Hz, i.e. one-half and one-third of the stimulation frequency, 200Hz) in Fig 9Bii. For real systems, given the temporal stochasticity of ACh release and the inherent variation in quantal size at the time of release, variations in both pulsatile and background stimulus amplitudes would be expected. This would increase the critical pulsatile stimulus amplitude required for reliable firing at the target frequency; that amplitude would more likely be reached further along the top of the staircase (e.g. at one of the grey arrowheads) than in a noise-free simulation.

Na⁺ entry for different stimulus regimes

We further wondered how different combinations of background and pulsatile synaptic stimulation would affect Na⁺ entry at frequencies spanning 200 to 600 Hz, the species EOD range (we consider APs of a fixed value, V_peak = 13 mV; gNa_max was adjusted accordingly). Lewis et al. [6] determined EOD-linked O₂ consumptions at the whole animal level, not at the level of electrocytes. Here, we stress that from the perspective of whole animal energetics, a Na⁺ ion entering an electrocyte through a Nav channel (gNa) versus a Na⁺ ion entering an electrocyte through an AChR almost certainly presents a different (whole animal level) cost. Given an appropriate Na⁺ gradient, a Nav channel simply has to open in response to depolarization to allow Na⁺ to enter, but for Na⁺ entry via an AChR channel, ACh must first be manufactured and packaged and delivered (at a cost) then released via presynaptic depolarization (at a cost). Accordingly, both total and pathway-specific Na⁺ entry into the electrocyte was tallied. Table 2 lists outcomes for APs firing at 200 Hz, 500 Hz and 600 Hz. The frequency 500 Hz was included to show how Na⁺ entry/AP evolves as the system approaches its upper limit.

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Table 2. Na⁺ entry for various synaptic stimulus regimes^∆.

https://doi.org/10.1371/journal.pone.0196508.t002

The stimulus extremes, i.e. purely pulsatile and purely steady-state (0 Hz) stimulation are represented by rows 1, 7, 13 and rows 6, 12, 18 respectively. The former reiterates Fig 6B, i.e., synaptic components essentially constant across frequencies while Na⁺ entry/AP through Nav channels increases nonlinearly with frequency. For the latter (rows 6, 12, 18), total Na⁺ entry for 0 Hz syn_clamp is considerably higher than for pure pulsatile, most markedly so at the high frequency end. Taken together with the argument that 0 Hz syn_clamp in vivo would not on its own generate reliable clock-like EODs, it is difficult to conceive of any benefit of a pure 0 Hz regime.

For AP target frequencies 200 Hz, 500 Hz, 600 Hz, the intermediate rows explore situations with some level of [ACh]_synaptic persisting between pulsatile stimuli, i.e., combinations of syn_clamp0Hz >0 plus a pulsatile syn_clamp at some level. For each target frequency, two values of syn_clamp0Hz were tested, one subthreshold, the other strongly suprathreshold. Subthreshold was represented by 80% of the 0 Hz threshold (rows 2, 3, 8, 9, and 14, 15), and suprathreshold was represented by the 0 Hz syn_clamp amplitude that elicits firing at ~70% of the target frequency (rows 4, 5, 10, 11, and 15, 16). In Table 2, footnotes list how, as target frequency increases, the syn_clamp0Hz threshold decreases (this occurs because the gNa_max imposed to maintain V_peak = 13 mV yields a more excitable system).

To provide a basis of comparison in the intermediate (mixed stimuli) rows, each of the two chosen 0 Hz syn_clamp amplitudes is first tested in conjunction with the “control” pulsatile amplitude (first column of rows 1, 7, 13) needed for 13 mV AP trains at the target frequency (i.e. rows 2, 8, 14), albeit with gNa_max slightly reduced (to maintain V_peak = 13 mV). Next (rows 3, 9, 15), with the same subthreshold background, a pulsatile amplitude was found that slightly exceeds the minimum needed to achieve the target AP frequency. For 200 Hz for example, this is slightly rightward of the left margin of the “devil staircase”plateau (Fig 9Bii; syn_clamp0.34_200Hz was used here, row 3). For the “0 Hz suprathreshold-to-70%-target” cases, the Na⁺-entry results are in rows 4 and 5 for 200 Hz, 10 and 11 for 500 Hz, and 16 and 17 for 600 Hz. Thus, across all frequencies, total Na⁺-entry is minimal if there is no steady-state component to the stimulus (rows 1, 7, 13) but if this is not achievable, then the most cost-effective approach appears to be maintaining background synaptic stimulation as low as possible.

The question of which combination of depolarizing Na⁺ entry is most economical will depend on how much more expensive synaptic Na⁺ entry is compared to Nav Na⁺ entry. For example, if Na⁺-entry through AChRs incurred 10X the whole-body metabolic cost of Na⁺ entry through Nav channels, then regimes that reduced entry through AChRs (e.g., rows 3, 5, 6 for 200 Hz) would be beneficial. There, the reduced pulsatile component yields a marked saving in pulsatile ACh, though at higher frequency (rows 9, 15) the relative benefit diminishes. As Table 2 shows, the pulsatile syn_clamp component required to override the natural firing frequency of the purely fixed amplitude stimulation is larger than for the subthreshold background, and increases with frequency. Nevertheless, at 500 Hz, if the AChR entry pathway was 10X more expensive (as per our suggestion here) then, for rows 10 and 11, row 11 might be less expensive at the whole-animal level, even though row 10 shows the smaller total Na⁺-entry. In other words, if clearing the synapse of ACh between pulses was not possible, then minimizing the pulsatile amplitude would be beneficial.

Overall, the results of Table 2 provide some important insights. Between stimulatory pulses (especially at higher frequencies), if electrocyte synapses could not fully remove ACh, an energy efficient system would need to optimize the ratio of background-to-pulsatile amplitude to minimize Na⁺ entry through AChRs. There is no penalty for background ACh provided it is subthreshold. In fact, at low frequency, there is even a saving from a subthreshold background provided the pulsatile component is reduced. Even with a suprathreshold background, the system can fire at reasonable cost, but with costs rising more steeply with frequency than in the case of a purely pulsatile stimulus or with a combined pulsatile and subthreshold background. Even with the caveat that the entry pathway specific differential is as yet unknown, Table 2 suggests that a moderate background ACh is not detrimental, but that fish generating EODs at the higher end of the frequency range would have relatively little latitude for lowering synaptic costs by increasing the background.

Calculated electrocyte costs versus whole-animal measurements

Lewis et al. [6] provided whole animal O₂ consumption data for nine fish generating EODs at frequencies > 300 Hz and < 500 Hz. The data were fit to an exponential. With Epm, we calculated electrocyte Na⁺ entry at frequencies from 200 Hz to 600 Hz. At 200 Hz and below, all Epm parameters return to their pre-stimulus values prior to the next stimulus (in other words, from 0 to 200 Hz, a plot for ATP/AP(Hz) is a flat line. Above that, the values rise non-linearly, though not exponentially. We nevertheless fit an exponential to our Epm-calculated points (for entry through both Nav and AChR channels) at 300, 400 and 500 Hz, and that exponential predicts a 1.17-fold increase between 300 and 500 Hz whereas the Lewis et al. [6] fit over that same range gives a 6.05-fold increase. With Na+ entry/AP through the AChR channels essentially constant across all frequencies (Fig 6), the small but distinct increase with frequency is attributable to influx into the electrocyte through the increasing numbers of Nav channels. However, while Nav channel activation depends only on the voltage of membranes in which the channels are embedded, AChR channel activation depends on a plethora of “upstream” processes whose costs would contribute to whole-animal O2 consumption, but not to the cost of electrocyte ion homeostasis. Therefore, the very large frequency dependent discrepancy (slopes of 6.05 vs 1.17) indicates that the very high cost of EODs at the high end of Eigenmannia’s frequency range cannot reasonably be attributed to electrocyte excitability. Instead it suggests that neural events preceding the electrocyte-based APs, along with costs of circulatory processes that help ensure that electrocytes maintain ion homeostasis are substantially responsible. Perhaps this helps explain why the very high frequency Apteronotid species have a 'neurogenic' electric organ in which neurally-derived electrocytes receive direct electrotonic inputs without a chemical synapse[42].