A model for studying the energetics of sustained high frequency firing

Regulating membrane potential and synaptic function contributes significantly to the energetic costs of brain signaling, but the relative costs of action potentials (APs) and synaptic transmission during high-frequency firing are unknown. The continuous high-frequency (200-600Hz) electric organ discharge (EOD) of Eigenmannia, a weakly electric fish, underlies its electrosensing and communication. EODs reflect APs fired by the muscle-derived electrocytes of the electric organ (EO). Cholinergic synapses at the excitable posterior membranes of the elongated electrocytes control AP frequency. Based on whole-fish O2 consumption, ATP demand per EOD-linked AP increases exponentially with AP frequency. Continual EOD-AP generation implies first, that ion homeostatic processes reliably counteract any dissipation of posterior membrane ENa and EK and second that high frequency synaptic activation is reliably supported. Both of these processes require energy. To facilitate an exploration of the expected energy demands of each, we modify a previous excitability model and include synaptic currents able to drive APs at frequencies as high as 600 Hz. Synaptic stimuli are modeled as pulsatile cation conductance changes, with or without a small (sustained) background conductance. Over the full species range of EOD frequencies (200–600 Hz) we calculate frequency-dependent “Na+-entry budgets” for an electrocyte AP as a surrogate for required 3Na+/2K+-ATPase activity. We find that the cost per AP of maintaining constant-amplitude APs increases nonlinearly with frequency, whereas the cost per AP for synaptic input current is essentially constant. This predicts that Na+ channel density should correlate positively with EOD frequency, whereas AChR density should be the same across fish. Importantly, calculated costs (inferred from Na+-entry through Nav and ACh channels) for electrocyte APs as frequencies rise are much less than expected from published whole-fish EOD-linked O2 consumption. For APs at increasingly high frequencies, we suggest that EOD-related costs external to electrocytes (including packaging of synaptic transmitter) substantially exceed the direct cost of electrocyte ion homeostasis.


Introduction
Analysis of mammalian brain energetics identifies electrical signaling as the major consumer of ATP with molecular processes underlying synaptic transmission incurring the highest costs [1][2][3]. For brain, the costs of low frequency electrical signalling arising from various aspects of ion homeostasis have been analyzed and summarized (see Table 2A in Ref. [3]). Changes in ion concentrations due to synaptic inputs and action potentials (APs) are reversed by the action of energy-consuming ion pumps. This direct link allows energy consumption to be estimated from Na + -entry [1,4] which drives the 3Na + /2K + -ATPase activity (at 1 ATP/3 Na + expelled). Under the low-firing frequency conditions (4 Hz) typical of mammalian cortex, Na + (and Ca +2 ) entry through postsynaptic receptor channels is estimated to be much larger than that through AP-related voltage-dependent Na+ (Nav) channels. Synaptic processes are thus thought to dominate energetic costs in these conditions, but whether this is the case for other firing regimes is not known.
The weakly electric fish, Eigenmannia, generates a high-frequency electric organ discharge (EOD) for electric sensing (Fig 1A). The electric organ (EO), an array of electrocytes derived from skeletal muscle (but with no contractile machinery), is a relatively simple and homogeneous excitable cell tissue. Electrocytes are posteriorly innervated syncytial cells, each~1 mm long, with capillary beds at either end. EOD frequency is set centrally and conveyed to electrocytes via cholinergic nerve terminals at their posterior membranes. Regular and unremitting synaptic triggering of action potentials (APs) in synchronously firing electrocytes produces an externally detectable EOD at an individual-specific frequency within in the species range (200-600 Hz). For brain energetics in general, functional magnetic resonance imaging reports on localized regions of elevated neural activity via local/temporal variations in O 2 -rich blood flow [7,8]. However, tissue complexity and low spatial resolution impedes the study of discrete excitable cell activities, making interpretation of images problematic [9]. Eigenmannia is special in that whole-animal O 2 consumption (and thence ATP usage) can be linked to a uniform unremitting excitability process, the EOD [9]. Like skeletal muscle fibers (from which electrocytes derive), the EOD producing electrocytes have cholinergic synapses and capillaries that ramify extensively with fine invaginations of the plasma membrane [5,10,11]. Being non-contractile, electrocytes are physiologically simpler than skeletal muscle. It seems worth exploiting this simplicity to work out how cyto-and tissue architecture interact with the molecular machinery of excitability to so reliably yield the EOD. Although the spatiotemporal variations in electric potential surrounding the intact discharging fish has been well-described [12,13], it is not established where and for which specific processes ATP is used to maintain this oscillating discharge. Models are required to describe the underlying cellular and tissue level processes and then predict the known whole animal EOD pattern.
Monitoring EODs in vivo allows for precise quantitation of various aspects of Eigenmannia's high-frequency excitability [12,14]. Exploiting this, Lewis et al [6] found that the inferred ATP consumption per EOD-linked AP is not constant across all frequencies as generally assumed [1,15], but instead increases exponentially with EOD frequency (see Fig 1B). This indicates that the energetics of high frequency excitability need more attention. Given the potential utility of Eigenmannia as a tool, a sound mathematical model for its EOD-producing machinery is required. Building on a previously-proposed model of a single electrocyte [16], we propose a new model, "Epm", that includes synaptic input and is more robustly excitable. Over the range 200-600 Hz, Epm predicts that ATP consumption per AP is not constant. Using Epm to explore the source of Na + -entry at various frequencies leads to the prediction that Nav channel density but not AChR channel density should increase as a function of AP

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 .  [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: 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.
I NaP ¼ gNa max m 3 gðV À E Na Þ; ð3Þ where the kinetic variables j = m, h, and n obey the equation: while I L is given by: 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.
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 O 2 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. with: 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 clamp W or syn clamo W 0Hz will indicate a constant "cation permeability clamp" of magnitude W (for example syn clamp 0.5 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 a j ¼ k aj e Z aj V or b j ¼ k bj e Z bj V 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: For MKZ and Epm, Fig 2 plots the characteristics in Hodgkin-Huxley terms [26,27] of the voltage-gated conductances. In MKZ the m 3 (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 m 3 (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  Table 1. Left: equilibrium (i.e., t ! 1) values of the Hodgkin-Huxley parameters. As labeled, gNa and gK are proportional to m 3 and n 4 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 Pappone [29] for mammalian skeletal muscle).
The prefactor k αh was adjusted to yield a robust window conductance (see m 3 h 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.
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, Δ[Na + ] intracellular had no impact on E Na which is held constant in that model. Thus with or without the MKZ "Na-modulation" (asterisks ( Ã ) 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 Cland 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  Table 1. See text for further explanation.
https://doi.org/10.1371/journal.pone.0196508.g003 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 (n 4 (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 clamp 1) is based on synaptic currents in Torpedo [24] and lizard [25] see  Table 1. 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 clamp 1 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.
To maintain a constant V peak at higher frequencies, gNa max values must increase (as with MKZ; [6]). 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.   [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 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. The black dashed-line shows that syn clamp 1 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.
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 The question here: to produce a full sized V peak , could larger pulsatile stimuli (up to 10X syn clamp 1) 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 clamp 2, AP peaks are slightly more depolarized than for syn clamp 1 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 clamp 2 make it preferable to syn clamp 1? 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 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 clamp 1.0 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 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][38][39][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. 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.
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 Fig 9Aii shows that a threshold for AP generation occurs at~syn clamp 0.0092 0Hz (initial point in the plot). As syn clamp0Hz increases above this threshold, the AP frequency rises from 0 Hz until at syn clamp 0.05 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 clamp 0.40 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.
In The former (orange) is insufficient to achieve the target AP frequency (200 Hz) whereas the latter (blue) is sufficient. Over 4 cycles, the syn clamp 0.1 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 clamp 0.0074 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 clamp 0.32 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 O 2 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.
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 fre- quency (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 clamp 0.34 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 + - Syn clamp amplitude is non-dimensional (as explained in the section entitled "Epm responses to pulsatile synaptic current stimulation"), gNa max is in μS and Na + entry/AP is as defined for Fig 6B. As explained in the text, the 0 Hz background (or subthreshold level) for any set of conditions was chosen to be 80% of threshold for those conditions. Using 75% and 85% of threshold instead yields AP peak amplitudes identical to those for 80% to within 0.01 mV and Na + entry values identical to within 0.2%. Δ for each stimulus regime gNa max is adjusted until V peak = 13 mV (hence the threshold values given below vary with frequency) Ã syn clamp 0.0092 0Hz brings this system to firing threshold syn clamp 0.048 0Hz causes this system to fire at 200 Hz ÃÃ syn clamp 0.0080 0Hz brings this system to firing threshold syn clamp 0.28 0Hz causes this system to fire at 500 Hz ÃÃÃ syn clamp 0.0067 0Hz brings this system to firing threshold 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 O 2 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].

Discussion
Electrocyte action potentials (APs) underlie the wave-type electric organ discharge (EOD) of Eigenmannia. This continuous high frequency bioelectrical phenomenon is sustained over the lifespan of the fish. An explanation for how Eigenmannia unerringly maintains its EOD requires a better understanding of how the electrocytes produce APs at 200-600 Hz, of the metabolic demands this incurs, and of how subcellular, cellular and tissue geometry along with the associated molecular components are organized to meet these demands [6,42]. In addition, this knowledge should yield insights for energetic strategies in other highly active excitable tissues including brain and skeletal muscle [17]. Assessing the contribution of electrocyte APs to the whole-animal metabolic cost measured by Lewis et al. [6] requires a thorough accounting of the ionic and biophysical mechanisms of electrocyte APs and post-synaptic currents. Electrophysiology has revealed some of the ionic conductances contributing to electrocyte firing [6,16], and cyto-histochemistry is showing the locations of ion channels and pumps in the large and highly polarized electrocytes [5]. Here, a computational model for the excitable innervated posterior region of electrocyte membrane is updated to better reflect functional characteristics of electrocytes and to better depict molecular mechanisms of electrocyte excitability.
In the new model, Epm, addresses some limitations inherent in a previous computational model for Eigenmannia electrocyte APs. As done previously [16], however, we assumed here that across all EOD frequencies, APs attain the same peak depolarization (V peak ) and voltagegated channel kinetics are constant. Again, as previously, the model includes a delayed rectifier conductance (see Methods) even though one has not yet been found in Eigenmannia electrocytes. For Epm, we arbitrarily retained the same (frequency-invariant) gK max as previously, then tuned the Hodgkin-Huxley type kinetic parameters for gK(V) and gNa(V) to ensure that sustained APs were possible at 600 Hz, as per the Methods. A major change in the present model of the electrocyte's innervated posterior membrane is the inclusion of a biophysically appropriate post-synaptic cation conductance. Note that until these specific assumptions are tested, these models are predictive and as a "proof of principle".
Importantly, adjusting only one parameter in Epm enables the production of APs with consistent amplitude at frequencies up to 600 Hz, the upper end of the species' EOD frequency range. This parameter is gNa max and corresponds to Nav channel density. Epm exhibits more robust excitability than previous models; it produces all-or-none APs that start from a lower firing threshold, and it fires repetitively with moderate levels of constant current stimulation. It predicts, moreover, contrary to previous models, that electrocytes could achieve a transient jamming avoidance response (this entails a change in discharge frequency of approximately 10 Hz) with no requirement to alter channel properties or channel densities.
The updated model leads to two key predictions and questions for further experimental and computational work. Firstly, a strong prediction from Epm (gleaned from analysis of the distinct Na + -entry budgets for Nav and AChR channels and from exploring the costs of varying the densities of both types of channels) is that electrocyte Nav channel density should increase non-linearly with increasing EOD frequency, whereas AChR channel density is expected to be constant with frequency. Secondly, the analysis here indicates that the electrocyte AP itself is a relatively energy-efficient process and is not the major contributor to the whole-animal metabolic cost of EOD production. Instead, the Epm analysis predicts that a much greater proportion of the whole-animal costs associated with EOD production would arise from events "upstream" of the post-synaptic and AP currents. These upstream processes would be those necessary to bring about high frequency activation of the electocytes' post-synaptic AChRs. Included here would be the continual presynaptic manufacture and vesicular packaging of ACh molecules as well as the costs associated with producing high frequency electromotor neuron and central neuronal pacemaker action potentials [42]. Insofar as the electrocyte's extensive capillary beds contribute to maintenance of extracellular Na + and K + levels, part of the metabolic costs of continually producing high frequency EOD will also be associated with remote sites such as kidney and gills where circulatory levels of these ions are regulated.
Thus, while Epm applies only to the posterior excitable membrane and assumes that dissipation of E Na occurs only across that membrane, this should not be taken to mean that homeostatic mechanisms that maintain posterior membrane E Na and E K occur exclusively at the posterior membrane. What it implies is that the overall complex (electrocyte/electric organ/ circulatory system) is able to regulate the posterior membrane E Na and E K values within very narrow bounds. An important part of the system that has yet to be explored is the posterior and anterior regions' K-selective inward rectifier channels and the extensive fine t-tubule-like invaginations (given the evolution of electrocytes from skeletal muscle; [32,43,44]. Whereas dissipation of electrocyte E Na is a posterior membrane process, dissipation of electrocyte E K is surely an electrocyte-wide process. Ongoing efforts are devoted to developing a full model of the Eigenmannia electrocyte that incorporates whole-cell ion homeostasis and the simultaneously occurring processes at both anterior and posterior membranes. As indicated earlier, Epm includes a necessary "place holder" conductance (the gK(V)) whose physiological role is to achieve fast repolarization of the AP. Ongoing experimental work seeks the molecular identity of this necessary entity. One possibility is an as-yet unidentified K Na channel with rapid voltage dependent kinetics and/or extremely high Na + sensitivity. Another possibility is a fast voltage-gated Clchannel.
It is self-evident that continual high frequency electrocyte firing with near-zero variance requires near-perfect ion homeostasis and accordingly Epm assumes invariant E Na and E K . In principle, the sine qua non for perfect ion homeostasis is perfect reciprocal-transport: for each Na + ion that enters, one Na + ion is pumped out. Thus even with no explicit 3Na + /2K + -ATPase "sodium pump" the model predicts excitability-imposed ATP requirements (and by extension, the required O 2 consumption) by tallying all excitability-related Na + -entry. We again stress, however, that what is not self-evident, is the question of where (i.e., across which membranes and with what geometry) 3Na + /2K + pump activity would be best deployed to maintain invariant E Na and E K at the posterior membrane. E K or E Na dysregulation can readily produce pathological patterns of excitability [18] yet astonishingly little is known about the role of cytogeometry in the Na + /K + homeostatic "near-perfection" of high-frequency excitable systems. Epm is therefore a step towards a longer term goal of learning precisely where Eigenmannia spends ATP to sustain EODs. Since the 3Na + /2K + -ATPase is electrogenic, if homeostatic pumping occurred entirely across the same membrane site that generates the high-frequency APs, the dissipative process (ion channel currents) would have to perfectly balance every 3 incoming Na + with 2 outgoing K + . Any departure from this pump-imposed constraint would lead to membrane polarization and thus jeopardize the precision of high frequency AP production. If instead, the work of ion homeostasis is partly off-loaded to non-excitable electrocyte regions (e.g., anterior membrane, tubular invaginations of the posterior membrane) and/ or to remote organs such as kidney, there would be latitude for diverse flux machinery to offset pump induced polarization (e.g. cation leak channels, inward rectifiers, sodium-activated gK, etc) without jeopardizing the posterior membrane APs. With Epm therefore, we set explicit homeostatic pumping aside to make it possible to quantify (via posterior membrane Na + entry) what the electrocyte achieves as it maintains perfect homeostasis.
Reardon et al [45], examined EOD frequency and amplitude in Eigenmannia under reduced water oxygenation. In these experiments hypoxia did not alter EOD frequency, but extreme hypoxia is associated with decreased EOD amplitudes (independent of EOD frequency). This suggests that under these experimental conditions the metabolic cost of EOD signalling may not increase with frequency as dramatically as was observed in later work [6]. This is in line with our present results on AP-related Na+ costs. Note however that Lewis et al measured the direct cost of changing EOD frequency during a jamming responses while EOD amplitude was maintained. So alternatively, EOD frequency could be more vigorously protected (than EOD amplitude) under environmental challenges like hypoxia.
Voltage-clamp data from electrocytes of the lower frequency gymnotid, Sternopygus, led to the intriguing suggestion that Nav and Kv channel kinetics are jointly modulated along a continuum such that lower frequency EODs arise from slower channels and higher frequency EODs from faster channels [46,47]. No confirmatory current clamp data showing AP shapes were obtained, but the idea is to generate at all frequencies, appropriate-shaped APs for yielding sinusoidal EODs. We did not test this as a feature in Epm because, in conjunction with an appropriate pacemaker output, it implies the existence of extraordinarily complex (and perhaps biologically implausible) co-regulation of the kinetics of two discrete channel protein complexes. In any event, use of an invariant set of kinetic parameters in Epm yielded APs whose shape differed little between 200 Hz and 600 Hz. For electrocytes from fish of known EOD frequency, it would be instructive to measure V m while driving the input neuron at different frequencies and to perform current clamp experiments as suggested in Fig 3.

Conclusion
The two core questions driving the current study are broadly applicable to excitable cells in general: 1) what mechanisms enable sustained high-frequency firing? and, 2) what are the relative contributions of membrane excitability, synaptic processes, and organismal-level processes to the metabolic costs of high-frequency firing? The electric organ discharge of Eigenmannia is a powerful and accessible system where experimental work on the physiology of EOD production and computational modeling of these physiological processes can form a synergistic and mutually informative approach to answering these questions. An intriguing example of the potentially broad cell physiological relevance of Eigenmannia comes to light in the neural processing of high frequency auditory signals [48]. Energetics issues faced by Eigenmannia electrocytes arise also for auditory neurons, but to our knowledge how pumps are deployed to ensure post-synaptic homeostasis at high frequency auditory synapses is unknown. Transmission electron microscopy of both Eigenmannia electrocytes and auditory synapses reveal abundant but unexplained post-synaptic membrane "vesicles" of unknown function and topology. Perhaps these vesicles are tortuous invaginations that could contribute to ion homeostasis as noted above. For Eigenmannia, we showed that this region stains intensely for Na + /K + pumps [5] and in skeletal muscle, pumps are an important component of t-tubular membrane. Pursuing the role of these membranous structures in Eigenmannia electric organ energetics could, we suspect, shed light on the cellular energetics of high frequency firing in the auditory system.
And looking beyond cell physiology/cell biology, a complete account of the mechanisms and energetics of high frequency firing will have far-reaching implications for understanding processes as varied as the evolution of nervous systems [49,50], tradeoffs in sensorimotor systems [51] and neuroecological adaptations in animal behavior [52]. Morris.