Figures and data in The electrogenicity of the Na+/K+-ATPase poses challenges for computation in highly active spiking cells

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8 figures, 2 tables and 2 additional files

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Figure 1

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In the weakly electric fish electrocyte, Na⁺/K⁺-ATPase electrogenicity requires compensation, which comes at the cost of a more constrained ion channel composition and sub-optimal energetic efficiency.

(A) The weakly electric fish electrocyte is an excitable cell that locks to high-frequency input from an upstream pacemaker (left). Maintenance of ionic homeostasis is carried out by the Na⁺/K⁺-ATPase (gray), which exchanges three intracellular sodium ions for two extracellular potassium ions and thereby generates a net outward current (center). The compensation of this relatively strong pump current is crucial for faithful synchronization to pacemaker inputs (right). (B) High firing rates require significant pump activity, generating a significant hyperpolarizing pump current. (C) Increased pump activity, and thus an increased hyperpolarizing pump current, reduces cell excitability because a larger inward input current is needed to activate voltage-gated channels. (D) For a fixed physiologically relevant input current (0.63 μA, see Methods – ‘Stimulation in the mean-driven regime’) that generates tonic firing in an excitable cell, an increase in pump current `silences' the cell. (**E–G**) Sodium leak channels facilitate a depolarizing current that balances out the hyperpolarizing pump current. If Na⁺/K⁺-ATPase and sodium leak channels are co-expressed (E), the impact of increased pump activity on cell excitability is minimized (**F, G**). This is reflected in the impact of the pump current on the tuning curve (F) and the impact of the pump current on the firing rate for a fixed, physiologically relevant input current of 0.63 μA (G). (**H–J**) Comparison of action potentials and underlying currents for a constant and physiologically relevant synaptic drive ( ${syn}_{clamp}$ =0.13, see Methods – ‘Stimulation in the mean-driven regime’) for a model with and without compensated pump current. (H) Action potentials are similar in size. (I) The additional inward sodium current (dark red) required to balance the outward pump current (gray) results in a simultaneous flow of equally charged ions in opposite directions, decreasing energetic efficiency. (J) Effectively, due to this redundancy, more sodium ions per action potential have to be pumped against the gradient.

Figure 2

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Homeostatic feedback loops based on Na⁺/K⁺-ATPase activity affect firing responses through altered pump currents.

(**A, B, C**) Synaptic input suppression (A) initially silences the cell (B). The Na⁺/K⁺-ATPase adjusts to the reduction in energetic demand through reduced activity. This reduces the pump current (C), which increases cell excitability and results in spontaneous firing without synaptic inputs (B, black). Without a pump current (magenta), spontaneous firing is not induced. (**D, E, F**) Increased synaptic inputs (D) initially increase firing rates (E). The Na⁺/K⁺-ATPase adjusts to the increase in energetic demand through increased activity. This increases the pump current (F), which decreases cell excitability and results in reduced firing rates (E, black).

Figure 3

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The electrocyte operates in a mean-driven regime, and its mean-driven properties affect its entrainment to periodic inputs from the Pacemaker Nucleus (PN).

(A) The input current to the electrocyte stemming from the PN (top, left) sets the high-frequency firing rate of the electrocyte (bottom, left). Constant input currents (top, right) also elicit tonic high-frequency firing (bottom, right). (B) Mean input currents stemming from physiologically relevant PN inputs of 200–600 Hz. (C) High-frequency electrocyte firing is realized for constant input currents that lie within the mean of the input currents that are generated in the behaviorally relevant regime (200–600 Hz) (gray box, B). (D) There is a frequency mismatch between the pacemaker firing rate and the mean-driven electrocyte firing rates, which influences signal entrainment.

Figure 4

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Homeostatic feedback loops on Na⁺/K⁺-ATPase activity impede chirp generation in electrocytes and can be mitigated through extracellular potassium buffering.

(A) Schematic illustration of the chirp setting. Left: the electrocyte (black) is coupled to the pacemaker nucleus (PN, green) with an excitatory synapse. A potassium buffer (blue) regulates extracellular potassium concentrations. Right: PN spikes (green, top) induce chirps in the electrocytes through cessation of inputs and thereby temporarily shut off the Electric Organ Discharge (EOD, middle). When chirps are properly generated, instantaneous firing rates (bottom) of the electrocyte (black) equal those of the PN (green). (B) The pacemaker generates 10 consecutive chirps, indicated by green arrows and instantaneous PN firing rates $\frac{1}{{ISI}_{PN}}$ . This lowers the mean firing rate of the electrocyte (black, top) and thereby its energetic demand. Consequently, the pump current decreases over time (bottom). This decreased pump current increases cell excitability, which over time (in this paradigm after 400 ms) leads to a mismatch between PN and electrocyte firing rates (top). (C) Electrocyte (black) and PN (green) spikes (top) and electrocyte membrane voltage (bottom) during chirps before (left) and after (right) a significant decrease in the excitability-altering pump current. After such a deviation in pump current, electrocyte firing occurs during chirps (right). (**D, E**) Same as (**B, C**) with extracellular potassium buffering. Extracellular potassium buffering extends the timescale of the homeostatic feedback loop of Na⁺/K⁺-ATPase activity on energetic demand, which reduces the effect of transient firing-rate deviations on the pump current (D, bottom). This minimizes the deviation in pump current to the extent that chirps can be reliably generated (D, top), (E).

Figure 5

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Homeostatic feedback loops on Na⁺/K⁺-ATPase activity impede the generation of frequency rises and can be mitigated through strong synaptic coupling.

(A) Schematic illustration of the generation of frequency rises. Left: the electrocyte (black) is coupled to the pacemaker nucleus (PN, green) with an excitatory synapse. Right: Frequency rises are generated through a rapid increase in PN firing rates which exponentially decay back to baseline rates (green, top). As the electrocytes are entrained by the PN (bottom), their firing rates mimic that of the PN and also show a frequency rise (black, top). (B) The generation of consecutive frequency rises by the pacemaker (green) increases the mean firing rate of the electrocyte (black, top) and thereby the energetic demand of the electrocyte, which is fed back into an increased pump current (bottom). This increased pump current decreases cell excitability, which over time (in this paradigm after 15 s) leads to a mismatch between PN and electrocyte firing rates during the frequency rises (top). Overall, however, synchronization is very stable, which is reflected in the synchronization index $\bar{R}$ (Equation 29, Methods). (**C, D**) Electrocyte (black) and PN (green) spikes (top) and electrocyte membrane voltage (bottom) during frequency rises before (C) and after (D) a significant increase in excitability-altering pump current. After a significant deviation in pump current, not all PN spikes are reproduced in the electrocyte which leads to `missing' spikes (D). This is reflected in the synchronization index $\bar{R}$ , which decreases with increasing pump current deviation. (**E–G**) Same as (**B–D**) with strong synaptic coupling. Strong synaptic coupling attenuates the effect of altered pump currents on electrocyte entrainment and enables reliable production of frequency rises (E, top), (**F, G**).

Figure 6

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Ideal voltage dependence of the Na⁺/K⁺-ATPase for energy-efficient action potentials and minimal firing-rate adaptation.

(A) Action potential current contributions to the total in- and outward currents for a constant and physiologically relevant synaptic drive ( ${syn}_{clamp}$ =0.13, see Methods - Stimulation in the mean-driven regime) with a Na⁺/K⁺-ATPase without voltage dependence (left) and with an optimal voltage dependence that mimics potassium channels (right). The voltage-dependent pump current takes on the role of a potassium channel and contributes significantly to the net current at the AP downstroke. (B) Total amount of sodium (red) and potassium (blue) ions, and net ion transfer of the pump (gray) that are relocated per AP for a cell with a voltage-insensitive pump (left) and a voltage-dependent pump (right). (C) The effect of pump density on the tuning curve is minimal for ideal voltage-dependent pumps (bottom) compared to a non-voltage-dependent pump (top). (D) Signal generation in a cell with Na⁺/K⁺-ATPases with optimal voltage dependence. Synaptic input suppression (top left) silences the cell (center left) and reduces the pump current (bottom left). Firing rates are, however, not changed, and the cell remains silent. Synaptic input doubling (top right) increases firing rates (center right) and increases time-averaged pump currents (bottom right). Firing rates are, however, not affected. Note that the instantaneous pump current (bottom, gray) varies on the timescale of action potentials, which is highly compressed in this 100 s time window.

Appendix 1—figure 1

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Spike amplitudes decrease with increasing input current, but current contributions per action potential (AP) remain the same.

(A) Spike amplitudes decrease with increasing input currents. Without any compensatory mechanisms (i.e. co-expression of sodium leak channels, left), the pump current shifts the influence of input current on spike amplitude similarly to that on the frequency-vs-input curve (Figure 1C). Pump currents that are regulated either through co-expressed channels (center) or a voltage dependence of the pump (right) have little influence on the relation between input current and spike amplitude. (B) A comparison of all currents that contribute to the AP between constant small (top, syn = 0.05), and large (bottom, syn = 0.5) synaptic inputs which result in low and high firing rates respectively. (C) AP shapes of electrocyte models without and with a constant or voltage-dependent pump current are indistinguishable. (D) Ion exchanges per AP, and thus pump rates, are similar for low (left) and high (right) synaptic inputs.

Appendix 2—figure 1

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A deviation in pump current $Δ I_{pump}$ alters mean-driven electrocyte properties and thereby its entrainment region.

(A) Mean-driven electrocyte frequency $r_{e}$ as a function of $Δ I_{pump}$ . (B) Phase Response Curves (PRCs, $Z (ϕ)$ ) as a function of $Δ I_{pump}$ . (C) The entrainment range, $[r_{{pn}_{min}}, r_{{pn}_{max}}]$ which is a function of mean-driven electrocyte properties (A, B, Equations 37; 38), changes upon deviations in pump current. For very strong deviations in $I_{pump}$ , the pacemaker frequency $r_{pn}$ falls out of the entrainment range, which means that the electrocyte will not lock to the pacemaker in this regime.

Tables

Table 1

Updated model parameters.

Parameter	Value reported in Joos et al., 2018	Updated value
$[{Na}^{+}]_{in}^{0}$	1.35 mM	13.5 mM
$g_{{Na}_{max}}$	700 μS	900 μS
$T$	not reported	293.15 K
$ω_{in}$	n. a.	0.434 mm³ (Ban et al., 2015)
$\frac{ω_{in}}{ω_{out}}$	n. a.	2
$v_{onset}$	n. a.	–76 mV
$ϵ$	n. a.	1

Table 2

All baseline stimuli and tuned parameters presented in this article.

$g_{{Na}_{max}}$ was tuned to maintain a spike amplitude of 13 mV, and $I_{pump}^{0}$ and ${[{Na}^{+}]}_{in}^{t = 0}$ were tuned to maintain ion homeostasis (and thus steady state values) for the specified stimuli. Empty cells correspond to standard parameter values as reported in Joos et al., 2018 and Table 1.

Figure	Stimulus (at baseline)			Tuned parameters
	${syn}_{clamp}$ [arb. u.]	$r_{pn}$ [Hz]	$ϵ$ [arb. u.]	$g_{{Na}_{max}}$ [μS]	$I_{pump}^{0}$ [μA]	$[{Na}^{+}]_{in}^{t = 0}$ [mM]
Figures 1H-J and 6A-B (left)	0.13	-	-	-	1.888	-
Figure 2	0.10	-	-	-	1.630	-
Figure 4	-	600	-	1300	4.074	-
Figure 5B-D	-	260	0.5	881	1.368	-
Figure 5E-G	-	260	-	763	1.258	-
Figure 6A-B (right)	0.13	-	-	-	-	-
Figure 6D	0.10	-	-	-	-	13.517

Additional files

MDAR checklist: https://cdn.elifesciences.org/articles/103781/elife-103781-mdarchecklist1-v1.pdf
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Source code 1 All results presented in this article can be reproduced via this source code.: https://cdn.elifesciences.org/articles/103781/elife-103781-code1-v1.zip
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Liz Weerdmeester
Jan-Hendrik Schleimer
Susanne Schreiber

(2025)

The electrogenicity of the Na⁺/K⁺-ATPase poses challenges for computation in highly active spiking cells

eLife 14:RP103781.

https://doi.org/10.7554/eLife.103781.4

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In the weakly electric fish electrocyte, Na+/K+-ATPase electrogenicity requires compensation, which comes at the cost of a more constrained ion channel composition and sub-optimal energetic efficiency.

Homeostatic feedback loops based on Na+/K+-ATPase activity affect firing responses through altered pump currents.

The electrocyte operates in a mean-driven regime, and its mean-driven properties affect its entrainment to periodic inputs from the Pacemaker Nucleus (PN).

Homeostatic feedback loops on Na+/K+-ATPase activity impede chirp generation in electrocytes and can be mitigated through extracellular potassium buffering.

Homeostatic feedback loops on Na+/K+-ATPase activity impede the generation of frequency rises and can be mitigated through strong synaptic coupling.

Ideal voltage dependence of the Na+/K+-ATPase for energy-efficient action potentials and minimal firing-rate adaptation.

Spike amplitudes decrease with increasing input current, but current contributions per action potential (AP) remain the same.

A deviation in pump current ΔIpump alters mean-driven electrocyte properties and thereby its entrainment region.

Updated model parameters.

All baseline stimuli and tuned parameters presented in this article.

MDAR checklist

Source code 1

Downloads (link to download the article as PDF)

Open citations (links to open the citations from this article in various online reference manager services)

Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)

In the weakly electric fish electrocyte, Na⁺/K⁺-ATPase electrogenicity requires compensation, which comes at the cost of a more constrained ion channel composition and sub-optimal energetic efficiency.

Homeostatic feedback loops based on Na⁺/K⁺-ATPase activity affect firing responses through altered pump currents.

Homeostatic feedback loops on Na⁺/K⁺-ATPase activity impede chirp generation in electrocytes and can be mitigated through extracellular potassium buffering.

Homeostatic feedback loops on Na⁺/K⁺-ATPase activity impede the generation of frequency rises and can be mitigated through strong synaptic coupling.

Ideal voltage dependence of the Na⁺/K⁺-ATPase for energy-efficient action potentials and minimal firing-rate adaptation.

A deviation in pump current $Δ I_{pump}$ alters mean-driven electrocyte properties and thereby its entrainment region.