Homeostatically relevant Na+/K+-ATPase electrogenicity comes at the cost of a more constrained ion channel composition and sub-optimal energetic efficiency.

(A) The Na+/K+-ATPase (grey) exchanges 3 intracellular sodium ions for 2 extracellular potassium ions against their gradients, which generates a net outward current. (B) Increased pump activity, and thus an increased hyperpolarising pump current, reduces cell excitability as a bigger inward input current is needed to activate voltage gated channels. (C) For a fixed input current that generates tonic firing in an excitable cell, an increase in pump current ‘silences’ the cell. (D) High firing rates require significant pump activity, which generates a significant hyperpolarising pump current. (E, F, G) Sodium leak channels facilitate a depolarising current that balances out the hyperpolarising pump current. If Na+/K+-ATPase and sodium leak channels are co-expressed (E), the impact of increased pump activity on cell excitability is minimised (F, G). (H) The additional inward sodium current (dark red) that is required to balance the outward pump current (grey) creates a redundancy in current flows which hampers energetic efficiency. Currents are plotted as contributions to the total in- and outward currents (top) and as separate currents (bottom). (I) Action potentials are similar in size when generated with a (balanced out) pump current and without a pump current. (J) For the former however, the redundancy in current flows requires more sodium ions to be pumped against the gradient per action potential.

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 (blue), 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).

The weakly electric fish electrocyte: an example system with strong Na+/K+-ATPase activity and strict entrainment requirements.

(A) and (B) are modified from (49). (A) The Electric Organ Discharge (EOD) creates a weakly electric field that can be use to detect objects and communicate with conspecifics. (B) The weakly electric organ consists of a line of excitable cells called electrocytes. Every electrocyte is innervated by the spinal motor neuron that delivers signals from the pacemaker nucleus (PN). (C) The input current to the electrocyte stemming from the PN (top) sets the high frequency firing rate of the electrocyte (bottom). (D) Constant input currents (top) also elicit tonic high frequency firing in the electrocyte (bottom). (E) Tonic electrocyte firing is realised for constant input currents that lie within the mean of the input currents that are generated in the behaviourally relevant regime (200-600 Hz) (F). (G) There is a frequency mismatch between the pacemaker firing rate and the mean driven electrocyte firing rates, which influences signal entrainment.

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

(A) Schematic illustration of the chirp setting. Left: the electrocyte (black) is coupled to the pacemaker nucleus (PN, blue) with an excitatory synapse. Right: PN spikes (blue, 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 (blue). (B) The generation of consecutive long chirps by the pacemaker (indicated by black arrows and instantaneous PN firing rate slowers the mean firing rate of the electrocyte (black, top) and thereby the energetic demand of the electrocyte, which is fed back into a decreased pump current (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 (blue) spikes (top) and electrocyte membrane voltage (bottom) during chirps before (left) and after (right) a significant decrease in excitability-altering pump current. After a significant deviation in pump current, electrocyte firing occurs during chirps (right). (D,E) Same as (B, C) with extracellular potassium buffering. Extracellular potassium buffering reduces 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 pump current (D, bottom). This reduces the deviation in pump current to an extend that chirps can be reliably generated (D (top), E).

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, blue) with an excitatory synapse. Right: Frequency rises are generated through a rapid increase in PN firing rates which exponentially decay back to baseline rates (blue, top). As the electrocytes are entrained by the PN (bottom), their firing rates mimick that of the PN and also show a frequency rise (black, top). (B) The generation of consecutive frequency rises by the pacemaker (blue) increases the mean firing rate of the electrocyte (black, top) and thereby the energetic demand of the electrocyte, which is fed back into a increased pump current (bottom). This increased pump current decreases cell excitability, which over time (in this paradigm after 15 seconds) leads to a mismatch between PN and electrocyte firing rates (top). (C, D) Electrocyte (black) and PN (blue) spikes (top) and electrocyte membrane voltage (bottom) during frequency rises before (D) and after (E) 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 , 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) but require more ATP-demanding pumping (E, bottom) compared to weak coupling (B, bottom).

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

(A) Current contributions to the total in- and outward currents (top) and absolute current flows (bottom) for one electrocyte action potential 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 tranfer of the pump (grey) 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 affected, 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, grey) varies on the timescale of action potentials, which is highly compressed in this 100 second time-window.

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

(A) Mean-driven electrocyte frequency ωe as a function of ΔIpump. (B) Phase Response Curves (PRCs, Z(ϕ)) as a function of ΔIpump. (C) The entrainment range , which is a function of mean-driven electrocyte properties (A, B, eqs 33, 32), changes upon deviations in pump current. For very strong deviations in Ipump, the pacemaker frequency ωpn falls out of the entrainment range which means that the electrocyte will not lock to the pacemaker in this regime.