Persistent sodium and A-type potassium conductances serve as linearizing mechanisms over limited and different voltage ranges. This research investigates the relationship between time constants and the metabolic cost (here total potassium current IK) of such linearization. This metabolic cost is a window into explaining the 40% energy use by postsynaptic elements of the brain .
We consider neurons under constant synaptic bombardment spending much of their time in a range of -62 to -58 mV with threshold around -55 to -52 mV. For this subthreshold voltage range, the A-type potassium (gA)  and the persistent sodium (gNaP) [3,4] are the most relevant linearizing conductances. Here 'linear' means that, within a certain voltage range, each additional active synapse makes the same depolarizing contribution, in contrast to the sublinear contributions occurring in purely passive dendrites.
Steady-state voltages and currents are evaluated for a single-compartment dendritic model under synaptic bombardment. There are three conductances in each analysis: the resting dendritic conductance gd with a reversal potential of -72 mV; the synaptic conductance gs with a reversal potential of 0 mV; and a voltage-dependent conductance, either gNaP or gA, with reversal potentials of +55 mV and -95 mV respectively. The assumed capacitance of this collapsed dendritic field is 1 nF.
Table 1 shows that, in the presence of an appropriate amount of active conductance (gA or gNaP), there is 1) a constant voltage range of linearization across time constants and 2) there exists a direct relationship between time constant and total cost. Indeed as the time constant speeds up, the metabolic cost in terms of coulombs/sec increases as dictated by higher total conductance. To conclude: 1) faster computing is linearly increasing in metabolic cost; 2) changing inhibitory tone appears to require dynamic control of the available linearizing conductance if threshold is unchanged.
Table 1. Sample results
Agrawal N, Hamam BN, Magistretti J, Alonso A, Ragsdale DS: Persistent sodium channel activity mediates subthreshold membrane potential oscillations and low-threshold spikes in rat entorhinal cortex layer V neurons.