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Cost of linearization for different time constants
BMC Neurosciencevolume 9, Article number: P52 (2008)
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 I K ) 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 (g A )  and the persistent sodium (g NaP ) [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 g d with a reversal potential of -72 mV; the synaptic conductance g s with a reversal potential of 0 mV; and a voltage-dependent conductance, either g NaP or g A , 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 (g A or g NaP ), 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.
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