Skip to main content

Cost of linearization for different time constants

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 [1].

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 ) [2] 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.

Table 1 Sample results


  1. 1.

    Attwell D, Laughlin SB: An energy budget for signalling in the grey matter of the brain. J Cereb Blood Flow Metab. 2001, 21 (10): 1133-1145. 10.1097/00004647-200110000-00001.

    Article  CAS  PubMed  Google Scholar 

  2. 2.

    Hoffman DA, Magee JC, Colbert CM, Johnston D: K+ channel regulation of signal propagation in dendrites of hippocampal pyramidal neurons. Nature. 1997, 387 (6636): 869-875. 10.1038/43119.

    Article  CAS  PubMed  Google Scholar 

  3. 3.

    Magistretti J, Alonso A: Biophysical properties and slow voltage-dependent inactivation of a sustained sodium Current in entorhinal cortex layer-II principal neurons. J Gen Physiol. 1999, 114 (4): 491-509. 10.1085/jgp.114.4.491.

    PubMed Central  Article  CAS  PubMed  Google Scholar 

  4. 4.

    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. Neuroscience. 2001, 102 (1): 53-64. 10.1016/S0306-4522(00)00455-3.

    Article  CAS  PubMed  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to William B Levy.

Rights and permissions

Open Access This article is published under license to BioMed Central Ltd. This is an Open Access article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Morel, D., Levy, W.B. Cost of linearization for different time constants. BMC Neurosci 9, P52 (2008).

Download citation


  • Voltage Range
  • Reversal Potential
  • Metabolic Cost
  • Inhibitory Tone
  • Synaptic Conductance