Volume 9 Supplement 1

Seventeenth Annual Computational Neuroscience Meeting: CNS*2008

Open Access

Cost of linearization for different time constants

BMC Neuroscience20089(Suppl 1):P52

DOI: 10.1186/1471-2202-9-S1-P52

Published: 11 July 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 [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

g d

g s

g NaP

Voltage range

g total

Time constant

Total I K

6.25 nS

0.68 ± 0.5 nS

0.16 ± 0.15 nS

-61.8 ± 7.2 mV

7.09 ± 0.6 nS

141 ± 1.4 ms

0.14 ± 0.04 nA

12.5 nS

1.37 ± 1.1 nS

0.33 ± 0.31 nS

-61.8 ± 7.2 mV

14.18 ± 1.4 nS

70.5 ± 7.2 ms

0.29 ± 0.07 nA

25.0 nS

2.75 ± 2.2 nS

0.66 ± 0.63 nS

-61.8 ± 7.2 mV

28.36 ± 2.8 nS

35.2 ± 3.6 ms

0.58 ± 0.14 nA

50.0 nS

5.5 ± 4.5 nS

1.32 ± 1.26 nS

-61.8 ± 7.2 mV

56.72 ± 5.7 nS

17.6 ± 1.8 ms

1.16 ± 0.29 nA

g d

g s

g A

Voltage range

g total

Time constant

Total I K

6.25 nS

4.7 ± 1.2 nS

3.54 ± 0.35 nS

-51.1 ± 5.2 mV

14.5 ± 1.5 nS

68.0 ± 7.6 ms

0.44 ± 0.05 nA

12.5 nS

9.4 ± 2.5 nS

7.09 ± 0.70 nS

-51.1 ± 5.2 mV

29.0 ± 3.2 nS

34.0 ± 3.8 ms

0.88 ± 0.10 nA

25.0 nS

18.8 ± 5.1 nS

14.18 ± 1.40 nS

-51.1 ± 5.2 mV

58.0 ± 6.4 nS

17.2 ± 1.9 ms

1.76 ± 0.21 nA

50.0 nS

37.7 ± 10 nS

28.36 ± 2.81 nS

-51.1 ± 5.2 mV

116.0 ± 12.8 nS

8.6 ± 0.9 ms

3.5 ± 0.42 nA

Authors’ Affiliations

(1)
Department of Physics and Astronomy, James Madison University
(2)
Department of Neurosurgery, University of Virginia

References

  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.View ArticlePubMedGoogle Scholar
  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.View ArticlePubMedGoogle Scholar
  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 CentralView ArticlePubMedGoogle Scholar
  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.View ArticlePubMedGoogle Scholar

Copyright

© Morel and Levy; licensee BioMed Central Ltd. 2008

This article is published under license to BioMed Central Ltd.

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