Encyclopedia of Computational Neuroscience

Living Edition
| Editors: Dieter Jaeger, Ranu Jung

Electrotonic Length, Formulas and Estimates

  • William R. HolmesEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7320-6_480-1

Definition

The electrotonic length of a cylindrical neurite is its physical length divided by its space constant. The electrotonic length of a dendritic tree is estimated with formulas that assume the dendritic tree can be approximated as an equivalent cylinder.

Detailed Description

Rall showed that the complex branched morphology of a dendritic tree can be reduced to an equivalent cylinder given a certain set of assumptions (see entry “Equivalent Cylinder Model”). An important property of the equivalent cylinder is its electrotonic length, L, which equals the physical length of the cylinder, ℓ, divided by λ, the space constant. An estimate of L will tell us how far distal synapses are electrotonically from the soma, and this will tell us whether these synapses can be effective in driving the cell to threshold.

For a cell represented as an equivalent cylinder, Rall (1969) used separation of variables to derive the transient solution for voltage in the cylinder as
$$ V\left(X,,,T\right)={\displaystyle \sum_{n=0}^{\infty }{B}_n} \cos \left(\frac{ n\pi X}{L}\right) \exp \left[-\left(1+{\left( n\pi /L\right)}^2\right)T\right] $$
where X is normalized distance along the cylinder, defined by X = x/λ (x is physical distance), and T is normalized time, defined by T = t/τ m where τ m is the membrane time constant. The B n are determined from the initial conditions V(X, 0). For n = 0, the exponential part of the solution is just exp(−t/τ 0) where τ 0 = τ m = R m C m , the membrane time constant. For n > 0, the coefficient of t in the exponential term is [1 + (/L)2]/τ 0 where again τ 0 = τ m . Rall defined the “equalizing” time constants as τ n = τ 0/[1 + (/L)2] and let C n = B n cos(nπX/L) to simplify the solution to
$$ V\left(x,,,t\right)={\displaystyle \sum_{n=0}^{\infty }{C}_n}{\operatorname{e}}^{-t/{\tau}_n} $$
where we have also changed back from dimensionless variables X and T to the actual variables x and t. The C n will depend on the initial conditions. If one applies a current pulse or step and then records the voltage decay once this current is turned off, then this voltage decay is described by
$$ V\left(x,,,t\right)={C}_0 \exp \left(-t/{\tau}_0\right)+{C}_1 \exp \left(-t/{\tau}_1\right)+\dots . $$
The coefficients and time constants can be estimated from the experimental voltage trace by standard curve fitting procedures. In cells that have significant electrical extent, two time constants are usually obtained, although three are sometimes possible. We are particularly interested in the time constants, because we can rearrange the expression for the equalizing time constants above to get the formula
$$ L=\frac{\pi }{\sqrt{\tau_0/{\tau}_1-1}} $$

Thus, if we have τ 0 and τ 1, we can get an estimate of the cell’s electrotonic length L = ℓ/λ. One must be careful to fit as many terms as possible and then determine if adding a term results in a statistically significant improvement in the fit. Also one must make sure that the values of the coefficients, C i , are all significantly different from 0.

Rall (1969, 1977) noted that many formulas were possible:
$$ {L}_{\tau 0/\tau 1}=\pi /{\left({\tau}_0/{\tau}_1-1\right)}^{1/2} $$
$$ {C}_0/{V}_0= \tanh \left({L}_{C0}\right)/{L}_{C0} $$
$$ {L}_{C0/C1}=\pi /{\left(2{C}_0/{C}_1-1\right)}^{1/2} $$
$$ {L}_{vc}=\pi /\left[2{\left({\tau}_0/{\tau}_{vc1}-1\right)}^{1/2}\right] $$
$$ {L}_{vc2}=\pi {\left(9{\tau}_{vc2}-{\tau}_{vc1}\right)}^{1/2}/\left[2{\left({\tau}_{vc1}-{\tau}_{vc2}\right)}^{1/2}\right] $$

In these formulas, the subscript of L merely distinguishes the different L estimates. C 0, C 1, τ 0, and τ 1 are the first two coefficients and time constants from the general solution given above estimated from the voltage transient at the soma for a current step applied to the soma. The τ vc1 and τ vc2 represent time constants estimated from the voltage clamp current transient for voltage clamp at the soma. The V 0 in the second formula equals IRN the product of the applied current and the input resistance.

The first formula is the one recommended by Rall and is the one used most often; it works well for both brief and long current steps. The second and third formulas rely on coefficient estimates that are prone to error because of floating baseline problems caused by voltage-dependent conductances. The fourth formula requires time constants from both current clamp and voltage clamp experiments, but it may be difficult to get both, particularly in small cells. The last formula requires two voltage clamp time constants, but the second is usually too small to be measured accurately.

These formulas have been applied to data in hundreds of studies, and in most cases L estimates have been 1.0 or less. However, problems can arise with these L estimates if cells cannot be approximated well as an equivalent cylinder. In an equivalent cylinder, the τ 1 (and subsequent τ n ) can be interpreted as equalizing time constants over the length of the cylinder, but if the cell has dendrites that end at different electrotonic distances, the actual τ 1 represents equalization along the longest tip-to-tip path in the cell. The hope is that the estimated τ 1 will reflect equalization between the soma and dendritic tips, but whether or not this happens depends on the values of the coefficients, many of which have very small values. A second problem is soma shunt with electrode penetration. In this case, the formulas will tend to overestimate L. These issues are discussed in length in series of papers by Holmes et al. (1992), Holmes and Rall (1992), and Major et al. (1993a, b).

References

  1. Holmes WR, Rall W (1992) Electrotonic length estimates in neurons with tapering or somatic shunt. J Neurophysiol 68:1421–1437PubMedGoogle Scholar
  2. Holmes WR, Segev I, Rall W (1992) Interpretation of time constant and electrotonic length estimates in multicylinder or branched neuronal structures. J Neurophysiol 68:1401–1420PubMedGoogle Scholar
  3. Major G, Evans JD, Jack JJB (1993a) Solutions for transients in arbitrary branching cables: I. Voltage recording with a somatic shunt. Biophys J 65:423–449PubMedCentralPubMedCrossRefGoogle Scholar
  4. Major G, Evans JD, Jack JJB (1993b) Solutions for transients in arbitrarily branching cables: II. Voltage clamp theory. Biophys J 65:450–468PubMedCentralPubMedCrossRefGoogle Scholar
  5. Rall W (1969) Time constants and electrotonic length of membrane cylinders and neurons. Biophys J 9:1483–1508PubMedCentralPubMedCrossRefGoogle Scholar
  6. Rall W (1977) Core conductor theory and cable properties of neurons, chap 3. In: Handbook of physiology. The nervous system. Cellular biology of neurons, Sect 1, vol I, parts 1 & 2. American Physiology Society, Bethesda, pp 39–97Google Scholar

Further Reading

  1. Rall W, Burke RE, Holmes WR, Jack JJB, Redman SJ, Segev I (1992) Matching dendritic neuron models to experimental data. Physiol Rev 72:S159–S186PubMedGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Biological SciencesOhio UniversityAthensUSA