Encyclopedia of Computational Neuroscience

Living Edition
| Editors: Dieter Jaeger, Ranu Jung

Space (Length) Constant, Lambda, in Neuronal Signaling

  • William R. HolmesEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7320-6_36-1

Definition

The space (length) constant λ with λ = (Rmd/(4Ra))1/2 is a measure of steady-state voltage decay with distance in a cell. Quantitatively λ is the distance over which the steady-state voltage decays to 1/e or 37 % of its value at the origin in a semi-infinite cable.

Detailed Description

As given in the definition above, the space constant λ with λ = (Rmd/(4Ra))1/2 is a measure of steady-state voltage decay in a cell. Quantitatively λ is the distance (usually expressed in cm or μm) over which the steady-state voltage decays to 1/e or 37 % of its value at the origin in a semi-infinite cable. This can be seen by solving the steady-state cable equation (Rall 1977)
$$ {\uplambda}^2{\mathrm{d}}^2\mathrm{V}/{\mathrm{d}\mathrm{x}}^2-\mathrm{V}=0 $$
with boundary conditions of voltage clamp to V0 at the origin and voltage bounded at x = ∞ to get
$$ \mathrm{V}\left(\mathrm{x}\right)={\mathrm{V}}_0 \exp \left(-\mathrm{x}/\uplambda \right). $$
The above equation is useful to describe voltage displacements from rest. A more general solution for voltage decay with distance that includes resting potential explicitly is
$$ \mathrm{V}\left(\mathrm{x}\right)={\mathrm{V}}_{\mathrm{rest}}\hbox{--} \left({\mathrm{V}}_{\mathrm{rest}}\hbox{--} {\mathrm{V}}_0\right) \exp \left(-\mathrm{x}/\uplambda \right) $$
where now V0 is actual voltage instead of a difference from rest.

The importance of λ is that it determines how spatially separated inputs are integrated (spatial summation). If λ is small, then spatially separated inputs are unlikely to sum because voltage will decay significantly over the distance between the inputs, but if λ is large, inputs will not decay as much with distance allowing summation to occur. Given the formula for λ, spatial summation is more likely to occur when diameter or membrane resistivity is large or axial resistivity is small.

One must be careful not to equate quantitatively the steady-state voltage decrement with distance for the semi-infinite cylinder with voltage decay in finite cylinders. For finite cylinders steady-state voltage decay with distance will depend strongly on both the boundary conditions and the length of the cylinders (see entry “Cable Equation”), and for transient inputs voltage decay with distance will be very different. Conductance changes associated with transient inputs may make λ variable in time; depending on the level of background synaptic activity, λ may be large at one moment and small at another causing effective spatial integration to vary widely.

The definition of λ given above is for the special case where extracellular voltage is assumed to be isopotential. In the derivation of the cable equation, a generalized λ is defined as
$$ \uplambda =\sqrt{\frac{{\mathrm{r}}_{\mathrm{m}}}{{\mathrm{r}}_{\mathrm{i}}+{\mathrm{r}}_{\mathrm{e}}}} $$
where rm is membrane resistance of a unit length of membrane in Ωcm and ri and re are the intracellular and extracellular resistances per unit length in Ω/cm. In single cell models, it is usually appropriate to neglect re unless ephaptic interactions are a concern. If we neglect re then
$$ \uplambda =\sqrt{\frac{{\mathrm{r}}_{\mathrm{m}}}{{\mathrm{r}}_{\mathrm{i}}}}\kern0.5em \mathrm{or}\ \mathrm{in}\ \mathrm{terms}\ \mathrm{of}\ \mathrm{specific}\ \mathrm{resistivities}\ \uplambda =\sqrt{\frac{{\mathrm{R}}_{\mathrm{m}}\mathrm{d}}{4{\mathrm{R}}_{\mathrm{a}}}} $$
Setting re to 0 in the above expression for λ leads to another insight when the equation is expressed as λri = rm/λ. What this equation says is that λ corresponds to the length of the core conductor for which the core resistance (λri) equals the resistance (rm/λ) across the membrane.

Reference

  1. Rall W (1977) Core conductor theory and cable properties of neurons. In: Handbook of physiology. The nervous system. Cellular biology of neurons, sect 1, vol I, pt 1, chap 3. American Physiological Society, Bethesda, pp 39–97Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Biological SciencesOhio UniversityAthensUSA