## DC space constant

chang

### DC space constant

Hello,

Before my question, i have to say that i read several posts concerning cable theory and even space constant . Since those posts were old, i opened a new one as space constant.

what i learned from previous posts are :Theoretically, length constant can be determined by : lambda=sqrt(d*Rm/Ra/4) (where, d: section diameter, Rm: membrane resistance and Ra: cytoplasmic resistance )and since this formula is only correct for a section with constant diameter and infinite length therefore, to determine space constant we have to find the distance distance where action potential amplitude drops to 0.37 of maximum amplitude at the initiation site.

My question is: i'm wondering why for a multi-compartment modeld cell in different papers, usually people speaks about "one" value for space constant (or electrotonic length), i think since the AP attenuation could be different on different path(from soma to dendritic section),then we have to define space constant for each section, rigth?(My guess is , each dendritic section has its own space constant, and the respoted value probably is the average value )

2- The way to define space constant on the model is calculating the distance at which the AP amplitude falls to 0.37 of AP in soma (if we consider the initiation sit at soma), is it rigth? what is the practical way, to define the space constant for each path (from soma to a dendritic section)
ted
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### Re: DC space constant

This question is not really about NEURON, so I have moved it to
"General questions and discussions about computational neuroscience"
chang wrote:to determine space constant we have to find the distance distance where action potential amplitude drops to 0.37 of maximum amplitude at the initiation site.
By definition, an action potential, or spike, is "actively conducted." That is, it regenerates itself. Unlike subthreshold fluctuations of membrane potential (such as synaptic potentials, or small perturbations of membrane potential caused by injecting current), an action potential's amplitude does not decay as it propagates along a neurite.
My question is: i'm wondering why for a multi-compartment modeld cell in different papers, usually people speaks about "one" value for space constant (or electrotonic length)
Fuzzy (incorrect) thinking. The same mistake is often made by experimentalists.

There are two problems here. One is with the notion of "space constant," which is not a very useful way to think about the spread of electrical signals in cells. The second problem is that, even if space constant were a useful idea, most information processing in neurons seems not to be static (steady state), but instead is dynamic, involving fluctuating signals whose time courses range from 0.1 ms to a few seconds. So even if space constant were a useful idea, _DC_ space constant would be far less important than _AC_ space constant.

The concept of "space constant" is useful for characterizing the spread of electrical signals in three cases:
1. Infinite cables with constant diameter and uniform electrical properties
2. Unbranched neurites or cells that are _very_ long and have uniform electrical properties
3. "Equivalent cylinder" models of neurons

Case 1 doesn't exist. I am not aware of any published example of Case 2, although it might apply to subthreshold spread of signals in some skeletal muscle cells. As far as case 3 is concerned, there is no class of neuron or any other kind of cell that satisfies the conditions for reduction to an equivalent cylinder (which are:
A. all dendritic terminations must be electrically equidistant from the soma. But no such neuron has been discovered.
B. all diameters must be cylindrical. But neurites are never cylindrical--diameters vary along the length of every neurite that has ever been measured. And if you ever look at electron micrographs, or 3D reconstructions from confocal scanning laser microscopy, you'll see that most neurites have very irregular, noncircular outlines which change along the length of each branch.
C. at all branch points, the "3/2 power law" must be satisfied. That is, if the diameter of the parent branch is dp and the diameters of the child branches are d1 and d2, then dp^(3/2) = d1^(3/2) + d2^(3/2). Every time this has been tested on real morphometric data, it has been found to be untrue--not even within 10 or 20% of being true.

As bad as this is, the biggest problem with the notion of "space constant" is that it encourages sloppy thinking. It tempts people to think that signals spread equally well in both directions along a neurite. This is incorrect.

Here is the key truth about the spread of electrical signals in cells:
in any cell of finite size, attenuation depends on the direction of signal propagation. This cannot be captured by a single number, like "space constant."

For a better way to think about these issues, see this paper
Carnevale, N.T., Tsai, K.Y., Claiborne, B.J., and Brown, T.H. The electrotonic transformation: a tool for relating neuronal form to function. In: Advances in Neural Information Processing Systems, vol. 7, edited by G. Tesauro, D.S. Touretzky, and T.K. Leen. Cambridge, MA: MIT Press, 1995, p. 69-76. http://www.neuron.yale.edu/neuron/paper ... psfin.html
and then work through this exercise
http://www.neuron.yale.edu/neuron/stati ... zclass.htm