Classical Cable Theory

[Meena]

Hi!

I was just looking at the paper:
http://www.neuron.yale.edu/neuron/paper ... psfin.html

I have a question on defining electrotonic distance in the classical cable theory and in NEURON.

1. When we say electrotonic distance do we mean the distance at which
the voltage has reduced to 1/e of its initial value?

But I thought thats the definition of space constant?

Also, when we say a dendrite has electrotonic distance of 1.5, do we mean that the length of the dendrite is 1.5*lambda??

So, then ideally for no attenuation at all we want a dendrite to have electrotonic distance of "0" or numbers very close to zero??

2. In the classical cable theory, voltage decay along an infinite cylindrical
cable can be defined as V(x)=V0*e^-(x/lambda)

Then, it follows that x/lambda = ln(V0/V(x))???

Okay this maybe an extremely basic question, but why in a infinitely long cable does the electrotonic distance X become x/lambda???

I mean I understand that if a cable is infinitely long, then the signal will
never reach the end. And since it can never reach the end, whatever
at the end will not affect it?

But how does X equal to x/lambda?

I really do appreciate all your time and efforts.

Thanks,
Meena

[csh]

I have a question on defining electrotonic distance in the classical cable theory and in NEURON.

An excellent introduction to cable theory is given in chapter 4 of Johnston & Wu, "Foundations of Cellular Neuropysiology", MIT Press, Cambridge, 1995. All your questions are answered within a few pages to which I will refer here.

1. When we say electrotonic distance do we mean the distance at which
the voltage has reduced to 1/e of its initial value?

No. The electrotonic distance X is defined as X = x / lambda, where x is the physical distance along a cable, and lambda is the space constant (p. 76). Think of it as distance "in units of lambda".

But I thought thats the definition of space constant?

Only in the case of an infinite cylinder. In general, the steady-state space constant (sometimes called length constant) lambda is defined as lambda = sqrt((r*R_m)/(2*R_i)), where r is the radius of the cylinder, R_m is the specific membrane resistance, and R_i is the specific intracellular (axial) resistivity (p. 64). In a cable with sealed ends, steady-state voltage will have decayed to values that are greater than 1/e at lambda (i.e. less attenuation), while it will have decayed to values that are smaller than 1/e in a cable with open ends (Fig. 4.15, p. 79).

Also, when we say a dendrite has electrotonic distance of 1.5, do we mean that the length of the dendrite is 1.5*lambda??

I suppose you meant electrotonic "length" rather than distance? The electrotonic length L is defined as L = l / lambda, where l is the total length of the cylinder (p. 76). Note that this is different from the definition of electrotonic distance given above, which can be at any position along the cable (not just the end). This might have caused your confusion. The second part of your question is correct.

So, then ideally for no attenuation at all we want a dendrite to have electrotonic distance of "0" or numbers very close to zero??

We want it to have a small eletrotonic "length" (rather than "distance").

2. In the classical cable theory, voltage decay along an infinite cylindrical
cable can be defined as V(x)=V0*e^-(x/lambda)
Then, it follows that x/lambda = ln(V0/V(x))???
Okay this maybe an extremely basic question, but why in a infinitely long cable does the electrotonic distance X become x/lambda???
I mean I understand that if a cable is infinitely long, then the signal will
never reach the end. And since it can never reach the end, whatever
at the end will not affect it?
But how does X equal to x/lambda?

X = x / lambda is a definition for any kind of cable. It's just a useful placeholder to simplify equations. Hence, you can write
X = ln(V0/V(x)) instead of writing x/lambda = ln(V0/V(x))
You probably confounded electrotonic length and electrotonic distance. x can be any position along the cable, not just the end. Thus, the electrotonic length equals the electrotonic distance for x=l
Hope that helps,
Christoph

[rllin]

Meena wrote:

Also, when we say a dendrite has electrotonic distance of 1.5, do we mean that the length of the dendrite is 1.5*lambda??

I suppose you meant electrotonic "length" rather than distance? The electrotonic length L is defined as L = l / lambda, where l is the total length of the cylinder (p. 76). Note that this is different from the definition of electrotonic distance given above, which can be at any position along the cable (not just the end). This might have caused your confusion. The second part of your question is correct.

Lambda is the square root of a ratio of resistances. These are easily defined in a model. My question then is if you were to have a dendrite of 1 lambda long, this would mean electrotonic distance is 1 and 1*lambda is your length? Thanks!

[ted]

Lambda is the square root of a ratio of resistances.

No, because length constant has units of distance, and a ratio of resistances has dimensionless units.

if you were to have a dendrite of 1 lambda long, this would mean electrotonic distance is 1 and 1*lambda is your length?

If you were to say that a dendrite is the same length as the length constant of an infinitely long cylindrical cable with specified electrical properties of membrane and cytoplasm, then I would know what you mean. But the term "electrotonic distance" has no meaning when applied to a structure of finite length, because it doesn't tell anything about how well electrical signals spread along that structure.