I'm slightly confused about the speed at which "electricity" flows through neurite (i.e. dendrite etc) as per cable theory (http://en.wikipedia.org/wiki/Cable_theory). In standard electrical circuits, we have two velocities - the drift velocity of the electric charge and the velocity of the electric field (http://en.wikipedia.org/wiki/Speed_of_electricity). In the case of neurite we have a resitance to the flow of charge through the cytosol and presumably attenuation of the electric field is an important factor. In any case, all of the resources I've come across describe the flow of electric charge but none mention the electric field. For standard electric circuits the opposite is true - the speed of the electric field is the most important factor. So what about the electric field in a neuron? Is it unimportant compared to the physical flow of charge? Why is it neglected?
In standard circuit theory, the speed of light is infinite. This is why the potential (V) on both sides (x, y) of a capacitor (C) change at the same time (t) even though they are spatially separated, and we simply write current I=Cd(V(x)-V(y))/dt. There is a more careful way of stating this as an approximation of the full Maxwell's equations under the assumption of low frequency and long wavelength. Another way to see this in passive neurite theory is that there is no true propagation speed. http://www.jhu.edu/motn/coursenotes/cable.pdf: "This value can be though of as the speed of spread of electrotonic disturbances. Of course, it is not a true propagation speed, in the sense of the action potential propagation speed, because there is no fixed waveshape that is propagating, i.e. this is not a true wave." An action potential in Hodgkin-Huxley theory has a true speed, because the cable is not passive, but has voltage dependent sodium and potassium channels.
There have been several papers written on attentuation in dendrites (reflecting the change in shape of a traveling wave) using a value called "electrotonic distance" and interestingly enough, the electrotonic distance is different one way down the dendrite then it is the other. I believe there were actually longer electrotonic distances backwards through the dendrites, implying an important role for reflection waves off the soma. From Molecules to Networks by John Byrne has a chapter about it (and the passive electrical properties of neurite in general) with references.
Thanks for the answers. Yes there is a type of a "wave" given as a solution to the cable equation which attenuates through dendritic filtering as it propagates. I suppose I'm still having some trouble conceptualising what is actually travelling down the dendrite. Mathematically it is the voltage which is propagating, but the restistance is claimed to be the resistance of the flow of ions through the cytosol. Does the propagation of voltage exactly coincide with the propagation of ions or is there some electric field and hence voltage which travels ahead and at a different speed to the physical charge?
I wouldn't evens say the voltage is traveling. It's the perturbation (the change in voltage) that's traveling. So if you have a line and you whip it, you send a wave down it. The y-location of each segment of the rope is what's changing, but the y-location isn't "traveling" only the disturbance (the energy) is traveling. So it's as if somebody started "the wave" through the crowd at a stadium. The ions all take a turn standing up and sitting down, but they don't actually travel.
Sure its the flow of ions through the membrane that causes the change and voltage at a given location but the cytosol resistance is the resistance to flow of ions along the x-direction and in general the electric field can propagate faster than the physical flow of charge. It is not simply a case of flow in the y-direction in this case. There is more than one velocity to consider here - the physical flow of charge along the x-direction and the resultant propagation of the electric field.
It's analogous to standard circuit theory. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmmic.html "Even the electron speeds are themselves small compared to the speed of transmission of an electrical signal down a wire, which is on the order of the speed of light, 300 million meters per second."
The analogy to circuit theory was the motivation for my original question. I was pretty surprised when i found out how slowly a signal propagates through a neuron. I expected it to be close to the speed of light. I'm not clear on why the propagation of the signal (in the subthreshold case at least) seems to be given by the physical flow of charge which is clearly in juxtaposition to the classical circuit theoretical case.
The cable equation is derived using circuit theory, in which the speed of light is infinite. The whole situation is well-approximated by a scalar potential of quasi-electrostatics, whereas in the full case with finite speed of light, a vector potential is needed. So in that sense the signal propagates very quickly (infinite speed of light). The rise time should just be seen as the charging of a capacitor, which has a finite time constant even for a "point element" capacitor with infinite speed of light. However, since we are asking not about any signal, but only signals big enough to record say a 0.5 mV EPSP, then the question of speed is the question of the speed of suprathreshold signals, which is modelled by the Hodgkin-Huxley equations (which also assume infinite speed of light). For reference, the cable equation has the same form as the heat equation, which has infinite propagation speed http://www.mth.pdx.edu/~marek/mth510pde/notes 2.pdf "It is important to point out here that one of the drawbacks of the heat equation model is - as evident from the form of the fundamental solution - that the heat propagates at infinite speed. Indeed, a very localized heat source at y is felt immediately at the entire infinite bar because the fundamental solution is at all times nonzero everywhere."
It's not the physical flow (drift velocity) of charge which sets the speed. And as was stated, the speed of light does not enter ordinary circuit theory since there are no electromagnetic fields and circuits have zero physical extent. The slow propagation is simply what the circuit elements are arranged to do in order to be analogous to the neuron. Here's a simple example: consider a 573 Farad capacitor with a 1 volt charge that is connected to a 573 Henry inductor. This LC circuit oscillates at one cycle per hour. Much much slower than a neuron.