# Current along an axon

When an action potential is induced on a neuron, the local transmembrane potential jumps from $E_{\mbox{rest}}$, the resting potential of the neuron, to $E_{\mbox{eq}}$, the equilibrium potential of an ion or group of ions in the vicinty. Current thus travels in and out of a part of the neuron's cell membrane as ions are exchanged during an action potential. This effect repeats itself as the neighboring regions of the membrane are depolarized, causing the action potential to propagate along the neuron's axon.

Take $\Delta Q$ to be the total charge exchanged locally during an action potential, $C$ to be the membrane capacitance, and $\Delta t$ to be the duration of the action potential. While we can approximate the local average current through an neuron's membrane by taking $$I = \frac{\Delta Q}{\Delta t} = C\frac{\Delta V}{\Delta t} = C\frac{E_{\mbox{eq}} - E_{\mbox{rest}}}{\Delta t}$$ Is there a way to approximate current traveling along the axon of the neuron? Is there even such a thing?

• I might be wrong here, but do you mean to obtain the current in units of amperes or similar? I presume that's going to be difficult as the capacitance (or inversely resistance) will vary from cell to cell due to different densities and will never be accurate enough for generalization. Thats why perhaps people measure the voltage rather than current? May 23, 2015 at 21:07
• @RoverEye, yes, current in amperes. Please feel free to make assumptions about average capacitance or inverse resistance. I'm just looking for a very rough way to model this. Thanks! May 23, 2015 at 21:39
• Well, the thing is you really cant model a neuron, as there are quite a few types of varying sizes and lengths... there is a ppt online which I found, which my be helpful to you? columbia.edu/cu/biology/courses/w3004/Lecture5.pdf May 23, 2015 at 21:44
• Can you clarify what you mean by "current along the axon" by describing a possible measurement that would detect the phenomenon? There isn't really a current travelling down the axon. The charges that rush in and out aren't being carried down the axon. They're part of the mechanical system that makes the wave move, but the speed of the wave is not fundamentally related to the amount of charge apart from how the whole thing works; you could do a similar thing with a pressure wave, etc. May 24, 2015 at 2:53

As far as I was able to find there exists two models of the action potential by using RC circuits. Do note that there is no flow of electrons per se across a tube, but rather a migration of potential. See this.

The Hodgkins Huxley model :

The Hodgkin–Huxley model, or conductance-based model, is a mathematical model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear differential equations that approximates the electrical characteristics of excitable cells such as neurons and cardiac myocytes, and hence it is a continuous time model

http://en.wikipedia.org/wiki/Hodgkin%E2%80%93Huxley_model

The Fitzhugh-Nagumo model:

The FitzHugh–Nagumo model is a simplified version of the Hodgkin–Huxley model which models in a detailed manner activation and deactivation dynamics of a spiking neuron. In the original papers of FitzHugh, this model was called Bonhoeffer–van der Pol oscillator...

http://en.wikipedia.org/wiki/FitzHugh%E2%80%93Nagumo_model

http://en.wikipedia.org/wiki/Quantitative_models_of_the_action_potential

Hodgkin, Allan L., and Andrew F. Huxley. "Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo." The Journal of physiology 116.4 (1952): 449-472.

Now, I have not dealt with differential equations in quite a while, and hence may not be the best source for explaining the model itself, but I'll try and update the post with what (if anything) I understand.

• It would be very helpful if you could explain the terms in the H-H equation in a more physiological context. I am from a non-math background and find it difficult to understand the equation in a bioological perspective May 24, 2015 at 10:34

Although it is not a flow of electrons like in an electronic circuit, it is nonetheless current flow which drives action potential propagation. It could be seen as a positive current flow (outward Na+), followed by a negative one (inward K+) going from dendrite to synapse.

In fact, a neuron can be modeled by a a basic electronic circuit with resistors (representing ion channels), capacitors (representing the membrane capacitance) and batteries (representing the various ion charge/concentration gradients that power the action potential), see Fig. 1. Fig. 1. Model of a neuron. Source: Yale university

• Could you please explain the direction of current flow I(Na) and I(K ) in the figure above ? May 24, 2015 at 10:35
• Of course - Because it is an electronic-circuit model, both Na and K currents flow from + to -, like electrons from a battery.
– AliceD
May 24, 2015 at 10:44
• Usually in a circuit, current flow is from the positive to negative terminal of battery, right ? But here, you have depicted current flow from negative (small bar) to positive (large bar) end ? I may be wrong since I am from a non-phy/math background . May 24, 2015 at 10:54
• In reality, electrons go from - to +. Often it is still drawn from + to - because it was decided early on (say, in the 1600's) that current goes from + to 1 (electrons and their charge were unknown). Hence, the convention is persistent and + to - is often still in use. I don't like it :)
– AliceD
May 24, 2015 at 11:14
• + to - I meant to say, not + to 1 :)
– AliceD
May 24, 2015 at 13:32

The electrical impulse through the axon is a transverse wave of ions moving in and out. There is no net change in charge to the axon itself. It isn't an electric current at all; it is a transverse chemical wave powered by electric charges.

• I taught the OP was asking how to model the neuron, not the neuron itself. May 24, 2015 at 9:50
• @RoverEye I agree with this answer, since it is meaningless to calculate the current if there is no net current flow along the neuron. May 24, 2015 at 9:55
• @MarchHo Yes, I know no "current" flows through the axon per se. But it has been modelled by using circuits by Huxley (for his nobel prize) May 24, 2015 at 10:00
• @RoverEye Why do you say that there is no net current flow along the axon ? May 24, 2015 at 10:03
• @Curious This Quora answer explains it much more clearly than I can. May 24, 2015 at 10:07