More About the Math:

The Traveling Spike (Part 3)

Again, our equation for the axon voltage V (x; t) is:
\frac{1}{R} \frac{\partial^2 V(x,t)}{\partial x^2} = C \frac{dV(x,t)}{dt} + \bar{g}_{Na} m(x,t)^3 h (V(x,t)-V_{Na}) - \bar{g}_K n(x,t)^4 (V(x,t)-V_K) - g_L (V(x,t)-V_L)

The key challenge here is that this equation mixes changes in space x and time t, that is, the rate of change of voltage in time depends on the rate of change of voltage over space! This means that to figure out how a solution evolves in time at one point x we need to keep track of, hence evolve in time, the solution at all of the other spatial points. Fortunately, there is a way to sidestep this difficulty. Remember what we're looking for: a spike of fixed shape that travels down the axon at a fixed speed c. That is,

Note that the value of voltage at some point x at time t = 0 (V (x; 0)) is the same as the voltage at the point x + ct at time t (V (x; t)). That means that our solution V (x; t) is really just a function of a (x - ct): V (x; t) = V (x - ct). So, we have this crucial fact:

\frac{\partial^2 V(x,t)}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 V(x,t)}{\partial t^2} \;.

Therefore, our evolution equation is:
\frac{1}{Rc^2} \frac{\partial^2 V(x,t)}{\partial t^2} = C \frac{dV(x,t)}{dt} + \bar{g}_{Na} m(x,t)^3 h (V(x,t)-V_{Na}) - \bar{g}_K n(x,t)^4 (V(x,t)-V_K) - g_L (V(x,t)-V_L)

with only derivatives with respect to time (the same is true of equations for m(x; t), h(x; t), and n(x; t)). So, we can solve these equations using the methods we started out with for the space-independent Hodgkin-Huxley equations. CLICK here to take this final step.

Brain visualizations courtesy of Chris Johnson and Nathan Galli, Scientific Computing and Imaging Institute, University of Utah