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Dimension 1: Waves on a ring

animation

Define $t(x)$ to be the time of arrival of an action potential at position $x$. Then

\begin{displaymath} APD(x) + RT(x) = t(x) - t(x-L), \end{displaymath}

for a ring of length $L$. Further, $APD(x) = a(RT(x-L))$, and $C(x) = c(RT(x)).$ Now we use that
\begin{displaymath} t(x) - t(x-L) = \int_{x-L}^x \frac{dt}{dx} dx = \int_{x-L}^x \frac{1}{c(RT(x)) } dx \end{displaymath}

to find the integral delay equation for $RT(x)$,
\begin{displaymath} RT(x) = \int_{x-L}^x \frac{1}{c(RT(x)) } dx - a(RT(x-L)). \end{displaymath}

stable pulse animation

unstable pulse animation

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