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Appendix A
Constructing the Hines Matrix
A.l Current Implementation
Applying finite difference schemes to the spatial derivatives involved in the cable
equation and the boundary conditions is a straightforward procedure, but care must
be taken to properly account for coupling at junctions in order to maintain an accurate
numerical solution.
The Hines ordering implies that compartments are labeled in increasing order from
most distal to most proximal on a branch, starting with the most proximal branch
and ending with the soma. The latter is measured by called a branch’s depth, which
is defined as the number of branches it must traverse to reach the soma, including
itself. Hence the soma compartment has depth = 0, roots have depth = 1, and so
on. Without loss of generality, assume further that each branch has N compartments
numbered 1 through N. This gives 6 possible compartment types whose entries in
the Hines matrix we must compute:
1. Interior compartment