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With this partition in place, it is necessary to reorder the branches in the strong
and weak parts. This means that we assign local branch indices according to decreas-
ing dendritic depth with respect to a specific reference point. For the strong part
the soma is the reference point, and for the weak system it is the transition point
xτ- This reordering allows for construction of a coupled “Hines” matrix for which
Gaussian Elimination can efficiently be used. As a consequence of the reordering, the
transition point xτ lies at the proximal end of branch βw = Bw in the local ordering of
the weak part, but xτ lies at the distal end of branch βs ∈ [1, Bs]. Thus the absolute
voltages for the strong and weak parts of branch bτ in the local indexing are denoted
by Vβs,s and Vβw,w> and the radii of these components are denoted by aρs,s and apw>w.
The boundary conditions, initial conditions, and soma conditions for the strong-
weak model are the same as those given in (2.19)-(2.26), but we must include condi-
tions at the transition point to account for the strong-weak coupling. The voltage at
the transition point xτ is given by
Vτ(t) = Vρsts(xτ,t) = Vρwtiυ(xτ,t), Vτ(t) = Vτ(t)-Vt- (4.18)
We assume that the dendritic radii are not discontinuous in space, and thus continuity
of current at the transition point requires
∂xVβs,s(xτ,t) = ∂xυ0wrw(xτ,t).
(4.19)