A sparse reformulation of the Green's function formalism allows efficient simulations of partial differential equations on tree graphs

We prove that when a class of partial differential equations, generalized from the cable equation, is defined on tree graphs, and when the inputs are restricted to a spatially discrete, well chosen set of points, the Green's function (GF) formalism can be rewritten to scale as $O(n)$ with the number $n$ of input locations, contrary to the previously reported $O(n^2)$ scaling. We show that the linear scaling can be combined with an expansion of the remaining kernels as sums of exponentials, to allow efficient simulations of equations from the aforementioned class. We furthermore validate this simulation paradigm on models of nerve cells and explore its relation with more traditional finite difference approaches. Situations in which a gain in computational performance is expected, are discussed.