Determining distributed parameters in a neuronal cable model on a tree graph

Abstract

For graph domains without cycles, we show how unknown coefficients and source terms for a parabolic equation can be recovered from the dynamical Neumann‐to‐Dirichlet map associated with the boundary vertices. Through use of a companion wave equation problem, the topology of the tree graph, degree of the vertices, and edge lengths can also be recovered. The motivation for this work comes from a neuronal cable equation defined on the neuron's dendritic tree, and the inverse problem concerns parameter identification of k unknown distributed conductance parameters. Copyright © 2017 John Wiley & Sons, Ltd.

Source and coefficient identification problems for the wave equation on graphs

“…The specifics of the Neumann problem were discussed in detail in Section 2 for the one‐interval graph. For the general tree, all main results of the work of Avdonin and Nicaise can be extended in a similar way (see also the work of Avdonin et al).…”

“…Example applications include propagation of electromagnetic waves in optical fiber networks, heat flow in wire meshes, electron flow in quantum mechanical circuits, and mechanical vibrations of multilinked flexible structures. Recently, the identification problem was extended to a neurobiology application, where was replaced by a parabolic equation …”

“…Recently, the identification problem was extended to a neurobiology application, where (1) was replaced by a parabolic equation. 2,3 Graphs with differential equations on the edges satisfying compatibility conditions at the interior vertices have become known as quantum graphs. While the term "quantum graph" is relatively new (see, eg, the work of Berkolaiko and Kuchment 4 ), the literature on the theory of partial differential equations on graphs is extensive (see, for example, related books, [5][6][7][8][9] and the surveys of Kuchment 10 and Avdonin 11 ).…”

Reconstruction of the coefficients of a star graph from observations of its vertices

“…x u (j) (x, t) − q (j) (x)u (j) (x, t) for 0 < x < a, t > 0, u (1) (0, t) = u (2) (0, t) = u (3) (0, t) = 0, (a) u (1) (a, t) = u (2) (a, t) = u (3) (a, t),…”

“…(b) ∂ x u (1) (a, t) + ∂ x u (2) (a, t) + ∂ x u (3) (a, t) = 0, (c) u (j) (x, 0) = f (j) (x) for j = 1, 2, 3.…”