A Graphical Approach to a Model of Neuronal Tree with Variable Diameter

We propose a simple graphical approach to steady state solutions of the cable equation for a general model of dendritic tree with tapering. A simple case of transient solutions is also briefly discussed.


Introduction
The function of many physiological systems depends on branched structures that exist both at the tissue (e.g. nervous plexi, lungs, and the vascular and lymphatic systems) and the cellular level (e.g. neurons). Of particular interest, local and global propagation of electrical signals within the nervous system depends on the integration, processing, and further generation of electrical pulses that travel through neurons. In turn, the tree-like morphology of neurons facilitates simultaneous signaling to cells located in different places and over long distances.
Theoretical research involving realistic neuronal morphologies is typically done by numerically solving systems of cable equations defined on cylinders with different radii, and assuming that voltage and current are continuous functions of space and time. To the best of our knowledge, graphical methods seem have not been widely applied yet in the mathematical modeling of neurons. Graphical methods are very useful and popular in different branches of modern physics. It is worth noting, for example, Feynman diagrams in quantum mechanical or statistical field theory [3], [4], [10], [25], [26], [27], [32], [37], [63], Vilenkin-Kuznetsov-Smorodinskii approach to solutions of ndimensional Laplace equation [38], [42], [54], [55], applications in solid-state theory, etc. A goal of this paper is to make a modest step in this direction (see also [1], [14] and references therein). We use explicit solutions from recent papers on variable quadratic Hamiltonians in nonrelativistic quantum mechanics [12], [15], [16], [17], [18], [35], [38], [56], [58] to describe steady state and transient solutions to linear cable equations modeling neurites with non-necessarily constant radius.

Cable Equation with Varying Radius
At a closer view, neurites can be regarded as volumes of revolution, defined by rotating a smooth function r = r(x) representing the local radius of the neurite where x represents distance along the neurite. As a result, the cable theory implies the following set of equations [31], [47]: Here, V represents the voltage difference across the membrane (interior minus exterior) as a deviation from its resting value, I m is the membrane current density, I = I a is the total axial current, R m is the membrane resistance, R i is the intercellular resistivity and C m is membrane capacitance (more details can be found in [31], [45] and [47]). Differentiating equation (2.3) with respect to x and substituting the result into (2.1) with the help of (2.2) one gets which is the cable equation with tapering for a single branch of dendritic tree (see [19], [31], [47] and [57] for more details).
We shall be particularly interested in solutions of the cable equation (2.4) corresponding to termination with a "sealed end", namely, when at the end point x = x 1 the membrane cylinder is sealed with a disk composed of the same membrane. In this case, the corresponding boundary condition can be derived by setting I a = πr 2 I m , (2.5) at x = x 1 . Then, in view of (2.2)-(2.3), one gets [45]: In a similar fashion, at the somatic end one gets where R s is the somatic resistance and C s is the somatic capacitance [20]. We shall use these conditions for the steady-state and transient solutions of the cable equation. (Later we may impose similar boundary conditions at the points of branching.) In this Letter, we shall first concentrate on steady-state solutions of the cable equation, when ∂V /∂t ≡ 0. Then This boundary value problem can be conveniently solved (by a direct substitution for each branch of the dendritic tree) in terms of standard solutions of this second order ordinary differential equation as follows where C (x) and S (x) are two linearly independent solutions of the stationary cable equation (2.8) that satisfy special boundary conditions C ( with the corresponding current density/voltage ratio function B (x) given by in term of the standard solutions C (x) and S (x) . Throughout this Letter, we shall refer to a case, , as the case of weak tapering. An opposite situation, when B (ξ) = 0 at certain point x 0 < ξ < x 1 and an inverse of the current may occur, shall be called a case of the strong tapering. (A case of strong tapering has been numerically discovered in [28].)

Tapering with Analytic, Asymptotic and/or Numerical Solutions
In this Letter, we consider a general model of a dendrite as a (binary) directed tree (from the soma to its terminal ends) consisting of axially symmetric branches with the following types of tapering.
This special case of tapering is integrable in terms of elementary functions [30] (see also [12] and [17] for a similar problem related to a model of the dumped quantum oscillator). For the steady-state solutions one obtains the following equation The corresponding two linearly independent solutions, namely, can be verified by a direct substitution for an arbitrary parameter γ.
The required steady-state solution of the boundary value problem is given by where (See [30] for more details.) 3.4. General Case of Axial Symmetry. One can use numerical methods and/or WKB-type approximation in order to obtain standard solutions. For example [41], where (See [31] and [47] for further details.)

A Graphical Approach
Graphical rules for steady-state voltages and currents in a model of dendritic tree with tapering are as follows.

4.1.
Single Axially Symmetric Branch with Arbitrary Tapering. For a single branch with tapering voltage and current density/voltage ratio are given by respectively (see Figure 1).

Junction of Three Branches with Different Types of Tapering.
The internal potential and current are assumed to be continuous at all dendritic branch points and at the soma-dendritic junction [45]. We consider a general case when each branch has its own tapering, say r = r (x) , r 1 = r 1 (x) and r 2 = r 2 (x) (see Figure 2). Then The total ratio constant B (x 12 ) at the branching point x 12 is given by the following expression .
with the coefficient B (x 12 ) found by the previous formula (4.5).

4.3.
Junction of (n + 1)-Branches. In a similar fashion, at the branching point x α , one gets In order to find voltage at a point x of the dendritic tree, follow the path x 0 → x and multiply the initial voltage V (x 0 ) by consecutive corresponding factors from formula (4.1) changing at each intersection of the tree. The ratio of voltages V (x α ) and V (x β ) at two terminal points, can be determine in a graphic form by the previous rule applied to the shortest path x β → x β .

Examples
Our formulas (4.5)-(4.6) define ratio coefficients B (x α ) for all vertexes for the standard node on Figure 2. For the corresponding voltages, one can write and where α is a separation constant. The boundary condition at the sealed end (2.6) takes the form A general solution of this problem can be conveniently written (for each branch of the dendritic tree) as follows where A is a constant and C (x, ω) and S (x, ω) are two linearly independent standard solutions of equation (6.2) that satisfy special boundary conditions C (x 1 , ω) = 1, C ′ (x 1 , ω) = 0 and S (x 1 , ω) = 0, S ′ (x 1 , ω) = 1. Then the boundary condition (2.7) at the somatic end x = x 0 , namely, results in a transcendental equation for the eigenvalues ω. (There are infinitely many discrete eigenvalues [13] and [29], Ince?.) The corresponding eigenfunctions U n = U (x, ω n ) = A n u n (x) are orthogonal (u m , u n ) = δ mn (u n , u n ) (6.7) with respect to an inner product that is given in terms of the Lebesgue-Stieltjes integral [13] (see also Appendix and [33], [51] and [52]): A formal solution of the corresponding initial value problem takes the form where V (x, ∞) is the steady-state solution, ω = ω n are roots of the transcendental equation (6.6) and the corresponding eigenfunctions are given by Coefficients A n can be obtained by methods of Refs. [13] and [20] with the help of the modified orthogonality relation (6.7) as follows Substitution of (6.11) into (6.9) and changing the order of summation and integration result in is an analog of the heat kernel. Infinite speed of propagation. Method of images.

Summary
In this Letter, we propose a simple graphical approach to steady state solutions of the cable equation for a general model of dendritic tree with tapering. A simple case of transient solutions is also briefly discussed.
is defined in terms of the Lebesgue-Stieltjes integral [34]. The modified orthogonality relation (A.6) holds also in the case of a piecewice continuous derivative k ′ on the interval [x 0 , x 1 ] .
The junction of three branches (see Figure 2) can be considered in a similar fashion. Suppose that with k = 0, 1, 2 for three corresponding branches, respectively, and boundary conditions are given by at the terminal ends. Introducing integration over the whole tree T by additivity, and applying the Green formula (A.4) for each branch, one gets T (vLu − uLv) dx = k 0 (x 12 ) (v 0 (x 12 ) u ′ 0 (x 12 ) − u 0 (x 12 ) v ′ 0 (x 12 )) (A.11) . We shall assume that the following continuity conditions: u 0 (x 12 ) = u 1 (x 12 ) = u 2 (x 12 ) , (A.12) k 0 (x 12 ) u ′ 0 (x 12 ) = k 1 (x 12 ) u ′ 1 (x 12 ) = k 2 (x 12 ) u ′ 2 (x 12 ) hold at the branching point x 12 . In view of of the boundary conditions (A.9), the modified orthogonality relation takes the form The case of junction of (n + 1)-branches (see Figure 3)is similar. In general, for an arbitrary tree, one may conclude that only the terminal ends shall add additional mass points to the measure, if the corresponding boundary and continuity conditions hold. Further details are left to the reader.