Bulletin of Mathematical Biology Vol. 53, No. 3, pp. 457-467, 1991. Printed in Great Britain.

0092-8240/91 $3.00 + 0.00 Pergamon Press pie © 1991 Society for Mathematical Biology

A GENERALIZED TAPERING EQUIVALENT CABLE MODEL FOR DENDRITIC NEURONS ROMAN R. POZNANSKI Research School of Biological Sciences The Australian National University Canberra, A.C.T. 2601 Australia A mathematical model has been developed which collapses a dendritic neuron of complex geometry into a single electrotonically tapering equivalent cable. The modified cable equation governing the transient distribution of subthreshold membrane potential in a branching tree is transformed, becoming amenable to analytic solution. This transformation results in a Riccati differential equation whose six solutions (expressed in terms of elementary functions) control the amount and degree of taper found in the equivalent cable model. To illustrate the theory, an analytic solution (in series form) of the modified cable equation is obtained for a voltage-clamp present at the soma of a quadratically tapering equivalent cable whose distal end is sealed.

Introduction. A particular class of dendritic tree can be represented by a single equivalent cylinder, if several symmetry requirements hold (for a comprehensive summary see Rail (1977, 1989) and the monograph by Jack et al. (1975)). The equivalent cylinder concept was formulated by Rail over two decades ago and it provided neurophysiologists with tremendous insight into the role dendrites play in neuronal functioning (see e.g. Redman, 1976). Most neurons are not of the equivalent cylinder class, but instead show a decline in the dendritic trunk parameter DTP, caused by a deviation from the 3/2 power law at branch points. Consequently, the concept was extended by incorporating the notion of exponential taper into the model (see Rail, 1962a,b; Gutman, 1984; Poznanski, 1988). It is the aim of this paper to provide other forms of taper for neurons which show a fall in the DTP that is different from an exponential profile. Only recently, Schierwagen (1989) has formulated a general branching condition that incorporates some of these tapers. Theoretical works dealing with tapers have been published by several authors using nonequivalent cable models (Goldstein and Rail, 1974; Strain and Brockman, 1975; Keller and Lal, 1976; Brockman, 1981; Rose and Dagum, 1988) in order to understand the role changing geometry plays in controlling neuronal activity. In these models the number of dendritic branches parameter is set equal to unity. 457

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Condition for Reduction to a Generalized Tapering Equivalent Cable. The following parabolic partial differential equation describes passive membrane potential distribution in a dendritic tree with noncylindrical branches (Rall, 1962a, equations (20) and (23); Jack et al., 1975, equation 7.42):

OV

~2V

OT-OZ 2

v

aVFFdzT-~ d

{

[

[drl211/4~l

In r~/~,~ 1 + i ~ j ] jj,

÷~-ZL[~xxj

(1)

where the radius (r) of all branch segments and the number of dendritic branches (n) are both functions of actual distance (x) from the soma, and Z = fo X ds

(2)

~,.pe~

defines the electrotonic distance for situations where there is a continuously changing characteristic length parameter (/~taper):

f/'11/2[ 1 + ~d/'121-1/4

2t,pcr=2 o - -

Lrod

Ldx] ]

'

(3)

where 20 = [R,,ro/2Ri] 1/2 is the characteristic length parameter for a cylinder with radius r o (taken as the initial radius at x = 0). The generalized electrotonic distance and characteristic length Z and 2t,por respectively, are concepts first introduced by Rail (1962a,b) and illustrated in some detail for an exponentially tapering core conductor by Goldstein and Rall (1974, equations (16) and (17)). Rall (1962a, p. 1079) has shown mathematically that there exists a particular class of branching pattern, involving tapering of branches which would allow a dendritic tree to be electrotonically equivalent to a single one-dimensional cylinder. The following condition between r and n must hold for such a branching pattern (see also Jack et al., 1975, equation (7.43)):

[ Fd l=l

nr ~/2 1 +Ldxj ]

=constant.

However, since the primary aim of this paper is to determine combinations of tapering and branching which will allow the dendritic tree to be electrotonically equivalent to a tapering cable, the following condition between r and n must hold (cf. Rail, 1962a, equation (21) for the special case of F(Z; K)= e x p ( K ( Z - Zo)); and also Jack et al., 1975, equation (7.59), for the special case of F(Z; K) = exp(KZ)):

n~/~r[ l + LPQg'" dxjj = r~/anoF(Z; K),

(4)

G E N E R A L I Z E D T A P E R I N G E Q U I V A L E N T CABLE M O D E L

459

where n o is the number of branches at x = 0; F(Z; K) is the geometric ratio, which imposes a taper on the equivalent cable; and the parameter K ( < 0 ) determines the amount or rate of taper present. As a result, the coefficient of OV/~3Z in equation (1) simplifies to F-t(dF/dZ) and the partial differential equation describing passive membrane electrotonus in a branching dendritic tree (with noncylindrical branches) reduces to the following modified cable equation (provided dendritic branching satisfies equation (4)):

OV 02V ~T

OZ 2

v+

1 dF ~V F dZ OZ"

(5)

It has been shown that the class of dendritic trees which are represented as equivalent cylinders, satisfy the property that the rate of increase of dendritic surface area, with respect to x, remains proportional to the rate of increase of electrotonic distance, with respect to x (Rall, 1962a, equation (25); Jack et al., 1975, equation (7.45)): dA d Z --OC dx dx " However, for the larger class of dendritic trees represented by tapering equivalent core-conductors, the following property must hold:

/ FdZ]

dA

Ld _] '

or alternatively dA - - o c F(Z; K),

dZ

which has been quoted in Rall (1969, p. 1505) for the special cse of

F(Z; K) = exp(KZ). If for convenience, the analysis can be restricted to the case where each dendritic branch is represented by a cylinder of constant diameter [dr/dx = 0] and each individual branch at any given value of x or Z is characterized by a different diameter, then after replacing radius with diameter, together with the above assumptions, the condition on r and n governed by equation (4) becomes (cf. Rall, 1962b, p. 149 for the special case F(Z; K ) - e x p ( K Z ) ) :

I

a2/2 F(Z; K)= 2 d2/2, Lj=I

(6)

j= *' 1

which defines the combined DTP, where dj represents the diameter of the jth

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branch at distance x from the soma, with Z = 0 when x = 0. Alternatively, if x represents actual distance measured along successive branch points and branching Occurs at distances 0 = x o < . . . < xp with n i branches between xi and x~+ 1, where xg ~