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A New Computational Method for Cable Theory Problems

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A New Computational Method for Cable Theory Problems A new computational method for cable theory problems Bulin J. Cao and L. F. Abbott Physics Department and Center for Complex Systems, Brandeis University, Waltham, MA 02254 USA ABSTRACT We discuss a new computational procedure for solving the linear cable equation on a tree of arbitrary geometry. The method is based on a simple set of diagrammatic rules implemented using an efficient computer algorithm. Unlike most other methods, this technique is particularly useful for determining the short-time behavior of the membrane potential. Examples are presented and the convergence and accuracy of the method are discussed. 1. INTRODUCTION Cable theory is the primary tool used to relate the geo- used here, require us to assume that over the voltage metric form of a neuron to its electrical function ( 1-3). range being considered, the membrane conductance is The basic problem of cable theory is to compute the approximately constant (although see (18)). In addi- membrane potential everywhere on a complex neuron tion, the non-compartmental methods typically treat as a function ofthe external and synaptic currents enter- synapses as sites ofcurrent injection, ignoring the accom- ing the cell. Much work has been done for neurons with panying synaptic conductance changes (although see (8, restricted dendritic structures (4-7) satisfying, for exam- 19-23)). This is only valid if the synaptic contribution ple, Rall's 3 /2 power rule (8). In addition, several power- to the total membrane conductance is small. The fact ful and practical techniques have been developed to that the compartmental approach does not require these solve the cable equation for neurons with dendritic trees assumptions makes it an attractive and powerful alterna- ofarbitrary geometry (9-17). Because dendritic trees are tive. The biggest disadvantage of the compartmental typically so elaborate, it is essential that any general method is that it does not provide a continuous descrip- computational method be relatively simple to imple- tion ofthe potential as a function ofposition but rather a ment, even for complex trees, and easy to code for com- discrete approximation to it. Consider the potential that puter calculations, which are essential for all but the sim- results from a localized spike of current injected into a plest structures. Particularly noteworthy among the dendritic structure. Shortly after the spike is injected, the various methods available are diagrammatic rules for potential on the tree varies rapidly as a function ofposi- computing the Laplace transform of the cable potential tion so the compartmental approximation is poor and a developed by Butz and Cowan (9) and, perhaps the most continuous description is desirable. As time passes, the widely used method, compartmentalization (13, 14) potential becomes more uniform and the validity ofthe with efficient computer implementation ( 15). On the compartmental approximation increases. To comple- basis ofa path integral approach, a new method for solv- ment the compartmental method, we need an approach ing cable theory problems has recently been developed that provides a continuous description and is accurate at by E. Farhi, S. Gutmann, and one of us ( 16, 17). The short times when the compartmental technique breaks method is based on a remarkably simple set ofdiagram- down. This is precisely what the method presented here matic rules which, in contrast to virtually all other gen- provides. It is ideally suited for computing rise and re- eral methods, are ideally suited for investigating the sponse times and other short-time phenomena. short-time behavior ofthe membrane on a den- potential When we use a voltage-independent approximation dritic tree. These rules have been derived elsewhere ( 16, for the membrane conductance, the electrotonic charac- 17). Here we will show how the new diagrammatic ap- teristics of a dendritic tree can be parameterized by three proach for solving cable problems can be efficiently im- quantities: the resistivity of the intracellular fluid r, plemented on a computer and will exhibit the results for the membrane capacitance per unit area C, and the mem- several sample structures. We will pay particular atten- brane conductance per unit area we tion to the convergence and accuracy of the method. which denote by 1 We will assume that these three parameters are Of all the methods for analyzing the potential on a IR. the same on all branches of the tree (the case of a spatially complex dendritic structure, compartmentalization is varying membrane the most powerful and general. This is because it is the conductivity is considered in (16) only method that allows a full treatment of the voltage- but this case is much more difficult to compute). The dependent membrane conductances and time-depen- radius ofa given branch ofthe tree will be denoted by aseg dent synaptic conductances found in real neurons. Non- where seg labels a particular segment of the tree. (All compartmental cable theory methods, including the one segments are taken to be cylinders, a tapering cable can be treated as a sequence ofcylindrical segmentsjoined at non-branching nodes.) The formulas ofcable theory are Address correspondence to Dr. L. F. Abbott, Department of Physics, simpler if we measure all distances along segment seg in Brandeis University, Waltham, MA 02254, USA. units of the electrotonic length constant (Raseg/2r)1/2 Biophys. J. Biophysical Society 0006-3495/93/02/303/11 $2.00 303 Volume 64 February 1993 303-313 6 We include the exponential factors in Eq. 1.2 to simplify the definition of the Green's function G(x, y, t). The advantage of using the Green's function is that it de- pends only on the structure of the tree and not on the injected current. Therefore, it only has to be computed x once for a given dendritic structure. Ifthe tree is initially at its resting potential, V(x, 0) = 0, the first term in can In this case, y, is 5 Equation 1.2 be ignored. G(x, t)e-t 3 the potential measured at time t at the point x in re- sponse to an instantaneous spike of current of unit inte- FIGURE I A dendritic tree with labeled nodes and terminals. The gral strength (a delta function) injected at the point y at point marked x indicates where the potential is measured and y indi- time zero. This provides a physical interpretation of the cates the site ofcurrent injection. The points marked 2 and 4 are nodes and 1, 3, 5, and 6 are terminals. meaning of the Green's function. Once G is known, Eq. 1.2 allows the potential to be constructed for any initial condition and distribution of currents. and all times in units ofthe membrane time constant RC and we will do this throughout. The geometric structure 2. THE DIAGRAMMATIC RULES of the trees we consider is completely general. A tree is program we have developed and results made of a number of segments that meet at nodes and The computer end at terminals. For example, in Fig. 1, the points we will present are based on a simple diagrammatic algo- the Green's function for any den- marked 2 and 4 are nodes and 1, 3, 5, and 6 are termi- rithm for computing The basic idea is to express the Green's func- nals. An arbitrary number of segments can meet at a dritic tree. tion as a sum over ( 16, 17 Each trip represents node and there is no restriction on their radii or on their "trips" ). lengths. one term in a series giving the exact Green's function. Specifically, for any dendritic structure Let V( x, t) be the membrane potential at time t mea- sured at point x. (Since dendrites have multiple G(x, y, t) = z AtipGo(LtiP t) (2.1) branches, we really should identify the particular seg- trips ment along which the point x is located, by using a sub- where the sum is over all possible trips constructed using script, for example (see ( 16, 17)), but we will ignore this the rules given below, L,,jp is the length of the trip being complication, except where it is unavoidable.) We define summed and the potential Vso that V = 0 is the resting potential ofthe neuron. V satisfies the cable equation GO(LtriP, t) = 4 ex )tp (2.2) (1.1) are at = Ox2 - + I(X, t) Also given below the rules for determining the coeffi- cients Atrip which depend on the particular trip being where I(x, t) is the current (in appropriate units) being summed. injected at the point x at time t. In addition to solving A trip is a path along the tree that starts at the point x this equation, the potential Vmust satisfy boundary con- (the point where the potential is being measured) and ditions at all ofthe nodes and terminals ofthe tree. At a ends at the point y (the point where the current is being node, V must be continuous as we cross from one seg- injected). Trips are constructed in accordance with the ment to another and current must be conserved. At a following rules: terminal, various different boundary conditions can be imposed. We will require that no current flows through a * A trip may start out from x by traveling in either terminal, the so-called "sealed end" condition. A trivial direction, but it may subsequently change direction modification of the diagrammatic rules allows the only at a node or a terminal. A trip may pass "killed end" condition to be treated as well ( 16, 17) but through the points x and y an arbitrary number of we will not consider this case here. times but must begin at x and end at y.
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