Equivalent Cylinder Model (Rall)

William R. Holmes*Department of Biological Sciences, Ohio University, Athens, OH, USA

Definition

The equivalent cylinder model is a means to reduce the complex branching structure of a dendritictree to a simple cylinder by making a set of assumptions about the morphological and electrotonicproperties of the dendrites, allowing tractable mathematical analyses that can provide useful insightsinto dendritic function.

Detailed Description

It is difficult to gain mathematical insight into the function of complex branched dendritic trees.Analysis requires that the cable equation be solved for each dendritic segment, given boundaryconditions at the ends of the segments, plus an initial condition. Mathematical solutions becomeunwieldy, even with small numbers of dendritic segments (see “▶Cable Equation” entry). In 1962,Rall showed that with a few assumptions complex dendritic morphology could be reduced toa simple cylinder. This simplification is known as Rall’s equivalent cylinder model, and itsapplication has provided much insight into neuron function.

MotivationThemathematical arguments for reduction of a complex tree to a cylinder are given in Rall (1962a, b,1964). Here, we will motivate this reduction by considering the simple case of a cable that branchesinto two daughter segments and showing how this can be reduced to a single cable (Fig. 1). Theboundary condition at the end of the parent cable is a leaky end with a leak conductance GL equal tothe sum of the input conductances of the two daughter cables G1 + G2. Can we construct a singlecable having input conductance G1 + G2 but also having the same diameter as the parent cable?

Recall that the input conductance, GN, for a finite cylinder is (see entry “▶Cable Equation”)

GN ¼ G1tanh ‘=lð Þ ¼ tanh ‘=lð Þd3=22=pð Þ ffiffiffiffiffiffiffiffiffiffiffi

RmRap :

Then, the sum of the input conductances of the two daughter cables (1 and 2 in Fig. 1) is

GN1 þ GN2 ¼ p2

ffiffiffiffiffiffiffiffiffiffiffi

RmRap tanh

‘1l1

� �

d3=21 þ tanh‘2l2

� �

d3=22

� �

:

The input conductance of a single cable (3 in Fig. 1) is

*Email: holmes@ohio.edu

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GN3 ¼ p2

ffiffiffiffiffiffiffiffiffiffiffi

RmRap tanh

‘3l3

� �

d3=23

� �

:

Now suppose that Rm and Ra are the same in all cylinders and suppose that the electrotonic lengthsof the cylinders are also the same, that is, ‘1/l1 ¼ ‘2/l2 ¼ ‘3/l3. Then, GN3 ¼ GN1 + GN2 whend3

3/2 ¼ d13/2 + d2

3/2. This is called the “3/2 rule.” Thus, we can construct a single cable equivalentto two cables that also has the same diameter as the parent cable if the above assumptions aresatisfied, the “3/2 rule” is satisfied, and d3 equals the parent cable diameter.

Rall’s modelRall (1962a, b, 1964) formally stated the conditions for the reduction of a branched dendritic tree toan equivalent cylinder as follows:

1. Rm and Ra are the same in all branches.2. All terminal branches end with the same boundary condition (typically sealed end).3. All terminal branches end at the same electrotonic distance, L, from x ¼ 0 (L¼∑ ‘i/li is the same

for all paths from x ¼ 0 to all tips).4. At every branch point, the diameter of the parent branch and its daughter branches must satisfy the

relationship d3/2parent ¼ ∑ d3/2daughters.5. Any dendritic input at a particular location must be delivered proportionally to all branches at the

same electrotonic distance.

It should be noted that the daughter branches do not have to have the same diameters and thatbranches do not have to occur at the same electrotonic distances as shown, for convenience, by thesymmetrical branching pattern in Fig. 2. Furthermore, the conditions above do not require branchesto have the same physical length. Importantly, as can be seen from the equations following Fig. 1, theconditions above preserve both membrane area and input conductance of the original dendritic treein the equivalent cylinder.

Do dendritic trees satisfy these constraints? The first condition that Rm and Ra are the same in allbranches is reasonable as a first approximation although Rm and to a lesser extent Ra may benonuniform. The second condition that all terminals end with the same boundary condition isreasonable. The third condition that all terminal branches end at the same electrotonic distance islikely to be violated. For example, the short basilar and long apical dendrites of pyramidal cells donot end at the same electrotonic distance. Motoneurons, which might appear at first glance to begood candidates for reduction to equivalent cylinders, have some branches terminating sooner thanothers, making a cylinder followed by a tapering cable a more appropriate representation. The fourthcondition, the “3/2 rule”, was never claimed by Rall to be a rule of nature, but it happens to besatisfied in many dendritic trees, but also not in others. If the “3/2 rule” is satisfied, there is

Fig. 1 Can we replace segments 1 and 2 with an equivalent segment 3? GL is the leak conductance at the end of theparent segment, which must equal the sum of the input conductances of the daughter segments

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impedance matching at branch points and no possible “reflections.”Voltage attenuation in the soma-to-distal direction proceeds smoothly. If the “3/2 rule” is not satisfied, then there are abrupt changesin attenuation at the branch point in the soma-to-distal direction. If daughter diameters are thinnerthan the “3/2 rule,” then voltage attenuation will be less steep before the branch point and more steepafter the branch point than would be the case if the rule were satisfied. If daughter diameters arethicker than the “3/2 rule,” the opposite occurs; voltage attenuation will be steeper before the branchpoint and less steep after the branch point than when the “3/2 rule” is satisfied. This might haveimportant implications for the relative effectiveness of inhibition delivered at the soma for changingthe voltage at distances away from the soma. The fifth condition is required only when we considerinputs not at the soma. It is clearly not going to be satisfied generally. Nevertheless, superpositionmethods have been used in conjunction with the equivalent cylinder model to deal appropriatelywith inputs on a single branch (Rall and Rinzel 1973, 1974). Despite potential violations of theunderlying assumptions, the equivalent cylinder model provides an excellent first approximation ofthe structure of dendritic trees and has proved to be an exceptional model for providing insights intoneuronal function.

References

Rall W (1962a) Theory of physiological properties of dendrites. Ann NYAcad Sci 96:1071–1092Rall W (1962b) Electrophysiology of a dendritic neuron model. Biophys J 2:145–167Rall W (1964) Theoretical significance of dendritic trees for neuronal input-output relations. In:

Reiss RF (ed) Neural theory and modeling. Stanford University Press, Stanford, pp 73–97Rall W, Rinzel J (1973) Branch input resistance and steady attenuation for input to one branch of

a dendritic neuron model. Biophys J 13:648–688Rinzel J, Rall W (1974) Transient response in a dendritic neuron model for current injected at one

branch. Biophys J 14:759–790

1.0

1.6

2.5

4.0

6.3

10

0 0.2 0.4

EQUIVALENT CYLINDER

ELECTROTONICDISTANCE

DENDRITIC TREE

dj3/2 = CONSTANT

j

0.6 0.8 z

Σ

Fig. 2 Collapsing a dendritic tree to an equivalent cylinder (Reproduced from Rall (1964))

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