1. Compartmental mdoesl were used to compute the time constants and coefficients of voltage and current transients in hypothetical neurons having tapering dendrites or soma shunt and in a serially reconstructed motoneuron with soma shunt. These time constants and coefficients were used in equivalent cylinder formulas for the electrotonic length, L, of a cell to assess the magnitude of the errors that result when the equvialent cylinder formulas are applied to neurons with dendritic tapering or soma shunt. 2. Of all the formulas for a cylinder (with sealed ends), the most commonly used fromula, which we call L(tau0/tau1) (the formula uses the current-clamptime constants tau0 and tau1), was the most robust estimator of L in structures that tapered linearly. When the diameter at athe end of the cylinder was no less than 20% of the initial diameter, L(tau0/tau1) underestimated the acutal L by at most 10%. 3. The equivalent cylinder formulas for a cylinder were applied to neurons modeled as a cylinder with a shunted soma at one end. The formula for L based solely on voltage-clamp time constants gave an exact estimate of L. However, the second voltage-clamp time constant cannot be reliably obtained experimentally for neurons studied thus far. Of the remaining formulas, L(tau0/tau1) was again the most robust estimator of L. This formula overestimated L with the size of the overestimates depending on beta, rho(beta = 1), and the actual L of the cylinder, where beta is the soma shunt factor, and rho(beta = 1) is the dendritic-to-somatic conductance ratio when beta = 1 (no shunt). When the actual L was 0.5 and the soma shunt was large, this formula overestimated L by two- and threefold, but when the actual L was 1.5, the overestimate was only 10-15% regardless of the size of the shunt. 4. In neurons modeled as two cylinders with soma shunt, the L(tau0/tau1) value computed with the actual tau0 and tau1 values overestimated the average L by two to six times when soma shunt was large. However, the L(tau0/tau1) estimates computed with tau0 and tau1 values estimated with the exponential fitting program DISCRETE from voltage transients computed for these neuron models were never this large because of inherent problems in estimating closely spaced time constants from data. When the electrotonic lengths of the two cylinders differed by <10%, the L overestimates were about the same as the for a single cylinder (with a shunted soma at one end) having an L equal to the average L. This 10% criterion may be relaxed to greater-than-or-equal-to 20% when only two exponentials can be resolved from the data. 5. When the morphology from a serially reconstructed motoneuron was used in the model, the L(tau0/tau1) estimates computed with the actual (theoretical) tau1 were very large, especially when there was a large shunt (Fig.9). Such large values are never obtained in practice. When time constants were estimated from computed voltage transients with the program DISCRETE, the formula L(tau0/tau1) overestimated L in a manner similar to that found for the single cylinder with shunt at one end. The L estimates were two to three times toolarge when the actual L was 0.8 and beta was large but were only 0-60% too large when the actual L was 1.7 for the same large beta. However, the size of the overestimates depended critically on the estimated value of tau1, and this varied widely depending on the nuber of exponentials that were fit to the data. 6. A formula for L for a linearly tapering dendrite is derived (Eq. 6). Its usefulness depends on the ability to estimate tau1 accurately and on knowledge of the degree of taper. 7. New formulas for a cylinder with soma shunt at one end and for multiple cylinders with soma shunt are also given (Eqa.9-13). Unlike previous formulas that take into account rho, the dendritic-to-soma conductance ratio, or formulas derived specifically for soma shunt, these new formulas do not rely on estimates of the parameters tau1, C0, or C1 obtained from a voltage transient. However, they do depend on knowledge of the morphology of the cell studied and the cell input resistance. 8. If values fo input resistance and tau0 are measured for the same neuron under two sets of experimental conditions, without and with a soma shunt, and if that neuron can be approximated as a cylindre with the soma at aone end, then the value of L; can be estimated graphically. When shunted and nonshunted values of input resistnace and tau0 differ by factors of five and four, respectively, L is approximately 0.5. Before accepting this conclusion, one should be careful to eliminate all possible sources besides soma shunt that might cotnribute to the differences int he two sets of values. 9. In conclusion, the equivalent cylinder formula L(tau0/tau1) underestimates L when dendrites taper and overestimates L when soma shunt is present; however, uncertainties about the tau1 estimates make electrotonic length estimates based on teh formula L(tau0/tau1) difficult to interpret. Our results suggest that estimates of L that have been computed with this formula for the past 20 years have overestimated the acutla L by a factor of two in electrotonically short cells (e.g., L = 0.5) and by 0-20% in electrotonically long cells (e.g., L = 1.5). This expians why electrotonic length estimates have not differed very much among different cell types, even when differences could be expected. In most cases, approaches that do not rely on tau1 of C1, such as the formulas presentd here or the compartmental model approach described in the next paper, can provide improved estimates of L.
