ESTIMATING THE ELECTROTONIC STRUCTURE OF NEURONS WITH COMPARTMENTAL-MODELS

1. A procedure based on compartmental modeling called the "constrained inverse computation" was developed for estimating the electrotonic structure of neurons. With the constrained inverse computation, a set of N electrotonic parameters are estimated iteratively with use of a Newton-Raphson algorithm given values of N parameters that can be measured or estimated from experimental data. 2. The constrained inverse computation is illustrated by several applications to the basic example of a neuron represented as one cylinder coupled to a soma. the number of unknown parameters estimated was different (ranging from 2 to 6) when different setws os constraints were chosen. the unkiowns were chosen from the following: dendritic membrane resistivity R(md), soma membrane resistivity R(ms), intracellular resistivity R(i), membrane capacity C(m), dendritic membrane area A(D), soma membrane area A(S), electrotonic length L, an dresistivity-free length, rfl(rfl = 2l/d1/2 where l and d are length and diameter of the cylinder). The values of the unknown parameters were estimated from the values of an equal number of known parameters, which were chosen from the following: the time constants and coefficients of a voltage transient tau0, tau1,..., C0, C1,..., voltage-clamp time constants tau(vc1), tau(vc2,..., and input resistance R(N). Note that initially, morphological data were treated as unknown, rather than known. 3. When complete morphology was not know, parameters from voltage and current transients, combined with the input resistance were not sufficient to completely specify the electrtonic structure of the neuron. For a neuron represented as a cylinder coupled to a soma, there were an inifinte number of combinations of R(md), R(ms), R(i), C(m), A(S), A(D), and L that cold be fitted to the same voltage and current transients and input resistance. 4. One reason for the nonuniqueness when complete morphology was not specified is that the R(i) estimated is intrinsically bound to the morphology. R(i) enters the inverse computation only in the calculation of the electrotonic length of a compartment. the electrotonic length of a compartment is l[4R(i)/(dR(md)]1/2, where l and d are the length aqnd diameter of the compartment. Without complete morphology, the inverse computation cannot distinguesh between a change in d or l and a chagne in R(i). Even when morphology is known, the accuracy of the R(i) estimate obtained by any fitting procedure is affectd by systematic errors in length and diameter measurements (i.e., tissue shrinkage); the R(i) estimate is inversely proportional to the length measurement and proportional to the square root of the diameter measurement. 5. Another source of nonuniqueness when morphology is not completely known is that R(md), R(ms), C(m), A(S), and A(D) are interrelated in the computation. For a neuron represented as a soma coupled to a cylinder with electrotnic length L, one can multiply C(m) by a constant and divide R(md), R(ms), A(S), and A(D) by the same constant, and the new electrotonic parameters will give exactly the same transients and input resistance as before. 6. When the dimensions of the cylinder were treated as known, the inverse computationconverged to the same values of the for unknown parameters, R(md), R(ms), C(m), and R(i), for a wide range of starting conditions, gfiven four known parameters consisting of tau0 and R(N) plus two parameters chosen from C0, tau1 adn tau(vc1). The sensitivity of thresults to inaccuracies in theknown parameters was small in some rregions of the solution space, but large in others. For examples studied, this sensitivity was relatively small when the membrane area ratio A(S)/A(D), was small; however, this sensitivity was much larger for the biologically less relevant case where the ratio, A(S)/A(D), was large. 7. The constrined inverse computation was also applied to a motoneuron (withknown morphology( assuming auniform R(m) model and a soma shunt model. We first generated a reduced morphology and applied the inverse computation toget good neuron, only R(N), tau0, and tau1 were available from experimental data. We could not make use of tau1, however, because the tau1 estimated from an experimental transient is rarely equal to the actual (theoretical) tau1 of a mulitpolar or branched neuron. Thus we could only estimate R(m) and R(i) for fixed values of C(m), or estimate R(md) and R(ms) for fixed values fo C(m) and R(i). 8. Estimates fo electrtonic parameters in cells having dendritic spines may be highly erroneous if spines are ignored. Fortunately, the effect of dendritic spines can be included in compartmental models in either fo two ways. A dendritic segment with spines can be modeled as asimple dendritic segment if either 1) the segment membrane resistivity, R(m), is reduced and the membrane capacity, C(m) is increased in accordance with spine membrane area; or 2) the segment dimensions are changed, keeping l/d2 constant, to incorporate spine membrane area. 9. In conclusion, theconstrained inverse computation provides a useful means to study the electrotonic structure of neurons. A useful set of parameters (from experiments) is tau0, R(N), and C0; tau(vc1) is useful parameter for full morphology models, and tau1 is sometimes a useful parameter for reduced moedls. However, a complete and unique specification of the electrotonic structure is not possible without complete specification of the morphology. Estimates will also depend ont he constrints one applies to the model (i.e., uniform R(m) model, soma shunt model, linear increse in dendritic membrane resistivity with distance, etc.). Nevertheless, the inverse computation provides a means to explore these various models, and, if other measurements can be made, the contrsined ivnerse computation proveide a means to explore these various models, and, if other measurements can be made, the constrianed inverse computation provides the appropriate tool for distinguishing among the different models.