Abstract
Electrical parameters of spinal motoneurons were estimated by optimizing the parameters of motoneuron models to match experimentally determined impedance functions with those of the models. The model was described by soma area, somatic and dendritic membrane resistivities, and the diameter of an equivalent dendritic cable having a standard profile. The impedance functions of motoneurons and optimized models usually differed (rms error) by <2% of input resistance. Consistent estimates for most parameters were obtained from repeated impedance determinations in individual motoneurons; estimates of dendritic resistivity were most variable. The few cells that could not be fit well had reduced impedance phase lag consistent with dendritic penetrations. Most fits were improved by inclusion of a voltagedependent conductance G_{V} with time constant τ_{V}. A uniformly distributed G_{V} with τ_{V} >5 ms provided a better fit for most cells. The magnitude of this conductance decreased with depolarization. Impedance functions of other cells were adequately fit by a passive model or by a model with a somatic G_{V} and τ_{V} <5 ms. Most of these neurons (7/8) had resting potentials positive to −60 mV. The electrotonic parameters ρ, τ, and L, estimated from model parameters, were consistent with published distributions. Most motoneuron parameters obtained in somatic shunt and sigmoidal models were well correlated, and parameters were moderately affected by changes in dendritic profile. These results demonstrate the utility and limitations of impedance measurements for estimating motoneuron parameters and suggest that voltagedependent conductances are a substantial component of resting electrical properties.
INTRODUCTION
The importance of a neuron's electrical characteristics in synaptic integration has motivated considerable effort directed at methods for their determination (reviewed in Jack et al. 1983; Rall 1977; Rall et al. 1992). Typically, system time constant (τ), electrotonic length of the dendritic tree (L), and the dendritictosomatic conductance ratio (ρ) are determined from the voltage transients produced by injection of pulses or steps of current. However, these estimates may be skewed by deviations from idealized neuron properties, like nonuniform membrane resistivity, tapering dendritic trees, and dendritic branches of unequal length (Holmes and Rall 1992a; Holmes et al. 1992; Rose and Dagum 1988). Approaches exist for estimating parameters in neurons with somatic shunts (Durand 1984; Kawato 1984), but obtaining useful estimates often requires the availability of complete morphological information (Clements and Redman 1989; Fleshman et al. 1988; Major et al. 1994; Thurbon et al. 1998) and appropriate constraints (Holmes and Rall 1992b). This requirement limits analysis to relatively few neurons given the long time needed for complete reconstructions.
If reasonable assumptions can be made regarding selected morphological and electrical characteristics of a neuron, the remaining parameters of a suitable neuron model can be determined from recorded responses, providing estimates of electrical properties. This approach is well suited for the use of frequency domain methods. Rall (1960) and Nelson and Lux (1970) explored frequency domain methods theoretically and experimentally (see also Lux 1967), but other attempts to characterize spinal motoneurons in this manner have not been made. The parameters of other neurons have been identified after determination of their impedance and admittance functions (e.g., Moore and Christensen 1985; Moore et al. 1988; Saint Mleux and Moore 2000a,b; Tabak et al. 2000; Weckström et al. 1992; Wright et al. 1996).
Changes in the impedance function of a motoneuron can be used to determine the magnitude and location of a sustained conductance change (Maltenfort et al. 2004a,b). The accuracy of these determinations can be improved if estimates of the neuron's electrotonic parameters are available. This paper describes experimentally determined impedance functions of motoneurons and their use to estimate the electrical parameters of motoneurons.
METHODS
Measurement of impedance
Intracellular recordings were made from motoneurons in pentobarbital anesthetized cats, as described in Maltenfort et al. (2004b). Membrane potentials were recorded during injection of a quasiwhite current consisting of a series of evenly spaced sinusoids (2.44–732 Hz). Injected current amplitudes (rms) ranged from 0.2 to 1.7 nA (mean of 1.2 ± 0.5), producing voltage responses (rms) of 0.5 to 1.5 mV (mean of 0.90 ± 0.3). Motoneurons in the current analysis include the cells described in the previous paper plus cells not analyzed in that study because characteristics of recurrent inhibitory postsynaptic potentials (RIPSPs), cumulative impedance change (cuΔZ), and its variance failed to meet acceptance criteria. Three motoneurons from a separate experiment also were included.
We used discontinuous current clamp (DCC) to minimize the contribution of electrode characteristics to the recordings. Before each motoneuron was impaled, the monitor output of the amplifier was inspected during DCC use. Capacitance compensation and sampling rate were adjusted to maximize the settling rate of the electrode response without overshoot and allow the response to settle during each cycle of current injection. Minor adjustments were made if needed after impalement. Sampling rates ranged from 2.4 to 7.2 kHz, and most (24/32) were >4 kHz (mean of 4.9 kHz). The antialiasing filter was adjusted to minimize noise while minimizing the error from residual electrode voltage caused by current injection (cf. Finkel and Redman 1984). Although these measures would reduce the effect of electrode characteristics on impedance estimations, some residual effect was likely.
The impedance function—the transfer function between injected current and measured voltage—was estimated from the autospectrum of the voltage G_{vv}(f) and the crossspectrum of the voltage and current G_{vi}(f) (Bendat and Piersol 1986) (1) The impedance estimates were not particularly sensitive to the particular choice of tapering window used to suppress distortion of the power spectra attributed to finite record length (cf. Oppenheim and Schafer 1989; Press et al. 1992). We used a Welch window (Press et al. 1992), which has the advantage of a narrow main lobe in the frequency domain, to minimize overlap between successive spectral points.
The phase of the impedance function was corrected for the delay between A/D channels (onehalf the sample period) and the delay introduced by the use of DCC. The later delay depends on DCC sample rate and neuron characteristics and was corrected empirically. Our simulations show that the phase of motoneuron models changes linearly with respect to log (frequency) at the larger frequencies used in our study (Maltenfort et al. 2004a; Fig. 3). A set of phase spectra was constructed for each recorded cell by removing the A/D delay and a range of DCC delays at 10μs intervals (corresponding to 2.5° at 700 Hz). The most linear phase spectrum from this set between 200 and 700 Hz (based on least error in leastsquares linear fit) was used in parameter fits.
The model
Step (somatic shunt) and sigmoidal models were based on 6 morphologically and physiologically identified motoneurons (Cullheim et al. 1987; Fleshman et al. 1988), as described by Maltenfort et al. (2004a). The step model, which we used more often, had a low somatic resistivity R_{ms}, and a larger uniform dendritic resistivity R_{md}, which in the sigmoidal model increased from the value of R_{ms} proximally to larger values more distally, proportional to cumulative dendritic area. The dendritic tree was represented by a cable with standard dimensions in all motoneurons: diameter (D_{eq}) was constant within 2.5 mm from the soma, beyond which it tapered linearly over 4 mm for a total length of 6.5 mm. Somatic area (A_{s}) was a fourth free parameter. Values for intracellular resistivity and specific membrane capacitance were fixed at 70 Ωcm and 1.0 μF/cm^{2}, respectively.
Model impedance functions were determined using cable equations when the dendrites had uniform resistivity and no voltagedependent conductances, or using equivalent circuits to represent each compartment when dendritic membrane resistivity was nonuniform and/or contained voltagedependent conductances (Maltenfort et al. 2004a). Dendritic compartments had electrotonic lengths of 0.05 (cable equations) or 0.01–0.0125 (equivalent circuits); these lengths were chosen to provide a close match between models computed with cable equations and equivalent circuits.
Somatic and dendritic compartments in many models included a voltagedependent conductance (see results), represented by a firstorder transfer function approximating a Hodgkin–Huxleytype ionic current (Guttman et al. 1974; Moore and Christensen 1985) (2) where Y_{C}(ω) is the admittance (1.0/impedance) of the compartment, Y_{P}(ω) is the admittance of the passive compartment, G_{V} is magnitude of the voltagedependent conductance, and τ_{V} is the time constant describing the kinetics of this conductance. As discussed by Koch (1984), this firstorder voltagedependent term may represent one of several processes, such as activation of an outward current or the inactivation of an inward current with depolarization.
Fitting parametric models to impedance functions
The model parameters D_{eq}, A_{s}, R_{md}, β (R_{md}/R_{ms}, yielding R_{ms} from R_{md}/β), G_{V}, and τ_{V} were determined to provide a match between the experimental and model impedance functions. The use of complex impedance functions, including both magnitude and phase, was critical in these determinations. Preliminary simulations and parameter estimations that neglected phase information showed that A_{s} and R_{ms} affect impedance magnitude in a similar manner, so that these 2 parameters are difficult to fit independently based on magnitude alone.
Parameter estimation used a combined optimization approach to minimize the leastsquare error between the experimental and model impedance functions. First, simulated annealing from multiple starting points in the parameter space provided an initial randomized search, avoiding local minima that can trap gradientbased optimization. Simulated annealing alone is computationally expensive with convergence criteria not easily defined (Kirkpatrick and Sorkin 1995; Press et al. 1992), so the second phase used a gradientbased optimization to search for a global minimum for the parameter space, a best fit between the model and the experimental impedance function. We used the function “amebsa” (Press et al. 1992) for the simulated annealing algorithm, followed by the gradientbased algorithm “frprmn” (Press et al. 1992).
The starting points for simulated annealing were based on systematically perturbed parameter values of model motoneurons based on data of Fleshman et al. (1988); there were 8 different starting points for each of the 6 models. A moderate tolerance (1e4) was used in these fits. The 12 parameter sets out of the 48 that provided the best fits (leastsquared error) were selected, and parameter averages from all sets of 5 in this group of 12 were determined. Leastsquared errors between model and experimental impedance functions were recalculated using these parameter averages, and the set of averages that provided the best fit was accepted as the set of final model parameters. This strategy achieved good fits while minimizing computation time with multiple starting points. Parameter variability within the set of 5 used for this average was determined to ensure that parameters were from a neighborhood around the bestfit averages.
Upper and lower bounds were placed on each parameter to ensure that unphysiological parameter values were not selected. To implement these boundary conditions, the parameter set was mapped to a sigmoid function (3) In this equation, p_{i} is the value of the ith parameter (e.g., R_{ms}, D_{eq}, etc.); b_{i} is the minimum allowable value of the parameter p_{i}; a_{i} is the difference between the maximum and minimum values of p_{i}; and y_{i} is the value actually adjusted in the optimization routines. This mapping converts a constrained optimization problem (a_{i} ≤ p_{i} ≤ b_{i}) into an unconstrained problem (−∞ < y_{i} < ∞) and permits a parameter to approach its bound without encountering a discontinuity, thus ensuring stability (for a related approach, see D'Aguanno et al. 1986).
Bounds for R_{md} were based on the estimates of Fleshman et al. (1988) and Clements and Redman (1989). Because this data set is small (12 cells), the bounds were extended 2fold to 3.5–70 kΩcm^{2}. These studies report R_{ms} estimates that are much smaller than R_{md} (β ≫ 1), but values of β closer to 1 have been observed (Campbell and Rose 1997). Thus we set the lower bound of β to 1, or R_{ms} = R_{md}, and the upper bound to 999. When using sigmoidal models, ranges of 50–6000 Ωcm^{2} were used for R_{ms} and initial R_{md}, and 12.5–108 kΩcm^{2} for final R_{md}. These values were extended from those reported by Fleshman et al. (1988).
D_{eq} bounds (16.5 to 62.3 μm) were set to match reported dendritic surface areas (Cullheim et al. 1987; Ulfhake and Cullheim 1988). Somatic surface areas ranged from 2.8 × 10e5 cm^{2} (Burke et al. 1982) to 29 × 10^{−5} cm^{2}. This upper bound was extended from previously estimated values (Burke et al. 1982; Ulfhake and Kellerth 1983) because A_{s} is physiological rather than anatomical and may include part of the proximal dendrites.
Voltagedependent conductance magnitude G_{V} was given broad limits: from 0 to 5 times somatic conductance, if somatic, or to 600 μS/cm^{2}, if uniformly distributed. The time constant of the voltagedependent conductance τ_{V} was given a range from 0.1 to 75 ms. These limits were determined by the bandwidth of the noise used and the duration of segments of data analyzed. Using a relaxed upper bound for τ_{V} (500 ms) in our initial estimations sometimes yielded long values of τ_{V} with large G_{V} values and unrealistically low impedances at frequencies <1–2 Hz. When a pair of conductances was used (see results), τ_{V} limits were set at 0.1–10 and 10–75 ms, respectively.
The optimizations were implemented using a C program on Pentiumbased personal computers. Parameter optimization for each impedance record typically took 50 to 60 min for a model with a uniformly distributed voltagedependent conductance (circuit equations) or 30–40 min for a model with a somatic voltagedependent conductance (cable equations) run on a computer with a Pentium III processor.
RESULTS
Parametric fits to impedance functions
Parametric fits were attempted for 44 impedance functions obtained from 32 motoneurons. Acceptable fits were obtained for 36 impedance functions (25 motoneurons), with error <40 kΩ or <2.5% input resistance (rms error, squareroot of average squared difference between model and measured impedance functions). A 4parameter passive model was less satisfactory for many neurons than models that included a voltagedependent conductance. The impedance magnitude of passive models declines monotonically with frequency, but the magnitude of experimental impedance functions often (23 of 44) exhibited a short rise before starting to decline at 10–20 Hz (Fig. 1A). The phase of impedance functions with this characteristic also displayed a small lead at low frequencies (Fig. 1B), unlike the passive models. These characteristics were adequately fit using a model that included a voltagedependent conductance (G_{V}; dotted lines in Fig. 1). The impedance function of this model, the impedance function of this model with the effect of G_{V} removed (dasheddotted line), and the impedance function fit to a passive model (dashed line) coincided at frequencies >100 Hz. Impedance estimates were less certain at the low frequencies in which G_{V} effects were greatest, as indicated by lower coherence values (Fig. 1C), but the common occurrence of these effects indicates they are genuine features of motoneuron impedance. The remaining impedance functions tended to be flatter, with slight leads or smaller lags at low frequencies than predicted by the passive models, and models with G_{V} better described many of these.
Figure 1 gives an example of one of the best fits (cell 11, type FF, rms error = 4.7 kΩ, 0.7% of R_{in}). The mean rms error for acceptable fits was 11.9 kΩ (1.0%). Examples of impedance magnitude and phase for an average fit (cell 25, type FF, rms error = 11.3 kΩ, 1.1%) and for one of the worst acceptable fits (cell 22, type FF, rms error = 19.5 kΩ, 2.1%) are shown in Fig. 2, A and B and C and D, respectively.
Acceptable fits were not obtained for 8 impedance functions from 7 motoneurons, even with models that included one or 2 voltagedependent conductances. These impedance functions were distinguished by larger impedance magnitudes, smaller phase lags at higher frequencies, A_{s} estimates near the lower boundary (2.8e5 cm^{2}), and smaller R_{ms} estimates. Similar impedance functions were observed in simulations of dendritic recordings, using a model with 2 dendritic cables in which input impedance was determined in the first dendritic compartment of the smaller cable.
Impedance functions with acceptable fits included 10 (measured from 8 motoneurons) with minimum A_{s} values. The mean phase lag at 450 Hz in this group was −33.9 ± 4.5°, compared with −39.7 ± 5.1° for other motoneurons with acceptable fits and −23.7 ± 4.8° for cells without good fits. Cells with impedance functions that required minimal A_{s} values also had small R_{ms} estimates and large R_{md} estimates, possibly as a result of adjustments by the optimation algorithms to compensate for limiting values of A_{s}. These cells were excluded from the following analysis, which was based on 26 impedance functions from 17 motoneurons.
We compared fits obtained with a passive model, a model with a voltagedependent conductance restricted to the soma, and a model with a voltagedependent conductance uniformly distributed through the neuron. A 10% reduction in error was set as a criterion for one fit to be better than another. Smaller differences did not appear to provide meaningful discriminations. Somatic voltagedependent models provided better fits than passive models for 22 of 26 cases, with an average improvement of 30.2 ± 21.2%. Uniform voltagedependent models provided better fits for 20 of 26 cases with an average improvement of 31.3 ± 23.8%. Errors for the somatic and uniform voltagedependent models were generally similar (Fig. 3). The time constant of the voltagedependent conductance (τ_{V}) was longer (9.5–71 ms) for the 8 cases in which the uniform models provided better fits, and shorter (0.9–3.3 ms) for the 3 cases in which the fit was better using the somatic model. We proceeded with the assumption that motoneurons with τ_{V} <5 ms were described better by the somatic model, whereas motoneurons with τ_{V} greater than this value were described better by the uniform model.
Models with 2 voltagedependent conductances, one with a short time constant confined to the soma and one with longer time constant that was uniformly distributed, did not provide better fits than models with a single voltagedependent conductance, except in one case (reducing rms error by 17%). In the set of 26 impedance functions accepted for full analysis, passive models were most appropriate for 4, models with uniformly distributed voltagedependent conductances for 18, models with somatic voltagedependent conductances for 3, and a dual conductance model for one.
Distribution of model parameters
The 4 parameters describing the passive electrical structure of the motoneuron, were skewed to lower values (Fig. 4), D_{eq} presenting the most normal distribution (Fig. 4A). A_{s} estimates (Fig. 4B) included values higher than suggested by the published anatomical measurements, ranging from 1.25 to 8.9% of dendritic area (mean of 3.52 ± 1.86%). These relatively large somatic area estimates probably include juxtasomatic regions of proximal dendrites that are effectively isopotential with the soma. R_{ms} values (Fig. 4C) tended to be low, although one large estimate was obtained, in a type S, soleus motoneuron. R_{md} estimates (Fig. 4D) ranged widely and included values higher than those of Fleshman et al. (1988), but comparable to the estimates of Clements and Redman (1989). The distributions of R_{md} and A_{s} were quite similar for motoneurons of different motor unit type (as determined from input resistance and rheobase). D_{eq} tended to be smaller in FR than in FF units (24.4 ± 5.3 vs. 27.9 ± 6.7 μm), and R_{ms} tended to be smaller in FF than in FR units (181 ± 102 vs. 258 ± 138 Ωcm^{2}), but these tendencies were not significant (t = 1.39, P = 0.177 and t = 1.50, P = 0.149, respectively).
The only passive parameter correlated with resting potential was R_{ms}, which tended to increase with depolarization (Fig. 5A; r = 0.42, t = 2.22, P = 0.04). This correlation suggests the presence of a somatic voltagedependent conductance with very short time constant, indistinguishable from passive conductance with the bandwidth of injected current used in these experiments. An inward current activated with depolarization, like subthreshold sodium current or persistent sodium current, would produce the observed correlation between R_{ms} and resting potential. Although an increase in R_{ms} with activation of a conductance seems paradoxical, an inward current would decrease the slope conductance at depolarized membrane potentials, yielding larger R_{ms} estimates.
G_{V} estimates for uniformly distributed conductances ranged from 36 to 247 μS/cm^{2} (mean of 102.4 ± 59.1; Fig. 5B). These values were large in relation to dendritic conductance, averaging 196 ± 151% of 1/R_{md}. Somatic G_{V} values associated with short time constants were also substantial, ranging from 57 to 221 nS (mean of 131 nS; not shown). The uniformly distributed G_{V} decreased in size with depolarization (r = −0.51, t = −2.40, P = 0.03). Moreover, 3 of the 4 cells described by passive models and each of the cells with τ_{V} values <5 ms had resting potentials more positive than −60 mV (Fig. 5C). τ_{V} values (in all cells with G_{V} terms) were not significantly correlated with resting potential (Fig. 5C; r = −0.31, t = −1.44, P = 0.17). Overall, this analysis indicates that the experimentally determined impedance functions include the contributions of one or more uniformly distributed conductances at hyperpolarized resting potentials that inactivate with depolarization.
The consistency of parameter estimates can be judged by repeated estimates obtained from fits to different impedance functions from the same motoneuron, linked by lines in Fig. 5, A–C. (The separate records used to compute different impedance functions were obtained during tests of different sources of recurrent inhibition; Maltenfort et al. 2004a.) The same model (i.e., the same G_{V} distribution) provided the best fit in 6 of 9 of these comparisons. In the other 3 comparisons the best model changed from a passive to a somaticconductance model, or from a uniformconductance to a passiveconductance model. Each of these 3 cases was associated with depolarization to a resting potential more positive than −60 mV.
Most changes in A_{s}, D_{eq}, and R_{ms} between samecell estimates were small compared with the range of population values and are consistent with the small changes (5–10% of magnitude) observed between 2 impedance functions from single motoneurons (Fig. 6, A and B). R_{md} estimates show greater variability (Fig. 6B). Figure 6 demonstrates a correlation between A_{s} and D_{eq} (r = 0.61, t = 3.72, P = 0.001) and suggests a negative correlation between R_{ms} and R_{md} (r = −0.38, t = 2.03, P = 0.05 for all cells; r = −0.52, t = −2.90, P = 0.008 without the outlier).
Distribution of electrotonic parameters (τ, ρ, and L)
Electrotonic parameters (Fig. 7) were determined from the passive model parameters. Dendritictosomatic conductance ratio, ρ, was the ratio between the steadystate (f = 0 Hz) impedance of the somatic compartment and that of the dendritic cylinder [i.e., ρ = Z_{s}(0)/Z_{d}(0)]. The system time constant τ was based on the membraneweighted average of somatic and dendritic membrane time constants, with an empirical correction for ρ (Fleshman et al. 1988). L was defined by the location of the dendritic compartment where the cumulative dendritic membrane area reached 97% of the total dendritic membrane area, corresponding to the conductanceweighted electrotonic length L_{G}, of Fleshman et al. (1988). Corresponding values from the studies of Fleshman et al. (1988) and Clements and Redman (1989) are plotted in Fig. 7 for comparison. The ranges of ρ, τ, and L were similar to those found in previous studies, although we found a broader range of values. Several estimates of L exceeding 2 were associated with lower values of R_{md}. All but one ρ value was <1; this value (2.9) occurred in the type S motoneuron with large R_{ms}. Differences between repeated estimates of ρ, τ, and L for the same motoneuron (not shown) varied by an extent similar to that found for repeated R_{md} estimates (Fig. 6B).
No significant differences were found between different neuron types in τ (FF: 7.8 ± 0.9 ms; FR: 8.5 ± 1.0 ms; S: 6.1 ± 0.3 ms), ρ (FF: 0.29 ± 0.06; FR: 0.34 ± 0.06 ms; S: 1.67 ± 1.26), or L (FF: 1.65 ± 0.13; FR: 1.85 ± 0.16 ms; S: 2.25 ± 0.15). Comparisons were limited by the small sample of 2 type S motoneurons.
Dependency of parameters on assumed motoneuron electrical and morphological structure
Parameters also were determined for sigmoidal models and for models with altered cable structures to assess the effect of model assumptions on parameter estimates. Sigmoidal models, in which membrane resistivity increases monotonically from soma through dendrites, can produce electrical behavior identical to that of step models (Fleshman et al. 1988; Segev et al. 1990). Two sigmoidal models were used, one with a uniformly distributed voltagedependent conductance, the other with a voltagedependent conductance that was proportional to 1/R_{md}. The model with a uniform G_{V} provided better fits, on average, than the proportional model, although this difference was not significant (paired ttest; t = 2.02, P = 0.05). The proportional model produced a better fit (rms error reduced by >10%) in only one cell; the best stepmodel description of this cell had a somatic voltagedependent conductance.
Step models provided slightly better fits than sigmoidal models with uniform voltagedependent conductances, on average, with 91% of the rms error of the latter (paired ttest, t = 2.43, P = 0.03). Estimates of each of the 4 passive parameters obtained with the 2 models were strongly correlated: A_{s} (r = 0.98), D_{eq} (Fig. 8A; r = 0.99), R_{md} (Fig. 8B; r = 0.82), and R_{ms} (r = 0.90). As found by Fleshman et al. (1988), R_{ms} in the sigmoidal model was approximately 2fold larger. Sigmoidal estimates of A_{s} and D_{eq} were approximately 90 and 110%, respectively, of the corresponding stepmodel estimates. Estimates of L (Fig. 8C) and ρ were well correlated (r = 0.89, r = 0.86, respectively); correlations between τ estimates were weaker (r = 0.65). Both G_{V} and τ_{V} estimates in the 2 models were highly correlated (r = 0.97 and r = 0.95, respectively).
Clements and Redman (1989) reported that parameter estimates were sensitive to changes in the length of the equivalent cylinder, particularly estimates of R_{md}. We examined the effect on parameter estimates of models with altered dendritic profiles in a subset of 6 motoneurons selected to provide a range of parameter values. Dendritic profiles were altered by shortening the uniform or the tapered portion of the equivalent cable by 15% (1 mm), decreasing membrane area by 31 and 19%, respectively. For 5 of 6 motoneurons, R_{md} estimates were less in shortened dendrites (average decrease of 22%; Fig. 8D), although results were variable (range +12 to −67%). These results are generally consistent with those of Clement and Redman (1989), who observed changes as large as 20–30% in dendritic resistivity when the constantdiameter dendritic region was reduced by 0.1 mm. This variability in R_{md} resembles that found for repeated estimates in individual motoneurons (cf. Fig. 6). In contrast, most estimates of equivalent diameter (Fig. 8E), somatic area, and R_{ms} obtained in shortenedcable models were within 10% of standard values. Changes in G_{V} were moderate and variable, but τ_{V} estimates increased by an average of 83% (range of 26 to 230%) when the constantdiameter was shortened. Changing the taper produced smaller effects (−4 to 44%).
Estimated time constant was decreased by 23% (range 2 to −51%) when the constantdiameter region was shortened and by 19% (range −11 to −29%) when the tapering region was changed, similar to changes in R_{md}. ρ consistently increased with shortened tapers but was less predictable when the constantdiameter segment was shortened (Fig. 8F). Changes in L were variable but usually <20% of the standard value.
DISCUSSION
The results of this study demonstrate the feasibility of estimating neuron parameters using impedance functions determined from in vivo recordings. Frequencydomain techniques for parameter estimation provide some advantages over more frequently used timedomain methods, including greater immunity to the effects of noise and nonlinearities (Fu et al. 1989; Wright et al. 1996). Several findings support the reliability of parameter estimates in this study, including their similarity to estimates of previous studies, their relative insensitivity to choice of model, and the dependency of voltagedependent terms on resting potential. However, the use of this approach is subject to several limitations.
Methodological issues
Our estimated parameters are subject to some uncertainty attributable to incomplete compensation for electrode properties. Accurate compensation for electrode capacitance and adjustment of sampling parameters pose significant difficulties (Wilson and Park 1989). One approach to this problem is to include electrode parameters in the optimization procedures (e.g., Saint Mleux and Moore 2000a; Wright et al. 1996). With the potential for electrode polarization and complex electrode characteristics when recording from relatively deep tissue, we elected to use discontinuous current clamp (DCC) despite its attendant uncertainties (cf. Campbell and Rose 1997). With the sampling rates in this study (mean of 4.9 kHz), part of the motoneuron response may have decayed during each cycle of current injection, reducing impedance estimates throughout the frequency range. The parameters most directly affected by this error would be D_{eq} and A_{s}, given that size parameters and membrane capacitance are the primary determinants of impedance at higher frequencies. Incomplete settling of the electrode response would, on the other hand, increase estimated impedance. Use of higher sampling rates and electrodes with faster settling characteristics, such as shielded electrodes (e.g., Finkel and Redman 1983), would provide more accurate estimates of each of these parameters.
The use of DCC added an additional phase delay that appeared to depend on sampling rate and cell characteristics. Correction for this factor is necessary because phase information is needed to distinguish the effects of R_{ms} and A_{s} on impedance and estimate these parameters (see methods). We used an empirical approach, subtracting a delay sufficient to produce a phase profile at high frequencies that resembled model impedance phase functions. This procedure undoubtedly left some error, but the similarity in phase profiles of recorded and model neurons (cf. Figs. 1 and 2 with Fig. 3 of Maltenfort et al. 2004b) suggests that this error is relatively small. Any errors would affect R_{ms} directly, given that R_{ms} determines the frequency at which phase is −45° (Maltenfort et al. 2004b), and A_{s} indirectly, to compensate for the R_{ms} error.
R_{md} estimates displayed the greatest variability in repeated estimates of individual cells. White et al. (1992) observed that the determination of dendritic parameters in a neuron with a somatic shunt is ill posed, sensitive to small errors. The variability of repeated R_{md} estimates of model motoneurons with physiological amounts of noise is substantial and consistent with this observation (T. M. Hamm, unpublished observations). Although accuracy can be improved by anatomical constraints, R_{md} estimates are inherently the least reliable. Consequently, estimates of τ, ρ, and L, dependent on R_{md}, are subject to error.
Model assumptions
Spinal motoneurons possess several features that violate assumptions used in Rall's original procedures for estimating electrotonic parameters in neurons (Rall 1969, 1977), including somatic shunts (Barrett and Crill 1974; Clements and Redman 1989; Fleshman et al. 1988; Iansek and Redman 1973a; Rose and Vanner 1988; cf. however, Nitzan et al. 1990), and by tapering dendrites and dendrites of unequal length (Barrett and Crill 1974; Bras et al. 1987; Cameron et al. 1985; Fleshman et al. 1988; Kernell and Zwaagstra 1989; Redman and Walmsley 1983; Rose et al. 1985; Ulfhake and Kellerth 1981). Consequently, we based our estimates on a model that incorporated these features, as other investigators have in estimation and modeling studies (e.g., Durand 1984; Holmes and Rall 1992a; Kawato 1984; Powers and Binder 1996). However, substantial errors can be introduced by variation of these characteristics (Holmes et al. 1992; Rose and Dagum 1988; White et al. 1992). In general, the electrotonic parameters of a neuron cannot be uniquely determined without a complete morphological description (Holmes and Rall 1992b), and several investigators have noted that similar fits to experimental data can be provided with different parameters (e.g., AliHassan et al. 1992; Rose and Dagum 1988). Our parameter estimates may be influenced by departures from model assumptions.
Our models used a standard cable structure based on tapering dendritic profiles found by Fleshman et al. (1988) and Clements and Redman (1989). These profiles were rather similar within the small number of motoneurons in each study, but differed between the 2 studies. Although dendritic geometries of motoneurons in different muscle systems vary (Rose et al. 1985), lumbosacral motoneurons exhibit rather similar geometries with few exceptions (Ulfhake and Kellerth 1983). This uniformity of dendritic geometry in different lumbosacral motoneuron pools is consistent with the assumption of a standard dendritic profile, but electrotonic profiles of the dendrites of different species of motoneurons have not been compared. Moreover, the dendritic trees of type F motor units are more expansive that those of type S units (Cullheim et al. 1987; Gustafsson and Pinter 1984; Ulfhake and Kellerth 1982), raising the possibility that motoneuron of different types may be better represented by models with different dendritic profiles.
We observed a moderate sensitivity of parameter estimates to changes in cable profile (Fig. 8). The greater sensitivity observed by Clements and Redman (1989) may result from differences in the 2 studies: the shorter dendritic profile used by Clements and Redman, the use of timedomain rather than frequencydomain methods, or the use of a fixed soma as opposed to estimating soma area with the other parameters. Regardless of the reason for the different sensitivities, both studies indicate that parameter estimation should be accompanied by an analysis of parametric sensitivity to model assumptions.
Parameters obtained with step and sigmoidal models were strongly correlated. The choice of model affected A_{s} and D_{eq} estimates by about 10%, in addition to the expected influence on R_{ms}. Step models provided slightly better fits on average than sigmoidal models, suggesting that the combination of somatic shunt with uniform voltagedependent conductance is a better representation for most motoneurons than a sigmoidal distribution of resistivity and uniform G_{V}. The similarity of fit provided by different models implies that alternative models, G_{V} distributions weighted toward the soma or dendrites, for example, are unlikely to improve the goodness of fit or substantially affect the parameters.
Parameter estimates are also dependent on R_{i} and C_{m}. Previous studies support the values of 70 Ωcm (Barrett and Crill 1974; Clements and Redman 1989; Stuart and Spruston 1998; Thurbon et al. 1998) and 1 μF/cm^{2} (Major et al. 1994; Ulrich et al. 1994; Wright et al. 1996) used for R_{i} and C_{m}, respectively. However, several studies have provided larger estimates of R_{i} in different kinds of neurons (e.g., Major et al. 1994) as well as a range of C_{m} values (Barrett and Crill 1974; Nitzan et al. 1990; Thurbon et al. 1998). Our parameter estimates are subject to these uncertainties.
Distribution of motoneuron parameters
Motoneurons of different type and size differ in some parameters, including membrane resistivity and system time constant (Burke et al. 1982; Fleshman et al. 1988; Gustafsson and Pinter 1984; Kernell and Zwaagstra 1981; Zengel et al. 1985). We found that R_{ms} tended to be greater in FR than in FF motoneurons, consistent with previous work. R_{md} did not differ between cell types; the uncertainty in this parameter may have obscured any differences, if present.
The low R_{ms} values found in this study are characteristic of neurons with a somatic shunt. Uniform membrane resistivities are sufficient to characterize whole cell recordings from ventral horn neurons in slice (Thurbon et al. 1998), suggesting that the somatic shunt arises fully from damage. However, R_{md}/R_{ms} is decreased in cesiumloaded motoneurons (Campbell and Rose 1997), indicating that voltagedependent potassium channels contribute to the somatic shunt. Although inclusion of voltagedependent parameters in the identification process should reduce the effect of voltagedependent conductances on estimates of membrane resistivity, our estimates may have been affected by conductances active at rest. The dependency of R_{ms} on resting potential found in this study (Fig. 5A) suggests that voltagedependent channels contribute to somatic resistivity.
A voltagedependent conductance was required for most cells, and G_{V} was often substantial. The uniformly distributed G_{V} found in most cells decreased with depolarization, and τ_{V} averaged 39 ms, although the range of values was broad. The hyperpolarizationactivated mixed cation current, I_{h}, which is present in motoneurons (Barrett et al. 1980; Bayliss et al. 1994; Chandler et al. 1994; Takahashi 1990a,b), has properties consistent with G_{V} and likely contributes to this conductance. I_{h} channels are distributed in the dendrites as well as the soma of neonatal rat motoneurons (perhaps with a dendritic dominance; Kjaerulff and Kiehn 2001). Motoneurons with shorter afterhyperpolarizations (AHP; larger cells) exhibit greater amounts of sag (Gustafsson and Pinter 1984), attributable to I_{h}. Considering AHP durations in motoneurons of different type (Zengel et al. 1985), differences in sag and I_{h} should be greater between types F and S than between types FF and FR motoneurons. Uniformly distributed G_{V} averaged 130 ± 68 and 82 ± 43 μS/cm^{2} (t = 1.72, P = 0.11) in the FF and FR cells of our sample, respectively.
Several neurons with resting potentials > −60 mV were best described by models having a fast voltagedependent conductance localized to the soma. Potassium conductances like I_{A} and I_{Kdr} are present in motoneuron somata (Safronov and Vogel 1995; cf. Campbell and Rose 1997) and could contribute to this somatic conductance, although these conductances appear to be present in dendrites as well (Clements et al. 1986). Characteristics of these conductances are consistent with results of this study, but firm conclusions cannot be made. Impedance measurements at a single mean membrane potential have limited ability to characterize voltagedependent currents; more information can be obtained from measurements at multiple potentials (Saint Mleux and Moore 2000a,b; Tabak et al. 2000).
Other conductances may have contributed to the characteristics of the impedance functions, including Ca^{2+}activated K conductances (Barrett et al. 1980; Takahashi 1990a,b; Umemiya and Berger 1994) and persistent inward Na^{+} conductance (Chandler et al. 1994; Lee and Heckman 2001; Powers and Binder 2003). The size of the contribution made by voltagedependent terms in our estimates implies that voltagedependent conductances make substantial contributions to subthreshold behavior in lumbosacral motoneurons. The abundance of mechanisms for modulating these conductances in motoneurons (Powers and Binder 2001; Rekling et al. 2000) implies considerable potential for control of synaptic integration in motoneurons.
GRANTS
This work was supported by National Institute of Neurological Disorders and Stroke Grant NS22454 to T. M. Hamm and NS07309 to the University of Arizona—Barrow Neurological Institute Motor Control Neurobiology Training Program. M. G. Maltenfort received support from NS10341.
Acknowledgments
We thank T. Fleming for technical assistance and Drs. R.E.W. Fyffe and P. K. Rose for comments on an early draft of this work. We also thank the journal's anonymous referees for constructive, helpful comments.
Present address of M. G. Maltenfort, Department of Neurobiology and Anatomy, Drexel University College of Medicine, 2900 Queen Lane, Philadelphia, PA 19129.
Footnotes

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