INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION CONCLUSION REFERENCES

The electrophysiological properties of sympathetic postganglionic neurons of guinea pigs show characteristic differences between three subgroups or classes differentiated on the basis of their discharge properties (Fig. 1A). Phasic (Ph) neurons fire with a transient burst on depolarization, tonic (T) neurons fire throughout a maintained depolarization, and long afterhyperpolarizing (LAH) neurons fire only once due to a prolonged outward current. These three classes of neuron express distinct populations of voltage- and Ca²⁺-dependent K⁺ conductances and contain different neuropeptides (Cassell and McLachlan 1986, 1987; Keast et al. 1993). They also show characteristic differences in the dimensions of soma and dendrites and dendritic branching patterns (Fig. 1B), confirming they are distinct neuronal phenotypes (Boyd et al. 1996). These peripheral neurons are multipolar, with relatively simple arbors of dendrites that are of finer diameter than those of central neurons of comparable somatic dimensions. Cell input capacitance, derived from voltage responses to small hyperpolarizing steps from resting potential (Fig. 1C), has previously been found to differ between the classes roughly in proportion to the differences in total estimated surface area of neuronal membrane (Boyd et al. 1996).

View larger version (29K): [in this window] [in a new window]   Fig. 1. Three classes of sympathetic postganglionic neuron with different electrophysiology and morphology and distinct passive properties. A: records of membrane potential (top) during depolarizing and hyperpolarizing current steps (bottom) showing 3 distinct patterns of action potential discharge. Phasic (Ph) neurons fire in an initial burst, tonic (T) neurons fire throughout the depolarization, and long afterhyperpolarizing (LAH) neurons fire only once due to a prolonged outward current (shown in the top inset). Calibrations apply throughout. B: examples of biocytin filled neurons of each class. Ph neurons (top) have small cell bodies and intermediate dendritic arbors. T neurons (middle) have large cell bodies and the most extensive dendritic arbors, whereas LAH neurons (bottom) have large cell bodies with the least extensive dendrites. To the right are average representations of the soma (black ellipse) and dendrogram of each neuron class. C: examples of normalized voltage transients in response to current steps (timing shown below middle trace). These responses were averaged for 5 neurons in each class, a subset of the 58 cells included in the present analysis. To the right, the voltage transients are plotted semilogarithmically with lines of best fit superimposed. (Note that A and B are reproduced from Boyd et al. 1996 and the traces in C are illustrative only).

The passive electrical properties of multipolar neurons reflect the distribution of ion channel conductances throughout the distributed capacitance provided by the lipid bilayer of the plasma membrane. These properties determine the integrative behavior of neurons during synaptic bombardment (Jack et al. 1975; Rall 1977; Rall et al. 1992; Redman 1976; Segev et al. 1995). To investigate the electrical behavior of central neurons, the morphology of the soma and dendritic tree has been revealed after intracellular dye injection and correlated with detailed analyses of passive voltage and current transients recorded from the soma of the same cell (Major et al. 1994; Stuart and Spruston 1998; Thurbon et al. 1998). Computational models based on such experimental data have been generated that simulate the electrotonic voltage responses (see e.g., Rall et al. 1992; Segev et al. 1995), and more complex models that incorporate voltage-dependent behavior have also been developed (Cook and Johnston 1997; McCormick and Huguenard 1992).

The first electrotonic models were mathematically simplified by reducing the entire dendritic tree to a single "equivalent cylinder" linked to an R-C circuit representing the soma. This is acceptable provided that the diameters of successive levels of dendritic branching follow the "d^3/2" rule, which describes progressive tapering of the dendritic tree, as for alpha-motoneurons (Clements and Redman 1989), but this rule does not apply to most neurons. The existence of dendrites of unequal electrical lengths places an even greater limitation on the suitability of the single equivalent cylinder model. More recently, compartmental models have been used in which the dimensions of each dendritic branch are precisely defined (e.g., Rall et al. 1992; Stuart and Spruston 1998).

Manipulating the parameters of these models can predict the possible distributions of channels underlying a neuron's behavior. In electrotonic models, only three properties can be varied: specific membrane resistivity (R_m), specific membrane capacitance (C_m), and specific resistivity of the axoplasm (R_i). The effects on model behavior are therefore determined primarily by the structural components of the model. It has usually been assumed that the distribution of membrane conductance is uniform, although this is clearly not always the case (Campbell and Rose 1997). Such a model applied to sympathetic neurons should predict the input capacitance from the morphological characteristics provided that membrane resistivity is the same for neurons of all classes.

A common finding in such studies is that the cell input capacitance, derived by dividing the cell input time constant, ₀, by the cell input resistance, R_N, is larger than that calculated by multiplying the estimated surface area of the filled neuron by C_m, assuming the widely accepted value of 1 µFcm² (Jack et al. 1975). In studies where C_m has been derived rather than assumed, the common finding has been that C_m > 1 µFcm² (Lux et al. 1970; Nitzan et al. 1990; Rapp et al. 1994; Thurbon et al. 1998). Several possible explanations have been put forward. One is that C_m is actually higher than 1 µFcm² (Thurbon et al. 1998). Another explanation is that there may be a somatic shunt conductance, i.e., the somatic membrane is much leakier than the dendritic membrane, either because of a higher density of leak channels in the soma than in the dendrites or as an artifact of microelectrode impalement (Barrett and Crill 1974). It is also possible that membrane surface area (SA) has been underestimated, and studies that attempt to account for this tend to agree that C_m = 1 µFcm² (Gentet et al. 1999; Stuart and Spruston 1998).

Here we have compared the measured passive electrical properties of each class of sympathetic neuron with those calculated from cell dimensions, initially assuming C_m = 1µF cm². We used a compartmental model (Hines and Carnevale 1997) to predict the voltage transients that would be recorded in the soma. By modifying the parameters of the model in feasible ways, the electrotonic voltage transients recorded experimentally were mimicked. The resting membrane was assumed to be passive and of uniform resistivity, i.e., its impedance was linear without voltage- or time-dependent conductances. The adequacy of some assumptions has been assessed, and the effect of a possible somatic shunt conductance has been investigated. The results imply that the values of resting membrane conductance differ between the three classes of sympathetic neuron.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION CONCLUSION REFERENCES

Electrophysiology

Details of the electrophysiology experiments have previously been described (Boyd et al. 1996). Sympathetic ganglia were dissected from young adult guinea pigs perfused with physiological saline under deep urethane anesthesia (1-1.5 g/kg ip). Neurons were impaled in vitro at 35°C with high-resistance microelectrodes (80-150 M) containing 2% biocytin and studied with single electrode current and voltage-clamp techniques. Biocytin filled the cells during current switching. Neurons were classified as T, Ph, or LAH by their discharge characteristics during graded depolarizing current steps (Fig. 1A) and by the presence or absence of certain voltage- and Ca²⁺-dependent conductances (Boyd et al. 1996; Keast et al. 1993; McLachlan and Meckler 1989).

Passive electrical properties were determined from recordings in current clamp of small amplitude (<10 mV) responses that avoided activating voltage-dependent conductances in these neurons (Cassell and McLachlan 1987; Cassell et al. 1986). Hyperpolarizing current steps were used rather than brief current pulses (Jack et al. 1975) because of current-passing limitations of high-resistance electrodes. Voltage and current were filtered (1-3 kHz) and digitized at 1 kHz.

The cell input time constant, ₀, was derived by fitting a single exponential to the averaged response between 20 and 80% of the steady-state value. A single exponential provides a close fit to these data (Fig. 1C). The cell input resistance, R_N, was calculated from the slope of the steady-state current-voltage relation in the low-conductance voltage region just negative of resting membrane potential (RMP) (Boyd et al. 1996). The values of R_N and ₀ from 58 neurons (Fig. 2) were pooled by class, and means and SE were calculated (Table 1A) for comparison with the computed output of the model.

View larger version (18K): [in this window] [in a new window]   Fig. 2. Plot of cell input time constant (0) against cell input resistance (RN) determined for 22 tonic, 18 phasic, and 18 LAH neurons in guinea pig lumbar sympathetic ganglia. The means ± SE of these data have been used as the "targets" in the model analysis. Regression lines fitted through the points (and the origin) have r2 = 0.84 (tonic), 0.88 (phasic), and 0.89 (LAH).

Morphology

Ganglia were fixed, reacted to localize biocytin, dehydrated, and cleared (Boyd et al. 1996). The stained cells were traced in the horizontal plane (Fig. 1B) at 500× using NEUROLUCIDA (MicroBrightField, Colchester, VT). No z axis correction was applied since tissue shrinkage was greatest in that direction (Boyd et al. 1996), and the dendritic arbor of these cells was generally limited in this plane.

Average morphologies for each class were previously derived (Boyd et al. 1996) (Fig. 1B; Table 1B) based on the following parameters: 1) soma major and minor axes; 2) axon length (7 mm within the preparation) and diameter; 3) total dendritic length and number of primary, branching, and nonbranching dendrites; 4) number of dendritic terminations (set to within average ± 1); 5) number and length of short (<50 µm) and long (50 µm) nonbranching dendrites; 6) number and length of branched dendrite segments of each branch order, using both centripetal and centrifugal branch order numbering (MacDonald 1983); and 7) dendritic diameter, measured approximately 100 µm from the soma.

Modeling

To focus on class differences, the aim was to match the R_N and ₀ simulated in the models with the average electrical properties and morphologies derived experimentally (Boyd et al. 1996) rather than analyzing data or fitting voltage transients from single neurons.

The average morphologies described above (see also Fig. 1B) were incorporated into compartmental models using the program NEURON (Hines and Moore 1999), which uses cylindrical representations of the soma and processes. The cell body was a cylinder with equal diameter and length, set to the average of the measured cell body major and minor axes (Table 1B), so it had the same SA as a sphere with this average diameter. The axon was given constant diameter and length equal to the average measured values for each class (Table 1B), and the dendrograms in Fig. 1B were used with the average branching patterns for each class fully implemented (see Boyd et al. 1996). The dendrites of each class had constant diameters that were equal to the measured average for that class (Table 1B).

As a compromise between accuracy and time for each simulation (Hines and Carnevale 1997), each morphological segment was subdivided into only five variable length compartments using the NEURON software (Hines and Moore 1999). The inaccuracy associated with representing long segments of axon as a single compartment was estimated by performing some simulations using 1-µm-long compartments. The differences were less than the SE of the experimental passive properties (Table 1A) and have been ignored.

Simulations were performed with a sampling frequency of 1 kHz, which matches the slowest experimental sampling. The models were entirely passive, with no voltage- or time-dependent impedances. The model parameters were assumed to be uniform throughout the cell, except where specified.

The models had five parameters: 1) C_m, the specific membrane capacitance, taken as 1 µFcm², but values between 0.75 and 2.5 µFcm² were tested (Cole 1968; Thurbon et al. 1998); 2) R_i, the cytoplasmic resistivity, the same for all three classes with values between 70 and 2,500 cm tested (Barrett and Crill 1974; Cole 1968; Stuart and Spruston 1998; Surkis et al. 1996); 3) R_m, the specific membrane resistance, uniform over the entire cell, except in simulations with a somatic shunt, and allowed to vary between classes in steps of 1 kcm²; 4) SA corrections implemented by scaling diameter or SA directly (see next paragraph); and 5) g_shunt, a somatic shunt conductance.

The constant diameter cylinder representation fails to allow for surface irregularities and for the possibility of histological shrinkage. One correction (A) was implemented by equally scaling cell body, axon, and dendrite diameters. For example, multiplying all cylinder diameters by 1.25 scaled the modeled neuron SA by 1.25, a 25% SA increase. A second method of SA correction (B) did not change diameters but artificially scaled the SA directly (as in Stuart and Spruston 1998) by increasing C_m and decreasing R_m by a common factor, thereby maintaining _m, the membrane time constant.

g_shunt accounts for a possibly leaky cell body (Clements and Redman 1989; Pongracz et al. 1991; Staley et al. 1992) and was modeled by reducing the somatic R_m uniformly. Such a shunt might, in reality, consist of a transmembrane conductance and/or a leak around the microelectrode.

Voltage responses at the soma to hyperpolarizing current steps of 250 ms duration, as in the electrophysiology measurements (Fig. 1C), were simulated. Values of ₀ and R_N were derived by fitting a single exponential to the first 100 ms of this response using the praxis curve-fitting module within NEURON. Single exponentials provided very good fits to the modeled responses. It should be noted that, in neurons with branched processes, ₀ is not necessarily the same as _m (Holmes et al. 1992).

The aim of the simulations was to find model parameter combinations and ranges that gave voltage responses with R_N and ₀ that matched the corresponding average experimental values, R_av and _av, for each neuron class. The difference between modeled and experimental passive properties was standardized to remove bias of R_N over ₀ because of larger numerical values. The values that provided the closest match were those that minimized the expression {[(R_N  R_av)/R_sem]² + [(_0  av)/_sem]²}, where R_sem and _sem are the SE of R_N and _0, respectively.

To investigate the impact of the correlation between R_N and ₀, some simulations were performed using R_N and cell input capacitance, defined as C_in = ₀/R_N, as the model outputs. The results were more or less indistinguishable from those obtained for R_N and ₀ and lay within much less than 1 SE of the experimental values (Table 1A).

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION CONCLUSION REFERENCES

Models with uniform R_m and uncorrected morphology

Initial simulations were performed with uniform R_m and R_i of 100 cm (Barrett and Crill 1974; Burke et al. 1994; Cole 1968; Stuart and Spruston 1998). The R_m and C_m values obtained were smallest for the LAH class (12 kcm² and 1.25 µFcm²), intermediate for the Ph class (18 kcm² and 1.5 µFcm²), and largest for the T class (23 kcm² and 1.75 µFcm²). The ranges of R_m and C_m that gave modeled R_N and ₀ within ±1 SE of the mean experimental values were also determined (Table 2). However, all neurons are expected to have the same C_m. Using a median C_m value of 1.5 µFcm², the simulated values matched the mean experimental R_N and ₀ for all classes within ±1 SE, with little change in R_m, but an exact match was only possible for the Ph class.

View this table: [in this window] [in a new window]   Table 2. Derived values of Rm and Cm with Ri = 100 cm and uniform Rm

The possibility of C_m being as high as 2.5 µFcm² was explored (Thurbon et al. 1998). The models could be solved with C_m > 1.5 µFcm² by increasing R_i (Table 3; Fig. 3A). For example, the Ph model could be solved with C_m = 1.75 µFcm² and an acceptable R_i of 200 cm (Rall et al. 1992) (Table 3). For C_m = 2.5 µFcm², the Ph and T responses could be matched using a biologically extreme R_i of 1,000 cm (Clements and Redman 1989; Rall et al. 1992), but the LAH responses required an improbable R_i of 2,500 cm. C_m in these neurons must be <2 µFcm², purely on the basis of unrealistically high R_i.

View larger version (18K): [in this window] [in a new window]   Fig. 3. Relation between Ri and Cm and the effects of surface area (SA) correction. The lines in this and subsequent figures interpolate between simulated data points for each class. A: combinations of Cm and Ri that provided solutions to the uncorrected models with uniform Rm. Only the LAH curve is shown because this limit applies to all 3 classes (see Table 3). Hatching illustrates the combinations of Cm and Ri that do not provide solutions for any model, and the arrows indicate the combinations that can theoretically match the experimental data for all classes when a SA correction, as in B and C, is applied. B: combinations of Cm and SA correction factor (correction A) that provided solutions to the uncorrected models with uniform Rm and constant Ri. Curve illustrates the case for Ri = 100 cm. Note that there are solutions for all 3 classes only when Cm < 1.25 µFcm2. C: combinations of Ri and SA correction factor (correction A) that provided solutions to the uncorrected models with uniform Rm and constant Cm. Curve illustrates the case for Cm = 1 µFcm2. Error bars in this and subsequent figures represent the values derived to match the mean passive properties ± SE (see Fig. 4).

There is mounting evidence that C_m = 1 µFcm² for almost all cells (Gentet et al. 1999; Okamoto et al. 1977; Sukhorukov et al. 1993). Using C_m = 1 µFcm², the LAH response could not be matched exactly, but it could be matched within ±1 SE when R_i =70-100 cm. However, a match was not possible for the T or Ph classes when C_m < 1.25 µFcm², except using unrealistically low R_i (<20 cm). This conflicts with the idea that C_m = 1 µFcm² in all neurons. It can be concluded that a systematic error exists that is greater for the T and Ph than for the LAH classes.

Surface area correction is justified

A likely source of systematic error is SA underestimation. Two possible sources of SA underestimation are histological specimen shrinkage and unmapped surface convolutions.

HISTOLOGICAL SHRINKAGE Tissue shrinkage was primarily in the z axis (Boyd et al. 1996). Cell shrinkage was not measured directly in these preparations. However, it is likely to be small (<5%) and independent of tissue shrinkage, because the histochemical staining of intracellular biocytin was undertaken prior to dehydration (Grace and Llinas 1985). A correction for histological shrinkage was therefore considered unnecessary.

UNDETECTED/IGNORED SURFACE AREA Sympathetic neurons deviate from the computer model comprised of constant diameter cylinder segments in several ways. To varying extents, neurons of all three classes have irregular somatic surfaces and several types of dendritic irregularity, including varicosities, lumps, and fine branches (Boyd et al. 1996). As indicated in a summary of the fine surface morphology of these neurons (Table 4), a correction for ignored SA is likely to be necessary. Further, the different classes probably require different corrections. On average, T neurons had the most irregularities and surface convolution, and LAH neurons had the least (Table 4). Determination of the complete extent of surface membrane invaginations would have required complete reconstructions of individual neurons using the electron microscope, which was beyond the scope of this study.

Models with uniform R_m and surface area correction

To account for SA underestimation, simulations were repeated using a variable SA correction applied (A) by increasing the diameters of all morphological segments and (B) by increasing SA directly (see METHODS).

Increasing diameter of each compartment

INTERACTION BETWEEN SURFACE AREA CORRECTION A AND C_M Increasing SA decreased the required values of C_m (Fig. 3B). With R_i = 100 cm, solutions were obtained using C_m values that decreased by about 0.05-0.2 µFcm² for each 10% of SA increase. The T class required the most correction and the LAH class required the least. The LAH class could not be solved with C_m > 1.25 µFcm² because this would require a negative correction. For larger R_i, the effect on C_m was more pronounced.

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View larger version (22K): [in this window] [in a new window]   Fig. 4. Matching the passive electrical properties of compartmental models of each neuron class, with experimental RN and 0, by varying Rm and increasing SA. Hatched boxes are target regions for each model, centered on the average experimental RN and 0 and extending from the center by ± experimental SE for each neuron class. Cm = 1 µFcm2, Ri = 100 cm, and uniform Rm. Optimal Rm and SA corrections are derived from those simulation points nearest the target region centers, and upper and lower limits are derived from the simulation points nearest the target region edges (see Table 5). Lines link simulation points derived using the same Rm but different SA corrections. Rm is distinct, and SA corrections differ for each neuron class.

INTERACTION BETWEEN SURFACE AREA CORRECTION A AND R_I Increasing SA with C_m fixed increased the required values of R_i (Fig. 3C), as was the case when C_m was increased when there was no SA correction (Table 3). With C_m = 1 µFcm², a 10-fold increase in R_i (from 100 to 1,000 cm) corresponded to an increase in the SA correction of about 30% (Fig. 3C). With C_m < 1 µFcm², this relationship was less sensitive to R_i, and for C_m > 1 µFcm², it was more sensitive. However, for all C_m, changing R_i over a wide range had relatively little effect on the SA correction (see Fig. 3C). These models tolerate almost any value of R_i. R_i is limited by what is biologically realistic rather than any constraint of the modeling.

UPPER LIMITS OF SURFACE AREA CORRECTION A The solutions for C_m = 1 µFcm² and R_i = 100 cm already described (Table 5) suggest that 65-80% [= (1 + SA correction)¹] of the neuron SA is accounted for by an uncorrected cylindrical representation. However, larger SA corrections could be accommodated by a corresponding large increase in R_i and/or reduction in C_m. The lowest realistic C_m value (0.75 µFcm², e.g., Almers 1978) and highest imaginable R_i value (1,000 cm) were used to calculate an upper limit to the SA correction (Fig. 3, B and C). With these values of C_m and R_i, the SA corrections were 100% (LAH), 120% (Ph), and 150% (T). In other words, in the most extreme case, the cylindrical representation accounted for only 40-50% of the neuron SA.

Direct scaling of SA

The axial resistance of a cylindrical cable with diameter, d, is proportional to R_i/d², so it is possible that some of the modeled insensitivity to high R_i values using correction method A (Figs. 3C and 5B) is a consequence of the increase in axon and dendrite diameters. The second SA correction method was implemented by scaling the SA independently of diameter.

Simulations for all three classes using the preferred R_i and C_m values: R_i = 100 cm and C_m = 1 µFcm² yielded values of R_m for the three classes that were within 1 kcm² of the values obtained using correction A. However, about 10% greater SA corrections were necessary with correction B. The T model again required the largest correction, and the LAH model the least, as for correction A.

Thus the two methods of SA correction gave basically the same results, suggesting that increasing the diameter (correction A) may only be inappropriate if R_i is unrealistically high. Method A is also likely to be unsuitable for estimating electrotonic attenuation along the dendrites (Rall et al. 1992), because of underestimation of axial resistance.

Differences in R_m between neuron classes

As already stated, R_m values and corresponding R_m ranges were derived for R_i = 100 cm and C_m = 1 µFcm² by applying SA corrections (Fig. 4). The optimal R_m values were 40, 27, and 15 kcm² for T, Ph, and LAH classes, respectively. These were obtained using SA corrections (A) of 55, 45, and 25%, respectively (cf. Table 5 and Table 4). The R_m for each class was different, and the R_m ranges were distinct.

EFFECT OF VARYING C_M AND R_I ON DERIVED R_M The effects on R_m, of varying C_m and R_i from 1 µFcm² and 100 cm, respectively, were investigated to make comparisons with other studies. The derived R_m decreased as C_m increased (Fig. 5A). In contrast, R_m, derived for large values of R_i (2,500 cm), differed little from the values derived at lower R_i (Fig. 5B). Even though the predicted R_m changed when C_m was varied, the ranges of R_m for each class remained distinct for all values of C_m (0.75-2.5 µFcm²) and R_i (70-2500 cm), and the relative R_m differences between classes were preserved (Tonic R_m > Phasic R_m > LAH R_m).

View larger version (15K): [in this window] [in a new window]   Fig. 5. Effects of varying Cm and Ri on derived Rm. A: effect of varying Cm with Ri = 100 cm. Derived Rm decreases as Cm is increased. Note there are solutions for all 3 classes only when Cm < 1.25 µFcm2 (see Fig. 3, A and B). B: effect of varying Ri with Cm = 1 µFcm2. Derived Rm is insensitive to Ri, consistent with these neurons being either isopotential or electrotonically compact.

Models with a somatic shunt conductance

The models with uncorrected morphology could not be solved with C_m approximately 1 µFcm². SA underestimation successfully accounted for the discrepancy, but an alternative or contributing error may have resulted from a somatic shunt conductance, g_shunt. A somatic shunt conductance, whether intrinsic (due to higher conductance of the soma membrane) or artifactual (introduced by the microelectrode), would reduce the measured R_N and lead to a lower R_m estimate. A high value of C_m (as determined in many studies) would then be required to match the measured ₀.

EFFECTS OF G_SHUNT ON C_M AND R_I g_shunt affected the models with uncorrected morphology in a complex manner. The relationship between C_m and R_i (Fig. 3A) became skewed to lower C_m but never <1.5 µFcm², and only for R_i  200 cm and g_shunt > 3 nS. Otherwise, the only effect of g_shunt was to increase R_m (see next paragraph). Models with g_shunt could not be solved at all for C_m = 1 µFcm² unless a SA correction was also applied.

2

2

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EFFECT OF G_SHUNT ON R_M. The main effect of a shunt was to increase the R_m required to solve the models (Fig. 6). R_m increased dramatically as g_shunt was increased toward the mean observed input conductance for each class [G_N = 1/(average R_N) = 7 nS (T), 6 nS (Ph), and 10 nS (LAH)]. For the models with corrected SA, the R_m ranges for each class were distinct for g_shunt  1 nS, but for g_shunt > 3 nS, the R_m ranges for the Ph and T classes overlapped. With g_shunt  5 nS and either or both of R_i >1 00 cm and C_m > 1 µFcm², the R_m derived from the Ph class exceeded that from the T class. However, the range of R_m for the LAH class remained distinct regardless of the shunt size. R_m could be the same for all classes only if g_shunt < 1 nS for the T neurons, g_shunt = 1-2 nS for the Ph neurons, and g_shunt > 4 nS for the LAH neurons.

View larger version (19K): [in this window] [in a new window]   Fig. 6. Influence of a somatic shunt conductance, gshunt on the predicted Rm ranges for each neuron class. Cm = 1 µFcm2, Ri = 100 cm. Introduction of a shunt increases Rm in each case, particularly as gshunt approaches the experimental input conductance (~6 nS). Points at gshunt = 0 nS (uniform Rm) correspond to the data in Table 5 and Fig. 4. Rm ranges derived for all classes are distinct if gshunt  1 nS. Rm range derived for the LAH model is distinct from those derived for the other classes, for all sizes of somatic shunt.

SIZE OF G_SHUNT There is, however, a real limit on the size of g_shunt. The experimental R_N ranged 317 M for Ph, 339 M for T, and 263 M for LAH cells, so the corresponding minimum input conductances, G_N, were 3.2, 2.9, and 3.8 nS, respectively. If g_shunt is similar for all neurons (individually and by class), then g_shunt must have been <3 nS. When this is the case, R_m again remains distinct for all classes.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION CONCLUSION REFERENCES

This study has shown that voltage transients in sympathetic neurons can be modeled using commonly accepted biophysical parameters, resulting in discrete ranges of specific membrane resistance (R_m) for each of three phenotypic classes (T, Ph, and LAH). The derived values of R_m were only about 15% higher in each case than values derived assuming that the cells were isopotential. This implies that the neurons are electrotonically compact. It was also concluded that a large proportion of the total SA of these neurons exists as fine surface features.

Measured values of R_m range from 1 kcm² in squid giant axon (Cole and Hodgkin 1939) to 22 and 9 kcm² in patches of somatic membrane from brain stem motoneurons of neonatal and juvenile rats, respectively (Singer et al. 1998). Calculations based on simulations similar to the present ones have given values ranging from 0.1 kcm² in cat spinal motoneuron somata (Clements and Redman 1989) to 200 kcm² in hippocampal pyramidal neurons (Major et al. 1994). Even the most extreme values achieved in the present study were well within this range.

Differences in R_m between neuron classes

The difference in R_m between the classes was robust and could not be invalidated by varying uniform model parameters. The differences cannot be ascribed to morphological differences as these were the basis of the models. Although sympathetic neurons are all derived from the same kind of neuroblast (Leblanc and Bronner-Fraser 1992), the conclusion is that the complement and number of channels open at RMP constitute another phenotypic difference between the three classes of sympathetic neuron defined on the basis of their discharge characteristics.

It is known that certain types of channel active at resting potential are expressed differentially across the three classes of neuron.

1) In LAH neurons, a Ca²⁺-dependent K⁺ conductance (gKCa²) that is responsible for the long afterhyperpolarization is active at rest (Davies et al. 1999); this conductance is similar to the "K-creep" current that accounts for 10-20% of the resting conductance in some enteric neurons (North and Tokimasa 1987). Consistent with this, nifedipine block of L-type Ca²⁺ channels that virtually abolishes the slow afterhyperpolarization affects passive properties only in LAH neurons, depolarizing them slightly and reducing resting conductance by approximately 15% (Davies et al. 1999).

2) In T neurons, the voltage characteristics of inactivation of A-type K⁺ channels are such that a proportion of these channels are open at RMP (Cassell et al. 1986). Thus blockade of these channels with catechol depolarizes T neurons, reducing their resting conductance and increasing the input time constant (Inokuchi et al. 1997). Neither catechol nor block of Ca²⁺ entry have significant effects on the resting conductance of Ph neurons (Davies et al. 1999; Inokuchi et al. 1997).

3) Fewer than 10% of guinea pig sympathetic neurons exhibit time-dependent rectification (I_H) (Inokuchi et al. 1997), and the time course and amplitude of the afterhyperpolarization in any cell class are not affected by the addition of 2 mM Cs⁺ to block I_H (Davies et al. 1999).

Most of these channels will not be affected by the small amplitude hyperpolarizations of the soma used to determine R_m. For example, no more A channels would have been opened than were already open at RMP. Thus it is clear that distinct populations of active channels may contribute to the resting membrane characteristics in each class of sympathetic neuron.

It must be kept in mind that the distribution of these channels is probably not uniform over the somatic and dendritic membrane. For example, the voltage-dependent Ca²⁺ channels activated during the action potential in rat sympathetic neurons have high thresholds and are located at a site electrically distant from the soma (Hirst and McLachlan 1986), and this may be reflected in the location of Ca²⁺-activated K⁺ channels. In hippocampal neurons, Ca²⁺-activated K⁺ channels have been localized to the proximal dendrites (Poolos and Johnston 1999). At this stage, the precise location of the various voltage- and Ca²⁺-activated channels in each class of sympathetic neuron is not yet known.

Although it was assumed here, for simplicity in the modeling, that R_m was spatially uniform over the entire dendritic tree, there is evidence that this is not the case in sympathetic neurons. Leaky distal dendrites were identified in rat paravertebral neurons by the disproportionately larger effect of pharmacological K⁺ channel blockade on R_N than on ₀ (Redman et al. 1987). A similar result was obtained in T neurons following blockade of A-type K⁺ channels, in that R_N increased by 30%, whereas ₀ increased by only 23% (Inokuchi et al. 1997). This is consistent with the location of these channels in the dendritic membrane, as has been shown for hippocampal neurons (Magee et al. 1998). For neurons, such as sympathetic neurons, with voltage transients that relax with a single relatively large time constant (Fig. 1C) and with distal dendritic membrane that is "leakier" than the proximal dendritic segments, it has been calculated (London et al. 1999) that the total resting conductance is higher than that expected if R_m is uniform. In other words, there are likely to be more ion channels open at RMP, particularly for T neurons, than suggested by the results of the uniform R_m modeled in this study. However, removal of this nonuniformity by blockade of open A channels left T neurons with even lower conductance than the other classes of sympathetic neuron (see 2 above), indicating that the presence of active channels in the resting membrane cannot account for the differences in R_m between classes determined here.

Surface area estimates

Compartmental electrical models incorporating simple cylindrical segments based on gross morphological measurements were unable to mimic the electrical properties of the cells using reasonable values for C_m and R_i, unless surface irregularities were also accounted for. One correction (A) involved increasing SA by scaling all segment diameters by a variable but uniform amount. The corrections applied differed between classes in accord with the observed details of fine surface morphology, including infolding of the soma surface, dendritic varicosities, short fine processes, and lumpy or bulbous dendrites (Boyd et al. 1996; Gibbins et al. 1998; Kawai et al. 1993).

SAs recently estimated from three-dimensional confocal imaging of sympathetic neurons were much larger than the values estimated here (Anderson et al. 2001; Ermilov et al. 2000). This might have been expected from the higher resolution of the imaging technique. However, because of the limitations of the light microscope, even this technique is unlikely to detect fine surface irregularities at the sub-micron scale. In fact, the somatic SAs and estimates of the dendritic lengths derived in these and other studies (Jobling and Gibbins 1999; Miller et al. 1996) match ours very well. The major difference is in estimates of dendritic SA. It is known that SA estimates based on reconstruction of digital images are prone to errors of bias, particularly for single projection images of nonisotropic objects and for images of nonuniform intensity (Roberts et al. 2000), which were the case for these confocal measurements of sympathetic neurons. In addition, the SA estimates derived from confocal images depend on section thickness and pixel size so that, at low magnifications such as those used to capture the dendrites, the imaged object can be markedly distorted. Finally, neuronal SAs as large as derived from these images cannot be supported by our modeling, unless the electrical parameters are allowed to vary to unrealistic values. The requirements would be that C_m be well below 0.75 µFcm², R_i be beyond 2,500 cm, and R_m be at least twice as high as in our study (Fig. 3).

We have taken R_i = 100 cm as the best estimate for mammalian neurons. However, if some of the dendrites of these neurons are densely packed with mitochondria, as recently noted for these (Gibbins et al. 1998) and other neurons (Surkis et al. 1996), effective R_i might be much higher, especially in the distal dendrites. This might explain some of the SA discrepancy above. If mitochondrial crowding were a factor, dendritic R_i would be important in determining the electrotonic attenuation of synaptic inputs.

Contribution of somatic shunt conductance

As mentioned above for the case of leaky dendritic membranes (London et al. 1999), higher R_m values were derived when a somatic shunt conductance was included in the models. However, the results provided no evidence either way to confirm or deny the presence of a shunt, as was the case in at least one previous study (Major et al. 1994). Some modeled and experimental data agree more closely if a somatic shunt is incorporated (Clements and Redman 1989; Thurbon et al. 1994, 1998). A somatic shunt might result if the somatic R_m was intrinsically lower than the dendritic R_m or if there was a real but artifactual shunt produced by insertion of the microelectrode. Any microelectrode shunt conductance would probably have to include a transmembrane current, rather than just a leak through a hole surrounding the microelectrode, because the latter would be expected to have had a larger effect on RMP than was observed (Clements and Redman 1989; Pongracz et al. 1991; Staley et al. 1992). Sympathetic neurons have similar firing rates when recorded either intracellularly (McLachlan et al. 1997) or extracellularly (Häbler et al. 1994) in vivo, suggesting that RMP and the effectiveness of synaptic inputs are not significantly changed by microelectrode penetration.

It has been suggested that a somatic shunt conductance is not present in whole cell recordings made with patch electrodes (Thurbon et al. 1998). However, dialysis through whole cell patch electrodes has been shown to increase cell input resistance, probably by reducing second messenger systems that control K⁺ channels (Robinson and Cameron 2000). Similarly, in vivo whole cell recording of sympathetic postganglionic neurons (Gola and Niel 1993) demonstrated "pacemaker" firing properties that have never been observed with either microelectrodes (Cassell et al. 1986) or extracellular recording techniques in vivo (e.g., Jänig et al. 1991). In fact, the reported values for passive electrical properties are quite similar when recorded with either high resistance "sharp" microelectrodes or patch pipettes, at least for sympathetic neurons (cf. Keast et al. 1993; Vanner et al. 1993), provided allowance is made for the absence of dendrites following dissociation and the neurons are derived from animals of comparable age. Finally, when K⁺ channels in these neurons are pharmacologically blocked, the input resistance measured with an intracellular microelectrode can rise as high as R_N = 1 G (G_N = 1 nS; unpublished observations). If g_shunt is considered to be entirely due to leakage around the microelectrode rather than being a transmembrane conductance, blocking K⁺ channels should not change the shunt. G_N = 1 nS implies that g_shunt < 1 nS. Thus while the possibility of microelectrode impalement artifact cannot be discounted, it seems likely that such artifacts were small in the present experiments.

Whereas the physical effects of impalement are unlikely to differ systematically between classes of sympathetic neuron, g_shunt might differ if the cell bodies of neurons of each class express different resting leak conductances as our data suggest. In the models, if g_shunt was allowed to differ between classes, it became possible to derive a uniform dendritic R_m of about 35-40 kcm² for all neuron classes. The corresponding somatic shunts were g_shunt = 4-5 nS for LAH neurons, g_shunt = 1-2 nS for Ph neurons, and g_shunt <1 nS for T neurons (Fig. 6). This is consistent with a somatic shunt of 1-2 nS (R_shunt = 0.5-1 G) in all neurons and an additional shunt of about 3 nS on the LAH cell bodies due to activation of the Ca²⁺-activated K⁺ conductance. Activation of an additional K⁺ conductance by the influx of Ca²⁺ at the time of impalement would be expected to hyperpolarize the LAH neurons, but their RMPs are not more negative than those of other classes of neuron (Davies et al. 1999). Overall, it seems more likely that there is a real difference in the density of passive leak channels between the three neuronal phenotypes.

	CONCLUSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION CONCLUSION REFERENCES

The present modeling study has revealed another distinct difference in the electrophysiological characteristics of the three phenotypes of sympathetic neuron. Not only are different populations of voltage- and Ca²⁺-dependent conductance present in each class, but characteristic nonuniformities in the distribution and type of membrane channels (including passive leak channels) contribute to the resting conductance. This property is of prime importance for the integration of subthreshold synaptic responses. However, in Ph neurons in rat SCG, activity in vivo arises almost exclusively from large suprathreshold ("strong") preganglionic inputs (McLachlan et al. 1997), and summation of subthreshold ("weak") inputs is rare. The same is probably true for LAH neurons that also receive a large strong input (McLachlan and Meckler 1989). However, T neurons in prevertebral ganglia receive many subthreshold synaptic inputs from the intestine which must summate, together with relatively small amplitude responses arising from the preganglionic inputs, to activate the cells (McLachlan and Meckler 1989). These neurons also receive peptidergic inputs during distension of the intestine which modulate resting conductance and amplify the fast inputs (Kreulen and Peters 1986). Combined with the higher effective R_m, this synaptic repertoire provides T neurons with considerable flexibility in their mechanisms for integration. Thus the different classes of sympathetic neuron are well designed to operate variously as relays (strong inputs to neurons with relatively low R_m) or by integration (modulation of summed weak inputs and a high R_m).