Abstract
The mean location of Renshaw synapses on spinal motoneurons and their synaptic conductance were estimated from changes in impedance magnitude produced by sustained recurrent inhibition. Motoneuron impedance was determined by injecting quasiwhite noise current into lumbosacral motoneurons of pentobarbitalanesthetized cats. Synaptic location and conductance were estimated by comparing observed impedance changes to simulation results obtained using standard motoneuron models and compartmental models fit to each impedance function. Estimated synaptic locations ranged from 0.10 to 0.41λ, with a mean of 0.19 or 0.24λ, depending on the estimation method. Average dendritic path length was 262 μm. Average synaptic conductance was 23 to 27 nS (range: 6.7 to 57.9 nS), corresponding to conductance changes of 78 to 88% of resting membrane conductance. Estimated accuracy was supported by consistency using different estimation methods, agreement with Fyffe's 1991 morphological data, and comparisons of observed and simulated recurrent IPSP amplitudes. Synaptic location, but not synaptic conductance, was correlated with rheobase, a measure of motoneuron excitability. Synaptic conductance did not depend on synaptic location. A regression analysis demonstrated that synaptic conductance and cell impedance were the principal factors determining recurrent IPSP amplitude. Simulations using the observed values and locations of Renshaw conductance demonstrate that recurrent inhibition can require as much as an additional 14 to 18% sustained excitatory synaptic conductance to depolarize motoneurons sufficiently to activate somatic or dendritic inward currents and recruit motoneurons or amplify excitatory synaptic currents.
INTRODUCTION
Recurrent inhibition by Renshaw interneurons is an apparently simple spinal circuit, and its anatomy and physiology have been extensively documented (Windhorst 1996). Despite this, its specific function is still under debate. The recurrent inhibitory input to motoneurons is small, as assessed by either the amplitude of postsynaptic potentials (e.g., Eccles et al. 1961; Hamm 1990; Hultborn et al. 1971) or the magnitude of synaptic current reaching the soma (Lindsay and Binder 1991). However, these measures do not consider the potential of recurrent inhibition to inhibit by shunting excitatory synaptic inputs originating in the dendrites. Morphological studies show that Renshaw synapses on motoneurons are dendritic, averaging 0.25 electrotonic length constants from the motoneuron soma (Fyffe 1991). This position would allow Renshaw synapses to shunt excitatory synaptic inputs that are located more distally on those dendrites.
Estimates of the conductance change produced by recurrent inhibition have shown negligibly small increases or paradoxical results. Sustained recurrent inhibition produces only a small decrease in input resistance measured at the soma (Lindsay and Binder 1991), and measurements during transient recurrent inhibitory postsynaptic potentials (RIPSPs) have yielded insignificant changes in input resistance or increases in neuron impedance (Friedman et al. 1981; Smith et al. 1967). However, in the companion paper (Maltenfort et al. 2004) we have shown that relatively large synaptic conductances on the dendritic tree (e.g., a doubling of membrane conductance over a span of 0.15 electrotonic length constants, applied equally to all dendrites) produce only small changes (<5%) in the input resistance measured at the soma.
Following the work of Fox (1985), Maltenfort et al. (2004) demonstrated that changes in motoneuron impedance can be used to estimate the electrotonic location and relative magnitude of synaptic conductances. The measured impedance at the motoneuron soma is increased at higher frequencies by an active synapse on the dendritic tree, and the frequency at which this takes place, the reversal frequency, is inversely proportional to the mean distance of the synaptic conductance from the soma. For a known electrotonic position, the relative change in impedance below this frequency increases linearly with the relative magnitude of the synaptic input.
In the current study, RIPSPs and resulting impedance changes in motoneurons were measured. The goal was to estimate the location and magnitude of conductance changes produced by recurrent inhibition, and to assess the ability of recurrent inhibition to shunt excitatory synaptic inputs to the dendritic tree of motoneurons. Our estimates of the location of Renshaw synapses obtained using impedance measurements are in good agreement with the estimates made by Fyffe (1991) from morphological data. The estimated conductance of recurrent inhibition causes a modest shunting of somatic and dendritic potentials produced by excitatory synaptic inputs. Preliminary accounts of the work were previously presented (Hamm and McCurdy 1992; Hamm et al. 1993).
METHODS
Experimental preparation
Experiments were performed in 10 adult pentobarbitalanesthetized cats of either sex. The first 6 animals were used in the development ofthe experimental techniques; the data presented in this paper were obtained from the last 4 animals in this series. All experimental procedures were reviewed and approved by the Institutional Animal Care and Use Committee at St. Joseph's Hospital and complied with principles from the Guide for the Care and Use of Laboratory Animals (National Research Council 1996). Anesthesia was induced in a chamber with 5% isoflurane mixed with nitrous oxide and oxygen. After induction, 3% isoflurane was delivered by a mask while a tracheal cannula was inserted. After insertion of a tracheal cannula, isoflurane was maintained between 1.7 and 2.3% as needed. A cannula placed in one carotid artery was used to monitor blood pressure.
Pentobarbital was given gradually, intravenously, to replace the isofluorane and periodically supplemented throughout the experiment. The initial dosage was adjusted to effect a deep surgical anesthesia after complete removal of isoflurane (29–43 mg/kg). A neuromuscular blocker (gallamine triethiodide) was given during recording to provide mechanical stability. Paralysis was allowed to subside after each administration to permit assessment of reflex signs and muscle tone in addition to heart rate and blood pressure for administration of supplemental anesthesia.
Expired CO_{2} was monitored and maintained at 3–4% throughout the experiment. Cats were infused with a slow drip infusion of a buffer solution (5% glucose, 0.85% NaHCO_{3}) or lactated Ringer solution throughout the experiment. Flaps of skin were tied up to form pools of mineral oil around the exposed spinal cord and dissected muscle nerves in the hindlimb. Core temperature and temperatures of the spinal and hindlimb pools were maintained at 37°C with a heating pad and radiant heat. The muscle nerves to anterior and middle biceps femoris (ABF), lateral gastrocnemius (LG), medial gastrocnemius (MG), and soleus (Sol) were sectioned and placed on bipolar hook electrodes for stimulation. The dorsal roots (L4–S2) were sectioned so that RIPSPs could be produced by stimulation of muscle nerves.
Recording procedures
Intracellular recordings and current injection were performed with glass micropipettes with beveled tips (3–7 MΩ) filled with 2 M potassium citrate. Recordings were made with an intracellular amplifier with a highimpedance probe (Axoclamp 2A or 2B) and a sampleandhold amplifier, which provided amplification without loss of low frequency components in the recording of motoneuron potentials. Electrode position was controlled by a stepping motor or piezoelectric microdrive. After penetration of each motoneuron, the following recordings were made: antidromic action potential, for determination of its amplitude and conduction velocity; heteronymous RIPSPs produced by stimulation at 3 times threshold for the motor volley (and the homonymous RIPSP if the threshold for antidromic invasion was high enough to allow recording of a near maximum RIPSP without antidromic invasion); responses to injection of current pulses, for the determination of input resistance and rheobase; and the voltage responses to injection of a mixture of sinusoidal currents, with and without concurrent stimulation of a muscle nerve at 200 Hz to produce recurrent inhibition, for calculation of the impedance function of the motoneuron. A stimulus frequency of 200 Hz was selected to produce a Renshaw conductance as steady as possible. The responsiveness (spikes/cycle) of Renshaw cells to vibratory orthodromic and repetitive antidromic stimuli decreases at higher frequencies, but the average Renshaw cell should be responsive to each stimulus at 200 Hz (Pompeiano et al. 1975; cf. Lindsey and Binder 1991).
If the resting potential was stable throughout the recording, then a second record was made of the same motoneuron receiving a heteronymous RIPSP from a different muscle nerve. Of the 30 cells examined in this study, recording conditions in 14 cells were stable enough to allow more than one heteronymous nerve to be used for stimulation during separate impedance tests. The electrode was withdrawn from the cell at the end of the final record to determine the net resting potential. Records were accepted for analysis only if the net resting potential was >50 mV.
Input resistance was determined from the responses to injection of 4–6 rectangular current pulses of 50 ms each (range of ±2 nA). Voltage responses were measured after adjustment of capacitance compensation to minimize the initial capacitive transient, and adjusting the bridge balance to remove the offset produced by electrode resistance. Input resistance was estimated as the slope of the regression line between the amplitudes of the current steps and the corresponding voltage responses. These values were similar to impedance magnitude at 10 Hz (Z(10) = 0.270 + 0.968 × R_{N}, r^{2} = 0.840), although the impedance values were larger on average (1.44 vs. 1.21 MΩ, P < 0.0005, paired ttest). Differences between input resistance and Z(10) may have arisen from the use of bridge mode versus discontinuous current clamp (see following text), respectively, for the 2 measurements and small errors in the adjustments needed for each recording mode (e.g., bridge balance).
Recordings were made of RIPSPs produced by 1 to 2Hz stimulation of each heteronymous nerve and the homonymous nerve (if the threshold for antidromic activation was sufficiently high). The nerve producing the largest average heteronymous RIPSP (16–32 samples) was selected for stimulation during the determination of impedance in 26 of 44 records. In the remaining cases, the nerve producing the smaller RIPSP was selected for use in this test to obtain a more representative distribution of RIPSP amplitudes. RIPSP amplitudes produced by 200Hz stimulation during the impedance tests were also measured. Mean amplitudes were determined for: the peak of the RIPSP; the sustained RIPSP just before the onset of white noise; and the sustained RIPSP just after the end of white noise. The sustained RIPSPs were determined from averages of 125 points (50 ms) in each record.
Classification of motoneurons
Motoneurons were classified by type based on the ratio of rheobase to input resistance (Zengel et al. 1985), with neurons classified as FF if this ratio was above 18, as S if the ratio was below 5.6, and as FR at intermediate values. For the 30 cells in this study, 5 were classified as S, 14 as FR, and 11 as FF. Rheobase was not measured in 3 cells with blocked spikes; these were classified as FR on the basis of input resistance and conduction velocity (0.86, 1.69, 1.30 MΩ; 102, 98.9, 91.2 m/s).
Zengel et al. (1985) based their classification on MG motoneurons. Whether this classification is valid for all motor nuclei has not been established, although Dum and Kennedy (1980) found that input resistance increased in the order FF < FR < S for motoneurons associated with tibialis anterior and extensor digitorum longus. The distribution of rheobase and input resistance values for our mixed set of motoneurons was similar to that found by Zengel et al. (1985).
Estimates of impedance
A discontinuous current clamp was used to inject a quasiwhite current into the motoneuron for impedance determinations. Each current waveform consisted of a sum of 300 equalamplitude sinusoids of 2.44–732, each an integer multiple of 2.44 Hz. Fifty unique current waveforms were used; the phase of each sinusoid in each waveform was determined by a pseudorandom number generator. Each waveform was used twice, once during synaptic activation, as shown in Fig. 1, and once without synaptic activation. The power in the current waveforms was constant across trials, usually with an amplitude of 1 nA^{2}.
Before each motoneuron was impaled, the voltage monitor of the amplifier was used to adjust capacitance compensation to optimize the settling rate of the electrode and to choose a suitable sampling rate for the discontinuous current clamp. Sampling rates ranged from 2.4 to 7.2 kHz; for records accepted for complete analysis, rates ranged from 2.9 to 7.2 kHz (mean of 5.4 kHz). In each trial, the noise waveform (lasting 820 ms) was injected 600 ms after reset of the sampleandhold amplifier. In trials with recurrent inhibition, a muscle nerve was stimulated for 1.5 s commencing within 10 ms after the sampleandhold reset. With this timing inhibition approached a nearconstant level before current injection. Voltage and current were lowpass filtered (800 Hz, fourth order Butterworth) and sampled at a rate of 2.5 kHz.
The impedance functions, Z_{n}(f), were estimated as the transfer functions between voltage and current (Bendat and Piersol 1986). Coherence functions (squared magnitude of the cross spectral density divided by the product of the power spectral densities of current and voltage) were used to determine whether noise or nonlinearities affected the impedance estimates. If the estimated coherence function [γ^{2}(f), analogous to an r^{2} value in the time domain; see Bendat and Piersol 1986] between measured voltage and injected current did not reach 0.95 in the first 30 Hz of the power spectrum, the record was excluded from the study.
For each voltage or current record, the first 512 points (205 ms) of white noise trials were discarded so that initial transients would not affect the power spectra. The remaining 1,536 points were used to form 5 overlapping (by 50%) 512point records. This procedure provided effectively 4.1 segments per record for calculating impedance (Press et al. 1992). Each record was multiplied by a Welch (parabolic) window (chosen to minimize loss of spectral resolution; Press et al. 1992) to remove the “picketfence” effect introduced by the finite length of the time series (Bendat and Piersol 1986; Press et al. 1992).
The power of the voltage response to current injection was analyzed in each set of trials to ensure that the response was not affected by progressive changes in electrode or cell impedance. A reverse arrangements test (Bendat and Piersol 1986) was used to identify a statistically stationary (P < 0.05) subset of trials, which was used to calculate the impedance estimates.
Quantifying impedance changes
The change in impedance caused by an active synapse was quantified using a cumulative normalized, frequencyweighted measure of the change in impedance magnitude at frequencies less than the reversal frequency, F_{r} (Maltenfort et al. 2004) (1) where ΔZn is the normalized change in impedance magnitude [Z(f) − Z_{syn}(f)]/Z(f_{0}); f_{0} is the lowest nonzero frequency in the spectrum; Z(f) and Z_{syn}(f) are impedance magnitudes without and with Renshaw conductance, respectively; and Δf/f is the frequency interval divided by the frequency at each point, the sum of which is taken from f_{0} to F_{r}. The statistical errors in the impedance function estimates, although small relative to the impedance functions, were large compared with ΔZn. The cumulative sum in Eq. 1 was chosen to minimize the variance in the estimate of the impedance change, but it is still sensitive to noise in ΔZn at low frequencies, which contribute most to cuΔZ. To reduce further the variance of this estimate, ΔZn was smoothed with a 5point running average and the summation of Eq. 1 was performed using the trapezoidal rule to approximate integration.
The reversal frequency was estimated as the frequency f = F_{r} at which Z_{syn} exceeded Z. To determine F_{r}, ΔZn was smoothed using median filters (Hämäläinen et al. 1994) of width 34–100 Hz (7–21 spectral points). Each ΔZn record was inspected using filters of several widths, starting with a narrow filter and progressively increasing filter width. The filtered ΔZn waveforms (Fig. 3) and apparent reversal frequencies were compared to determine the narrowest filter that provided a clear and consistent reversal frequency. A median filter was preferable to weighted averaging windows for determining reversal frequency. Although the median filter may distort the dynamics of a signal, for our purposes it was sufficient to identify the transition across zero, that is, the frequency at which more than half the points in the window were positive.
The reliability of estimates of synaptic location and conductance deteriorates as the SD of cuΔZ (σ_{cuΔZ}) increases in relation to cuΔZ (Maltenfort et al. 2004). To assess the reliability of each estimate, σ_{cuΔZ} was estimated for each set of recordings. The variance of impedance magnitude, σ_{Z}^{2} was computed from the squared product of impedance magnitude and the normalized random error of the impedance magnitude E_{Z}, where E_{Z} is determined by the coherence between the injected current and voltage response and the number of samples used to calculate the impedance function (Maltenfort et al. 2004; Eq. 4). The variances of the impedance records with and without synaptic activation were added to compute the variance of the change in impedance magnitude σ_{ΔZ}^{2}. The normalized change in impedance σ_{ΔZn}^{2} was approximated by (2) Equation 2 neglects the contribution of variability in Z(f_{0}) to σ_{ΔZn}^{2}. However, these contributions are small and contributed no more than 2–3% in the present data. σ_{ΔZn}^{2} was determined for each point of ΔZn from the lowest frequency to the reversal frequency. The variance of cuΔZ was then computed by summing these terms, weighting each by a factor based on the number of times each term was used in the summation of Eq. 1 (determined by the use of the moving average and trapezoidal rule) and the Δf/f term. The square root of the variance σ_{cuΔZ} and the ratio cuΔZ/σ_{cuΔZ} were used to qualify records for analysis (Maltenfort et al. 2004; see results).
Estimation of synaptic locations and conductance magnitudes
Estimates of synaptic location and conductance magnitude were made using F_{r} and cuΔZ (Maltenfort et al. 2004). The absolute mean position of the set of Renshaw synapses on the equivalent dendritic cable or a compartmental model representing each motoneuron was determined from the reversal frequency (3) This equation was derived from simulated data from step (somatic shunt) and sigmoidal representations of 6 motoneurons in the data of Fleshman et al. (1988).
Dendritic path lengths from the soma to mean synaptic location (Fig. 5) were based on the position estimates from Eq. 3, and on the measurements by Cullheim et al. (1987; their Fig. 7) of mean dendritic diameter as a function of path length from the soma for motoneurons. Noting that the electrotonic length (l_{e}) of each dendritic compartment depends on the length of each dendritic (l_{d}) or cable compartment (l_{eq}), dendritic resistivity (R_{md}), and the diameter of the dendrites (D_{d}) or equivalent cable (D_{eq}) (4) the length of each dendritic compartment can be determined from the corresponding cable compartment length by rearranging Eq. 4 (5) Using a standard cable profile (Maltenfort et al. 2004) and the Cullheim data, Eq. 5 was applied iteratively starting at the soma to determine l_{d} for small increments of l_{eq} until the sum of the l_{eq} equaled the position on the equivalent cable indicated by Eq. 3. These calculations used a proximal cable diameter of 34.5 μm (the approximate diameters of the equivalent cables of FF motoneuron 41/2 and S motoneuron 36/4, and close to the average for the motoneurons of Fleshman et al. 1988). The relations between average dendritic diameter and dendritic path length for S and F motoneurons described by Cullheim et al. (1987) were applied to motoneurons classified in this study as S and FR or FF, respectively.
Reversal frequency and cuΔZ were used to determine electrotonic synaptic location and conductance changes by 2 methods. In the first, values of F_{r} and cuΔZ were normalized by the system time constant (τ) and the dendritictosomatic conductance ratio (ρ); estimates of normalized (electrotonic) synaptic location (X_{syn}) and relative conductance (expressed as a percentage, %ΔG) were determined from normalized grids using the following equations (6) (7) (8) These equations were determined from curve fits to normalized impedance grids determined for 6 step (somatic shunt) models from the data of Fleshman et al. (1988), as described in Maltenfort et al. (2004).
Estimates of τ and ρ for each neuron were made from the computed step response to a hyperpolarizing current pulse. This response was calculated for each neuron by multiplying each cell's impedance function by the Fourier transform of a 25ms current pulse. The impedance function was multiplied by a Welch window (in the frequency domain) before multiplication to compensate for its limited bandwidth and reduce the picketfence effect. The inverse Fourier transform of this product, equivalent to the convolution of the motoneuron's impulse response and the current step, provided the estimate of the neuron's step response. τ and ρ were then determined for the equivalent somatic shunt representation of the motoneuron using the method of Durand (1984) and Kawato (1984). The first 2 exponential components of the response derivative, with time constants τ_{0} and τ_{1}, were determined by exponential peeling (Rall 1969). A constant was added to the derivative of the response as needed to reduce the effect of sag on the response (cf. Fleshman et al. 1988). τ_{0} was taken as τ. The values of R_{md} (to determine τ_{md}) and L that provided the best fit between the measured values of C_{0} and C_{1}, the coefficients of the 2 exponential terms in the step response, and the values given by the Durand–Kawato equations were determined from a matrix of R_{md} and L values. ρ and the ratio of somatic to dendritic resistivity were then determined from τ_{0}, τ_{1}, τ_{md}, and L, as summarized by Rose and Dagum (1988).
Values of τ and ρ obtained in this manner were used to normalize F_{r} and cuΔZ and calculate mean electrotonic synaptic location and the relative change in membrane resistivity produced by activated Renshaw synapses. Absolute synaptic conductance was also obtained from ρ, L, and Z(10) (as an estimate of input resistance). Assuming that Renshaw synapses in each motoneuron are distributed over 0.15 length constants in the equivalent dendritic cable, synaptic area A_{syn} is given by (9) where A_{d} is the area of the equivalent dendritic cable. Based on the equation for input conductance of a uniformdiameter dendritic cable with sealed end (Rall 1977), A_{d} is given by (10) where G_{d} is dendritic conductance. The absolute synaptic conductance g_{syn} is given by the product of A_{d} and relative change in membrane conductance (11) The second method for determining %ΔG, g_{syn}, and synaptic location used models whose parameters were fit to match the impedance functions (Maltenfort and Hamm 2004). Parameters were fit to step models in which separate values were specified for somatic and dendritic resistivity. Most of these models included a linearized voltagedependent conductance distributed uniformly through the neuron, or localized to the soma. The choice of model was determined by the goodness of fit. τ and ρ were computed for each model and initial estimates of %ΔG and synaptic location were computed from cuΔZ and F_{r} using Eqs. 6–8. The user then determined cuΔZ and F_{r} for successive estimates of %ΔG and synaptic location in the bestfit model until a good match was found between model and experimental values of cuΔZ and F_{r}. The change in magnitude of Z produced by the synaptic conductance in the model was also computed for comparison to the measured function (see Fig. 3). Once %ΔG and location were determined, g_{syn} was computed as ΔG × A_{syn}/R_{md}, where A_{syn} was the area spanning 0.15λ of dendritic cable centered at the synaptic location. These computations were repeated using sigmoidal models, in which resistivity increased smoothly from soma through the dendrites, for most neurons. These models had linearized voltagedependent conductances distributed uniformly through each motoneuron.
Simulation of postsynaptic potentials in motoneurons
The time courses of transient and sustained excitatory postsynaptic potentials (EPSPs) were simulated to examine the effect on EPSP magnitude of an inhibitory conductance with locations and magnitude as observed in this study (Figs. 9 and 10). Using Simulink (v. 5, MathWorks 2002), compartmental models of cells from Fleshman et al. (1988) were constructed. These models incorporated the same tapering equivalent dendritic cables used for impedance calculations; the length of each compartment was 0.05 length constants. The voltage V at each compartment j (relative to resting potential) was defined by (12) where C_{m} and g_{m} are the capacitance and resting conductance of the membrane in each compartment; g_{j−1,j} is the axial conductance between the (j − 1)th and jth compartments; and g_{syn} and E_{syn} are the conductance and reversal potential associated with the synapse (if any) active on the compartment (Segev et al. 1989). E_{syn} was −10 mV for Renshaw synapses and +80 mV for excitatory synapses.
The amplitudes of sustained IPSPs (Fig. 8) were computed (MATLAB) using bestfit compartmental models from Maltenfort and Hamm (2004) and estimates of synaptic location and g_{syn} from the present study. Compartmental lengths were 0.0125 or 0.025λ. Setting dV/dt to zero in Eq. 12 and adding a voltagedependent conductance term, g_{v}, give the following equation for each compartment (13) This set of equations was represented by the following matrix equation (14) where G is a matrix of the conductance terms on the left of Eq. 13, V is a vector representing the potential in each compartment, and Y is a vector of the terms on the right of Eq. 13. The elements of V were determined by left division of G by Y in MATLAB, equivalent to the operation (15) E_{syn} was set to the difference between the recorded resting potential and −77 mV for the reversal potential for RIPSPs (Forsythe and Redman 1988; Lindsay and Binder 1991). The sustained recurrent PSP was given by the value of V in the somatic compartment.
Two sets of simulations were performed. One set assumed a uniform resting potential. A second series of sustained potentials were computed in which resting potential varied through the neuron to represent somatic depolarization associated with the somatic shunt. In these simulations, terms for reversal potential times conductance (for g_{m} and g_{V}) were added to the right side of Eq. 13 and to vector Y. Dendritic reversal potentials were −77, −75, and −70 mV for E_{syn}, E_{m}, and E_{gv}, respectively. Somatic reversal potentials were adjusted to match the observed resting potential. Equation 14 was solved with and without synaptic conductance terms, and the sustained RIPSP was determined from the difference in the solutions.
RESULTS
The change in impedance produced by recurrent inhibition was determined in 44 records from 30 ABF, MG, LG, and Sol motoneurons by antidromic activation of one of the heteronymous nerves in this group. A subset of these records was accepted for analysis of Renshaw synaptic location and conductance. The following criteria were used in this selection: 1) the estimated SD of cuΔZ, σ_{cuΔZ}, was <0.75% (approximately 1/3 of the range of cuΔZ); 2) the ratio cuΔZ/σ_{cuΔZ} was >1.0; 3) records in which the control impedance (no synaptic activation) changed progressively over time were excluded; 4) RIPSP amplitudes before and after noise injected differed by no more than 50%; 5) a compartmental model could be fit satisfactorily to the impedance function of each motoneuron. These criteria were selected to limit the uncertainty in estimates of F_{r} and cuΔZ, minimize nonstationarity in the records, and ensure that the analyzed records had been obtained in acceptable recording conditions. Based on these criteria, 19 records from 15 motoneurons (4 ABF, 4 MG, 5 LG, and 2 Sol) were used to estimate the location and conductance change of Renshaw synapses. Five of these motoneurons were classified (based on input resistance and rheobase) as FF, 8 as FR, and 2 as S.
The RIPSP produced by 200Hz stimulation increased rapidly to a peak and then declined to approach a sustained level within 250 ms from the beginning of stimulation (Fig. 1). RIPSP amplitudes measured after noise injection were often lower than those before noise injection, averaging 78% of the amplitude before noise injection in records accepted for full analysis. RIPSP amplitudes were assumed to move linearly from pre to postnoise values. This assumption has been supported by observations of the profiles of RIPSPs produced by 200Hz stimulation without noise injection (Maltenfort et al. 1999).
Figure 2A shows the relation between peak RIPSP and sustained RIPSP. In all cells, inhibition declined from a mean peak of 0.66 ± 0.35 mV to a sustained value of 0.21 ± 0.14 mV. Changes in the smaller set of 19 records were similar (peak of 0.80 ± 0.38 mV, sustained value of 0.27 ± 0.14 mV), although RIPSP amplitudes tended to be slightly larger in this group. For comparison, the mean amplitude of RIPSPs produced by applying single stimuli to the same muscle nerves was 0.52 ± 0.28 mV. Lindsay and Binder (1991) observed a similar decrease in RIPSP amplitude from an initial peak. They noted a greater relative decrease in motoneurons putatively classified as type FR or FF than S. This difference was not evident in the few (5) S motoneurons in our total sample.
RIPSP amplitude was not significantly correlated with any of the sizerelated parameters: conduction velocity, input resistance, or equivalent dendritic diameter (in neurons in which acceptable model fits were obtained; Maltenfort and Hamm 2004). However, RIPSP amplitude was correlated with rheobase, such that larger RIPSPs were found in more excitable motoneurons (Fig. 2B). A multiple regression was performed to determine whether this relation was dependent on resting potential. The regression slope of RIPSP amplitude versus rheobase was significant in both the total and smaller samples (P = 0.001 and P = 0.021, respectively), whereas the relation between RIPSP amplitude and resting potential was not significant.
Overall, characteristics of the final sample of 19 records were similar to those of the total sample with respect to input resistance, rheobase, and RIPSP characteristics, with the exception that the selection criteria tended to favor records with slightly larger RIPSP amplitudes.
Changes in motoneuron impedance produced by recurrent inhibition
The changes in impedance at the lowest frequencies provided a measure of the change in input resistance produced by recurrent inhibition. The average decrease in Z(10) was 1.9% for the total set of records in this study, and 2.7% for the final set of 19. For comparison, Lindsay and Binder (1991) estimated a decrease in input resistance of 3.5%.
Figure 3 illustrates the impedance changes observed in the current study. Figure 3A shows the unfiltered change in impedance, and Fig. 3B shows the result of applying the median filter to these data. There is a region of about 300 Hz (identified with arrows) where the difference in impedance function isclearly positive [i.e., the synaptic input from recurrent inhibition has increased impedance in accordance with model predictions (Maltenfort et al. 2004)]. The thick, smooth curve superimposed on this plot is a simulated change in impedance magnitude. This curve was generated using a bestfit model of the motoneuron represented in Fig. 3, A and B, in which synaptic location and relative conductance change were chosen to match the observed values of cuΔZ and F_{r}. The simulated impedance change matched the observed change reasonably well in this record and in all but one of 13 records in which F_{r} could be determined. Matches between simulated and observed records were judged acceptable if the records had similar profiles from intermediate (20 to 40 Hz) through higher frequencies.
Figure 3C shows one of 6 cases in which F_{r} could not be determined: the change in impedance produced by recurrent inhibition decreases with frequency but never becomes clearly positive. The observed change in impedance could be matched by selecting F_{r} greater than the highest frequency in the experimental impedance function for the example in Fig. 3C and in one other record. The change in impedance approached zero more rapidly, and ΔZ was smaller than expected for a somatic conductance change (cf. Fig. 4 in Maltenfort et al. 2004). These 2 records were judged to be produced by “juxtasomatic” synapses, with mean electrotonic locations <0.15λ from the soma. Synaptic locations were estimated in the 14 records with acceptable matches between the simulated and observed impedance records. Acceptable matches were not found in the other 4 records in which F_{r} could not be determined, nor in one of the 13 records in which F_{r} was evident. The observed change in impedance was flatter at intermediate and/or high frequencies than the simulated response in records without acceptable matches, as shown by the example in Fig. 3D. Estimated synaptic conductance values for records with poor matches were computed based on an assumed proximal location, given that conductance is relatively insensitive to location (cf. Fig. 6).
Estimates of Renshaw synaptic location and conductance change
Reversal frequencies were used to compute both absolute and normalized distances between the motoneuron soma and Renshaw synapses. Figure 4A shows that reversal frequencies tended to be greater in cells with lower rheobase values. Because F_{r} is a direct measure of synaptic location on a cable representing the dendrites of a motoneuron, this finding suggested mean synaptic location is closer to the soma in more excitable motoneurons. Equivalentcable locations were computed from the F_{r} values shown in Fig. 4A using Eq. 3 and are plotted against rheobase in Fig. 4B. Equivalentcable location was correlated with rheobase (r^{2} = 0.50, P = 0.005), supporting this suggestion.
Estimated dendritic path lengths from soma to mean synaptic location in branching dendritic arbors were computed from equivalentcable locations using data on dendritic diameters and path lengths from Cullheim et al. (1987) as described in methods. Dendritic path lengths were also directly correlated with rheobase (r^{2} = 0.53, P = 0.003). In Fig. 5, the distribution of path lengths is plotted with the morphological data of Fyffe (1991). The 2 distributions are very similar. The mean dendritic path length from soma to Renshaw synapses in the present data was 262 ± 90 μm (range: 128 to 412 μm), in comparison to a mean of 255 ± 171 μm reported by Fyffe.
The distributions of F_{r} and cuΔZ values for the selected set of 19 motoneurons are shown in Fig. 6, plotted on cuΔZ–F_{r} grids determined for the 6 “step” model motoneurons used in Maltenfort et al. (2004). The cuΔZ values of records in which reversal frequency could not be determined are plotted on the right of each graph; cases in which the simulated impedance change did not match the observed change are marked by ×'s. The left side of each figure shows the observed values of F_{r} and cuΔZ, whereas the right side shows F_{r} and cuΔZ values after normalization by τ and ρ (Maltenfort et al. 2004). Comparison of the data points to the grids in Fig. 6 shows that Renshaw conductance ranged from roughly 25 to 150% of resting membrane conductance and that most mean synaptic locations were within 0.3λ of the soma.
Values of τ and ρ used for normalization in Fig. 6 were determined from a step response (computed from the control impedance function) using the method of Durand (1984) and Kawato (1984) for estimating electrotonic parameters of neurons with somatic shunts (see methods). In 2 cases, this estimation could not be made. In both cases, the derivative of the step response included an intermediate component that prevented accurate determination of the coefficients of the first 2 exponential terms of the response; the bestfit models of both neurons (Maltenfort and Hamm 2004) included large somatic voltagedependent conductances with short time constants. For the normalization used in Fig. 6, values of τ and ρ for these 2 neurons were based on the passive electrotonic properties of bestfit compartmental models.
Estimates of electrotonic synaptic location and conductance magnitude were made by 2 methods. One method used Eqs. 6–8 and normalized values of F_{r} and cuΔZ. This method gave an estimated average electrotonic location of 0.24 ± 0.10λ (range 0.10–0.41λ). The estimated relative conductance change was 78 ± 42% (range 19–158%), which corresponded to a synaptic conductance of 27.1 ± 14.5 nS (range 7.0–57.9 nS).
The second set of estimates was made by choosing synaptic location and conductance to obtain a match between observed and simulated values of F_{r} and cuΔZ, using a compartmental model of each neuron (Maltenfort and Hamm 2004). Electronic locations determined from bestfit models were somewhat closer to the soma, with a mean location of 0.19 ± 0.09λ (range 0.1–0.35λ). The bestfit estimate of the relative conductance change produced by Renshaw inhibition was 88 ± 53% (range 30–193%). The estimated synaptic conductance was 23.1 ± 11.3 nS (range 6.7–45.3 nS). The 2 sets of estimates were correlated (r^{2} = 0.58, P = 0.004, for synaptic location, Fig. 7A; r^{2} = 0.38, P = 0.008 for relative synaptic conductance, not shown; and r^{2} = 0.87, P < 0.0001, for synaptic conductance, Fig. 7B).
Comparisons were also made of synaptic location and conductance estimates obtained from step and sigmoidal models fit to each motoneuron (Maltenfort and Hamm 2004). Synaptic locations obtained using sigmoidal models, each with a uniformly distributed voltagedependent conductance, had a mean of 0.38 ± 0.16λ (range 0.15–0.71 λ). The values obtained with sigmoidal models were well correlated with those obtained using step models (X_{sig} = 0.05 + 1.72 × X_{step}, r^{2} = 0.84). Electrotonic synaptic locations were further from the soma in the sigmoidal models, consistent with their different electrotonic structure (Fleshman et al. 1988). Mean synaptic conductance obtained with the sigmoidal model was 24.0 ± 11.7 nS (range 6.9–46.3 nS). Synaptic conductance estimates obtained with sigmoidal and step models were also correlated (g_{sig} = 0.74 + 1.01 × g_{step}, r^{2} = 0.94, P < 0.0001).
We examined the distributions of both electrotonic synaptic location and synaptic conductance with respect to cell properties, using averages of the estimates provided by the normalized grids and bestfit models. Electrotonic synaptic locations, like dendritic path length and equivalentcable location, were directly correlated with rheobase (X_{syn} = 0.08 + 0.011 × I_{R}, r^{2} = 0.58, P = 0.001). Location was not related to input resistance or conduction velocity. Unlike synaptic location, synaptic conductance was not correlated with rheobase, nor with conduction velocity or input resistance. There was no relation between synaptic conductance and synaptic location (r^{2} = 0.09, P = 0.29).
Dependency of RIPSPs on synaptic parameters
RIPSP amplitudes were computed using the estimates of synaptic location and conductance magnitude just described, recorded resting potentials, and a compartmental model (step model) based on the impedance function of each motoneuron (Maltenfort and Hamm 2004). Figure 8A shows these predicted values plotted against the recorded sustained RIPSP amplitudes. The simulated RIPSP amplitudes were correlated with the observed values (r^{2} = 0.30, P = 0.02). Agreement between the predicted and observed amplitudes was reasonably good in most cases, although simulated amplitudes were substantially greater in a subset of these pairs. The mean simulated RIPSP amplitude was 0.36 ± 0.21 mV compared to the observed mean of 0.27 ± 0.14 mV.
A motoneuron with a somatic shunt could have a lower resting potential if a nonspecific leakage conductance contributes to the shunt. A second series of simulations were performed in which resting potential was nonuniform (Fig. 8B). The correlation between simulated and recorded amplitudes was similar to that found for simulations with a uniform resting potential (r^{2} = 0.34, P = 0.009), but the difference between simulated and observed amplitudes was slightly smaller with nonuniform resting potentials (paired t = 2.72, P = 0.01). The mean simulated RIPSP amplitude with nonuniform resting potential was 0.32 ± 0.19 mV.
These simulations also provided estimates of the current produced by Renshaw synapses at the synaptic site. Using a heterogeneous resting potential, the mean synaptic current was 0.32 ± 0.20 nA (range 0.09–0.80 nA). Synaptic current was not correlated with cell properties.
Predictions of RIPSP amplitude using a sigmoidal compartmental model (uniform resting potential) were quite similar to those from step models, with a mean of 0.36 ± 0.21 mV (range 0.07–0.80 mV). The regression equation that described the relation between the 2 estimates was RIPSP_{sig} = 0.001 + 1.00 × RIPSP_{step} (r^{2} = 0.96).
These comparisons suggested that analysis of impedance provided reasonable estimates of synaptic location and conductance. We then investigated the dependency of observed RIPSP amplitude on synaptic location, synaptic conductance, motoneuron impedance magnitude, and resting potential using a multiple regression analysis. We assumed that RIPSP amplitude was proportional to the product of 4 variables where V_{RIPSP} is the sustained RIPSP amplitude, X_{syn} is the electrotonic synaptic location, g_{syn} is the synaptic conductance, Z(10) is the impedance magnitude at 10 Hz, and ΔV is the synaptic drive potential (estimated as the difference between recorded rest potential and −77 mV). The multiple regression was performed after taking the logarithm of both sides The regression using this equation was significant, but only the dependency of V_{RIPSP} on g_{syn} was significant (P = 0.004), and V_{RIPSP} was unrelated to ΔV (P = 0.84). Repeating this analysis without the ΔV term provided the regression equation (r^{2} = 0.66, P = 0.01) The coefficients for ln (g_{syn}) and ln (Z(10)) were both significant (r = 0.79, P = 0.002 and r = 0.63, P = 0.03, respectively); the coefficient of X_{syn} was not (r = −0.52, P = 0.08). This analysis suggests that RIPSP amplitude in our sample was determined primarily by synaptic conductance and neuron impedance and perhaps to a lesser extent by synaptic location.
Estimated number of Renshaw cells per RIPSP
The number of Renshaw cells contributing to RIPSPs in each motoneuron was estimated based on the estimated synaptic conductance. RIPSPs are mediated by both glycine and GABA (Cullheim and Kellerth 1981; Schneider and Fyffe 1992). We assumed that both glycine and GABA are released from Renshaw cells based on evidence for corelease of these transmitters from unidentified ventral horn neurons to motoneurons (Jonas et al. 1998). The synaptic conductance produced by individual Renshaw cells in motoneurons is unknown. The glycinergic IPSP produced by individual Ia reciprocal interneurons in motoneurons has a peak transient conductance of 9.1 nS, with rise time and decay time constants of 0.3 and 0.8 ms, respectively (Stuart and Redman 1990). We assumed that the peak synaptic conductance produced by the average Renshaw cell had the same value, and that its glycinergic component had the same time course as that mediating the Ia reciprocal IPSP. The rise time and decay time constant for the GABAmediated component were set at 0.3 and 3 ms, respectively, based on the relative time course of glycine and GABAmediated synaptic currents observed by Jonas et al. (1998). The peak GABA conductance was set at values from 15 to 33% of the total conductance (cf. Cullheim and Kellerth 1981; Jonas et al. 1998; Schneider and Fyffe 1992). The conductance time course for activation at 200 Hz was simulated in MATLAB using these values, and the timeaveraged conductance was determined.
The average conductance per Renshaw cell ranged from 3.3 nS (GABA contribution of 15%) to 4.1 nS (GABA contribution of 33%). Based on these values and an average synaptic conductance of 20–23 nS, we estimate that the average RIPSP in our study was produced by 5 to 7 Renshaw cells. The smallest conductance values we observed (7 nS) would be produced by 1 or 2 Renshaw cells, whereas 11–16 Renshaw cells would be required for the largest conductance values (45–58 nS).
Assessment of shunting produced by recurrent inhibition
An assessment was made of the extent to which recurrent inhibition would shunt an excitatory synaptic input located on the dendritic tree using a Simulink model (see methods). Figure 9A illustrates an EPSP produced by a transient synaptic conductance in a compartment spanning 0.3–0.45λ (typical of Ia EPSP synaptic locations; e.g., Burke and Glenn 1996). This EPSP was simulated with and without a Renshaw conductance of 100% (34.6 nS) in the next most proximal compartment (from 0.15 to 0.3λ). Figure 9B shows a sustained depolarization produced at the same site, with and without activation of Renshaw conductance in the adjacent compartment. Each plot shows the EPSP alone (solid line), combined EPSP and IPSP (dotted line), and the EPSP with inhibitory shunting (dashed line; the IPSP amplitude has been subtracted from this record). The diminution of the transient EPSP produced by the proximal conductance change is negligible compared with the IPSP amplitude. The reduction in amplitude of the sustained depolarization is more substantial: the decrease in EPSP amplitude is 8.2%, compared with an IPSP amplitude that is 3.9% of EPSP amplitude.
Simulations like those illustrated in Fig. 9B were performed using excitatory conductances that produced depolarizations over the subthreshold range to determine the effect of recurrent inhibition on somatic and dendritic potentials. The threshold for spike initiation was assumed to be 15 mV (Gustafsson and Pinter 1984). Inhibitory conductances of 25 and 50 nS were placed 0.15 to 0.3λ from the soma, whereas excitatory conductances spanning 0.15λ were placed at various dendritic locations (ranging from the 0.15λ adjacent to the soma to 0.6–0.75λ from the soma). Figure 10, A and B show the effect of inhibitory conductance on somatic potentials. At excitatory conductances sufficient to depolarize the soma by 15 mV from each of the dendritic sites, inhibitory conductances of 25 and 50 nS reduced somatic potentials by 0.57–0.66 and by 1.11–1.28 mV, respectively. The reduction in depolarization was slightly greater when the excitatory conductance was located just distal (0.3–0.45λ) to or at the site of the inhibitory conductance. The inhibitory conductances alone yielded hyperpolarizations of 0.28 and 0.54 mV. The additional excitatory conductance required to depolarize the soma to 15 mV in the presence of inhibition was noted as an approximate measure of the effect of inhibition on motoneuron threshold. A 50nS inhibitory conductance increased the required conductance by 18.7 to 43.2 nS (9.7–14.7%), depending on the site of the excitatory conductance. Simulations with the other 5 model neurons gave similar results, with a 50nS inhibition producing depolarization decreases ranging from 0.85 to 1.40 mV and threshold elevations of 8.4 to 18.2%.
The right panels in Fig. 10, C and D show the effect of inhibition on the potentials at the dendritic sites of excitatory conductance. These simulations were conducted over the conductance range needed to depolarize the dendritic compartment to 15 mV, a depolarization sufficient to activate persistent inward currents (Lee and Heckman 2000; Powers and Binder 2000). Inhibitory conductances reduced membrane potential (at an excitatory conductance that produced 15 mV of depolarization) by 0.53 to 0.74 mV (25 nS) and by 1.03 to 1.43 mV (50 nS). A 50nS inhibitory conductance increased the conductance required for a 15mV dendritic depolarization by 9.9 to 19.1 nS (8.7–12.4%). Differences in the effect of recurrent inhibition on dendritic potentials at different sites were more pronounced than found for somatic potentials, being largest at 0.15–0.3λ and at 0.3–0.45λ. In the 6 models, a 50nS conductance reduced depolarization produced by excitatory conductance at 0.15 to 0.3λ by 0.99 to 1.64 mV and raised the excitatory conductance needed for 15 mV of depolarization by 8.4 to 14.3%.
DISCUSSION
Results from this study provide estimated locations of Renshaw synapses on lumbosacral motoneurons that are in excellent agreement with previous studies. Our data also provide estimates of Renshaw synaptic conductance needed to assess the potential of Renshaw cells to inhibit motoneurons through shunting of excitatory synaptic inputs. In the following discussion, we consider how motoneuron and synaptic properties may interact to determine the distribution of recurrent inhibition, and how assumptions and experimental limitations may affect our estimates.
Reliability of estimated synaptic location and conductance magnitude
The primary limitation of the estimates of our synaptic location and conductance magnitude is the reliance on values of cuΔZ and F_{r} determined from small, noisy changes in impedance magnitude. The greater the uncertainty in cuΔZ, the less reliable the estimate of synaptic conductance; estimates of F_{r} are severely affected as well (Maltenfort et al. 2004). For these reasons, we estimated the SD of cuΔZ and excluded records in which σ_{cuΔZ} or the ratio of cuΔZ to σ_{cuΔZ} was unacceptably large. This selection process should increase estimate reliability. However, because the change in impedance (or input resistance) produced by a synaptic conductance decreases with distance and is practically undetectable with distal synapses (Rall 1967), rejection of records with small values of cuΔZ/σ_{cuΔZ} could bias location estimates toward more proximal locations. The excellent agreement between our estimates and the data of Fyffe (1991) on locations of Renshaw synapses on motoneurons (Fig. 5) indicates that this bias, if present, is negligible.
Our estimates were based on comparison of experimental data to the results of simulations performed with compartmental motoneuron models. Estimates from normalized grids and equations fit to them were based on the 6 motoneurons described by Cullheim et al. (1987) and Fleshman et al. (1988). Although this model required simplifications, it accurately replicated many of the properties of the 6 motoneurons. Model characteristics were also consistent with the data of Clements and Redman (1989). Compartmental models fit to each motoneuron (Maltenfort and Hamm 2004) relied on a standard cable profile based on the Fleshman data, but these individual models incorporated linearized voltagedependent conductances, which can affect the estimates of synaptic parameters (Maltenfort et al. 2004). In addition, normalized grid estimates used values of τ and ρ based on the expected response of a neuron with a somatic shunt. The normalized grid estimates were independent in many respects of the assumptions of the individual models. Comparing these 2 sets of estimates provides a useful assessment of their reliability.
Agreement between estimates was generally good and was excellent with respect to the synaptic conductance estimates. Greater variability was found in estimates of synaptic location (normalized to λ) and relative synaptic conductance. These differences may be attributed to limitations in the 2 methods of identifying motoneuron parameters. The leastreliable parameter determined from fitting compartmental models to individual impedance functions is the specific dendritic resistivity R_{md}, estimates of which were quite variable (Maltenfort and Hamm 2004), ranging from 6.5 to 67.8 kΩcm^{2} in motoneurons with acceptable records. In contrast, estimates of motoneuron parameters from computed step responses yielded lower R_{md} estimates (5.7–18.7 kΩcm^{2}). The determination of motoneuron parameters from step responses is complicated by the presence of “sag” in the tail of the response, which shortens the apparent time constant (Rose and Dagum 1988). This effect can be only partly reduced by adding a constant to the response (Fleshman et al. 1988), as done in this study. Consequently, R_{md} values may be underestimated by this method. Both estimates of synaptic location (normalized to λ) and relative synaptic conductance are directly dependent on R_{md} estimates. Considering the variability and biases in the 2 sets of R_{md} estimates, the agreement between the 2 estimates of synaptic location and relative conductance is rather good.
Reliability of the estimates was supported by agreement between the step and sigmoidal models. Although it is clear that somatic resistivity is less than dendritic resistivity in spinal motoneurons recorded with sharp electrodes (Clements and Redman 1989; Fleshman et al. 1988; Rose and Vanner 1988), it is unknown whether this difference in resistivity is represented better by the step model (uniform R_{md}), or the sigmoidal model (increasing R_{md} from the soma). Despite the differences in these models, agreement between synaptic conductance estimates was excellent, and the differences between location estimates were consistent with the different electrotonic profiles of these representations (Fleshman et al. 1988). Considering the overall agreement between these sets, our estimates of synaptic location and conductance appear to be relatively insensitive to assumptions about motoneuron electrotonic structure.
Some assumptions were common to all motoneuron representations. Our simulations used uniform synaptic distributions over 0.15λ. Actual synaptic distributions can be expected to be nonuniform and have varying ranges, of course, as indicated by studies of Ia and Renshaw synapses on motoneurons by Burke and Glenn (1996) and Fyffe (1991), respectively. Estimations of synaptic location based on F_{r} are relatively insensitive to changes in the shape and width of the synaptic distribution (Maltenfort et al. 2004). Estimated relative conductance magnitude would be affected by how both the width and the shape of the actual distribution varied from that used in the model. If, for example, the actual synaptic distribution were uniform and twice the width of the assumed distribution, then estimated relative conductance would have to be reduced by a factor of 2. On the other hand, if most of the extra width in the real distribution occurred in a “tail” of the distribution where conductance is small, then the reduction in the estimate would be slight. In either case, estimates of absolute synaptic conductance, determined by the product of relative conductance and synaptic area, would be scarcely affected.
Our analysis assumes that antidromic activation of motor axons affects heteronymous motoneurons only through the action of Renshaw synapses. However, Renshaw cells inhibit both Ia reciprocal inhibitory interneurons (Hultborn et al. 1971) and other Renshaw cells (Ryall 1970). Either of these effects can produce a disinhibition. These effects, which can be readily demonstrated in an unanesthetized spinal cord (e.g., Turkin et al. 1998), would confound the estimates of synaptic location and conductance magnitude. We do not think that disinhibition significantly affected our estimates. First stimulation was limited to nerves in a set in which activation of Renshaw cells would inhibit Ia interneurons that project to the antagonists of these motor nuclei (Hultborn et al. 1971). Second, we used pentobarbital anesthesia, which tends to suppress much of the spontaneous activity of interneurons necessary for disinhibition to be demonstrated. This effect is evident in the different results obtained by Eccles et al. (1961) and Wilson et al. (1960).
A critical assumption of the estimation method is that the synaptic input does not activate or inactivate voltagedependent currents. RIPSPs were generally small in this study, but we did not test directly for activation of voltagedependent currents. However, if the synaptic parameter estimates are reasonably accurate, then the amplitudes of simulated RIPSPs based on these parameters should agree with recorded values. Simulated and actual RIPSP amplitudes (Fig. 8) agreed reasonably well. Simulated values were somewhat larger, and correlation between these values was weakened by several simulation values that were substantially higher than the recorded values. One possible factor contributing to this mismatch is that Renshaw synapses were confined to part of the dendritic arbor. An assumption of the cable model is that synapses are distributed on all dendritic branches at the same distance from the soma. However, Burke and Glenn (1996) found that the proportion of Ia synapses on motoneuron dendrites varied with dendritic orientation, and we consider it likely that Renshaw synapses are not distributed uniformly on motoneuron dendrites either. The effect of a nonuniform distribution of Renshaw synapses would be that synaptic density and relative synaptic conductance would be greater on dendritic branches with Renshaw synapses than our estimates would suggest, and hyperpolarization produced by Renshaw cells at these sites would be correspondingly greater. RIPSP amplitudes would then be less than predicted because of a decrease in synaptic drive potential (cf. Rall and Rinzel 1973). Considering this possibility and the inherent uncertainty in each of the parameters required to simulate RIPSPs, the results presented in Fig. 8 support the reliability of these estimates of synaptic location and conductance.
RIPSP amplitudes, synaptic location, conductance changes, and motoneuron excitability
This study was not designed to examine the distribution of recurrent inhibition within a motoneuron pool, but we did find the relation between RIPSP amplitude and rheobase (Fig. 2) that would be expected, based on the dependency of amplitude on motoneuron type demonstrated in previous studies (Friedman et al. 1981; Hultborn et al. 1988). These RIPSP amplitudes should be dependent on several factors, including synaptic conductance, resting potential (or synaptic drive potential), distance of the Renshaw synapses from the (presumed) somatic recording site, and the impedance, or input resistance, of the motoneuron. The dependency of RIPSP amplitudes in this study on these variables was examined in a multiple regression analysis. Significant relations were found between RIPSP amplitude and both synaptic conductance and impedance magnitude (measured at 10 Hz), but the regression coefficient for synaptic location was not quite significant, and no relation was found between RIPSP amplitude and the estimated synaptic drive potential. Although RIPSP amplitude is known to vary with membrane potential within a cell, error in estimated synaptic drive potential may be introduced by difficulty in obtaining an accurate extracellular potential at the end of a recording and by uncertainty in the membrane potential at the synaptic site. Such difficulties, in addition to the dependency of RIPSP amplitude on several factors, can obscure the dependency of IPSP amplitude on resting potential (Hamm et al. 1987; see also Lindsay and Binder 1991).
Both synaptic conductance and impedance influenced RIPSP amplitude in this analysis. Both Z(10) and input resistance were inversely correlated with rheobase in our sample (r^{2} = 0.19 and r^{2} = 0.21, respectively). Synaptic conductance was not correlated with rheobase, nor was the synaptic current determined from simulations with compartmental models. These results are consistent with those of Lindsay and Binder (1991), who found that effective synaptic current produced by recurrent inhibition (i.e., the amount of injected current just sufficient to counteract the sustained RIPSP at the soma) was distributed uniformly through medial gastrocnemius motoneurons, and the wellknown dependency of RIPSP amplitude on motoneuron properties was attributable to different input resistance values.
Both synaptic location and RIPSP amplitude were dependent on rheobase, yet the regression coefficient between location and RIPSP amplitude was not statistically significant. A dependency of postsynaptic potential amplitude on synaptic location would be expected based on the wellknown electrotonic properties of dendrites. The dependency of postsynaptic potential amplitude on synaptic location can be countered by a dependency of synaptic conductance on location, as suggested by the independence of individual Ia EPSP amplitudes from estimated synaptic locations (Iansek and Redman 1973; Jack et al. 1981) and observations in other cell types (Magee and Cook 2000). However, we found no relation between location and synaptic conductance in our estimates, indicating that properties of Renshaw synapses on motoneurons do not vary significantly with location. The failure to demonstrate a statistically significant dependency of RIPSP amplitude on synaptic location may be attributable to the small sample size and the relatively narrow range of synaptic locations. It is also possible that active dendritic properties may have influenced RIPSP amplitude. Considering these several factors, we suspect that synaptic location contributes to RIPSP amplitude, but this dependency remains to be demonstrated with a larger data base.
Estimates of shunting by Renshaw synapses
Our simulations of interactions between recurrent inhibition and excitatory synaptic inputs in a passive dendritic tree show that the decrease in depolarization produced by Renshaw conductance in the voltage region at which inward currents are activated is approximately double the size of the RIPSP at resting membrane potential (Fig. 10). For the largest Renshaw conductances observed in this study (∼50 nS), these somatic effects are small and indicate that recurrent inhibition should have relatively weak effects on recruited motoneurons, with effects on discharge consistent with those predicted from measurements of Renshaw effective synaptic current (Lindsay and Binder 1991). Recurrent inhibition also would reduce the number of motoneurons in the recruited pool. The additional excitatory conductance required to initiate discharge in the presence of recurrent inhibition, approximated by the conductance required to reach a 15 mV threshold, ranged from 8 to 18% in our simulations.
Figure 10 shows that the effect of Renshaw conductance on dendritic potentials is similar to the effect on somatic potentials, although these actions of recurrent inhibition may be more significant. Depolarizations sufficient to activate dendritic persistent inward currents substantially amplify excitatory synaptic currents in motoneurons (Bennett et al. 1998; Lee and Heckman 2000; Powers and Binder 2000; Prather et al. 2001). Inhibitory currents that prevent activation of or inactivate persistent inward currents eliminate this amplification (Bennett et al. 1998; Kuo et al. 2003; Powers and Binder 2000). This mechanism probably underlies the observation that recurrent inhibition is substantially more effective at reducing motoneuron discharge produced by dendritic synaptic inputs than by current injected at the soma (Hultborn et al. 2003). Our results (Fig. 10) indicate that a Renshaw conductance of 50 nS increases the excitatory conductance needed to activate persistent inward currents in a dendritic compartment, as approximated by the conductance required for a 15mV depolarization, by 8 to 14%. This level of recurrent inhibition would modulate activity in the motoneuron pool, in part, by reducing the fraction of the motoneuron pool in which synaptic amplification occurs.
Simulations that incorporate estimates of the magnitude and location of Renshaw conductance and those of excitatory inputs, combined with information on persistent inward currents, may provide a more complete assessment of the modulation of motoneuron discharge by recurrent inhibition during physiological patterns of motor activity. The largest values of Renshaw conductance magnitude reported here probably are less than the maximum occurring during physiological activity, based on several considerations. The amplitudes of the RIPSPs produced by our stimulus paradigm were small compared to those of other studies (e.g., Eccles et al. 1961; Hultborn et al. 1971; Lindsay and Binder 1991; Turkin et al. 1998). Estimates of the number of Renshaw cells contributing to RIPSPs in this study are also small compared with the size of the Renshaw cell pool (Carr et al. 1998). Also, motor activities like locomotion involving the activation of multiple motoneuron pools should provide a stronger activation of the Renshaw cell pool, considering the extended pattern of recurrent inhibition and the convergence on Renshaw cells from motoneuron pools (Eccles et al. 1961ab; Hultborn et al. 1971; Ryall 1981; see discussion in McCurdy and Hamm 1994). The increase in motoneuron discharge observed during fictive locomotion following block of cholinergic input to Renshaw cells (Noga et al. 1987) is consistent with this argument. The values of conductance in this study provide a starting point for simulations to assess the role of recurrent inhibition in dendritic integration and control of motoneuron discharge.
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 and C. A. Phillips received support from the Undergraduate Biology Research Program at the University of Arizona.
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 addresses: M. G. Maltenfort, Department of Neurobiology and Anatomy, Drexel University College of Medicine, 2900 Queen Lane, Philadelphia, PA 19129; M. L. McCurdy, Department of Kinesiology, University of Wisconsin, 2146 Medical Science Center, 1300 University Ave., Madison, WI 53706–1532; C. A. Phillips, Arizona Endocrinology, Diabetes, and Osteoporosis Center, 5130 W. Thunderbird Rd., #1, Glendale, AZ 85306.
Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
 Copyright © 2004 by the American Physiological Society