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Location and Magnitude of Conductance Changes Produced by Renshaw Recurrent Inhibition in Spinal Motoneurons

Division of Neurobiology, Barrow Neurological Institute, St. Joseph's Hospital and Medical Center, Phoenix, Arizona 85013

Submitted 8 September 2003; accepted in final form 13 April 2004

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 quasi-white noise current into lumbosacral motoneurons of pentobarbital-anesthetized 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 0.41, with a mean of 0.19 or 0.24, 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

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
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

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
Experimental preparation

Experiments were performed in 10 adult pentobarbital-anesthetized 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 CO₂ 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% NaHCO₃) 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 M) filled with 2 M potassium citrate. Recordings were made with an intracellular amplifier with a high-impedance probe (Axoclamp 2A or 2B) and a sample-and-hold 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 x R_N, r² = 0.840), although the impedance values were larger on average (1.44 vs. 1.21 M, P < 0.0005, paired t-test). 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 2-Hz 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 200-Hz 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 M; 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 quasi-white current into the motoneuron for impedance determinations. Each current waveform consisted of a sum of 300 equal-amplitude 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 nA².

View larger version (34K): [in this window] [in a new window]   FIG. 1. Examples of voltage and current records used to determine motoneuron impedance. These records were obtained from a Sol motoneuron. In this trial a recurrent inhibitory postsynaptic potential (RIPSP) was produced by stimulating the MG nerve at 200 Hz (immediately after a 1-mV calibration pulse at the start of the voltage record). A noise current, consisting of a sum of sinusoids, was injected 0.6 s after the start of muscle nerve stimulation. Trials were alternated with and without nerve stimulation to provide impedance estimates with and without recurrent inhibition. In this example, the power of the noise waveform was 1 nA2, with a bandwidth of 2.44–732 Hz. Peak RIPSP (based on an average of 30 trials) was 1.61 mV, and the sustained RIPSP was 0.56 mV. Arrowheads under the voltage record indicate times at which peak RIPSP amplitude ("peak") and sustained RIPSP amplitude ("RIPSP") were measured. Dashed line shows the baseline for the RIPSP.

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 [²(f), analogous to an r² 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%) 512-point 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 "picket-fence" 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, frequency-weighted measure of the change in impedance magnitude at frequencies less than the reversal frequency, F_r (Maltenfort et al. 2004)

(1)

where Zn is the normalized change in impedance magnitude [|Z(f)| – |Z_syn(f)|]/|Z(f₀)|; f₀ is the lowest nonzero frequency in the spectrum; |Z(f)| and |Z_syn(f)| are impedance magnitudes without and with Renshaw conductance, respectively; and f/f is the frequency interval divided by the frequency at each point, the sum of which is taken from f₀ to F_r. The statistical errors in the impedance function estimates, although small relative to the impedance functions, were large compared with Zn. 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 Zn at low frequencies, which contribute most to cuZ. To reduce further the variance of this estimate, Zn was smoothed with a 5-point 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, Zn was smoothed using median filters (Hämäläinen et al. 1994) of width 34–100 Hz (7–21 spectral points). Each Zn record was inspected using filters of several widths, starting with a narrow filter and progressively increasing filter width. The filtered Zn 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.

View larger version (26K): [in this window] [in a new window]   FIG. 3. Changes in impedance magnitude produced by recurrent inhibition. A: difference between the magnitudes of the impedance functions [|Zwith inhibition(f)| – |Zwithout inhibition(f)|]. These differences were small and correspondingly noisy. B: difference between magnitude functions in A after smoothing with a median filter (in this example, 11-point width; ±24 Hz); the approximately 300 Hz in which the difference was clearly positive is identified by the vertical arrows. Smooth, thicker line is the change in impedance magnitude produced in a best-fit model in which synaptic location and magnitude were selected to yield the same reversal frequency and cuZ values as observed in the recorded neuron. Differences between impedance magnitudes in 2 neurons in which reversal frequency could not be determined are illustrated in C [7-point filter (±15 Hz) used to smooth] and D (11-point filter; ±24 Hz). Agreement between the observed and modeled change in impedance magnitude was satisfactory in C but not in D.

cuZ

cuZ

Z

Zn

(2)

Equation 2 neglects the contribution of variability in Z(f₀) to _Zn². However, these contributions are small and contributed no more than 2–3% in the present data. _Zn² was determined for each point of Zn from the lowest frequency to the reversal frequency. The variance of cuZ 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 f/f term. The square root of the variance _cuZ and the ratio cuZ/_cuZ 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 cuZ (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.

View larger version (18K): [in this window] [in a new window]   FIG. 5. Distribution of estimated locations of Renshaw synapses on the dendritic tree. Filled bars in this histogram represent the dendritic path distances to individual Renshaw synapses on motoneuron dendrites identified by Fyffe (1991). Renshaw synapses were located 255 ± 171 µm (mean ± SD) from the soma. Shaded bars show the estimates of mean synaptic position in the current study, based on mapping reversal frequency to absolute position on the equivalent dendritic cable, and on the relationship between the equivalent cable and the actual path length of dendrites in a typical motoneuron. Average of these estimates was 262 ± 90 µm from the soma.

View larger version (15K): [in this window] [in a new window]   FIG. 7. Comparison of synaptic location and conductance estimates. This figure compares estimates of normalized synaptic location (A) and synaptic conductance (B) obtained by separate methods. Normalized-grid estimates obtained using values of Fr and cuZ after normalization by and (Eqs. 6–8, METHODS) are plotted on the abscissa in each plot. Best-fit estimates plotted on the ordinates were obtained by matching Fr and cuZ values from simulated impedance changes of compartmental models to the observed experimental values. Regression line in A is described by Xbf = 0.03 + 0.69Xng (r2 = 0.58). Regression line in B is described by gbf = 2.66 + 0.76gng (r2 = 0.87).

(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 and 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 25-ms 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 picket-fence 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. and 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 ₀ and 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). ₀ was taken as . The values of R_md (to determine _md) and L that provided the best fit between the measured values of C₀ and C₁, 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. and the ratio of somatic to dendritic resistivity were then determined from ₀, ₁, _md, and L, as summarized by Rose and Dagum (1988).

Values of and obtained in this manner were used to normalize F_r and cuZ 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 , 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 uniform-diameter 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 %G, 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 voltage-dependent conductance distributed uniformly through the neuron, or localized to the soma. The choice of model was determined by the goodness of fit. and were computed for each model and initial estimates of %G and synaptic location were computed from cuZ and F_r using Eqs. 6–8. The user then determined cuZ and F_r for successive estimates of %G and synaptic location in the best-fit model until a good match was found between model and experimental values of cuZ 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 %G and location were determined, g_syn was computed as G x A_syn/R_md, where A_syn was the area spanning 0.15 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 voltage-dependent 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.

View larger version (12K): [in this window] [in a new window]   FIG. 9. Effects of synaptic shunting on transient and sustained excitatory postsynaptic potentials (EPSPs). In each figure, the solid line shows the EPSP, and the dotted line shows the EPSP plus a sustained recurrent inhibition (EPSP + IPSP). Effect of shunting on the EPSPs is shown by the dashed line, in which the EPSP + IPSP curve is shifted by the IPSP alone. EPSP was produced by a conductance spanning 0.30–0.45 in a step model based on motoneuron 42/4 of Fleshman et al. (1988). An inhibitory conductance of 100% of resting membrane conductance (34.6 nS) was placed from 0.15 to 0.30. A: a transient EPSP produced by a transient conductance with time course (t/) x exp(–t/); = 0.1 ms. Peak EPSP conductance was 150 nS. B: a sustained EPSP produced by a conductance of 150 nS in the same motoneuron model.

View larger version (32K): [in this window] [in a new window]   FIG. 10. Reduction in somatic and dendritic depolarization produced by inhibitory conductances. This figure shows results from simulations performed with a compartmental motoneuron model based on cell 42/4 of Fleshman et al. (1988), as in Fig. 9. Simulations were run with total inhibitory conductances of 25 or 50 nS in dendritic compartments 0.15–0.3 from the soma. Excitatory conductance was placed in a dendritic segment of 0.15 at one of 5 sites (D1–D5), ranging from 0–0.15 to 0.6–0.75 from the soma. Left panels (A, B): effect of inhibitory conductances on somatic potentials. Right panels (C, D): effect of inhibitory conductances on dendritic potentials. Dendritic potentials were taken from the middle compartment of the dendritic segment with the excitatory conductance (e.g., from the compartment spanning 0.35–0.4 when the excitatory conductance was placed in D3, 0.3–0.45). Top panels: potentials without or with 25 or 50 nS of inhibitory conductance, as indicated in A. Dotted lines have been drawn at 15 mV to indicate the approximate threshold for activating action potentials (A) or dendritic persistent inward currents (C). Bottom panels: potential changes produced by inhibitory conductances of 25 and 50 nS. Dendritic location of the excitatory conductance is indicated at the end of each trace. With the exception of D, only 3 excitatory conductance sites are represented to maintain clarity. Results for all 5 sites are included in D to illustrate the effect of the inhibitory conductance on dendritic potentials produced by excitatory conductances at different sites.

(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.

View larger version (17K): [in this window] [in a new window]   FIG. 8. Comparisons of simulated and recorded RIPSP amplitudes. The amplitudes of simulated sustained RIPSPs are plotted vs. the observed values. Simulations used compartmental models fit to impedance functions (Maltenfort and Hamm 2004) with synaptic locations and conductances determined with these same models. A: simulations performed with uniform resting potentials. Solid regression line is described by RIPSPsim = 0.13 + 0.853 x RIPSP. B: simulations performed with nonuniform resting potentials. Regression in this case is RIPSPsim = 0.10 + 0.80 x RIPSP. Dashed lines are lines of identity.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 cuZ, _cuZ, was <0.75% (approximately 1/3 of the range of cuZ); 2) the ratio cuZ/_cuZ 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 cuZ, 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 200-Hz 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 200-Hz 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.

View larger version (15K): [in this window] [in a new window]   FIG. 2. Distribution of RIPSP amplitudes. A: distribution of peak RIPSP amplitudes produced by 200-Hz stimulation plotted against the sustained RIPSP amplitudes. B: sustained RIPSP amplitudes plotted against rheobase as a measure of motoneuron size. Data points are labeled as FF, FR, or S according to putative type established by input resistance and rheobase. Larger symbols represent cells that met all requirements for determination of synaptic location and conductance.

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 best-fit 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 cuZ 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  Z  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 <0.15 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).

View larger version (11K): [in this window] [in a new window]   FIG. 4. Relation between synaptic location and rheobase. A shows that reversal frequency (Fr) tended to be greater in more excitable motoneurons (i.e., those with lower rheobase values). Tendency for the location of Renshaw synapses to vary systematically with excitability is demonstrated in B, in which Fr values have been converted to location on the equivalent dendritic cable of a somatic shunt model. Relation between cable location, lc, and rheobase, Ir, is described by the regression lc = 310 + 30.7Ir (r2 = 0.495; P = 0.005).

View larger version (41K): [in this window] [in a new window]   FIG. 6. Distribution of reversal frequencies and cuZ values. Values of reversal frequency (Fr) and the cumulative, low-frequency impedance change (cuZ) determined in motoneurons putatively typed as FF (top), FR (middle), and S (bottom) are plotted on grids of cuZ and Fr produced by known conductance magnitudes and locations in 6 type-identified somatic-shunt (step) motoneuron models. Moving up from the bottom (increasing cuZ), the horizontal lines in each grid describe the effect of relative conductance changes of 10, 25, 50, 100, and 200%. Moving from left to right (increasing Fr), the vertical lines represent conductance changes centered at 0.75, 0.60, 0.45, 0.30, and 0.15 electrotonic length constants () from the soma. Each conductance change occurs over a span of 0.15. Left column: original values of cuZ and Fr plotted on model grids. Right column: values of Fr and cuZ normalized by multiplication by and by 0.33/0.46, respectively. Grids in the right column have also been normalized by these factors. Circles represent neurons in which the change in impedance magnitude determined in a model of that motoneuron provided a reasonable fit to the change observed experimentally. x symbols represent neurons in which a reasonable fit was not obtained. Symbols on the extreme right of each grid represent data from neurons in which Fr values could not be determined.

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. Equivalent-cable locations were computed from the F_r values shown in Fig. 4A using Eq. 3 and are plotted against rheobase in Fig. 4B. Equivalent-cable location was correlated with rheobase (r² = 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 equivalent-cable 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 (r² = 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 cuZ values for the selected set of 19 motoneurons are shown in Fig. 6, plotted on cuZ–F_r grids determined for the 6 "step" model motoneurons used in Maltenfort et al. (2004). The cuZ 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 x's. The left side of each figure shows the observed values of F_r and cuZ, whereas the right side shows F_r and cuZ values after normalization by and (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 0.3 of the soma.

Values of and 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 best-fit models of both neurons (Maltenfort and Hamm 2004) included large somatic voltage-dependent conductances with short time constants. For the normalization used in Fig. 6, values of and for these 2 neurons were based on the passive electrotonic properties of best-fit 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 cuZ. This method gave an estimated average electrotonic location of 0.24 ± 0.10 (range 0.10–0.41). 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 cuZ, using a compartmental model of each neuron (Maltenfort and Hamm 2004). Electronic locations determined from best-fit models were somewhat closer to the soma, with a mean location of 0.19 ± 0.09 (range 0.1–0.35). The best-fit 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 (r² = 0.58, P = 0.004, for synaptic location, Fig. 7A; r² = 0.38, P = 0.008 for relative synaptic conductance, not shown; and r² = 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 voltage-dependent conductance, had a mean of 0.38 ± 0.16 (range 0.15–0.71 ). The values obtained with sigmoidal models were well correlated with those obtained using step models (X_sig = 0.05 + 1.72 x X_step, r² = 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 x g_step, r² = 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 best-fit models. Electrotonic synaptic locations, like dendritic path length and equivalent-cable location, were directly correlated with rheobase (X_syn = 0.08 + 0.011 x I_R, r² = 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 (r² = 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 (r² = 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 (r² = 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 x RIPSP_step (r² = 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 V 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 V (P = 0.84). Repeating this analysis without the V term provided the regression equation (r² = 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 GABA-mediated component were set at 0.3 and 3 ms, respectively, based on the relative time course of glycine- and GABA-mediated 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 time-averaged 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 0.3–0.45 (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 0.3). 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 0.3 from the soma, whereas excitatory conductances spanning 0.15 were placed at various dendritic locations (ranging from the 0.15 adjacent to the soma to 0.6–0.75 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 (0.3–0.45) 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 50-nS 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 50-nS 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 50-nS inhibitory conductance increased the conductance required for a 15-mV 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 0.15–0.3 and at 0.3–0.45. In the 6 models, a 50-nS conductance reduced depolarization produced by excitatory conductance at 0.15 to 0.3 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

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 cuZ and F_r determined from small, noisy changes in impedance magnitude. The greater the uncertainty in cuZ, 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 cuZ