1 Department of Biomedical Engineering, Northwestern University; and 2 Department of Physiology, Northwestern University Medical School, Chicago, IL 60611
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ABSTRACT |
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Maltenfort, Mitchell G., C. J. Heckman, and W. Zev Rymer. Decorrelating actions of Renshaw interneurons on the firing of spinal motoneurons within a motor nucleus: a simulation study. J. Neurophysiol. 80: 309-323, 1998. A simulation of spinal motoneurons and Renshaw cells was constructed to examine possible functions of recurrent inhibition. Recurrent inhibitory feedback via Renshaw cells is known to be weak. In our model, consistent with this, motoneuron firing was only reduced by a few pulses per second. Our initial hypothesis was that Renshaw cells would suppress synchronous firings of motoneurons caused by shared, dynamic inputs. Each motoneuron received an identical pattern of noise in its input. Synchrony coefficients were defined as the average motoneuron population firing relative to the activity of selected reference motoneurons; positive coefficients resulted if the motoneuron population was particularly active at the same time the reference motoneuron was active. With or without recurrent inhibition, the motoneuron pools tended to show little if any synchronization. Recurrent inhibition was expected to reduce the synchrony even further. Instead, it reduced the variance of the synchrony coefficients, without a comparable effect on the average. This suggestssurprisingly
that both positive and negative correlations between motoneurons are suppressed by recurrent inhibition. In short, recurrent inhibition may operate as a negative feedback mechanism to decorrelate motoneurons linked by common inputs. A consequence of this decorrelation is the suppression of spectral activity that apparently arises from correlated motoneuron firings due to common excitatory drive. Without recurrent inhibition, the power spectrum of the total motoneuron pool firings showed a peak at a frequency corresponding to the largest measured firing rates of motoneurons in the pool. Recurrent inhibition either reduced or abolished this peak, presumably by minimizing the likelihood of correlated firing among pool elements. Renshaw cells may act to diminish physiological tremor, by removing oscillatory components from aggregate motoneuron activity. Recurrent inhibition also improved coherence between the aggregate motoneuron output and the common drive, at frequencies above the frequency of the "synchronous" peak. Sensitivity analyses demonstrated that the spectral effect became stronger as the duration of inhibitory synaptic conductance was shortened with either the magnitude or the spatial extent of the inhibitory conductances increased to maintain constant net inhibition. Overall, Renshaw inhibition appears to be a powerful way to adjust the dynamic behavior of a neuron population with minimal impact on its static gain.
The Renshaw cell is one of the few spinal interneurons that has been investigated intensively. Its location, firing behavior, and connections to other neurons and firing behavior are all well established (for review see Windhorst 1990) Overview of the model
At the outset, it is important to acknowledge that this is not a perfectly realistic model of motoneurons or of recurrent inhibition. Instead the goal of this work was to produce an observable system the individual components of which reproduce observed behaviors of their physiological counterparts. The behaviors include those relevant to the steady-state firing of a motoneuron pool receiving a synchronizing drive. With this system, it was possible to set the drive to the motoneuron pool and the strength of recurrent inhibition and to observe the behavior of any or all of the simulated motoneurons.
Parameterization of motoneurons
For each trial, each of the 256 motoneurons was assigned randomly a current threshold for steady firing using an exponential distribution
INTRODUCTION
Abstract
Introduction
Methods
Results
Discussion
References
. Despite this, its functional role in the nervous system remains a matter for debate.
; Granit and Renkin 1961)
. Another theory was that Renshaw cells may adjust the gain of the motoneuron pool (Hultborn et al. 1979a)
.
; Hultborn et al. 1988b
; Lindsay and Binder 1991)
. Furthermore, pharmacological blockade of Renshaw cell firing does not produce an increase of motoneuron activity (Redman and Lampard 1967
; Windhorst et al. 1978)
.
and so should a have similar impact on the interspike interval of active motoneurons. Previous reports have shown that Renshaw cell firing can affect motoneuron firing synchrony (Adam et al. 1978
; Windhorst et al. 1978)
. Motoneuron firing synchrony in turn can be linked to tremor (Allum et al. 1978
; Dietz et al. 1976)
.
; Heckman and Binder 1991)
. Although membrane properties of Renshaw cells are not as well known, there is enough information about their firing behavior and effects on motoneurons to build an acceptable functional model.
; Stein and Oguztöreli 1984)
to model this circuit have used smooth functions describing mean firing rates to represent the motoneuron and Renshaw cell populations. This approach does not allow for observation of synchronous firing between motoneurons, although Koelher and Windhorst (1985) could describe phase shifts between linear functions representing subpopulations of motoneurons.
used spiking neurons, but synaptic effects were mediated by voltage waveforms based on recorded postsynaptic potentials and not by active conductances. This may be a significant limitation, as the recurrent inhibitory current on motoneurons close to firing threshold may be twice that on motoneurons at rest (Lindsay and Binder 1991)
. The IPSP produced by a Renshaw cell may depend on both the driving potential on the motoneuron and the motoneuron's recent firing history; that is, whether the target motoneuron was depolarized or hyperpolarized at the time the Renshaw cell fires.
indicating simultaneous firings of motoneurons
would be affected. Contrary to our expectations, recurrent inhibition suppressed both positive and negative correlations, each to a comparable degree. This means that recurrent inhibition may operate as a negative feedback mechanism to decorrelate motoneuronal firing rather than merely desynchronizing motoneurons. A direct result of this mechanism is the suppression of spectral activity that apparently arises from phase-locked motoneuronal firing.
METHODS
Abstract
Introduction
Methods
Results
Discussion
References
; Heckman and Binder 1991
; Hultborn et al. 1988a
; Kernell 1979
; Zengel et al. 1985)
. The simulated Renshaw cells were homogeneous, as no data were available on variability of Renshaw cells within a pool. The Renshaw cell parameters were assigned to reproduce behavior observed by Hultborn and Pierrot-Deseilligny (1979). Similarly, the magnitude, time course, and spatial range of synapses between motoneurons and Renshaw cells were set to match electrophysiological data (Hamm et al. 1987b
; Hultborn et al. 1988b
; Lindsay and Binder 1991
; van Kuelen 1981) and anatomic studies (Cullheim and Kellerth 1978
; Lagerbäck and Kellerth 1985a)
. The motoneurons were arranged along a 4 × 64 grid (mediolateral by rostrocaudal). A single column of 64 Renshaw cells was arranged along the rostrocaudal axis of the motoneuron grid, so each Renshaw cell would interact with four motoneurons at the same rostrocaudal level. There are no published figures for the size of the Renshaw cell population involved with a given motor nucleus, so the decision to have 64 Renshaw cells was for mathematical convenience. The motoneurons were distributed along a tenfold continuum of rheobases (see the following text).
; MacGregor and Oliver 1974)
, defined as a "leaky integrator" with an exponentially decaying potassium conductance representing the afterhyperpolarization. The model neuron fires when the sum of its excitatory and inhibitory inputs crosses a voltage threshold, at which time synapses on target cells are fired and the afterhyperpolarization conductance is increased. The model was modified to explicitly include an input resistance parameter to model motoneurons of varying rheobases. While relatively simple, this model captured the firing behavior of neurons efficiently. With linear summation of afterhyperpolarization conductances, the MacGregor neuron model demonstrates a linear rate/current relationship in the steady-state (MacGregor and Oliver 1974)
.
where thri was the current threshold of unit i and ri was a random variable uniformly distributed between 0 and 1. This produced an exponential distribution of current thresholds in the range 4-40 nA. Half of the simulated motoneuron population have current thresholds of 4-14 nA, whereas the other half (the "large" cells) have thresholds ranging from 12 to 40 nA.
(1)
.
Input resistance was the voltage threshold divided by the current threshold
(2)
Motoneuronal time constants were proportional to resistance
(3)
Gustaffson and Pinter (1984)
(4)
noted that the systematic differences in voltage threshold might reflect cell-impalement injuries. The possible impact of these differences on the simulation results is considered in the sensitivity analysis at the end of the RESULTS.
. There is evidence that a linear relationship between the two parameters exist along the continuum of resistance values (Gustaffson and Pinter 1984
; Zengel et al. 1985)
. A linear relationship was assumed in this model for simplicity. The time constant controls the rate of change of the neuron membrane potential Ei in response to its inputs
where Ri and
(5)
i are as defined in Eqs. 3 and 4; the magnitude of afterhyperpolarization conductance GAHPi depends on the afterhyperpolarization parameters Bi and TGKi described later; EK is the equilibrium potential of the potassium conductance (defined as 10 mV below resting potential); the terms Gsyn and Esyn in the summation describe the conductance and equilibrium potentials of synapses exciting or inhibiting the neuron; and I is the current that is used to activate motoneurons (see further text).
.
; the minimum steady firing rate linearly increased with current threshold, from 8 to 20 Hz (Heckman and Binder 1991
; Kernell 1979)
; the slope of the firing rate/current relation for all motoneurons was ~1.5 pps/nA (Kernell 1979
; Schwindt 1973)
. For several motoneurons along the range of current thresholds, afterhyperpolarization parameters were determined empirically that meet these three criteria. Polynomial fits were used to interpolate Bi and TGKi, as functions of rheobase, and several interpolated values were examined to confirm good behavior. From the smallest to the largest motoneuron, the values of TGKi varied from 64.6 to 18.24 ms, whereas the values of Bi varied from 0.5 to 1.0 µS.
found that a motoneuron model required a substantial increase in input conductance during repetitive firing to produce a realistic firing response to transient inputs. In our model, an increased input conductance was provided by the afterhyperpolarization. The peak afterhyperpolarization conductance after a single motoneuron firing ranged from 62% (on the smallest-resistance motoneurons) to 126% (on the largest-resistance motoneurons) of the resting conductance. Similar relative magnitudes, though no size dependencies, were reported by Schwindt and Calvin (1973)
. The absolute magnitudes of peak afterhyperpolarization conductance (Bi; see preceding text) increased twofold from largest- to smallest-resistance motoneurons.
Parameterization of Renshaw cells
No direct measurements of Renshaw cell membrane properties were available at the time of this work. Renshaw cells are smaller than the smallest type S motoneuron and have fewer dendrites (Lagerbäck and Kellerth 1985a,b
), so it was assumed they would have a higher input resistance. Accordingly, 4 M
was used. The 30-ms afterhyperpolarization of Renshaw cells reported by Hultborn and Pierrot-Deseilligny (1979) indicates that the time constant of the Renshaw cell was ~8 ms. Afterhyperpolarizations of 30-ms duration and both transient and steady-state rate/current slopes agreeing with the observations of Hultborn and Pierrot-Deseilligny (1979) were produced by empirically determining appropriate values of B and
GK for the Renshaw cell, as described previously for motoneurons.
-c
) via descending excitatory pathways (Haase and van der Muelen 1961
; Kaneko et al. 1987
; Morales et al. 1988)
. Other synaptic and neuromodulatory inputs to Renshaw cells are reviewed by Baldissera et al. (1981)
, including the possibility of excitation from tonically firing gamma motoneurons.
Connectivity of pool
MAGNITUDE AND DURATION OF SYNAPSES. Each synapse was modeled as a conductance decaying exponentially from an instantaneous rise. Synaptic conductances arriving at different times or from different sources were assumed to sum linearly. The shape of the resulting postsynaptic potential depended on the membrane potential, time constant, and resistance of the target neuron, which might be modified by temporal and spatial summation of the synaptic conductances and by the afterhyperpolarization conductance.
; Hultborn et al. 1988b)
. On the basis of the distribution of motoneuron axonal collateral swellings with axon diameter (Cullheim and Kellerth 1978)
, the function used to match Renshaw cell excitation to motoneuron size was
where Gi was the average excitatory conductance to an Renshaw cell from motoneuron with current threshold thri, as described above (Eq. 1); and Gmin was the size of the excitatory conductance a Renshaw cell receives from the firing of the smallest (threshold 4 nA) motoneuron (Gi = 7-35 nS). This approximation was selected to match the quantitative proportions of axon collaterals belonging to motoneurons associated with slow-twitch (S), fast, fatigue-resistant (FR), and fast, fatiguable (FF) motor units.
(6)
. The long duration was attributed to synaptic barriers against the reuptake of acetylcholine, the neurotransmitter that excites Renshaw cells (Curtis and Eccles 1958)
. A 15-ms time constant was used for the decay of the excitatory conductance. The resulting EPSP on simulated Renshaw cells decayed to 10% of its peak value 55 ms after the activation of the excitatory synapse.
; Cleveland et al. 1981)
, but other studies contradict this (Hultborn et al. 1979b
; Ross et al. 1972
, 1982
). Orthodromic activation of Renshaw cells is less efficient in activating Renshaw cells (Hultborn et al. 1988b
; Ryall et al. 1972)
, indicating that naturally activated Renshaw cells would not be stimulated at the rates at which saturation occurred. Therefore, excitation from motoneurons was set so that Renshaw cells would fire below the 200 pps maximum observed by Cleveland et al. (1981)
. It should be noted that Cleveland et al. (1981)
attributed the saturation to shunting from the opening of excitatory synapses on the Renshaw cell membrane and that the same shunting is possible in the neuron model used in this study.
7.5 mV (relative to resting potential) and a decay time constant of 5 ms. The resulting average IPSP (across all motoneurons) resembled that reported by van Kuelen (1981).
Synaptic connection patterns
Synapses arising from both Renshaw cells and motoneurons decayed symmetrically along the rostrocaudal dimension of the grid. Along the mediolateral/dorsoventral dimension of the grid, where one Renshaw cell can synapse on four motoneurons, synapses were constant and extended over the entire row. This distribution of axonal connections assumes that decay with distance is only along the rostrocaudal dimension of the grid. This is reasonable given that motor nuclei in the cat are arrayed as rostrocaudally oriented columns (Romanes 1951).
where W was the weighting, d was the rostrocaudal distance between a firing neuron and its target, and dmax was the maximum distance for that neuron's axonal connections, both measured in rows of simulated neurons. K was selected so that the average value of W along the rostrocaudal distribution was 1.0. Each axon could reach target neurons along a total of 2 * dmax +1 rows, centered at the row occupied by the presynaptic neuron.
(7)
; Lagerbäck and Kellerth 1985a
; Windhorst and Kokkoroyiannis 1991)
. Histograms of the distribution of motoneuron axon collaterals (Cullheim and Kellerth 1978)
indicated this choice for motoneuron to Renshaw cell synapses was not inappropriate.
0.42 nA (Lindsay and Binder 1991)
, but that inhibition was produced by maximal stimulation of the heteronymous nerve. The difference in corresponding recurrent IPSPs can be twofold between the heteronymous and the homonymous case (Eccles et al. 1961)
so we set the magnitude in the simulation to be
0.84 nA. The maximum steady-state firing rate of Renshaw cells is ~200 pps (Cleveland et al. 1981)
, and we estimated the number of Renshaw cell synapses on motoneurons using the synaptic parameters
In the model, each Renshaw cell had a range of ±15 rows rostrocaudal (dmax = 15 in Eq. 7), so that each motoneuron would receive inhibitory input from 31 Renshaw cells. Assuming that the MG pool is 6-8 mm long rostrocaudally, this corresponds to a spatial range of ±1.6 mm as reported by Hamm et al. (1987b)
. The variations in IPSP from Renshaw cells due to rostrocaudal weighting and motoneuron properties created a 34-fold range of PSP amplitudes with the same maximum, minimum, and mean (55, 1.6, and 12.5 µV) reported by van Kuelen (1981).
; van Kuelen 1981). Each simulated motoneuron therefore synapsed on five Renshaw cells (dmax = 2). Making this estimate is tricky, given that there are four to six spikes in a Renshaw cell burst (Eccles et al. 1954
; van Kuelen 1981), and resulting recurrent IPSPs may not sum linearly. Measurements of population IPSPs (Eccles et al. 1954
; Friedman et al. 1981
; Lindsay and Binder 1991)
cannot be compared with the Hamm studies because each Renshaw cell may be activated by more than one motoneuron. Because of the columnar organization of the simulation, with 4 motoneurons along a row, each Renshaw cell could receive input from 20 motoneurons.
SIMPLIFYING ASSUMPTIONS. Certain features that may be present in the biological circuit were neglected deliberately in this study to make the simulation tractable within the limits of physiological observations (or lack thereof) and computing resources. The following are assumptions made beyond those previously stated or otherwise implicit in the use of a point neuron model and the data cited above for sub- and suprathreshold behaviors of this model.
also were excluded. Granit et al. (1960)
showed that increased inhibition from Renshaw cells could stop tonically firing motoneurons in a manner consistent with the disruption of plateau potentials. Although the inhibition was increased using ventral root shocks, which may be stronger than normal physiological activation, this role cannot be ruled out. A much more detailed motoneuron model would be required to generate plateau potentials, and we considered this beyond the scope of the present work.
. This is only a small fraction of the bulk of axonal swellings and may be on Renshaw cell dendrites that happen to travel into the motor nucleus region. In this simulation, excitatory linkages between motoneurons were considered to be absent.
; Ryall 1970
, 1981
; Ryall et al. 1971)
, but the existence of mutual inhibition between Renshaw cells activated by motoneurons belonging to the same pool is debated. It had been suggested that the pause after the Renshaw cell burst produced by a ventral root shock was the result of inhibition from other Renshaw cells excited by the same stimulus (Ryall et al. 1971)
. Yet this pause is not affected by glycine antagonists (Curtis et al. 1976)
and has too short a latency to be postsynaptic (Kokkoroyannis et al. 1989)
. In this study, it was assumed that inhibition between Renshaw cells innervated by the same motor pool is negligible. Possible effects of inhibition between two populations of Renshaw cells, each innervated by a separate (not necessarily antagonistic) (cf. Ryall 1981)
motor pool, are considered in Sensitivity analysis.
-c
). For a fairly constant level of motoneuron activity, the dynamic (nicotinic) response of Renshaw cells should be more important than the tonic firing. The use of a single synapse to represent Renshaw cell activation should not invalidate the simulation results.
-aminobutyric acid (GABA) as well as by glycine. Cullheim and Kellerth (1981)
reported both glycine and GABA antagonists could reduce IPSPs from Renshaw cells, but neither antagonist alone could completely abolish recurrent inhibition. The putative GABAergic component of recurrent IPSPs had a smaller magnitude but a longer time constant than the glycinergic component. The hypothesis underlying this study was that weak inhibition can influence timing of spike firings, so it was necessary to consider the effect of changing the relative magnitude and time course of recurrent inhibition. This issue is described further in Sensitivity analysis.
Activation of motoneuron pool
Motoneurons normally receive many synaptic inputs, so that even a steady input contains a significant degree of membrane noise superimposed on an average baseline current. In the simulations, we specified both a steady-state current to provide the baseline and added to this a stochastic input to simulate the noise. The steady-state input current was distributed to motoneurons according to size, following a distribution used by Heckman and Binder (1993). This distribution can produce realistic rate limiting of motoneurons, and one of the goals of this study was to observe motoneuron-Renshaw cell interactions under realistic conditions. The input current was divided into two components, a "low" component weighted on smaller motoneurons and a "high" component weighted on larger motoneurons. The rationale is that Ia excitatory inputs are preferentially weighted on smaller motoneurons and rubrospinal inputs preferentially excite larger motoneurons.
(8a)
The weights were based on the relative position of each motoneuron, noted by the value of ri as described in Eq. 1, along the exponential distribution of current thresholds. The weights were
(8b)
(9a)
where ri is the same value used to define the neuron's current threshold and related properties, and each motoneuron i receives excitatory current input of
(9b)
Weights that varied linearly with current threshold rather than motoneuron index produced comparable results to those presented in this paper.
(10)
. The scaling of the stochastic input was based on the observation that for a filtered white noise input, the variance is proportional to the mean (Papoulis 1984)
. The same modulation of the current drive was applied to all motoneurons simultaneously.
; 50 Hz, to represent the synchronizing effect of EPSPs, assumed to have a time course of a few milliseconds, arriving simultaneously on the same motoneurons; and 10 Hz, as an intermediate value. For comparison, Allum et al. (1978)
reported that the power spectra of low-frequency fluctuations in the net activity of the motor pool and higher-frequency activity of unfused or partly fused twitches of motor units overlapped at 6-12 Hz.
Number and length of trials
During a simulation run, 60 motoneuron pools were generated randomly, 10 at each mean activation level of interest (15, 18, 21, 24, 27, and 30 nA), according to the exponential distribution for current thresholds described previously. This eliminated the possibility that the observed effects of Renshaw cells were due to any particular arrangement of motoneurons within our grid. Each pool's behavior was examined before and after the recurrent inhibitory feedback loop was closed. The stochastic input to the motoneuron pools was different for each trial, although the same bandwidths and amplitudes were used.
Estimates of synchrony
After the initial portion of the simulation trial, 10 motoneurons that had fired at least four times were selected randomly as reference units. During the steady-state portion of the trial, individual firing times of those 10 units were saved by the program. These 10 units were considered the reference units for calculation of synchrony. At the end of the simulation, the summed firing activity of the motoneuron pool was filtered by a triangular averaging window of 11 ms width to detect the short-term synchrony observed by Dietz et al. (1976).
where T was the length of the steady-state portion of the trial (4.608 s, in 1-ms bins); mj(t) was the time series representing the total firing activity of the motoneuron pool minus the contribution of the reference unit j; Nj was the total number of times the reference unit had fired over the trial; and w(
(11)
j, t) is the averaging window centered at individual firing times
j of the reference unit. The coefficients of w were [1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1]/36.
.
Calculation of power spectra
If oscillatory components were present in the pool motoneuron activity, they would not necessarily be visible in the noisy time series. The power spectrum, a plot of average power in the time series versus frequency, would show periodic or quasiperiodic oscillations as peaks.
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RESULTS |
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Activation of Renshaw cells
Despite the substantial nonlinear scaling between motoneuron size and Renshaw cell excitation, average Renshaw cell firing rates varied linearly with average motoneuron firing rates across all examined levels of activation (r2 > 0.99, Fig. 1A). The greater effect of large motoneurons on Renshaw cells was apparent when recruitment alone was assessed. There was a systematic and visibly nonlinear increase of Renshaw cell firing with recruitment over that range where recruitment was increasing (Fig. 1B), but the linear correlation coefficient (r2 > 0.94) was still quite high. For total pool activity, defined as motoneuron firing rate times fraction recruited (Fig. 1C), the relation between Renshaw cell firing and pool activation was again highly linear (r2 > 0.99).
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, producing a less-than-linear increase in Renshaw cell firing with motoneuron activity that counteracted the more-than-linear effects due to scaling of Renshaw cell excitation with motoneuron size. It is consistent with observations that submaximally activated Renshaw cells are linearly dependent on the firing rate and number of stimulated motor axons (Ross et al. 1972
, 1982
).
Inhibition of motoneurons
INHIBITION OF THE POPULATION ACTIVITY.
Figure 2A plots the net change in average motoneuron pool firing rate produced by Renshaw feedback (ordinate) against the average firing rate observed without the feedback. The suppression of motoneuron firing rates increased with the average motoneuron firing rate. This is consistent with the experimental results of Granit and colleagues (Granit and Renkin 1961; Granit et al. 1960)
.
|
where A0 was the average motoneuron activity without recurrent inhibition and A1 was the average motoneuron activity with recurrent inhibition. Average motoneuron activity was defined as the average number of spikes fired per millisecond bin for all motoneurons, i.e., A = mean firing rate * number of motoneurons recruited.
(12)
0.01-0.17, 2 Hz; 0.07-0.16, 10 Hz; 0.07-0.12, 50 Hz). This loop gain appears to be too small to accomplish any of the tasks associated with negative feedback systems, such as adjusting the gain of the system, reducing the sensitivity to transients, or increasing the bandwidth of the system (Houk and Rymer 1981
; Marmarelis and Marmarelis 1978)
.
SIZE-DEPENDENT INHIBITION OF INDIVIDUAL MOTONEURONS. Recurrent inhibition had a greater effect on larger motoneurons (Fig. 3). Polynomials were fit to two data sets of motoneuron firing rate versus motoneuron current threshold (Fig. 3A), and the difference between them was used to estimate the decrease in firing rate from Renshaw cell feedback (Fig. 3B: heavy line). The greater inhibition on larger units might be a consequence of the scaling of voltage threshold with current threshold (see METHODS), but such a scaling would be linear with current threshold (Fig. 3B: thin, solid line). The function used to scale excitation of Renshaw cells with motoneuron size (Fig. 3B: thin, dashed line) shows a curvature qualitatively resembling the size-dependent inhibition of motoneurons.
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; Granit et al. 1960)
.
Renshaw cell effects on synchrony
POPULATION SYNCHRONY. The overall synchrony within a population was estimated using the coefficient of variation of the summed motoneuron firing. If the motoneurons fired synchronously, then the summed motoneuron firing would show peaks where the motoneurons fired and valleys where they did not. The coefficient of variation of the time series formed by the total firing of the motoneuron ensemble would indicate the effect of the Renshaw cell on synchronous firings across the population.
UNIT-TO-POPULATION SYNCHRONY.
The variance of summed motoneuron firing, used above as an overall population synchrony estimate, depends on two quantities. One is the sum of all possible covariances between pairs of active motoneurons, which we assume will be changed by recurrent inhibition. The other quantity is the sum of variances of individual motoneuron firing activity, which is not specifically of interest and which may be large enough to obscure the effect of recurrent inhibition. Therefore, we tested a synchrony measure (defined in METHODS) that was based solely on the cross-correlation between selected reference motoneurons and the rest of the motoneuron population. Each set of 10 trials at the same motoneuron activation regime (effective driving current, bandwidth, and presence or absence of recurrent inhibition) produced a population of 100 synchrony coefficients.
Renshaw cell effects on motoneuron power spectra
Decorrelation of motoneurons should produce a visible change in the power spectrum of aggregate motoneuron firing (cf. Allum et al. 1978)
Sensitivity analysis
INCLUSION OF DECORRELATING NOISE.
If the inputs to the motoneuron pool are routinely uncorrelated, the proposed actions of the Renshaw cells would become unnecessary. The random component of the noise would counteract phase-locked relationships between individual motoneurons. To examine this possibility, the simulation was run with the common drive input to the motoneurons reduced by one-half, and an equal random input (at the same bandwidth) was administered to each motoneuron individually. Each motoneuron saw an input with the same bandwidth and power as when the inputs to the pool were totally synchronizing. Because of the effects of averaging, this produced a substantial decrease in the overall variability of the aggregate motoneuron firing, even though the variability of firing of each individual motoneuron was still ~0.15.
SIMULATION OF THE SOLEUS MOTOR POOL.
Recurrent inhibition is known to be prominent in the soleus motor nucleus (Eccles et al. 1961) SIZE OF INHIBITORY CONDUCTANCE VERSUS SPATIAL EXTENT OR TIME COURSE.
To determine how the selected values for neuronal parameters impacted on the performance of the model, one of the three key parameters describing the inhibitory synapses from Renshaw cells EFFECTS OF INHIBITORY LINKAGES BETWEEN RENSHAW CELLS LINKING TWO SIMILAR MOTONEURON-RENSHAW CELL SYSTEMS.
Substantial mutual inhibition between Renshaw cells excited by separate motoneuron pools may exist (Ryall 1981)
The simulation results presented in this paper agree with experimental observations that motoneurons receiving correlating inputs become phase-locked in the absence of Renshaw cell activity (Adam et al. 1978 Competition between large and small units
The experimental findings of stronger Renshaw cell activation by larger motoneurons, and of larger IPSPs on smaller motoneurons, had suggested that recurrent inhibition may preferentially suppress smaller, weaker motor units as larger, stronger motor units were recruited (Friedman et al. 1981 Renshaw cells decorrelate, not desynchronize
Decorrelation and desynchronization are not the same thing. Synchronization of activity between two time series is indicated by a significantly positive correlation coefficient. The presence of any relationship between two time series is indicated by a correlation coefficient significantly different from zero, positive or negative.
Whitening of the power spectrum
As described in METHODS, the motoneuron pool activity studied in the simulation was depicted as the summed activity of 256 quasiperiodic units. Christakos (1986) Possible effects on the development of physiological tremor
Increased occurrences and larger magnitudes of positive correlations between motor unit firings have been associated with observations of large amplitude tremor in human subjects (Dietz et al. 1976) Will mutual inhibition between Renshaw cell populations synchronize motor nuclei?
If the role of Renshaw cells is to decorrelate motoneuron firings within a motor nucleus, it would follow that the role of inhibition between Renshaw cell populations is to produce correlations in population activity between their associated motor nuclei. When agonist motoneurons fire, antagonist motoneurons would be disinhibited and thus also tend to fire. Such inhibitory linkages between Renshaw cell populations have been found to occur between functional antagonists (Ryall 1981) The authors thank Drs. Tom S. Buchanan and Scott Delp for making UNIX workstations available for this study.
This work was supported by National Institutes of Health Grants NS-28076, NS-19331, and T32-HD-07418.
Present address of M. G. Maltenfort: Division of Neurobiology, Barrow Neurological Institute/St. Joseph's Hospital, 350 W. Thomas Rd., Phoenix, AZ 85013.
Address for reprint requests: C. J. Heckman, Dept. of Physiology M2111, Northwestern University Medical School, 303 E. Chicago Ave., Chicago, IL 60611. Received 9 April 1996; accepted in final form 24 March 1998.
vs.
). As the input bandwidth increased, the change resulting from recurrent inhibition became even smaller.
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FIG. 4.
Recurrent inhibition has little effect on population synchrony. A: population synchrony at 2 Hz is reduced by recurrent inhibition. Population synchrony is estimated as the coefficient of variation of the total number of motoneuron spikes in a 1-ms bin over the duration of the trial. Population synchrony without recurrent inhibition ( ) tends to be larger than that with recurrent inhibition (
) at the same firing rate. B: population synchrony at 10- and 50-Hz bandwidths is not as strongly affected. Tendency to synchronize, estimated from the magnitude of the population synchrony measure, also appears to increase with the modulation bandwidth. Population synchrony coefficients from trials at 10- and 50-Hz input bandwidth are plotted against coefficients from trials at 2-Hz input bandwidth and at the same level of activation of the motoneuron pool. Trials do not represent identical pools at each bandwidth; the individual trials are plotted instead of the averages to show the distribution of population synchrony coefficients.
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FIG. 5.
Effect of recurrent inhibition on unit-to-population synchrony coefficients. Left: motoneuron firing behavior during 2-Hz bandwidth synchronizing input; center: 10 Hz; right: 50 Hz. Unit-to-population synchrony coefficients for a specified level of motoneuron pool activation, synchronizing input bandwidth, and presence or absence of recurrent inhibition are averaged. Averages are plotted against average motoneuron firing rate (bottom 2 traces). At 2- and 10-Hz bandwidths, the average synchrony coefficient with the Renshaw cell loop closed (thick line) does not show substantial difference from the synchrony coefficient with the Renshaw cell loop open (thin line). At 50 Hz, recurrent inhibition increases synchrony coefficients, bringing them closer to 0 but still negative. When the SDs of the unit-to-population synchrony coefficients at each activation level were calculated, the closed-loop standard deviations (filled circles) tended to show a significant decrease from the open-loop standard deviations (open circles). Dashed lines bracket the 99% confidence region (n = 100 for both open and closed loop) around each estimate of the SD of the open-loop synchrony coefficients.
. The power spectrum describes how the variance of a time series is distributed across a range of frequencies. The variance calculated to produce the coefficients of variation plotted in Fig. 4 describes the net effect of the Renshaw cells across all frequencies, and this effect was seen to be small. This effect was found to be larger when considered in the context of a narrow, physiologically relevant frequency band.
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FIG. 6.
Recurrent inhibition suppresses peaks in power spectra of motoneuron population activity. Power spectra with (thick lines) and without (thin lines) during different levels of Renshaw cell activation for 2 Hz (left), 10 Hz (center), and 50 Hz (right) are plotted. (A: 15 pps mean firing rate, ~65% recruitment; B: 21 pps mean firing rate, ~90% recruitment; C: 30 pps mean firing rate, 100% recruitment). Recurrent inhibition consistently reduces or abolishes peaks in the power spectra. At higher input bandwidths for the same activity level of the pool, the magnitude of the peak is larger. At larger activations, the frequency at which the peak is largest moves to the right. This frequency is larger than the mean motoneuron firing rate ( ).
) consistently lagged the peak of the power spectrum. This relationship is explored in Fig. 7. The frequency of the highest amplitude point of the power spectrum was compared with both the average firing rate of the population and to the highest observed firing rates (Fig. 7A,
and *) observed during the averaged trials. Assuming that the frequency of the peak corresponded to motoneuron firing rates, the expected motoneuron firing rates for a given frequency are plotted as a dashed line. The better match was clearly between the frequency of the peak and the highest firing rates present. Furthermore, the units with the highest firing rates tended to be drawn from the middle of the active motoneuron population (Fig. 3A). This is because the smallest units were rate-limited by the input (see METHODS) and the largest units were recruited only recently. Therefore, the peak in the frequency spectrum probably represents the dominant firing rate of the motoneurons. Whether the spectral peak depends on rate-limiting is addressed when a homogeneous pool is considered in Sensitivity analysis.
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FIG. 7.
Spectral peak produced by periodic motoneuron firing varies with mean and peak motoneuron firing frequencies. A: frequency of greatest spectral amplitude varies with mean, peak motoneuron firing. Plot of the location of the spectral peak (before recurrent inhibition) as a function of mean firing rate is marked ( ). Plot of the location of the spectral peak vs. the largest firing rate of motoneurons (see Fig. 3) active in the pool is marked (*). For comparison, the line y = x is drawn as a dashed line. Although both measures of motoneuron firing rate vary linearly with the peak frequency, the peak firing rate of motoneurons is clearly a better match. Data for motoneuron pools receiving 2-Hz bandwidth synchronizing input, other bandwidths show identical relationships. B: coherence is improved by recurrent inhibition. A sample coherence plot for motoneuron pools active with (heavy line) and without (thin line) recurrent inhibition (10-Hz input bandwidth, 23 pps mean firing rate, 32 pps maximum observed firing rate, 100% recruitment in open-loop case; 22 pps mean firing rate, 29 pps maximum observed firing gate, 98% recruitment in closed-loop case). C: without recurrent inhibition, coherence dip occurs at the same frequency as the peak in the power spectrum. Without recurrent inhibition, the frequency at which the coherence dip reaches its minimum point is the same frequency at which the power spectrum of motoneuron activity has its maximum amplitude. Behavior for all 3 input bandwidths is plotted to show consistency (2 Hz,
; 10 Hz, +; 50 Hz, *).
; Windhorst et al. 1978)
that recurrent inhibition may improve the transmission of information through motoneuronal pathways. The decline of coherence at the frequency of the peak suggests that the peak arises due to intrinsic nonlinearities in the motoneurons, probably the transformation of inputs into discrete event trains. Decorrelation of motoneurons by Renshaw cells would diminish the effect of individual spike trains on the output power spectrum (see DISCUSSION), producing the increased coherence.
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FIG. 8.
effects of recurrent inhibition persist in presence of desynchronizing inputs. A-C: suppression of peaks in motoneuron power spectra 10-Hz synchronizing input is used, matched with an equal amount of desynchronizing noise (average coefficient of variation of interspike interval for motoneurons ~0.15). Magnitude of the peaks in the motoneuron pool power spectrum before recurrent inhibition is smaller (A-C, increasing activation as in Fig. 6), but as before, the peaks shift to the right with increasing activation level and are suppressed by recurrent inhibition. D: Renshaw cell effects on unit-to-population synchrony coefficients, as in Fig. 5.
, which has a much narrower distribution of motoneuron sizes than the medial gastrocnemius (Burke 1981
; Henneman and Mendell 1981)
. Recurrent inhibition presumably will have the same effect in both pools. A soleus motoneuron pool was approximated by keeping an exponential distribution of motoneuron sizes but restricting the range of current thresholds to 4.0-7.1 nA. All other aspects of the pool-number of neurons, statistical properties of input, distribution of excitation with motoneuron size, etc.
were the same as in previous simulations. As before, the effects of recurrent inhibition on spectral peaks and synchrony coefficients persisted.
either the magnitude or the time constant of the inhibitory conductance or the rostrocaudal distribution of synapses
was increased by 50%. One of the other two parameters then was decreased proportionally to preserve the average total recurrent inhibition. This was confirmed by calculating the normalized feedback gain as described earlier (under INHIBITION OF POPULATION ACTIVITY). This normalization made it possible to compare the relative effects of different synaptic parameters without altering the mean level of activity of either neuronal population.
, and an explanation of recurrent inhibition has to be able to account for these connections. In each trial, two motoneuron pools were generated randomly and each connected to a separate population of Renshaw cells. The two pools of Renshaw cells were linked by powerful, broad mutual inhibition. Each Renshaw cell in one pool could inhibit 63 Renshaw cells in the other pool, spatially weighted as described in METHODS. The inhibitory synapse between Renshaw cells was four times as large as the magnitude of conductance of the IPSPs on motoneurons. These parameters were based on the extent of Renshaw cell axons beyond the synaptic contacts on motoneurons (Jankowska and Smith 1973
; Ryall et al. 1971)
and on the greater number of glycinergic receptors on Renshaw cells than on motoneurons (Fyffe et al. 1993)
. The reversal potential and the time course of the inhibitory conductance were the same for inhibitory synapses from the Renshaw cells to the motoneurons and from the Renshaw cells to the opposite Renshaw cells. The mutual inhibition produced was enough to decrease Renshaw cell firing rates by one-third.
24 pps mean motoneuron firing rate, 99% recruitment) there was no increase in the spectral peak (Fig. 9B; note that the magnitude of the spectral peak is comparable to that seen in Fig. 6, for input frequencies of 10 Hz and intact recurrent inhibition.). This indicates that inhibitory links between Renshaw cells can produce significant phase-locking between the overall firing activity of two populations of motoneurons, without interfering with the decorrelating effect of recurrent inhibition within each motor nucleus. Possible functions of this phase-locking are considered in DISCUSSION.
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FIG. 9.
Pools linked by mutual inhibition between Renshaw cells show synchronization. A: coherence between pools is increased by mutual inhibition. Before the mutual inhibitory linkages are present between Renshaw cell pools (thin line), the coherence between firing behavior of the 2 motoneuron populations is relatively small (<0.13). When mutual inhibition between Renshaw cell pools is present (thick line), the coherence is increased substantially. Renshaw cell firing rates 90 pps without mutual inhibition, 60 pps with mutual inhibition. motoneuron average firing rates 20 pps, peak firing rates 25 pps (note peak in spectrum B), 89% recruitment (without mutual inhibition), 10 Hz input bandwidth. B: spectral peak of individual motoneuron pools is not necessarily increased by recurrent inhibition. Same trials and key as in A. Peak in power spectrum matches peak motoneuron firing rates of 25 pps, see also Fig. 7A and associated text.
DISCUSSION
Abstract
Introduction
Methods
Results
Discussion
References
; Windhorst et al. 1978)
and with the hypothesis that Renshaw cells should be able to affect firing times of motoneurons. These effects occurred even though the magnitude of recurrent inhibitory effects on motoneuron population firing was weak. The specific features of Renshaw cell effects on motoneuron firing synchrony were different from our expectations. We expected recurrent inhibition to desynchronize motoneurons by decreasing positive correlations at zero lag between spike trains. Instead, both positive and negative correlations were reduced so that the Renshaw cell effect is one of decorrelation, not just desynchronization. The net result was that recurrent inhibition had a potent impact on the power spectrum of the summed motoneuron output, greatly reducing a sharp peak phase locking of motoneuron firing rates.
; Hultborn et al. 1988a
,b
). When measured in terms of effective synaptic current, recurrent inhibition was approximately equal in all motoneurons (Lindsay and Binder 1991)
. We assume the variation in recurrent IPSP with motoneuron size is primarily due to the effect of differences in motoneuronal input resistance.
; Granit and Renkin 1961
; Ryall 1970
; Windhorst 1989)
, but it would be a relatively weak effect (only a few pps, see Fig. 3).
. This might occur as follows. A negative correlation is produced by a tendency for a simultaneous pauses in firing between two motoneurons. This pause would reduce recurrent inhibition, which then would tend to increase the likelihood that one or the other motoneurons would soon fire.
demonstrated that the combined activity of a population of independent, quasiperiodic random processes
such as neuronal firings
will show a spectral peak such as the one seen in these simulations. The power spectrum of the sum of random processes will be the sum of the power spectra of each process and of the real parts of all of the possible cross-spectra. If the processes in the ensemble are all quasiperiodic with the same mean frequency, it is not necessary for them to be correlated for a prominent peak in the power spectrum to emerge. Thus even though the overall synchrony among our simulated motoneurons was low, there was a sharp peak in the power spectrum for the sum of motoneuron firings patterns (e.g., Fig. 6).
1. If there is a 90° phase shift between them, the correlation coefficient is 0.)
saw in their model that recurrent inhibition, too small to impact the gain of motoneuron firing, would produce a phase-shift of 90° between S and combined FR and FF populations. When a homogeneous distribution of Renshaw cell inhibition was assumed, as in the current model, the FF lagged the S motoneurons by 90°, and the FR motoneurons showed an intermediate phase lag. In the current study, it was not feasible to calculate phase shifts between pairs of units. The reduction of the peak in the power spectrum of motoneuron population activity observed in the current study was consistent with the phase shifts predicted by the Koehler and Windhorst model.
. No inquiry has been made into the distribution of negative correlations under these conditions. This result is consistent with the findings of Dietz et al. (1976)
and makes a further prediction: if a similar experiment to that of Deitz et al. looked for statistically significant correlations between pairs of motoneurons and counted both positive and negative correlations, it would find that at higher levels of physiological tremor, both positive and negative correlations are larger and more frequent.
. The asynchronous stimulation attenuates the peak in a manner comparable to the reduction of the peaks in the motoneuron power spectra by Renshaw cells in the current model. If the motoneuron synchrony associated with tremor was reduced by recurrent inhibition in the manner indicated by this study, then less frequent significant correlations
positive or negative
would be observed.
observed that asynchronous stimulation of ventral root filaments produced both smoother and larger force output than did synchronous stimulation. This effect would be produced through nonlinear summation of motor unit twitches: adjacent muscle fibers, contracting simultaneously, might produce less total force than the sum of the individual twitch forces. It should be noted, though, that the division of ventral roots into ten or so filaments to produce "asynchronous" stimulation (Allum et al. 1978
; Rack and Westbury 1969)
still will produce substantial synchrony due to the number of simultaneously active motor units within each filament. Whether Renshaw cell effects can impact force generation directly may have to be tested by incorporating a muscle model incorporating the full range of muscle mechanical nonlinearities.
, which could counteract the low-pass filtering from muscle. Entrainment of Type Ia, Ib, or II afferents by quasiperiodic motor unit population activity may contribute to physiological tremor. The covariance of Ia afferents and recurrent inhibition has been noted elsewhere (Baldissera et al. 1981)
. The need to "prefilter" motor unit activity to prevent entrainment of afferents may be a reason for this covariance. Illert et al. (1996)
suggest, given that recurrent inhibition is weak or absent in distal motor nuclei and absent in amphibians, that the role of the Renshaw cell is to maintain posture. Stabilization of reflex loops, via suppression of the peaks in the motoneuron power spectra, would be a mechanism by which the Renshaw cell could assist in postural control.
.
, which would produce a further disinhibition of antagonists, may further contribute to the between-muscle correlating effect. Ia inhibitory interneurons were not considered in the current model because of the lack of information needed to model them in the same detail as the motoneurons and Renshaw cells.
ACKNOWLEDGEMENTS
FOOTNOTES
REFERENCES
Abstract
Introduction
Methods
Results
Discussion
References
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51: 417-426, 1985.[Medline]
0022-3077/98 $5.00 Copyright ©1998 The American Physiological Society