Lower Boundaries of Motoneuron 
Desynchronization via Renshaw Interneurons 
Mitchell Gii Maitenfort* 
Dept. of Biomedical Engineering 
Northwestern University 
Evanston, IL 60201 
C. J. Heckman 
V. A. Research Service 
Lakeside Hospital 
and Dept. of Physiology 
Northwestern University 
Chicago, IL 60611 
Robert E. Druzinsky 
Dept. of Physiology 
Northwestern University 
Chicago, IL 60611 
W. Zev Rymer 
Dept. of Physiology 
and Biomedical Engineering 
Northwestern University 
Chicago, IL 60611 
Abstract 
Using a quasi-realistic model of the feedback inhibition of motoneurons 
(MNs) by Renshaw cells, we show that weak inhibition is sufficient to 
maximally desynchronize MNs, with negligible effects on total MN 
activity. MN synchrony can produce a 20 - 30 Hz peak in the force 
power spectrum, which may cause instability in feedback loops. 
1 
INTRODUCTION 
The structure of the recurrent inhibitory connections from Renshaw cells (RCs) onto 
motoneurons (MNs) (Figure 1) suggests that the RC forms a simple negative feedback 
* send mail to: Mitchell G. Maltenfort, SMPP room 1406, Rehabilitation Insitute of 
Chicago, 345 East Superior Street, Chicago, IL 60611. Eraall address is mgm@nwu.edu 
535 
536 Maltenfort, Druzinsky, Heckman, and Rymer 
loop. Past theoretical work has examined possible roles of this feedback in smoothing or 
gain regulation of motor output (e.g., Bullock and Contreras-Vidal, 1991; Graham and 
Redman, 1993), but has assumed relatively strong inhibitory effects from the RC. 
Experimental observations (Granit et al.,1961) show that maximal RC activity can only 
reduce MN fh-ing rates by a few impulses per second, although this weak inhibition is 
sufficient to affect the timing of MN firings, reducing the probability that any two MNs 
will fire simultaneously (Adam et al., 1978; Windhorst et al., 1978). In this study, 
simulations were used to examine the impact of RC inhibition on MN fh-ing synchrony 
and to predict the effects of such synchrony on force output. 
Figure 1: Simplified Schematic of Recurrent Inhibition 
2 
CONSTRUCTION OF THE MODEL 
2.1 MODELING OF INDIVIDUAL NEURONS 
The integrate-and-fire neuron model of MacGregor (1987) adequately mimics specific 
fh-ing patterns. Coupled first-order differential equations govern membrane potential and 
afterhyperpolarization (AHP) based on injected current and synaptic inputs. A spike is 
fired when the membrane potential crosses a threshold. The model was modified to 
include a membrane resistance in order to model MNs of varying current thresholds. 
Membrane resistance and time constants of model MNs were set to match published data 
(Gustaffson and Pinter, 1984). The parameters governing AHPs were adjusted to agree 
with observations from single action potentials and steady-state current-rate plots 
(Heckman and Binder, 1991). Realistic faring behavior could be generated for MNs with 
current thresholds of 4 - 40 hA. 
Although there are no direct measurements of RC membrane properties available, 
appropriate parameters were estimated by extrapolation from the MN parameter set. The 
simulated RC has a 30 ms AHP and a current-rate plot matching that reported by 
Hultbom and Pierrot-Deseilligny (1979). Spontaneous firing of 8 pps is produced in the 
model by setting the RC firing threshold to 0.01 mV below resting potential; in vivo 
Lower Boundaries of Motoneuron Desynchronization via Renshaw Interneurons 537 
this fu'ing is likely due to descending inputs (Hamm et al., 1987a), but there is no 
quantitative description of such inputs. The RCs are assumed to be homgeneous. 
2.2 CONNECTIVITY OF THE POOL 
Simulated neurons were arranged along a 16 by 16 grid. The network consists of 256 
MNs and 64 RCs, with the RCs ordered on even-numbered rows and columns; as a result, 
the MN - RC connections are inhomogeneous along the pool. For each trial, MN pools 
were randomly generated following the distribution of MN current thresholds for a model 
of the cat medial gastrocnemius motor pool (Heckman and Binder, 1991). 
Communication between neurons is mediated by synaptic conductances which open when 
a presynaptic cell fires, then decay exponentially. MN excitation of Res was set to 
produce Re firing rates < 190 pps (Cleveland et al., 1981) which linearly increase with 
MN activity (Cleveland and Ross, 1977). MN activation of RCs scales inversely with 
MN current threshold (Hultborn et al., 1988). 
Connectivity is based on observations t_hat synapses from RCs to MNs have a longer 
spatial range than the reverse (reviewed in Windhorst, 1990). The IPSPs produced by 
single MN firings are 4 - 6 times larger than those produced by single Re firings 
(Hamm et al., 1987b; van Kuelen, 1981). In the model, each MN excites RCs within 
one column or row of itself, and each Re inhibits MNs up to two rows or columns 
away; thus, each MN excites 1 - 4 Res (mean 2.25) and receives feedback from 4 - 9 RCs 
(mean 6.25). 
2.3 ACTIVATION OF THE POOL 
The MNs are activated by applied step currents. Although this is not realistic, it is 
computationally efficient. An option in the simulation program allows for the addition 
of bandlimited noise to the activation current, to simulate a synchronizing common 
synaptic input. This signal has an rms value of 3% of the mean applied current and is 
low-pass filtered with a cutoff of 30 Hz. This allows us look at the effects due purely to 
Re activity and to establish which effects persist when the MN pool is being actively 
3 
EFFECTS OF RC STRENGTH ON MN SYNCHRONY 
3.1 DEFINITION OF SYNCHRONY COEFFICIENT 
Consider the total number of spikes fired by the MN pool as a time series. During 
synchronous firing, the MN spikes will clump together and the time series will have 
regions of very many or very few MN spikes. When the MNs are desynchronized, the 
range of spike counts in each time bin will contract towards the mean. It follows that a 
simple measure of MN synchrony is the the coefficient of variation (c.v. = s.d. / mean) of 
the time series formed by the summed MN activity. Figure 2 shows typical MN pool 
firing before and after RC feedback inhibition is added; the changes in "clumping" 
described above are quite visible in the two plots. 
538 Maltenfort, Druzinsky, Heckman, and Rymer 
3.2 "PLATEAU" OF DESYNCHRONIZATION 
The magnitude of the synaptic conductance from RCs onto MNs was changed from zero 
to twice physiological in order to compare the effects of 'weak' and 'strong' recurrent 
inhibition. At activation levels sufficient to recruit at least 70% of MNs in the pool 
(mean firing rate > 15 pps), a surprising plateau effect was seen. The synchrony 
coefficient fell off with RC synapfic conductance until the physiological level was 
reached, and then no further alesynchronization was seen. The effect persisted when 
synchronizing noise was added (Figure 3). At activation levels sufficient to show this 
plateau, this "comer" inhibition level was always the same. 
50 
Synchronized Firing (no RC inhibition) 
0 50 
100 150 
Desynchronized Firing (RC inhibition added) 
20 
0 
0 
50 100 150 200 
Time (ms) 
Figure 2: Comparison of Synchronous and Asynchronous MN Firing 
At this comer level, the decrease in mean MN firing rate was < 1 pps and not 
statistically significant. There was also no discernible change in the percentage of the 
MN pool active. The c.v. of the interspike interval of single MN firings during constant 
activation is < 2.5 % even with RCs active - this implies that the RC system finds an 
optimal arrangement of the MN firings and then performs few if any further shifts. When 
synchronizing noise is added, the RC effect on the interspike interval is swamped by the 
effect of the synchronizing random input. 
Figure 4 shows the effect of increasing MN activation on the synchrony coefficients 
before and after RC inhibition is added. The change is statistically significant at all 
levels, but is only large at higher levels as discussed above. As activation of the MNs 
Lower Boundaries of Motoneuron Desynchronization via Renshaw Interneurons 539 
increases, the "before" level of synchrony increases while the "after" level seems to move 
asymptotically towards a minimum level of about 0.35. This minimum level of MN 
synchrony, as well as the dependence of the effect on the activation level of the pool, 
suggests that a certain amount of synchrony becomes inevitable as more MNs are 
activated and fLre at higher rates. 
8 
1.4 
1.2 
1 
0.8 
0.6 
0.4 
synchronizing noise added 
no synchronizing noise 
I I I I I I 
0 0.002 0.004 0.006 0.008 0.01 
RC Synaptic Conductance (laSiemens) 
Figure 3: MN Firing Synchrony vs. RC Strength 
4 
EFFECTS OF MN SYNCHRONY ON MUSCLE FORCE 
4.1 MODELING OF FORCE OUTPUT 
Single twitches of motor units are modeled with a second-order model, f(t) = F-et -t/x, 
where the amplitude F and time constant  are matched to MN current threshold according 
to the model of Heckman and Binder (1991). A rate-based gain factor adapted from 
Fuglevand (1989) produces fused tetanus at high fh-ing rates. The tenfold difference in 
current thresholds maps to a fifty-fold difference in twitch forces. Twitch time constants 
range 30-90 ms. 
540 Maltenfort, Druzinsky, Heckman, and Rymer 
4.2 EFFECTS OF RECURRENT INHIBITION ON FORCE 
The force model sharply low-pass f'fiters the neural input signal (< 5 Hz). As a result, the 
c.v. of the force output is much lower than that of the associated MN input (< 0.01). 
Although the plot of force c.v. vs. RC strength during constant activation follows the 
curve in Figure 3, adding synchronizing noise removes any correlation between force .c.v. 
and magnitude of recurrent inhibition. The effect of recurrent inhibition on mean force is 
similar to that on the mean firing rate: small (<5 % decrease) and generally not 
statistically significant. 
1.6 
1.4 
1.2 
1 
0.8 
0.6 after 
0.4 
5 10 15 20 25 30 35 
Activation Current (nA) 
Figure 4: Effects of MN Activation on Synchrony Before and After Recurrent Inhibition 
When the change in synchrony due to RCs is large, a peak appears in the force power 
spectrum in the range 20 - 30 Hz. This peak is reduced by RCs even when the MN pool 
is being actively synchronized (Figure 5). Peaks in the force spectrum match peaks in the 
spectrum of pooled MN activity, suggesting the effect is due to synchronous MN firing. 
Although the magnitude of this peak is small (< 0.5% of mean force), its relatively high 
frequency suggests that in derivative feedback - where spectral components are multiplied 
by 2Jr times their frequency - its impact could be substantial. The feedback loop which 
measures muscle stretch contains such a derivative component (Houk and Rymer, 1981). 
Lower Boundaries of Motoneuron Desynchronization via Renshaw Intemeurons 541 
5 
DISCUSSION 
The preceding shows that the ostensibly weak recurrent inhibition is sufficient to sharply 
reduce the maximum number of synchronous firings of a neuron population, while 
having a negligible effect on the total population activity. This has a broad implication 
for neural networks in t_hat it suggests the existence of a "switching mechanism" which 
forces the peaks in the output of an ensemble of neurons to remain below a threshold 
level without significantly suppressing the total ensemble activity. 
One possible role for such a mechanism would be in the accommodation to a step or 
ramp increase in a stimulus. The initial increase synchronizes the neural signal from the 
receptor, which is then alesynchronized by the recurrent inhibition. The synchronized 
f'u-ing phase would be sufficient to excite a target neuron past its f'uSng threshold, but after 
that, the desynchronized neural signal would remain well below the target's threshold. 
0.2 
0.15 
0.05 
0 
0 
10 20 30 40 
Frequency (Hz) 
Figure 5: Recurrent Inhibition Reduces Spectral Peak. 95% confidence limit of means 
plotted, solid lines before recurrent inhibition and dashed lines after. 
Acknowledgments 
The authors are indebted to Dr. Tom Buchanan for use of his IBM RS/6000 workstation. 
This work was supported by NIH grants NS28076-02 and NS30295-01. 
542 Maltenfort, Druzinsky, Heckman, and Rymer 
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