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LETTERS TO THE EDITOR
Department of Biomedical Engineering
University of Alberta
Edmonton, Alberta, Canada
To the Editor:
In the May 2005 issue of the Journal of Neurophysiology, Moritz and coworkers report new findings on human motor unit properties during voluntary isometric contractions. A unique contribution of this paper is the extensive range of voluntary forces (4–85% MVC [maximal voluntary contraction]) used to explore the correlations between recruitment and discharge rates. Those familiar with isolating single units from needle/fine-wire EMG recordings will appreciate the difficulty of getting stable discrimination of units at these high contraction levels. The experimental data were then used to update some parameters in the Fuglevand motor unit model (Fuglevand et al. 1993
), resulting in better predictive value of the model when compared with the experimentally measured relationship between mean force and its variability (see Fig. 6E, Mortiz et al. 2005).
The definitive data set for the hypotheses addressed in the paper would require simultaneous recording of multiple motor units from a single subject across the full range of forces, and repeating this in a number of individuals. Because such heroism remains mythological, Moritz and colleagues take the next best approach: data from 38 units recorded from 18 subjects were normalized to % MVC. It is unfortunate that in Table 1 there is no indication whether any of the units were recorded from the same subject in a single experiment because this may well have added some anecdotal support to the conclusions.
Arguably, the most important changes to the model arising from the experimental data include: 1) setting minimal rhythmic firing rate proportional to recruitment threshold; 2) setting maximal firing rate proportional to recruitment threshold; and 3) making the coefficient of variation (CV) of the interspike intervals (ISIs) a function of recruitment threshold. The first modification is a refreshing innovation for the Fuglevand model that has a broad basis of support in previous animal and human literature (e.g., Erim et al. 1996; Gossen et al. 2003; Kernell et al. 1999).
With respect to the second modification, I believe a word of caution is warranted before modelers implement this change. As the authors state, their estimates of peak discharge rate are complicated by the ability to reliably discriminate single units over the full contraction range (Fig. 4 legend, Mortiz et al. 2005). A methodological bias could result in underestimation of the peak discharge rate in motor units with low recruitment thresholds arising in part from the rapidly deteriorating signal-to-noise ratio of intramuscular EMG with increased voluntary force. In addition to this potential methodological bias, the 12 units that the authors select for a clear plateau in discharge rate (Fig. 4, inset, Mortiz et al. 2005), are perhaps significantly correlated to recruitment threshold only as a result of the one data point near 60% MVC on the x-axis. Thus it may be prudent to await additional evidence of the linear relationship between peak discharge rate and recruitment threshold before the ubiquitous "onion-skin" phenomenon (De Luca and Erim 1994) is eliminated from the Fuglevand model.
The last modification is also of some interest because it addresses the issue of the relationship between variability and mean firing rate. It is often incorrectly asserted that firing rate variability decreases with an increase in mean firing rate and the naïve reader may find justification for this assertion in the ABSTRACT and DISCUSSION sections. Histograms of ISIs for many neurons recorded in vivo show a skewed probability distribution that is often fit to a Poisson or gamma function. This is also true for motoneurons firing near their minimal rhythmic firing rate. As the excitatory drive to a motoneuron is increased, the mean ISI decreases and the ISI histogram loses its skew. The ISI histogram is now better fit by a normal or Gaussian distribution (Clamann 1969; Fuglevand et al. 1993). With further increases in excitatory drive, the mean ISI decreases and the width of the histogram, quantified by the SD, also decreases. Typically a plot of SD versus mean ISI is used to illustrate this relationship (Person and Kudina 1972) and has been fit with a straight line indicating a constant coefficient of variation of about 20% (see Fuglevand et al. 1993). Perhaps intuitively it would seem that the low SD at small values of mean ISI, i.e., at faster firing rates, indicates that firing rate variability has decreased. However, by inverting the interval data to obtain the firing rate it can be verified analytically that this will result in the SD of firing rate increasing as the square root of the mean (see Stein et al. 2005). I suggest that when comparing statistical measures the best practice is to compare like with like—ISI variability-to-mean ISI or rate variability-to-mean rate—because it is helpful to have axes in the same units and it facilitates comparison to the literature on stochastic point processes.
The data from the three units in Fig. 3 (Moritz et al. 2005) are replotted here to illustrate that the data conform to the expected qualitative distribution for the statistical behavior of human motor units during isometric contractions. The explicit suggestion by Moritz and colleagues is that this relationship is better explained by an exponential fit (Eq. 1, Mortiz et al. 2005) rather than the typical linear regression. I would have preferred having Eq. 1 expressed with CVISI as a function of recruitment threshold expressed as excitatory input (RTE in Fuglevand et al. 1993) rather than force output. The latter formulation seems akin to putting the cart before the horse. Aside from this, by adding this subtle tweak the model output more closely approximates the experimental data (Fig. 6E, Mortiz et al. 2005).
If the nonlinear relationship between the SD versus mean ISI continues to be upheld experimentally, I speculate that the resulting increase in motor output variability at low forces may have broad impact for optimal control theories of sensorimotor systems (Scott 2004; Todorov 2005).
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1969 Clamann HP. Statistical analysis of motor unit firing patterns in a human skeletal muscle. Biophys J 9: 1233–1251, 1969.
1994 De Luca CJ and Erim Z. Common drive of motor units in regulation of muscle force. Trends Neurosci 17: 299–305, 1994.[CrossRef][ISI][Medline]
1996 Erim Z, De Luca CJ, Mineo K, and Aoki T. Rank-ordered regulation of motor units. Muscle Nerve 19: 563–573, 1996.[CrossRef][ISI][Medline]
1993 Fuglevand AJ, Winter DA, and Patla AE. Models of recruitment and rate coding organization in motor-unit pools. J Neurophysiol 70: 2470–2488, 1993.
2003 Gossen ER, Ivanova TD, and Garland SJ. The time course of the motoneurone afterhyperpolarization is related to motor unit twitch speed in human skeletal muscle. J Physiol 552: 657–664, 2003.
1999 Kernell D, Bakels R, and Copray JCVM. Discharge properties of motoneurons: how are they matched to the properties and use of their muscle units? J Physiol (Paris) 93: 87–96, 1999.
2005 Moritz CT, Barry BK, Pascoe MA, and Enoka RM. Discharge rate variability influences the variation in force fluctuations across the working range of a hand muscle. J Neurophysiol 93: 2449–2459, 2005.
1972 Person RS and Kudina LP. Discharge frequency and discharge pattern of human motor units during voluntary contraction of muscle. Electroencephalogr Clin Neurophysiol 32: 471–483, 1972.[CrossRef][ISI][Medline]
2004 Scott SH. Optimal feedback control and the neural basis of volitional motor control. Nat Rev Neurosci 5: 532–546, 2004.[Medline]
2005 Stein RB, Gossen ER, and Jones KE. Neuronal variability: noise or part of the signal? Nat Rev Neurosci 6: 389–397, 2005.[ISI][Medline]
2005 Todorov E. Stochastic optimal control and estimation methods adapted to the noise characteristics of the sensorimotor system. Neural Comput 17: 1084–1108, 2005.
1Department of Integrative Physiology
University of Colorado
Boulder, Colorado 80309-0354
To the Editor: The study by Moritz et al. (2005) demonstrated significant improvements in the performance of the Fuglevand model at matching the force fluctuations that occur during steady contractions by implementing a decreasing coefficient of variation (CV) for motor unit discharge with increases in muscle force. The exponential decline for discharge rate variability was derived from recordings of individual motor units discharging across a wide range of forces.
As suggested by Jones, it is appropriate to display variability on axes with the same units, and the figure in Panel A shows the relation between SD and mean interspike interval (ISI). As evident from the scatter in this plot, however, it is difficult to fit a single equation that describes this relation. Instead, by plotting the SD of ISI relative to the force above recruitment threshold (Panel B), the exponential relation becomes obvious. Consequently, the Fuglevand model was revised to include variability in motor unit discharge relative to force above recruitment threshold.
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Jones has also suggested that variability in motor unit discharge might have been better expressed in terms of the excitatory input delivered to the motor neuron pool. The advantage of the approach used by Moritz et al., however, is that finger force was directly measured experimentally, and the relation between excitation and force is not conserved in the model when motor unit number, recruitment range, or twitch force range are varied.
Regardless of how the data are presented, it should be noted that a linear relation between the SD and mean ISI does not necessarily indicate a constant coefficient of variation (CV = SD/mean). The relation between the SD and mean has to be linear and proportional for the coefficient of variation to be constant. For motor unit #6 re-plotted by Jones, the coefficient of variation for ISI was 2.5 times greater at long ISIs compared with short ISIs, despite a nearly linear relation. Furthermore, the Moritz et al. data do not display the purported relation in which the SD of discharge rate increased as the square root of the mean discharge rate (Stein et al. 2005).
The demonstration that discharge rate variability decreased rapidly as force rose above recruitment for each motor unit was the major factor resulting in a close match between simulated and experimentally measured force variability across the entire working range of the muscle. Of far less consequence to this aspect of the model's performance were the observations that minimal and peak discharge rate increased with recruitment threshold. Reliable recording of maximal discharge rate is difficult, and both the discrete isometric contractions used by Moritz et al. and the continuous ramp contractions used in other studies suffer from limitations. During ramp contractions to 40% or 80% MVC, for example, the force at the peak of the ramp may not be sufficiently high to elicit maximal discharge rates from high-threshold motor units, which could lead to the conclusion that high-threshold motor units have lower maximal discharge rates (De Luca et al. 1982). As emphasized by Moritz et al., the relation between maximal discharge rate and recruitment threshold remains an unresolved issue and probably depends on the task in which these measurements are made. Nonetheless, the recordings of single motor unit activity by Moritz et al. provided clear evidence of alterations in the relative variability of discharge rate within individual motor units.
REFERENCES
De Luca CJ, LeFever RS, McCue MP, and Xenakis AP. Behaviour of human motor units in different muscles during linearly varying contractions. J Physiol 329: 113–128, 1982.
Moritz CT, Barry BK, Pascoe MA, and Enoka RM. Discharge rate variability influences the variation in force fluctuations across the working range of a hand muscle. J Neurophysiol 93: 2449–2459, 2005.
Stein RB, Gossen ER, and Jones KE. Neuronal variability: noise or part of the signal? Nat Rev Neurosci 6: 389–397, 2005.[ISI][Medline]
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