Carlo J. De Luca

A practical method for calculating largest Lyapunov exponents from small data sets

Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. We present a new method for calculating the largest Lyapunov exponent from an experimental time series. The method follows directly from the definition of the largest Lyapunov exponent and is accurate because it takes advantage of all the available data. We show that the algorithm is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. Furthermore, one may use the algorithm to calculate simultaneously the correlation dimension. Thus, one sequence of computations will yield an estimate of both the level of chaos and the system complexity.

Physiology and Mathematics of Myoelectric Signals

The myoelectric (ME) signal is the electrical manifestation of the neuromuscular activation associated with a contracting muscle. It is an exceedingly complicated signal which is affected by the anatomical and physiological properties of muscles, the control scheme of the peripheral nervous system, as well as the characteristics of the instrumentation that is used to detect and observe it. Most of the relationships between the ME signal and the properties of a contracting muscle which are presently employed have evolved serendipitously. The lack of a proper description of the ME signal is probably the greatest single factor which has hampered the development of electromyography (EMG) into a precise discipline.

Filtering the surface EMG signal: Movement artifact and baseline noise contamination

The surface electromyographic (sEMG) signal that originates in the muscle is inevitably contaminated by various noise signals or artifacts that originate at the skin-electrode interface, in the electronics that amplifies the signals, and in external sources. Modern technology is substantially immune to some of these noises, but not to the baseline noise and the movement artifact noise. These noise sources have frequency spectra that contaminate the low-frequency part of the sEMG frequency spectrum. There are many factors which must be taken into consideration when determining the appropriate filter specifications to remove these artifacts; they include the muscle tested and type of contraction, the sensor configuration, and specific noise source. The band-pass determination is always a compromise between (a) reducing noise and artifact contamination, and (b) preserving the desired information from the sEMG signal. This study was designed to investigate the effects of mechanical perturbations and noise that are typically encountered during sEMG recordings in clinical and related applications. The analysis established the relationship between the attenuation rates of the movement artifact and the sEMG signal as a function of the filter band pass. When this relationship is combined with other considerations related to the informational content of the signal, the signal distortion of filters, and the kinds of artifacts evaluated in this study, a Butterworth filter with a corner frequency of 20 Hz and a slope of 12 dB/oct is recommended for general use. The results of this study are relevant to biomechanical and clinical applications where the measurements of body dynamics and kinematics may include artifact sources.

Common drive of motor units in regulation of muscle force

The neuromuscular system is responsible for all our interactions with our environment. Although recent decades have witnessed numerous discoveries that have shed light into various properties of this system, the basic principles underlying its overall operation still remain poorly understood. In this article, Carlo J. De Luca and Zeynep Erim discuss the concept of common drive of motor units that provides a possible scheme for the control of motor units, unifying various seemingly isolated findings that have been reported. According to this concept, a pool of motor units that makes up a muscle is controlled collectively during a contraction of that muscle. The unique firing patterns of individual motor units are effected, not by separate command signals sent to these units, but by one common drive to which motor units respond differently. The specific architecture of the system and the orderly gradation in the inherent properties of individual elements enable a single source to control the activities of all the motor units in a given pool. Such an arrangement relieves the CNS from the burden of monitoring and regulating each motor unit separately.

The role of plantar cutaneous sensation in unperturbed stance

Considerable evidence shows that sensation from the feet and ankles is important for standing balance control. It remains unclear, however, to what extent specific foot and ankle sensory systems are involved. This study focused on the role of plantar cutaneous sensation in quasi-static balance control. Iontophoretic delivery of anesthesia was used to reduce the sensitivity of the forefoot soles. In a follow-up experiment, subjects received intradermal injections of local anesthetic into the entire weight-bearing surface of the foot soles. Properties of the center-of-foot-pressure (COP) trajectories and ground reaction shear forces were analyzed using stabilogram–diffusion analysis and summary statistics. Effects of foot-sole anesthesia were generally small and mostly manifested as increases in COP velocity. Magnitude of COP displacement was unaffected by foot-sole anesthesia. Forefoot anesthesia mainly influenced mediolateral posture control, whereas complete foot-sole anesthesia had an impact on anteroposterior control. During bipedal stance, statistically significant effects of foot-sole anesthesia on COP were present only under eyes-closed conditions and included increases in COP velocity (11–12%) and shear force root-mean-square (13%), the latter indicating increases in body center-of-mass accelerations due to the foot-sole anesthesia. Similar effects were seen for unipedal stance in addition to an increase in anteroposterior COP median frequency (36%). Changes in stabilogram–diffusion parameters were confined to the short-term region suggesting that sensory information from the foot soles is mainly used to set a relevant background muscle activity for a given posture and support surface characteristic, and consequently is of little importance for feedback control during unperturbed stance. In general, this study demonstrates that plantar sensation is of moderate importance for the maintenance of normal standing balance when the postural control system is challenged by unipedal stance or by closing of the eyes. The impact of reduced plantar sensitivity on postural control is expected to increase with the loss of additional sensory modalities such as the concomitant proprioceptive deficits commonly associated with peripheral neuropathies.

Reconstruction expansion as a geometry-based framework for choosing proper delay times

Abstract The quality of attractor reconstruction using the method of delays is known to be sensitive to the delay parameter, τ. Here we develop a new, computationally efficient approach to choosing τ that quantifies reconstruction expansion from the identity line of the embedding space. We show that reconstruction expansion is related to the concept of reconstruction signal strength and that increased expansion corresponds to diminished effects of measurement error. Thus, reconstruction expansion represents a simple, geometrical framework for choosing τ. Furthermore, we describe the role of dynamical error in attractor expansion and argue that algorithms for determining τ should be considered as attempts at estimating an upper bound to the optimal delay.

Surface myoelectric signal cross-talk among muscles of the leg ☆

Surface myoelectric signals were detected from the skin surface above the tibialis anterior muscle, the peroneus brevis muscle, the soleus muscle and the tibial bone during selective maximal electrical stimulation of the tibialis anterior muscle in 12 normal subjects. The double differential technique developed by Broman et al. (1985) was used to determine if the detected signal was due to volume conduction from the tibialis anterior fibers. The peak-to-peak (PP), average rectified (ARV) and root mean square (RMS) amplitudes of the M waves were computed for each detection location. The values detected on the tibial bone, on the peroneus and on the soleus muscles were normalized with respect to those detected on the tibialis anterior and ranged from 4.8% to 33.0% (PP), 4.7% to 36.0% (ARV), and 7.7% to 37.4% (RMS) for the tibial bone area; from 4.0% to 20.0% (PP), 3.5% to 10.0% (ARV), and 3.0% to 10.0% (RMS) for the peroneus brevis muscle area; and from 3.0% to 8.0% (PP), 3.4% to 9.1% (ARV), and 2.0% to 9.8% (RMS) for the soleus muscle area. Neither peak-to-peak values, average rectified values nor root mean square values appeared to be correlated with leg size. It is concluded that a surface myoelectric signal detected on the skin above a leg muscle and having a peak-to-peak amplitude of up to 16.6% of a signal detected above a neighboring muscle may be due to cross-talk rather than to activation of the muscle below the electrode.

Use of the surface EMG signal for performance evaluation of back muscles.

For well over a decade my associates and I have been developing an objective, noninvasive technique to evaluate the performance of low‐back muscles, with emphasis on being able to distinguish between healthy and dysfunctioned backs. Our approach is based on the well‐known fact that the EMG signal undergoes a compression in the frequency domain during a sustained muscle contraction. In particular we track the median frequency of EMG signals detected from six muscles in the lower back during an isometric extension of the trunk. The measurements are taken with the Back Analysis System which consists of a postural restraining device, special electrodes for detecting the EMG signals, a muscle fatigue monitor which calculates the median frequencies, and the appropriate software. We have found that the pattern of fatigue exhibited by the six median frequency curves can be used to distinguish individuals who have low‐back pain from those who do not with an accuracy of at least 84%. An even more relevant and timely application of our technique is for quantifying the progression of the performance of low‐back muscles during a rehabilitation program. Although more work is required to explore the intricacies of the technique, present results provide a convincing indication that it is reliable and that it is ready to be placed into practice.

Effects of Aging on Motor-Unit Control Properties

It was hypothesized that the age-related alterations in the morphological properties of a motor unit would be accompanied by modifications in the control aspects of the motor unit, as either an adaptive or compensatory mechanism to preserve smooth force production. In specific, the objective of the study was to investigate the age-related alterations in the concurrent firing behavior of multiple motor units in the first dorsal interosseous (FDI) muscle in isometric contractions at 20 and 50% of the subjects voluntary contraction level. Analysis of the data collected from 10 young (24-37 yr of age) and 10 elderly (65-88 yr of age) subjects led to three novel observations regarding the firing behavior of aged motor units. 1) Among elderly subjects, there is a decrease in the common fluctuations that are observed among the firing rates of motor units in the young. 2) The relationship observed between the firing rate and recruitment threshold of young subjects is disturbed in the elderly. Although in young subjects, at any point in a given submaximal contraction, earlier recruited motor units have higher firing rates than later-recruited units; in aged subjects this dependency of firing rate on recruitment rank is compromised. 3) The progressive decrease observed in the firing rates of concurrently active motor units in constant-force contractions in the young is not seen in the aged. In addition to these original findings, this study provided support for earlier reports of 1) decreased average firing rates probably reflecting the slowing of the muscle, 2) a shift in recruitment thresholds toward lower force levels in line with the shift toward type I fibers, and 3) multiphasic action potential shapes indicative of the reinnervation process that takes place during aging. Taken as a whole, these findings indicate significant age-related modifications in the control properties of human motor units.

Median frequency of the myoelectric signal: effects of muscle ischemia and cooling

SummaryA study was performed to investigate the changes that occur in the median frequency of the myoelectric signal during local ischemia or reduction of intramuscular temperature produced by surface cooling. Data was obtained from experiments which involved the first dorsal interosseous muscle of 10 female and 16 male subjects. These subjects were asked to perform isometric constant-force abduction contractions of the index finger at 20% and 80% of maximal voluntary contraction level. The initial median frequency (IMF) of the myoelectric signal during the first 0.5 s of contraction was calculated. Results showed a significant reduction of the IMF in contractions performed under ischemic conditions; upon release, the IMF recovered quickly. At 80% maximal voluntary level of contraction, a greater decrease of the IMF was recorded. Similar results were demonstrated during reduction of intramuscular temperature with gradual recovery of the IMF after cooling. These results demonstrate that the median frequency of the myoelectric signal displays behavior similar to that reported for conduction velocity and this is consistent with the notion that accumulation of metabolic byproducts in muscle tissue causes a decrease in the conduction velocity of the muscle fibers.