On the time allometry of co-ordinated rhythmic movements

The focus is the power formulae relating periodic time in terrestrial locomotion and flight to mass and length. The periodic timing of limbs and wings oscillating comfortably in absolute co-ordination is viewed as the characteristic period tau 0 of a system in which the free, undamped oscillatory motion of a point mass m at a distance l from a fixed axis does work against two conservative forces. These forces are in the form of gravity g acting on the point mass and a spring of stiffness k acting at a distance b from the axis. The system's characteristic period can be expressed most simply as: tau 0 = 2 pi [ml2/(mlg + kb2)]1/2. In the biological instantiation of this hybrid mass-spring/simple pendulum system, muscular and other tissues function as the spring that elastically stores and releases mechanical energy. Regular oscillations are brought about and sustained by a muscular driving force that ordinarily is close to resonance. The resultant dynamical regime--basically, raising and lowering a mass at regular intervals with respect to gravity--is referred to as the pendular clocking mode of movement organization. The mode is investigated comfortably at a common period and a fixed phase. In absolute co-ordination, two wrist-pendulum systems can be interpreted physically as a virtual single system. The evidence suggests that the scalings of the periodic times of such systems to mass and to length follow directly from the dynamical properties inherent in the resonance equation of the pendular clocking mode. Recourse to biological constants to rationalize the time scale is unnecessary. Experiments on human wrist-pendular activity and detailed analyses of the mass and length dependencies of the locomotory cycle times of quadrupeds, large birds, small passerines, hummingbirds, and insects are performed with respect to the dynamical properties predicted for systems in the pendular clocking mode. The major conclusion is that all the time scales of terrestrial in locomotory time allometries follow systematically from differences in the length scale and differences in the relation of mass to length.