Weber's law and perceptual categories: Another teleological view

As an alternative to optimum-processor models in which sensors attempt to circumvent internal and external noise, a mechanism-independent argument is presented for Weber's law in vision and hearing. In vision, the argument is that categories of objects should be independent of the light intensity on these objects. In hearing, sound categorization should be independent of the distance from the sound source. An analogous desideratum for computer-based image segmentation is also presented.     

Radial distance determination in the rat vibrissal system and the effects of Weber's law

Rats rhythmically tap and brush their vibrissae (whiskers) against objects to tactually explore the environment. To extract a complex feature such as the contour of an object, the rat must at least implicitly estimate radial object distance, that is, the distance from the base of the vibrissa to the point of object contact. Radial object distance cannot be directly measured, however, because there are no mechanoreceptors along the vibrissa. Instead, the mechanical signals generated by the vibrissa's interaction with the environment must be transmitted to mechanoreceptors near the vibrissa base. The first part of this paper surveys the different mechanical methods by which the rat could determine radial object distance. Two novel methods are highlighted: one based on measurement of bending moment and axial force at the vibrissa base, and a second based on measurement of how far the vibrissa rotates beyond initial contact. The second part of the paper discusses the application of Weber's law to two methods for radial distance determination. In both cases, Weber's law predicts that the rat will have greatest sensing resolution close to the vibrissa tip. These predictions could be tested with behavioural experiments that measure the perceptual acuity of the rat.     

The effect of activity during early postnatal development on motor unit size

At early stages of neuromuscular development, motor unit territory is expanded, with each muscle fibre being supplied by several axons. During postnatal development, some synapses are eliminated, motor unit size decreases, and the adult distribution of motor unit sizes emerges. This process depends on activity, since it proceeds more rapidly when the nerve is activated and is slower when activity is reduced. Here we studied whether, in addition to influencing the rate of retraction of motor unit territory, activity during the critical period of development affects the final outcome of the distribution of motor unit sizes. The sciatic nerve of 8- to 12-day-old rats was stimulated daily. One week later the tension of the extensor digitorum longus muscle and that of its individual motor units was recorded. The sizes of individual motor units were calculated and compared with those from animals that received no stimulation. The distribution of motor unit sizes from stimulated muscles was not significantly different from those from control muscles. Therefore, we conclude that although activity increases the rate at which motor units attain their adult size, it does not influence the final outcome of motor unit size distribution.     

Amplitude cancellation reduces the size of motor unit potentials averaged from the surface EMG.

The purpose of the study was to evaluate the influence of selected physiological parameters on amplitude cancellation in the simulated surface electromyogram (EMG) and the consequences for spike-triggered averages of motor unit potentials derived from the interference and rectified EMG signals. The surface EMG was simulated from prescribed recruitment and rate coding characteristics of a motor unit population. The potentials of the motor units were detected on the skin over a hand muscle with a bipolar electrode configuration. Averages derived from the EMG signal were generated using the discharge times for each of the 24 motor units with lowest recruitment thresholds from a population of 120 across three conditions: 1) excitation level; 2) motor unit conduction velocity; and 3) motor unit synchronization. The area of the surface-detected potential was compared with potentials averaged from the interference, rectified, and no-cancellation EMGs. The no-cancellation EMG comprised motor unit potentials that were rectified before they were summed, thereby preventing cancellation between the opposite phases of the potentials. The percent decrease in area of potentials extracted from the rectified EMG was linearly related to the amount of amplitude cancellation in the interference EMG signal, with the amount of cancellation influenced by variation in excitation level and motor unit conduction velocity. Motor unit synchronization increased potentials derived from both the rectified and interference EMG signals, although cancellation limited the increase in area for both potentials. These findings document the influence of amplitude cancellation on motor unit potentials averaged from the surface EMG and the consequences for using the procedure to characterize motor unit properties.     

Detection of motor unit action potentials with surface electrodes: influence of electrode size and spacing

A model of the motor unit action potential was developed to investigate the amplitude and frequency spectrum contributions of motor units, located at various depths within muscle, to the surface detected electromyographic (EMG) signal. A dipole representation of the transmembrane current in a three-dimensional muscle volume was used to estimate detected individual muscle fiber action potentials. The effects of anisotropic muscle conductance, innervation zone location, propagation velocity, fiber length, electrode area, and electrode configuration were included in the fiber action potential model. A motor unit action potential was assumed to be the sum of the individual muscle fiber action potentials. A computational procedure, based on the notion of isopotential layers, was developed which substantially reduced the calculation time required to estimate motor unit action potentials. The simulations indicated that: 1) only those motor units with muscle fibers located within 10–12 mm of the electrodes would contribute significant signal energy to the surface EMG, 2) variation in surface area of electrodes has little effect on the detection depth of motor unit action potentials, 3) increased interelectrode spacing moderately increases detection depth, and 4) the frequency content of action potentials decreases steeply with increased electrode-motor unit territory distance.     

A macroevolutionary explanation for energy equivalence in the scaling of body size and population density

Across a wide array of animal species, mean population densities decline with species body mass such that the rate of energy use of local populations is approximately independent of body size. This "energetic equivalence" is particularly evident when ecological population densities are plotted across several or more orders of magnitude in body mass and is supported by a considerable body of evidence. Nevertheless, interpretation of the data has remained controversial, largely because of the difficulty of explaining the origin and maintenance of such a size-abundance relationship in terms of purely ecological processes. Here I describe results of a simulation model suggesting that an extremely simple mechanism operating over evolutionary time can explain the major features of the empirical data. The model specifies only the size scaling of metabolism and a process where randomly chosen species evolve to take resource energy from other species. This process of energy exchange among particular species is distinct from a random walk of species abundances and creates a situation in which species populations using relatively low amounts of energy at any body size have an elevated extinction risk. Selective extinction of such species rapidly drives size-abundance allometry in faunas toward approximate energetic equivalence and maintains it there.     

The resilience of the size principle in the organization of motor unit properties in normal and reinnervated adult skeletal muscles

Henneman's size principle relates the input and output properties of motoneurons and their muscle fibers to size and is the basis for size-ordered activation or recruitment of motor units during movement. After nerve injury and surgical repair, the relationship between motoneuron size and the number and size of the muscle fibers that the motoneuron reinnervates is initially lost but returns with time, irrespective of whether the muscles are self- or cross-reinnervated by the regenerated axons. Although the return of the size relationships was initially attributed to the recovery of the cross-sectional area of the reinnervated muscle fibers and their force per fiber, direct enumeration of the innervation ratio and the number of muscle fibers per motoneuron demonstrated that a size-dependent branching of axons accounts for the size relationships in normal muscle, as suggested by Henneman and his colleagues. This same size-dependent branching accounts for the rematching of motoneuron size and muscle unit size in reinnervated muscles. Experiments were carried out to determine whether the daily amount of neuromuscular activation of motor units accounts for the size-dependent organization and reorganization of motor unit properties. The normal size-dependent matching of motoneurons and their muscle units with respect to the numbers of muscle fibers per motoneuron was unaltered by synchronous activation of all of the motor units with the same daily activity. Hence, the restored size relationships and rematching of motoneuron and muscle unit properties after nerve injuries and muscle reinnervation sustain the normal gradation of muscle force during movement by size-ordered recruitment of motor units and the process of rate coding of action potentials. Dynamic modulation of size of muscle fibers and their contractile speed and endurance by neuromuscular activity allows for neuromuscular adaptation in the context of the sustained organization of the neuromuscular system according to the size principle.     

Signal transmission through a metabolic cycle follows the compression hypothesis or a weak Weber's law

Biological signal transduction often involves a metabolic cycle in which the flux at one point is driven by the input signal and the concentration of one of the metabolites of the cycle serves as the output signal. A kinetic analysis of such a metabolic cycle is made under an assumption that the law of mass action applies. The resultant kinetic model can produce a response that overshoots, quickens, and eventually saturates as the input intensity is increased. The possible model behavior ranges parametrically from non-adaptive (compression hypothesis) to weakly adaptive (limited Weber's law).     

Cognition of different length by Physarum polycephalum: Weber's law in an amoeboid organism

A true slime mold, the plasmodium of Physarum polycephalum has the ability to find the shortest route between two points in a labyrinth. To find the shortest route between two points, detection of the difference in lengths can be made from two aspects: the absolute difference between the lengths or the ratio of them. We found that the ratio of two lengths, rather than the absolute difference between the two lengths, was important in discriminating the difference in the two lengths by P. polycephalum. This finding indicates that an amoeboid organism detects differences in stimulus intensity as though it is constrained by Weber's law, suggesting that Weber's law is not reliant on the presence of a neural system and is used widely even in Amoebozoa.     

Weber's theory of the kernleiter

The potential distribution about a kernleiter is determined according to Weber's method. It is shown that the distribution reduces to the solution of a telegrapher's equation when the volume of the external medium is small. The velocity of propagation as a function of the external volume is determined approximately. This involves the solution of the equation

$$\frac{{\left[ {Y_0 (k\xi )} \right]^\prime }}{{\left[ {J_0 (k\xi )} \right]^\prime }} = \frac{{\left[ {\xi ^{ - a} Y_0 (\xi )} \right]^\prime }}{{\left[ {\xi ^{ - a} J_0 (\xi )} \right]^\prime }}$$     

A study of the motor unit action potential by means of computer simulation

In order to study the motor unit action potential a computer simulation model was developed. It is based on the superposition of single muscle fibre potentials of the fibres belonging to the motor unit. The parameters which characterize each fibre (spatial position, diameter, and a dispersion of arrival time of the potential at the electrode) are chosen from statistical distributions which can be derived from anatomical and physiological data. The electrode type, position and dimensions can be specified. Simulated motor unit action potentials are presented in the time and frequency domain. The simulation results refer to (1) the influence of the electrode position and dimensions with respect to the motor unit territory, (2) the meaning of this model for the study of pathological phenomena, (3) the variability of some parameters characterizing the motor unit, (4) the selectivity of uni- and bipolar electrodes and finally (5) the influence of the geometrical situation of the motor end-plates within the muscle, on the shape of motor unit action potentials.     

Physiological and developmental implications of motor unit anatomy

There is increasing evidence that the architectural design and arrangement of the fibers within a motor unit have important physiological and developmental ramifications. Limited data, however, are available to directly address this issue. In the present study the physiological properties of one motor unit in each of seven cat tibialis anterior (TA) muscles were determined. Each of these units then was repetitively stimulated to deplete the glycogen in all muscle fibers within the unit. Subsequently, the length, type of ending, and spatial distribution of fibers sampled from these physiologically and histochemically typed motor units were determined. Four fast fatigable (FF), one fast, fatigue resistant (FR), and two slow (S) motor units (MU) were studied. The samples consisted of all those glycogen-depleted fibers (9-27) contained within a single fascicle or a circumscribed area of each of the motor unit territories. The mean fiber lengths for the two slow motor units were 35.9 and 45.5 mm. The mean fiber lengths for the fast motor unit samples ranged from 8.8 to 48.5 mm. Some fibers of both the fast and slow units reached lengths of 58 mm. Most of the fibers in the slow units extended the entire distance between the proximal and distal musculotendinous planes, had relatively constant cross-sectional areas, and terminated at the tendon as blunt endings. In contrast, the majority of the fibers in the fast units terminated intrafascicularly at one end, and the cross-sectional area decreased progressively along their lengths, that is, showed a tapering pattern for a significant proportion of their lengths. Therefore, the force generated by units that end midfascicularly would appear to be transmitted to connective tissue elements and/or adjacent fibers. All fibers of a fast unit within a fascicle were located at approximately the same proximo-distal location. Thus, developmentally the selection of muscle fibers by a motoneuron would seem to be influenced by their spatial distribution. The architectural complexities of motor units also have clear implications for the mechanical interactions of active and inactive motor units. For example, the tension capabilities of a motor unit may be influenced not only by the spatial arrangement of its own fibers, but also by the level of activation of neighboring motor units.     

A dimensionless invariant for relative size at sex change in animals: explanation and implications

Recent comparative studies across sex-changing animals have found that the relative size and age at sex change are strikingly invariant. In particular, 91%-97% of the variation in size at sex change across species can be explained by the simple rule that individuals change sex when they reach 72% of their maximum body size. However, this degree of invariance is surprising and has proved controversial. In particular, it is not clear why this result should hold, given that there is considerable biological variation across species in factors that can influence the evolutionarily stable timing of sex change. Our overall aim here is to explain this result and determine the implications for other life-history variables. Specifically, we use a combination of approaches to formalize and make explicit previous analytical theory in this area, examine the robustness of the empirical invariance result, and carry out sensitivity analyses to determine what the empirical data imply about the mean value and variation in several key life-history variables.