The mathematical whisker: A review of numerical models of the rat׳s vibrissa biomechanics

Undulating Seal Whiskers Evolved Optimal Wavelength-to-Diameter Ratio for Efficient Reduction in Vortex-Induced Vibrations

Seals are well-known for their remarkable hydrodynamic trail-following capabilities made possible by undulating flow-sensing whiskers that enable the seals to detect fish swimming as far as 180 m away. In this work, the form-function relationship in the undulating whiskers of two different phocid seal species, viz. harbor and gray seals, is studied. The geometry and material properties of excised harbor and grey seal whiskers are systematically characterized using blue light 3D scanning, optical and scanning electron microscopy, and nanoindentation. The effect of the undulating geometry on the whiskers' vibration in uniform water flow is studied using both experimental (piezoelectric MEMS and 3D-printed piezoresistive sensors developed in-house) and numerical (finite element method) techniques. The results indicate that the dimensionless ratio of undulation wavelength to mean whisker diameter (λ/Dm ) in phocid seals may have evolved to be in the optimal range of 4.4-4.6, enabling an order-of-magnitude reduction in vortex-induced vibrations (compared to a similarly-shaped circular cylinder) and, consequently, an enhanced flow sensing capability with minimal self-induced noise. The results highlight the importance of the dimensionless λ/Dm ratio in the biomimetic design of seal whisker-inspired vibration-resistant structures, such as marine risers and wake detection sensors for submarines.

On the intrinsic curvature of animal whiskers

Facial vibrissae (whiskers) are thin, tapered, flexible, hair-like structures that are an important source of tactile sensory information for many species of mammals. In contrast to insect antennae, whiskers have no sensors along their lengths. Instead, when a whisker touches an object, the resulting deformation is transmitted to mechanoreceptors in a follicle at the whisker base. Previous work has shown that the mechanical signals transmitted along the whisker will depend strongly on the whisker's geometric parameters, specifically on its taper (how diameter varies with arc length) and on the way in which the whisker curves, often called "intrinsic curvature." Although previous studies have largely agreed on how to define taper, multiple methods have been used to quantify intrinsic curvature. The present work compares and contrasts different mathematical approaches towards quantifying this important parameter. We begin by reviewing and clarifying the definition of "intrinsic curvature," and then show results of fitting whisker shapes with several different functions, including polynomial, fractional exponent, elliptical, and Cesàro. Comparisons are performed across ten species of whiskered animals, ranging from rodents to pinnipeds. We conclude with a discussion of the advantages and disadvantages of using the various models for different modeling situations. The fractional exponent model offers an approach towards developing a species-specific parameter to characterize whisker shapes within a species. Constructing models of how the whisker curves is important for the creation of mechanical models of tactile sensory acquisition behaviors, for studies of comparative evolution, morphology, and anatomy, and for designing artificial systems that can begin to emulate the whisker-based tactile sensing of animals.

Of mice and monkeys: Somatosensory processing in two prominent animal models

Our understanding of the neural basis of somatosensation is based largely on studies of the whisker system of mice and rats and the hands of macaque monkeys. Results across these animal models are often interpreted as providing direct insight into human somatosensation. Work on these systems has proceeded in parallel, capitalizing on the strengths of each model, but has rarely been considered as a whole. This lack of integration promotes a piecemeal understanding of somatosensation. Here, we examine the functions and morphologies of whiskers of mice and rats, the hands of macaque monkeys, and the somatosensory neuraxes of these three species. We then discuss how somatosensory information is encoded in their respective nervous systems, highlighting similarities and differences. We reflect on the limitations of these models of human somatosensation and consider key gaps in our understanding of the neural basis of somatosensation.

Mechanics and Morphological Compensation Strategy for Trimmed Soft Whisker Sensor

Recent studies have been inspired by natural whiskers for a proposal of tactile sensing system to augment the sensory ability of autonomous robots. In this study, we propose a novel artificial soft whisker sensor that is not only flexible but also adapts and compensates for being trimmed or broken during operation. In this morphological compensation designed from an analytical model of the whisker, our sensing device actively adjusts its morphology to regain sensitivity close to that of its original form (before being broken). To serve this purpose, the body of the whisker comprises a silicon-rubber truncated cone with an air chamber inside as the medulla layer, which is inflated to achieve rigidity. A small strain gauge is attached to the outer wall of the chamber for recording strain variation upon contact of the whisker. The chamber wall is reinforced by two inextensible nylon fibers wound around it to ensure that morphology change occurs only in the measuring direction of the strain gauge by compressing or releasing pressurized air contained in the chamber. We investigated an analytical model for the regulation of whisker sensitivity by changing the chamber morphology. Experimental results showed good agreement with the numerical results of performance by an intact whisker in normal mode, as well as in compensation mode. Finally, adaptive functionality was tested in two separate scenarios for thorough evaluation: (1) A short whisker (65 mm) compensating for a longer one (70 mm), combined with a special case (self-compensation), and (2) vice versa. Preliminary results showed good feasibility of the idea and efficiency of the analytical model in the compensation process, in which the compensator in the typical scenario performed with 20.385% average compensation error. Implementation of the concept in the present study fulfills the concept of morphological computation in soft robotics and paves the way toward accomplishment of an active sensing system that overcomes a critical event (broken whisker) based on optimized morphological compensation.

The Euler spiral of rat whiskers

This paper reports on an analytical study of the intrinsic shapes of 523 whiskers from 15 rats. We show that the variety of whiskers on a rat's cheek, each of which has different lengths and shapes, can be described by a simple mathematical equation such that each whisker is represented as an interval on the Euler spiral. When all the representative curves of mystacial vibrissae for a single rat are assembled together, they span an interval extending from one coiled domain of the Euler spiral to the other. We additionally find that each whisker makes nearly the same angle of 47^∘ with the normal to the spherical virtual surface formed by the tips of whiskers, which constitutes the rat's tactile sensory shroud or "search space." The implications of the linear curvature model for gaining insight into relationships between growth, form, and function are discussed.

Quantification of vibrissal mechanical properties across the rat mystacial pad

Recent work has quantified the geometric parameters of individual rat vibrissae (whiskers) and developed equations that describe how these parameters vary as a function of row and column position across the array. This characterization included a detailed quantification of whisker base diameter and arc length as well as the geometry of the whisker medulla. The present study now uses these equations for whisker geometry to quantify several properties of the whisker that govern its mechanical behavior. We first show that the average density of a whisker is lower in its proximal region than in its distal region. This density variation appears to be largely attributable to the presence of the whisker cuticle rather than the medulla. The density variation has very little effect on the center of mass of the whisker. We next show that the presence of the medulla decreases the deflection of the whisker under its own weight and also decreases its mass moment of inertia while sacrificing <1% stiffness at the whisker base compared with a solid whisker. Finally, we quantify two dimensionless parameters across the array. First, the deflection-to-length ratio decreases from caudal to rostral: caudal whiskers are longer but deflect more under their own weight. Second, the nondimensionalized radius of gyration is approximately constant across the array, which may simplify control of whisking by the intrinsic muscles. We anticipate that future work will exploit the mechanical properties computed in the present study to improve simulations of the mechanosensory signals associated with vibrissotactile exploratory behavior. NEW & NOTEWORTHY The mechanical signals transmitted by a whisker depend critically on its geometry. We used measurements of whisker geometry and mass to quantify the center of mass, mass moment of inertia, radius of gyration, and deflection under gravity of the whisker. We describe how variations in these quantities across the array could enhance sensing behaviors while reducing energy costs and simplifying whisking control. Most importantly, we provide derivations for these quantities for use in future simulation work.

The Static and Dynamic Sensitivity of Magnetostrictive Bioinspired Whisker Sensor

Magnetostrictive bioinspired whisker is a new kind of sensor that can realize tactile and flow sensing by utilizing magnetoelastic effect. The sensitivity is a key technical indicator of whisker sensor. The paper presented a new magnetostrictive whisker based on Galfenol cantilever beam, as well as its operation principle. Then, the static and dynamic sensitivity of the whisker sensor was investigated by using a self-made experimental system. The results illustrated that the proposed sensor has a high sensitivity. Its static sensitivity is 2.2 mV/mN. However, its dynamic sensitivity depends on the vibration frequency. When working at the natural frequency of the cantilever beam, the dynamic sensitivity performs an obvious increase—1.3 mV/mN at 3.5 Hz (the first-order natural frequency) and 2.1 mV/mN at 40 Hz (the second-order natural frequency), respectively.

Effect of whisker geometry on contact force produced by vibrissae moving at different velocities

Rats and mice are able to perform a variety of subtle tactile discriminations with their mystacial vibrissae. Increasingly, the design and interpretation of neurophysiological and behavioral studies are inspired by and linked to a more precise understanding of the detailed physical properties of the whiskers and their associated hair follicles. Here we used a piezoelectric sensor (bimorph) to examine how contact forces are influenced by the geometry of individual whisker hairs. For a given point along a whisker, bimorph signals are linearly related to whisker movement velocity. The slope of this linear function, called velocity sensitivity (VS), diminishes nonlinearly as whisker diameter decreases. Whiskers differ in overall length, thickness, and proximal-distal taper. Thus VS varies along an individual whisker and among different whiskers on the mystacial pad. Thinner, shorter whiskers, such as those located rostrally in rats and those in mice, have lower overall VSs, rendering them potentially less effective for mediating discriminations that rely on subtle velocity cues. The nonlinear effect of diameter combined with the linear effect of arc length produces radial distance tuning curves wherein small differences in the proximal-distal location of impacts yields larger differences in signal magnitude. Such position-dependent cues could contribute to the localization of objects near the face. Proximal-to-distal changes in contact location during whisking sweeps could also provide signals that aid texture discrimination.NEW & NOTEWORTHY This study describes the geometry of facial whiskers distributed across the mystacial pad with emphasis on velocity encoding of object strikes. Findings indicate how the shapes, lengths, and thicknesses of individual hairs can contribute to sophisticated vibrissa-based tactile discrimination.

Variations in vibrissal geometry across the rat mystacial pad: base diameter, medulla, and taper

Many rodents tactually sense the world through active motions of their vibrissae (whiskers), which are regularly arranged in rows and columns (arcs) on the face. The present study quantifies several geometric parameters of rat whiskers that determine the tactile information acquired. Findings include the following. 1) A meta-analysis of seven studies shows that whisker base diameter varies with arc length with a surprisingly strong dependence on the whisker's row position within the array. 2) The length of the whisker medulla varies linearly with whisker length, and the medulla's base diameter varies linearly with whisker base diameter. 3) Two parameters are required to characterize whisker "taper": radius ratio (base radius divided by tip radius) and radius slope (the difference between base and tip radius, divided by arc length). A meta-analysis of five studies shows that radius ratio exhibits large variability due to variations in tip radius, while radius slope varies systematically across the array. 4) Within the resolution of the present study, radius slope does not differ between the proximal and distal segments of the whisker, where "proximal" is defined by the presence of the medulla. 5) Radius slope of the medulla is offset by a constant value from radius slope of the proximal portion of the whisker. We conclude with equations for all geometric parameters as functions of row and column position.NEW & NOTEWORTHY Rats tactually explore their world by brushing and tapping their whiskers against objects. Each whisker's geometry will have a large influence on its mechanics and thus on the tactile signals the rat obtains. We performed a meta-analysis of seven studies to generate equations that describe systematic variations in whisker geometry across the rat's face. We also quantified the geometry of the whisker medulla. A database provides access to geometric parameters of over 500 rat whiskers.