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

Abstract

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.

Figure 1.

Vibrissa coordinate system and description of variables relevant to radial distance determination. (a) Cylindrical coordinate system for three-dimensional object localization. (b) Example of the ‘angle-side-angle’ method for radial distance determination. (c) Forces and moments at the vibrissa base induced by contact with an object. (d) dM/d#x03B8; is monotonic with radial distance for the ‘rotational compliance’ method.

Figure 2.

Numerical simulations of vibrissa bending for large angles. (a) Illustrations of how the data were generated for three example force locations. (b) Moment as a function of contact force magnitude and location for a tapered vibrissa. (c) Moment as a function of contact force magnitude and location for a cylindrical vibrissa. (d) Radial distance can be uniquely determined by measuring only axial force and moment at the base for a tapered vibrissa. (e) Deflection angle can also be uniquely determined for a tapered vibrissa. (f) For a cylindrical vibrissa, there is a one-to-one relationship between axial force and moment, and determining radial distance or deflection angle from (M_z, f_x) is not possible. In all plots, radial distance is normalized by vibrissa length L, moment is normalized by EI/L and force is normalized by EI/L², where E is Young's modulus and I is the area moment of inertia at the base.

Figure 3.

The balance of moments method is based on how radial distance relates to how much the vibrissa rotates beyond initial impact. (a) For a given contact angle (60° shown here) and given radial distance (30, 50, 70 and 90% vibrissa length shown here), the angle of moment balance is defined as the maximum change in θ beyond initial impact (where the curves intersect). (b) The relationship between angle of moment balance and radial distance (for any given angle of impact) is monotonic, allowing the determination of radial distance. (c) Repeating these steps for a range of impact angles, we obtain a surface. The curve in (b) can be interpreted as a horizontal slice through the surface at θ_i = 60°. The white space below the dashed line indicates situations where the vibrissa flicked past the object, and the space above the line indicates where moment balance was not achieved owing to the limited range of whisking motion.

Figure 4.

(a) Just noticeable difference (JND) of radial distance as a function of radial distance for a tapered vibrissa (solid line) and a cylindrical vibrissa (dashed line) using the ‘rotational compliance’ method. (b) JND of radial distance as a function of radial distance and deflection angle, using the ‘moment and axial force map’ method. For both plots, both radial distance and JND are normalized by vibrissa length, and the JND also scales with the Weber fraction.