Simulations of a Vibrissa Slipping along a Straight Edge and an Analysis of Frictional Effects during Whisking

Abstract

During tactile exploration, rats sweep their whiskers against objects in a motion called whisking. Here, we investigate how a whisker slips along an object's edge and how friction affects the resulting tactile signals. First, a frictionless model is developed to simulate whisker slip along a straight edge and compared with a previous model that incorporates friction but cannot simulate slip. Results of both models are compared to behavioral data obtained as a rat whisked against a smooth, stainless steel peg. As expected, the frictionless model predicts larger magnitudes of vertical slip than observed experimentally. The frictionless model also predicts forces and moments at the whisker base that are smaller and have a different direction than those predicted by the model with friction. Estimates for the friction coefficient yielded values near 0.48 (whisker/stainless steel). The present work provides the first assessments of the effects of friction on the mechanical signals received by the follicle during active whisking. It also demonstrates a proof-of-principle approach for reducing whisker tracking requirements during experiments and demonstrates the feasibility of simulating a full array of vibrissae whisking against a peg.

Fig. 1

Simulations of whisker slip as a rat whisks against a peg or an edge. (a) Illustration of the slip problem. The trajectory of the whisker prior to making contact with the peg is assumed to follow the kinematics defined in [11]. In the left panal, the whisker is shown in blue and the initial contact point location of the whisker on the peg is indicated as a white dot. As the rat protracts further against the peg (right panal) the contact point could be at a wide range of locations along the length of the peg. Three plausible contact point locations are shown as white asterisks. The orange whisker represents the case with no friction. (b) Frames from Supplementary Video 1 show results of a simulation that used “edge mode” to model whisker slip along the edge of a triangular prism. The gray traces represent where the whisker would be had it not contacted the triangular prism, and the black trace illustrates the shape of the deflected whisker.

Fig. 2

Friction affects the shape of the deflected whisker as the rat whisks against a peg. The four vertical dashed lines in (b), (c), and (d) mark the times shown in (a). (a) Top and front views of four video frames from a 3,300 msec trial of contact whisking behavior. The lines show the experimentally-tracked whisker (black), whisker shape predicted using contact-point mode (dashed cyan), and whisker predicted using edge mode (purple dashed-dot). Simulations using contact point mode accurately match the experimentally-tracked whisker in both camera views. The accuracy of this match is seen as a striped black-cyan line because the two traces overlie each other nearly exactly. In contrast, when the simulation is run using edge mode, the predicted whisker shape does not always accurately match the experimentally-tracked whisker. The match is particularly poor in frames (ii) and (iv). (b) The mean error between the experimentally-tracked whisker and the two modes of simulation. (c) The vertical location of contact simulated using contact-point mode and edge mode. Traces are color coded as in (b). (d) The arc length of contact (s_applied) with the peg predicted by the two modes of simulation. Traces are color coded as in (b). (e) Equation (6) offers a good prediction for z_error, the difference in contact point location when computed by edge mode and contact point mode. The dots are semi-transparent to show the data density. The best fit line is plotted in red, and the identity line is plotted in black. (f) In this figure, values on the x-axis show the vertical position of contact of the whisker on the peg as measured by the 3D video. Values on the y-axis represent the predicted vertical point of intersection between the peg and the whisker as computed only with whisking kinematics, with no information about the actual contact point. The dots are semi-transparent to show the data density. The best fit line is plotted in red, and the identity line is plotted in black.

Fig. 3

The effects of friction on forces and moments at the whisker base. (a) Top: Schematic showing x-, y-, and z- directions in world coordinates. bottom: Forces and moments in world coordinates. Scale bar: F_x-world, F_y-world, and F_z-world: 1 mN; M_x-world: 5 µNm; M_y-world and M_z-world: 10 µNm. (b) Top: Schematic showing x-, y-, and z- directions in whisker-centered coordinates, as well as the definitions for F_T, M_B, F_D, and M_D. Bottom: Forces and moments in whisker-centered coordinates correspond to the mechanical signals directly transmitted to the follicle. One point in the F_D subplot was an outlier and is marked by an asterisk. Scale bar: F_x and F_T: 1 mN; F_D and M_D: 180°; M_x: 2.5 µNm; M_B: 10 µNm. (c) A plot of the value of the friction coefficient (μ) over time. Outlier values are plotted as red dots. Outliers larger than 3 (some as large as 70) are plotted at 3 for clarity. (d) A plot of normal force versus μ. The mean of μ across normal forces is plotted as a blue line, and the shaded region denotes the standard deviation above and below the mean. Normal forces below 0.1 mN, marked by the black vertical line, result in highly variable μ values. Therefore, any μ values resulting from normal forces below this are treated as outliers. (e) Histogram of all calculated values for the friction coefficient (μ), excluding outliers.

Fig. 4. Edge mode (frictionless) simulations can be used to predict how multiple whiskers will slip against a peg and to place error bounds on the resulting contact point location

(a) Four frames from Supplementary Video 3, which simulates a rat whisking against a vertical peg. Thick red whiskers are those currently in contact with the peg, and green whiskers are those that came in contact with the peg and subsequently pushed past it. Light gray whiskers never contact the peg. (b) The measured distance in contact point location between edge and contact point mode (z_error) is well predicted by equation (7). This equation thus allows error bounds to be placed on the contact point location calculated during simulations using edge mode. The dots are semi-transparent to show the data density. The inset excludes outliers as defined in the text. Measured z_error ranges between −0.76 and 0.96 mm, and predicted z_error ranges between −1.0 and 1.1 mm. In both Fig. 4b and the inset the identity line is plotted in black and the best fit line is plotted in red (see text for equations).