Modeling forces and moments at the base of a rat vibrissa during noncontact whisking and whisking against an object

Abstract

During exploratory behavior, rats brush and tap their whiskers against objects, and the mechanical signals so generated constitute the primary sensory variables upon which these animals base their vibrissotactile perception of the world. To date, however, we lack a general dynamic model of the vibrissa that includes the effects of inertia, damping, and collisions. We simulated vibrissal dynamics to compute the time-varying forces and bending moment at the vibrissa base during both noncontact (free-air) whisking and whisking against an object (collision). Results show the following: (1) during noncontact whisking, mechanical signals contain components at both the whisking frequency and also twice the whisking frequency (the latter could code whisking speed); (2) when rats whisk rhythmically against an object, the intrinsic dynamics of the vibrissa can be as large as many of the mechanical effects of the collision, however, the axial force could still generate responses that reliably indicate collision based on thresholding; and (3) whisking velocity will have only a small effect on the transient response generated during a whisker-object collision. Instead, the transient response will depend in large part on how the rat chooses to decelerate its vibrissae after the collision. The model allows experimentalists to estimate error bounds on quasi-static descriptions of vibrissal shape, and its predictions can be used to bound realistic expectations from neurons that code vibrissal sensing. We discuss the implications of these results under the assumption that primary sensory neurons of the trigeminal ganglion are sensitive to various combinations of mechanical signals.

Figure 1.

2D model of the vibrissa. A, The vibrissa is represented as a series of discrete nodes (black dots) connected by rigid links. Each link is a truncated cone with a point mass simulated at the center of mass. The vibrissa model is fixed to an anchor that is controlled during the simulation. The anchor can rotate and translate. B, The position of each node n is defined in terms of the local coordinate system of node n − 1. The (x, y) location of each node in local coordinates was calculated using a variable rotation about the local (out-of-plane) z-axis (R_z) and a constant translation along the local x-axis (T_x). The vibrissa coordinates start from the anchor coordinate frame. C, Each node of the model contains a torsional spring (constant k) and a torsional damper (constant b). The first link is modeled to lie within the follicle.

Figure 2.

Algorithm for computing vibrissal dynamics.

Figure 3.

Algorithm for resolving a collision.

Figure 4.

Schematic of the configuration state of the system when solving a collision. Discrete time steps are shown along the x-axis. At each time step, the system has an associated configuration state, q, represented on the y-axis. At time step t_k, the system crosses the boundary function (e.g., a wall). Three steps are required to resolve the collision. First, the time of the collision (t_A) must be found_. Second, the first postcollision configuration (q_k) must be computed. Finally, the second postcollision state (q_{k + 1}) must be computed.

Figure 5.

During simulations of slow deflections, the results of the dynamic model match those of a quasi-static model to within a median error of 0.005% or better. In these simulations, the vibrissa base is placed at the origin of a standard Cartesian coordinate system, and the vibrissa is assumed to rotate at 0.5°/s counterclockwise against a peg in the first quadrant. The peg is placed at two different radial distances: s = 10% (left column) and s = 95% (right column). Contact with the peg occurred at time t = 0. In all plots, thick lines indicate the results of the dynamic model (subscript “dynamic”); thin lines indicate results of the quasi-static model (subscript QS). A, B, Comparison of bending moment at the vibrissa base. C, D, Comparison of forces at the vibrissa base.

Figure 6.

Frequency response of the dynamic vibrissa model compared with experimental data from the D1 whisker. A, Displacement amplitude of base (dashed line) and tip (black). B, Magnification ratio of tip to base. Experimental data are from Hartmann et al. (2003), their Figure 7.

Figure 7.

During noncontact whisking, moment and transverse force reflect the driving frequency, while axial force reflects twice the driving frequency. The driving frequency was 8 Hz. A, Moment at the base vs time. B, Forces at the base vs time. Gray, F_y; black, F_x. In both A and B, the top subplot shows the driving angle, θ, with the same time axis as the lower subplot. C, Frequency content of moment at the base. D, Frequency content of forces at the base. Gray, F_y; black, F_x. PSD, Power spectral density.

Figure 8.

Energy losses due to the inelasticity of the collision are quickly matched by energy losses due to damping. The vibrissa was simulated to rotate against a peg at 705°/s. Each simulation time step was 2 ns. Collisions at two radial distances were simulated. Left column, Proximal collision (s = 40%). Right column, Distal collision (s = 85%). In all subplots, the black trace indicates a perfectly inelastic collision, and the gray trace indicates a perfectly elastic collision. A, B, Total energy of the vibrissa for proximal and distal collisions. Gray vertical lines at t = 0 indicate the time of collision. For the elastic collision, the energy loss at the collision time step is 0, but the system loses energy on subsequent time steps due to damping. For the inelastic collision, the energy loss at the collision time step is 0.216 nJ for the proximal collision, and 0.034 nJ for the distal collision. C and D illustrate the distance from the colliding link of the vibrissa to the object for proximal and distal collisions.

Figure 9.

The signals generated at the vibrissal base by an impact with an object scale with velocity at the time of collision; however, these dynamic effects are extremely small compared with the effects of bending and deceleration. A, B, Identical graphs illustrating the angle of the vibrissa (θ) as a function of time. The vibrissa was simulated to rotate with a half-sine profile against a point object (a peg). The frequency of the half-sine was 4 Hz (light gray trace), 8 Hz (dark gray trace), or 16 Hz (black trace). Each half-sine had a peak-to-peak amplitude of 20°, as the vibrissa was rotated from −14° to 6°. Collision occurred at t = 0. The peg was placed at one of two radial distances: proximal, s = 40% (middle row); and distal, s = 85% (bottom row). C, Moment at the vibrissa base vs time for a proximal collision (peg placed at s = 40%). The largest effect is a result of bending, while small ripples on each signal result from vibrissal dynamics. The inset expands the signals at the time of collision. D, Time derivative of moment at the vibrissa base for the proximal collision. The signal obtains its largest magnitude immediately following the collision event. E, Moment at the vibrissa base vs time for a distal collision (s = 85%). Dynamics due to the initial acceleration before impact are now of the same scale as the later portion of the signal that includes impact and bending. F, Time derivative of moment at the vibrissa base for the distal collision. Again, the signal obtains its largest magnitude immediately following the collision event, but there are other oscillations both precollision and postcollision that are the same order of magnitude.

Figure 10.

Peak bending moment generated after a collision as a function of the velocity of the vibrissa at the time of impact. Peak moments are shown for a vibrissa colliding with an object at s = 40% and s = 85%.

Figure 11.

With the exception of axial forces, mechanical signals at the vibrissa base become dominated by inertial effects as contact becomes more distal. Simulations represent whisking at 8 Hz against a point object. Peak-to-peak amplitude of the whisk was 20°. The time series of θ is shown above each subplot. The vibrissa was simulated to rotate 3°, 6°, or 9° against the object (black, dark gray, and light gray traces, respectively). The open vertical bracket on the right of each subplot indicates the magnitude of the signal generated during noncontact (free-air) whisking, as seen in Figure 7, A and B. The contact duration is plotted as a short horizontal bar at the top and bottom of each subplot. The color of the horizontal bar corresponds to the angle rotated into the object (3°, 6°, and 9°). A–C, Bending moment, transverse force, and axial force during whisking into an object placed proximally (s = 40% along the vibrissa arc length). D–F, Bending moment, transverse force, and axial force during whisking into an object placed distally (s = 85% along the vibrissa arc length).

Figure 12.

Comparison between dynamic and quasi-static models. A, Bending moment at the base of the vibrissa vs time for a θ_p of 6° at 8 Hz against a peg placed at s = 40% and s = 85%. These are the same data as presented in Figure 11. B, Total energy present in the model vs time. For A and B, the dynamic signal is shown in black, and the quasi-static signal is shown in gray. C, Percentage difference between the total energy in the dynamic and quasi-static signals. The equation for f within the subplot shows how this percentage was calculated. Different shades of gray indicate different values of θ_p, as shown in the legend.