The Mechanical Variables Underlying Object Localization along the Axis of the Whisker

Mice.

The behavioral procedures were similar to a previous study probing object localization in the anterior–posterior dimension (O'Connor et al., 2010a). Adult (> P60) male mice (C57BL/6Crl; Charles River Laboratories; n = 8) were housed individually in cages containing tunnels and bedding material in a reverse light cycle room. For 10 d before training and on days without behavioral testing, mice were maintained on 1 ml of water/d (O'Connor et al., 2010a). On days with behavioral sessions, mice obtained all water for the day during the session and performed until sated. The amount of water consumed was determined by weighing the mouse before and after the session. The volume consumed was ∼1 ml. The weight and health of the mice were monitored daily. All procedures were in accordance with protocols approved by the Janelia Farm institutional animal care and use committee.

A custom head post was mounted to the skull under isoflurane anesthesia (∼1.5–2% by volume in O₂). Bupivacaine or lidocaine HCl (10 μl, 0.5% s.c.) was applied at the incision site. The scalp and periosteum were removed from the dorsal surface of the skull. A thin layer of cyanoacrylate adhesive was applied to the skull and covered with dental acrylic (Lang Dental Jet Repair Acrylic 1223, catalog #1251546; Henry Schein). Head posts were fixed to the skull with dental acrylic. Buprenorphine HCl (0.05–0.10 mg/kg, i.p.; Bedford Laboratories) was used for postoperative analgesia. Ketoprofen (5 mg/kg) was administered at the time of surgery and postoperatively to control inflammation. Mice were allowed 10 d to recover from surgery before water restriction.

Apparatus.

Mice judged the distance to a thin steel pole (diameter 0.4 mm; Fig. 2A). In some experiments (see Figs. 9, 10), a cleaned, straightened rat whisker was substituted as a “compliant” pole, which was deformed by contact with the mouse whisker. A pair of stepper motors (catalog #NA08B30; Zaber) and linear sliders (catalog #NDN 2–50.40; Schneeberger) moved the pole in the horizontal (x–y) plane. This assembly was mounted on a pneumatic linear slider (SLS-10–30-P-A Mini slide, catalog #170496; Festo), which rapidly (∼250 ms) brought the pole into and out of reach of the whiskers along the z-direction, triggered by a computer-controlled solenoid valve (CPE 10-M1BH-5L-QS-6, catalog #196883; Festo). The entire behavioral apparatus was enclosed in a light- and sound-isolation box.

The apparatus was controlled by software (http://brodylab.princeton.edu/bcontrol). A real-time Linux system interfaced with valves and recorded licking responses using a PCI-6025E data acquisition board (National Instruments). The stepper motor was driven by its controller in response to commands sent through a serial port of a Windows computer running MATLAB.

A custom lickport, which provided the water reward and recorded licking, was placed within reach of the mouse's tongue. Licks were recorded as interruptions in the light path between an LED (860 nm, LN77L; Panasonic) and a phototransistor (L14G1; Fairchild Semiconductor). The phototransistor and LED signaled interruptions to the control computer.

Water was delivered by gravity into the lickport through a 1.3-mm-diameter steel tube under solenoid valve control (The Lee Company). Excess water was sucked out of the lickport through a tube (0.8 mm internal diameter [i.d.]) using a peristaltic pump (Rabbit Plus peristaltic pump; Rainin Instrument). Punishment was provided by puffs of compressed air (typically 10 psi) delivered through a small metal tube (∼2.3 mm inner diameter) pointed at the face from a distance of several centimeters. The air was controlled by a solenoid valve (The Lee Company) connected to a compressed air source.

During behavior, mice were crouched in a natural position in acrylic (2.9 cm i.d.; catalog #8486K433; McMaster) tubes such that their heads protruded out at the front and they could use their front paws to grip the tube's edge. The headpost was secured to the bench using a kinematic mount, guaranteeing precise positioning of the head across sessions. Whiskers were free to move around the space surrounding the heads.

Imaging whiskers.

A high-speed videography system using CMOS cameras (200 × 310 pixels, 500–1000 fps, 0.2 ms exposure, 8-bit depth, catalog #A504 K, Basler or Imaging Studio software and X-PRI camera, 180 × 250 pixels, 500 fps, 0.5 ms exposure, 8-bit depth, AOS Technologies) measured the position and shape of whiskers during behavior. Pixel size was either 0.07 or 0.06 mm. For every trial, a video of 1.5 s length was recorded starting 83 ms before the triggering of the pole descent (Fig. 2C). Illumination was with an infrared LED (940 nm, catalog #ELJ-940–211; Roithner Lasertechnik). The light was delivered through a diffuser and a condenser lens and pointed directly into the camera.

Whisker trimming and manipulation.

Whiskers were gradually (over 12–27 d) trimmed down to the C2 whisker under light isoflurane anesthesia. Whiskers were retrimmed every 2–4 d. To stiffen whiskers, three cut whiskers from the contralateral side were glued to the principal whisker with hairspray (göt2be glued Freeze Spray; Schwarzkopf). Two of these whiskers had their tips facing toward the base of C2 and the third in the same orientation as C2, producing a thickened whisker without taper. After the experiment, the whiskers and hair spray were removed with soap and water. For control experiments (two sessions per mouse), the three whiskers were glued to the contralateral C2. The stiffening procedure has unintended side effects in addition to stiffening and abolishing taper: the hairspray might alter the friction coefficient of the whisker and also increases whisker weight.

Behavior.

Head-fixed mice judged the distance to a metal pole that was presented at one of two positions (Fig. 2A,B). The sound of the pneumatic slider controlling the vertical pole position indicated trial start. The pole locations were arranged to lie along the whisker in the radial dimension. A proximal position was defined as the Go position; licking and breaking the beam of the lickport triggered a water reward (scored as a hit). One or several distal positions were defined as No Go positions; licking and breaking the beam triggered a time-out (2–10 s) and an airpuff as mild punishment (false alarm). Trials in which mice did not lick were neither rewarded nor punished in both the Go (miss) and No Go (correct rejection) trials.

Licking triggered reward or punishment only during the answer period, which began some time after the pole became available for sampling (sampling period; Fig. 2C). When there was no lick response (correct rejections and misses), the pole ascended and the trial was over. The sampling period was adjusted across and during sessions (typically between 0.5 and 1.25 s) to improve the mouse's performance. The answer period ended 2 s after the start of the pole descent. During a hit trial, the water valve opened, dispensing a drop of water (∼8 μl) and terminating the answer period. A drinking period was granted to allow the mouse to drink before the end of the trial. The probability of each of the two trial types was typically 50%, but was sometimes adjusted during training to suppress false alarms. To prevent discouraging mice with a succession of No Go trials, the number of successive trials of one type was limited to three.

Probing the mouse's ability to perform radial distance discrimination required that the azimuthal angle was as similar as possible for all object locations. The Go position (target) and No Go position (distracter) were selected based on the whisker video. The target was always 5 mm from the follicle. For the standard task, the distracter was 8 mm from the follicle (Fig. 2D). For the experiments shown in Figure 2, E and F, the distracter positions were in the range of 6–9 mm from the follicle; only one distracter position was used per session. For the experiments shown in Figure 7E–G, azimuthal jitter was introduced for both Go and No Go trials. In the absence of jitter, we defined a vector from the follicle to the point of contact between the whisker and pole (length r = 5 mm, Go; r = 8 mm, No Go). We then defined an arc by rotating the vector about the follicle by +3° and −3° (Fig. 7E). Pole positions across trials were then chosen randomly from points on that arc. For the experiments shown in Figures 9 and 10C, distracter positions were randomly chosen across trials from the range 7–13 mm, which generated a large range of bending moments and axial forces in the No Go positions. This large range of forces relaxed the demands on the properties of the flexible object for creating appropriate illusion trials.

To define the reaction time for a session (Figs. 2C, 6, 7, 8, 9, 10C), we constructed a histogram of the times of the first lick across trials. The reaction time was the time between the pole coming into reach (earliest frame of contact in the session, typically 250 ms) and the mode of the histogram. For individual hit and false alarm trials, the reaction time was defined as the time from earliest frame of contact in the session to the animal's first lick in that trial. Because mice did not lick in correct rejection and miss trials, we used the mean reaction time of the hit trials as a proxy. For correct rejection and miss trials, we defined the time point of the animal's decision as the time of first contact with the pole plus the mean reaction time on hit trials (time between first contact and first lick).

To determine the amplitude of whisking, we first filtered θ_base between 6 and 60 Hz (Butterworth filter) and then performed a Hilbert transformation on the filtered whisking signal (Huber et al., 2012). The amplitude component of the result is the whisking amplitude over time. For every trial, we determined the peak amplitude.

Measurement of whisker stiffness.

Mice performed all experiments with a single C2 whisker. After conclusion of the behavioral experiments, we analyzed the structure of three whiskers with light and electron microscopy (Fig. 5A,B). The whisker radius was measured as a function of distance from the follicle. The taper was remarkably linear (n = 3, r = −0.988) and the whisker can thus be modeled as a cone. The second moment of inertia is as follows (Birdwell et al., 2007): where a_base is the radius at the base of the whisker, s_p is the length along the contour of the whisker from the base to point p, and length_whisker is the total length of the whisker.

The whisker-bending stiffness was probed by pushing the whisker, which was mounted on a micromanipulator by its root, against a metal pole on a high-resolution balance (Mx5; Mettler Toledo; readability 1 μg, repeatability at full load, 0.9 μg) in 1 mm increments along the whisker. The measured weights ranged from <1 μg (0.1 mm displacement 1 mm from the tip) to 630 mg (0.5 mm displacement at 10 mm from the follicle); 87% of measurements were >20 μg, and therefore have expected errors ≤2.5%. Force on the whisker was calculated as follows: The whisker contour was extracted from photographs. For small displacements (<1 mm), force scaled linearly with deflection.

The bending stiffness (EI_p) was estimated by comparing measured deflections to a numerical model (Birdwell et al., 2007). Model whisker deflection was iteratively calculated from contact point back to whisker base in 1 μm segments using the following relationship (Solomon and Hartmann, 2006): For each set of deflections at a horizontal distance x from the base, numerical estimates of whisker shape were generated for a range of E. The model error was calculated as the ratio (A2/A1) of the area (A2), between the model deflection and actual whisker deflection from base to pole, and the area (A1) between the actual whisker deflection and the undeflected whisker from base to pole (Birdwell et al., 2007). The E that minimized the error at each x was tightly clustered around 5 GPa for whisker JF25395 (Fig. 5C), and ranged from 4 to 8.5 GPa for whisker JF25403. We used E = 5 GPa to calculate whisker forces.

Calculation of forces based on measurements from high-speed video.

In the main text, we derive the equations that allow forces to be calculated based on the idealized geometry of whisker and contacted object. However, in practice, critical details need to be considered in extracting these geometrical quantities from image data. In this section, we describe at the “algorithmic” level how we calculated forces, starting with the representation of the whisker as a sequence of points representing its medial axis (i.e., starting after “whisker tracking”).

Tracking whiskers in raw video images was performed as described previously (O'Connor et al., 2010a; Clack et al., 2012). Equations given below for angles and follicle coordinates reflect the specific videographic conditions of our experiments, in which the mouse face appeared at the top of the image and protraction involved rightward motion of the whisker within the image. After tracking, the whisker's medial axis is represented as a sequence of n points (x_i,y_i). To overcome discretization noise for computing derivatives, the shape was approximated as a parametric curve: where l ∈ [0,1] and x(l) and y(l) are fifth-degree polynomials. The polynomials were computed by fitting x_i and y_i, respectively, as a function of l_i, where, l₁ = 0 and: For subsequent evaluation of x(l) and y(l), l was evaluated at 100 evenly spaced points in the interval [0, 1].

Arc-length distance along the whisker (s; in mm) is derived from c(l) at each time point (i.e., for each video frame) as follows: Tracking of the face-most edge of the whisker can be unreliable due to movement of the fur on the whisker pad. In some early experiments, a small segment of whisker was obscured by the lickport. For these reasons we added a “mask” (Figs. 3, 4). For all analyses presented here, the mask consisted of a horizontal line chosen so that neither fur nor lickport would obscure the beginning of the whisker. This allowed the whisker to be effectively “truncated” at the intersection of a particular location on the whisker and the mask.

The arc-length distance at the intersection of the whisker and the mask was subtracted from s(l), so the arc-length origin becomes the intersection of the whisker and the mask. Therefore, considering the mask, s(l) was redefined as follows: where l_intersection is the value of l at which c(l) intersects the mask.

Angle (θ; in radians) at each point l along the whisker, for each time point, is as follows: where x′, y′ denote derivatives with respect to l.

θ gives the azimuthal angle in the horizontal plane. Protraction corresponds to increasing θ. θ = 0 is perpendicular to the midline of the mouse.

Angle at the whisker base (θ_base; in radians) at each time point is as follows: where l_base = arg min |s(l)|.

Angle at the point of contact between whisker and pole (θ_contact; in radians) at each time point is as follows: where l_c is the value of l at which the whisker is closest to the center of the contacted pole, calculated for each time point as follows: where (x₀,y₀) is the center of the contacted pole object. The point of finding the location along the whisker nearest the center of the contacted pole was: (1) to define operationally the location of contact during periods of contact and (2) to help identify such periods.

To help identify periods of whisker-pole contact, we calculated for each time point a quantity (d_pole; in mm) to estimate the distance between whisker and pole as follows: where barRadius is the radius in pixels of the pole presented to the mouse.

The coordinates of the contact point were defined as follows: Note that for the data presented here, (x_c, y_c) was not extrapolated to accommodate any failure of whisker tracking near the pole (the shadow of the pole could cause termination of whisker tracing at points very near the contact point). However, we have found this extrapolation useful in analysis of similar videography datasets and, to illustrate our general methods for extracting forces and moments from videography, we illustrate this step in Figure 3E. When extrapolated, Equations M11 and M13 were generalized to define the whisker-object contact point as the point along the whisker or its extrapolation that was closest to the center of the bar.

To define curvature, we could in principle use the “primary” fitted whisker, c(l) in Equation M4. However, there can be variations in the extent of the whisker actually tracked from frame to frame, mainly because: (1) the field of view does not necessarily capture the entire whisker and the whisker can therefore be more or less fully in the field of view, and (2) the shadow of the pole could terminate tracing of the whisker near the whisker-pole intersection, such that the whisker was not traced distal to the pole. Therefore, c(l) could be fitted to different segments of the whisker in different frames. This could generate subtle changes in the shape of c(l) even at a fixed point on the whisker and thus spurious changes in the apparent curvature (Fig. 4). To mitigate this problem in measuring curvature, we fitted a “secondary” parametric curve as follows: to a segment of the whisker falling within a constant region of arc length (Fig. 4D); that is, where x̂(l) and ŷ(l) are polynomials fitted to x_i and y_i, respectively, as a function of l_i, for i such that a ≤ s(l_i) ≤ b. Here we used second-degree polynomials fitted over the interval [a,b]. Therefore, ĉ(l) was fitted over a constant length of whisker, although subject to imperfections due, for example, to pixilation and to variability in tracking the whisker near the face. The interval [a,b] was chosen after inspection of the residuals between ĉ(l) and the raw (x_i,y_i). Over small intervals, second-degree polynomial fits were simple and accurate.

At each time point, we derived curvature (κ_p; in mm⁻¹) from ĉ(l) at a user-defined arc-length distance (in mm) along the whisker, s_p, as follows: for l_p such that s(l_p) = s_p.

Because whiskers can have intrinsic curvature, a more directly useful quantity (Δκ_p; in mm⁻¹) is the change in curvature (Birdwell et al., 2007) calculated as follows: where t_i is a set of N time points during which the whisker is not bent. Intrinsic curvature for the C2 whisker in the plane of the protraction was approximately 10-fold smaller than peak curvature changes and was thus ignored.

Whisker follicle position coordinates (x_f, y_f) were estimated by linearly extrapolating past the end of the tracked whisker, typically from the intersection with the mask; that is, proceeding from c(l_intersection). The extrapolation was based on θ_base. The distance to extrapolate (typically 1 mm) was estimated based on inspection of the video data as follows. where d_extrap is the distance to extrapolate and θ_extrap = θ_base.

Follicle position along the face (d_f) was estimated by taking the x-coordinate (anterior–posterior axis) value of follicle position, d_f = x_f.

The lever arm from location of curvature measurement to location of whisker-object contact, for each time point, is as follows: The angle (in radians) of this vector is as follows: Similarly, the lever arm from follicle to whisker-object contact location: And the angle (in radians) of this vector is as follows: At each time point, we calculated the magnitude of the force applied to the whisker (F; in 1 × 10⁻⁶ N) using Equation 4. Magnitude of moment at the follicle (M₀; in 1 × 10⁻⁶ N * 1 × 10⁻³ m) was calculated using Equation 5. Magnitude of the axial force (F_ax; in 1 × 10⁻⁶ N) pushing into the follicle along the axis of the whisker was calculated using Equation 6. Magnitude of the lateral force (F_lat; in 1 × 10⁻⁶ N) was calculated using Equation 7.

During whisking in air, the forces acting on the follicle are dominated by the whisker's moment of inertia and its angular acceleration. These forces are much smaller than the forces produced by contact between object and whisker and not relevant to object localization. The forces were therefore set to zero between contacts (Fig. 6B,D).

We excluded trials in which the whisker contacted the pole during retraction from behavior and whisker analyses (Figs. 6, 7, 8, 9, 10). We also excluded contacts after the decision.

Simulations with friction.

We used Equation M3 to compute whisker shape for a force applied at a point (x, y) along the whisker (arc-length s from the base; Fig. 5D–F). The force was of the form Fn̂ + μFt̂, where F is the force normal to the whisker, μ is the sliding friction coefficient, and n̂,t̂ are the unit vectors normal and tangential to the whisker, respectively. The model captures sliding friction; stick-slip events were not modeled. The force and the point along the contour where it acts were then varied until the whisker shape was consistent with contact with an object (i.e., the pole) at a particular point (methods are similar as those used in Solomon and Hartmann, 2006). A μ of 0.3 was used as a value typical for hair (Bhushan et al., 2005).

Calculating the protraction parameter.

Mice protract their whiskers into objects by changing the azimuthal angle of the whisker and by moving the follicle forward. To describe protraction with a single parameter, we combined azimuthal motion and follicle translation using the protraction parameter, θ_total (Fig. 8A). At a given follicle position d_f and θ_base, θ_total is the angle through which the whisker would have to be rotated around the fulcrum for it to detach from the pole. For the frame of first touch of each contact, we defined a line with origin in the follicle and along θ_touch. The intersection point between this line and a line along the anterior–posterior axis passing through the approximate contact point for the Go position (5 mm from the follicle) was determined. For each video frame of a contact, the angle between the base of the whisker (i.e., a line along θ_base) and a line connecting the follicle to the intersection point was calculated. This angle is θ_total. θ_total at first touch is thereby by definition 0.

Statistics.

Statistics were computed as indicated in the text. Comparisons of forces and bending moments for target and distracter locations (Figs. 6, 8, 9, 10) were based on bootstrap methods. For every session, we analyzed frames of contact in which θ_total was ≥2°. We restricted the analysis to these protraction amplitudes because whisker tracking could be unreliable near the moment of whisker-pole contact (corresponding to θ_total = 0°).

Analysis of whether the slope of M₀(θ_total), F_ax(θ_total), and F_lat(θ_total) was larger for the Go trials than for No Go trials (Fig. 8B–D) proceeded as follows, separately for each of three mice.

For M₀(θ_total), F_ax(θ_total), and F_lat(θ_total) time series, N_nogo trials were drawn with replacement 10,000 times from both the Go and No Go trials (after pooling trials across two behavioral sessions for one of the three mice), where N_nogo is the number of No Go trials. Therefore, each bootstrap sample comprised N_nogo randomly matched Go and No Go trials. For each bootstrap sample, we calculated the maximum likelihood estimate of the fraction of trial pairs in which the Go trial had a slope greater than that of the matched No Go trial (i.e., slope_go > slope_nogo) and the binomial 95% confidence interval for this fraction (MATLAB “binofit”). We then averaged the means and confidence intervals across all 10,000 bootstrap repetitions. For each of M₀(θ_total), F_ax(θ_total), and F_lat(θ_total), and for all mice, the mean 95% confidence intervals did not contain 0.5. Specifically, for M₀(θ_total), mean confidence intervals for the three mice were [0.59, 0.90], [0.88, 1.0], and [0.70, 0.94]; for F_ax(θ_total), they were [0.58, 0.90], [0.85, 1.0], and [0.71, 0.95]; and for F_lat(θ_total), they were [0.60, 0.91], [0.88, 1.0], and [0.71, 0.95]. Slopes were thus larger for Go trials at the α = 0.05 significance level.

Analysis of which force/moment cues could underlie performance on the illusion experiments (Fig. 9, Fig. 10) proceeded similarly, separately for each of two mice. For M₀(θ_total), F_ax(θ_total), and F_lat(θ_total) time series, N_illusion trials were drawn with replacement 10,000 times from both the illusion and No Go trials (after pooling trials across three behavioral sessions for each mouse), where N_illusion is the number of illusion trials. Therefore, each bootstrap sample comprised N_illusion randomly matched illusion and No Go trials. By bootstrapping in this way, we avoided the problem of having to decide which particular trials to compare between illusion and No Go trial types (because the number of trials of each type was not matched and there was no logical way to pair them). We restricted analysis to No Go trials in which the pole was in one of the two locations closest to the Go location. The rationale for choosing locations close to the Go location is that to achieve its level of illusion-trial task performance, the mouse must have been able to discriminate illusion trials (in which a flexible object was in the Go location) from even the most similar No Go trials. On the other hand, we chose the two closest No Go positions, rather than only the closest, in order to have more No Go trials for analysis (there was no obvious dependence of performance on the particular No Go location). For each bootstrap sample, we calculated the maximum likelihood estimate of the fraction of trial pairs in which the illusion trial had a slope greater than that of the matched No Go trial (i.e., slope_illusion > slope_nogo) and the binomial 95% confidence interval for this fraction. We then averaged the means and confidence intervals across all 10,000 bootstrap repetitions. For analysis of F_ax(M₀), the procedure was identical except that we estimated the fraction of trial pairs in which the illusion trial had a slope less than that of the matched No Go trial (i.e., slope_illusion < slope_nogo).

For the data shown in Figure 6G–I, we investigated whether performance depended on absolute force amplitudes. Calculations were done separately for each trial type of each session. We fitted a generalized linear model for binomial data with a logit link function (MATLAB “glmfit”) to force versus response. The reported p values are for the null hypothesis that response does not depend on force.

For decoding lick/no-lick behavioral choice using linear discriminant analysis (MATLAB “classify”), we pooled video time points across trials in each illusion behavioral session. We resampled illusion trials randomly with replacement such that they occurred at the same frequency as the normal (stiff-pole) Go trials. We then used linear discriminant analysis to predict the class membership (coming from a lick or a no-lick trial) for each time point for the indicated two-dimensional time series ({F_ax, M₀}, {M₀, θ_total}, {F_ax, θ_total}, or {F_lat, θ_total}). Decoder performance was quantified as the fraction of time points correctly assigned as coming from a lick or a no-lick trial.