## Abstract

The sense of touch is represented by neural activity patterns evoked by mechanosensory input forces. The rodent whisker system is exceptional for studying the neurophysiology of touch in part because these forces can be precisely computed from video of whisker deformation. We evaluate the accuracy of a standard model of whisker bending, which assumes quasi-static dynamics and a linearly tapered conical profile, using controlled whisker deflections. We find significant discrepancies between model and experiment: real whiskers bend more than predicted upon contact at locations in the middle of the whisker and less at distal locations. Thus whiskers behave as if their stiffness near the base and near the tip is larger than expected for a homogeneous cone. We assess whether contact direction, friction, inhomogeneous elasticity, whisker orientation, or nonconical shape could explain these deviations. We show that a thin-middle taper of mouse whisker shape accounts for the majority of this behavior. This taper is conserved across rows and columns of the whisker array. The taper has a large effect on the touch-evoked forces and the ease with which whiskers slip past objects, which are key drivers of neural activity in tactile object localization and identification. This holds for orientations with intrinsic whisker curvature pointed toward, away from, or down from objects, validating two-dimensional models of simple whisker-object interactions. The precision of computational models relating sensory input forces to neural activity patterns can be quantitatively enhanced by taking thin-middle taper into account with a simple corrective function that we provide.

- whisker bending
- quasi-static model
- mechanics
- stiffness
- somatosensation

## NEW & NOTEWORTHY

We find that mouse whiskers are thinner in the middle than a linearly tapered cone. They bend more than the standard model of whisker bending during touches in the middle of the whisker and less in distal locations. This taper affects touch-evoked forces and how whiskers slip past objects, which impacts the neural algorithms of object localization. We provide a corrective function that improves the quantification of touch-evoked forces in the whisker system.

the rodent whisker system is a canonical model for understanding how neural circuits process sensory information (Diamond et al. 2008; Kleinfeld et al. 2006). Its prominence stems from a number of experimental advantages: *1*) whiskers are primarily external sensors, so their instantaneous state can be determined by visual observation (Knutsen et al. 2006); *2*) the connectivity of circuits processing whisker input is well characterized (Kleinfeld et al. 1999, 2006); *3*) sensory processing in primary somatosensory cortex occurs in modules that uniquely map to individual whiskers (Woolsey and Van der Loos 1970); *4*) in mice, genetic lines allow experimental access to recording and manipulation of specific nodes within the neural circuit (Guo et al. 2014; O'Connor et al. 2013). Thus the system, from stimulus through response, provides the access needed for computational models of sensory processing to be experimentally constrained and conclusively tested.

Whisker shape strongly influences tactile sensation. Mechanoreceptors that drive primary sensory neurons are found within the follicle. Thus touch information must be mechanically transmitted by the whisker to the follicle. The relative flexibility along the length of the whisker affects the angle of touch-induced force and allows whiskers to slip past objects. Both of these are potential cues for object localization and identification. Touch begins with deformation of the whisker by an object. This deformation is mechanically transformed into stresses in the follicle, where sensory mechanoreceptors reside. Mechanoreceptors convert touch and whisker movement signals to action potentials in the trigeminal ganglion (Stuttgen et al. 2008; Szwed et al. 2003, 2006). This neural activity propagates to sensory cortex, as well as several lower-level nested loops, which evokes tactile sensation and perception. A quantitative description of the mechanical transformation from forces at the point of object contact to the stresses in the whisker base is essential to understand how this neural activity represents sensory input and drives behavior.

Whisker shape is evolutionarily specialized for an animal's natural environment and behaviors. For example, harbor seals have undulating whiskers that follow water vortices during prey tracking (Beem and Triantafyllou 2015). In nature, rodents live in dark, constrained spaces (Sofroniew and Svoboda 2015). Thus they rely heavily on their whiskers to sense their surroundings. This is highlighted by the large amount of cortical space devoted to whisker sensation. Unlike dogs or cats, the positioning of rodent whiskers is under active conscious control via sets of muscles that can pull them forward, backward, up, and down (Dorfl 1982; Haidarliu et al. 2010, 2011; Hill et al. 2008; Simony et al. 2010). Mice locate, investigate, and identify objects by actively sweeping their whiskers across them. Analysis of these “whisking” patterns has revealed computational mechanisms of tactile behaviors, including navigation (Sofroniew et al. 2014), exploration (Grant et al. 2009), texture discrimination (Hires et al. 2013; Lottem and Azouz 2008; Wolfe et al. 2008), azimuthal localization (Curtis and Kleinfeld 2009; O'Connor et al. 2010b), and radial localization (Krupa et al. 2001; Pammer et al. 2013; Solomon and Hartmann 2011). In turn, these insights have driven development of biologically inspired robotic systems that can perform similar tasks (Prescott et al. 2009; Schroeder and Hartmann 2012).

Mathematical descriptions of whiskers have generally adopted the cantilever beam model of Euler and Bernoulli (Euler 1744) for ideal linear cones, i.e., linear dependence of the whisker radius on the distance from the tip. Most of these models (Birdwell et al. 2007; Hires et al. 2013; Quist and Hartmann 2012; Williams and Kramer 2010) are based on the quasi-static approximation, which assumes that at each time point the whisker is in a steady state dictated by the boundary conditions at the base and at the contacted object. Intrinsic whisker curvature is usually modeled assuming parabolic whisker shapes (Hires et al. 2013; Quist and Hartmann 2012; Towal et al. 2011).

Here we compare how a two-dimensional (2D) quasi-static model with curved, linearly tapered conical whiskers fits experimentally determined whisker deformations during object contact at multiple whisker orientations and radial distances. Surprisingly, significant differences are observed between modeled and observed whisker shape. We demonstrate that nearly all of the model error can be accounted for by deviations of the whisker stiffness from that of a homogeneous cone. Furthermore, we show that this stiffness variation is primarily due to systematic deviations from a linear taper. We provide a corrective function that improves characterization of whisker sensory input from whisker videography, which generally improves computational models of sensory processing by the whisker system.

## METHODS

#### Experimental measurements.

Whiskers were plucked from 11 adult (4–12 mo old; 7 male, 4 female) C57BL/6 mice after humane euthanasia (University of Southern California IACUC protocol 20169, Janelia Research Campus IACUC protocol 11-71). A total of 33 whiskers were obtained, 12 from the C2 follicle and 3 each from A2, B2, D2, E2, C1, C3, and C4 (Table 1). Five C2 whiskers, denoted *W1–W5*, were mounted for dynamic imaging of galvo bending, while 28 whiskers were mounted for high-resolution measurement of whisker shape. For galvo whiskers, the portion of the whisker normally embedded in the follicle was fixed in a thin metal tube (30 gauge syringe, B-D) filled with Kwik-Cast (WPI). This tube was then slid snugly into a slightly larger metal tube mounted to a galvo scanner (Cambridge Technologies HP6800; max 20 μrad repeatability), and the junction between tubes was secured with Kwik-Cast. To ensure alignment between the whisker curvature and the plane of rotation, a mirror system projected two orthogonal perspectives of the whisker motion and object interaction onto two areas of a high-speed (500 frames/s) CCD camera (AOS Technologies) mounted with a telecentric lens (0.36X Gold Series, Edmunds Optics) (Hires et al. 2013). The video resolution was 32.8 μm/pixel, and the whiskers were traced with subpixel resolution with the Janelia Whisker Tracker (Clack et al. 2012). The galvo rotated in the *x*–*y* plane, such that θ_{0} (and also *x*_{0} and *y*_{0}) varied with time. The relative position of galvo mount and pole position was controlled via a three-axis manual micromanipulator. Rotation of the whisker was computer-controlled through a National Instruments IO card with NI-Max software.

#### Model of a whisker contacting a cylindrical pole.

Whiskers are modeled as quasi-static, truncated cones with length *L*_{w}, base radius *r*_{base}, and tip radius *r*_{tip} (Hires et al. 2013). The conical shape is virtually extended to a perfect cone of length *L*. The whisker is located in the *x*–*y* plane. The arclength along the whisker, *s*, is *s* = 0 at the base, *s* = *s*_{obj} at the point of object contact, *s* = *L*_{w} at the tip, and *s* = *L* at the virtual tip where the whisker radius is 0 (Fig. 1, *A* and *B*). The whisker base is located at point (*x*_{0},*y*_{0}), and the positions of a point along the whisker are (*x*(*s*),*y*(*s*)), 0 ≤ *s* ≤ *L*_{w} (Fig. 1*C*). The running angle between the whisker and the *x*-axis is θ(*s*), and θ(0) = θ_{0}. The whisker radius is *r*_{w} = (*L* − *s*)*r*_{base}/*L*, and the area moment of inertia is *I*(*s*) = π*r*_{w}^{4}/4. The Young's modulus is *E* = 3 GPa (Birdwell et al. 2007; Pammer et al. 2013; Quist et al. 2011).

The bending stiffness of the whisker is given by the product *T*(*s*)*EI*(*s*), where *T*(*s*) is an additional stiffness factor that may stem from an inhomogeneous Young's modulus, from the whisker geometry deviating from being purely conical, or from both. Unless otherwise stated we assume a conical whisker with uniform Young's modulus [*T*(*s*) = 1]. The intrinsic curvature of the whisker, in the absence of contact, is κ_{i}(*s*), which is extracted from the undeflected whisker. The object is a cylindrical pole oriented perpendicular to the *x*–*y* plane. Its projection on the *x-y* plane is a circle with radius *r*_{pole}, centered at (*x*_{cen},*y*_{cen}). Upon contact, the whisker touches the object at (*x*_{obj},*y*_{obj}), an arclength *s*_{obj} along the whisker, and an angle θ_{obj} = θ(*s*_{obj}). The object applies perpendicular force on the whisker at the contact point, where
(1)
where . The sliding friction coefficient of the whisker with the object is μ, and we use the typical value for hair, μ = 0.3 (Bhushan et al. 2005; Pammer et al. 2013). When the whisker protracts and moves toward the pole, the sliding friction force is
(2)
When the whisker retracts and moves away from the pole, is pointed in the opposite direction and is given by *Eq. 2* with negative multiplier (−0.3). The total force applied on the whisker is
(3)

At steady state, the shape of the whisker is determined by the solution of the static Euler-Bernoulli equation (Birdwell et al. 2007; Euler 1744; Love 1944; Williams and Kramer 2010):
(4)
where *M*_{z} is the component of the bending moment perpendicular to the *x-y* plane and , together with the equations
(5)
(6)
We seek a solution for *Eqs. 4–6* knowing the boundary conditions at the base (*x*_{0}, *y*_{0}, and *θ*_{0}) and that the whisker contacts the pole at an (initially unknown) arclength *s*_{obj}. While the boundary conditions in a follicle are debatable and dependent on viscoelastic properties of the tissue (Hill et al. 2008; Simony et al. 2010), here the whisker is firmly mounted to a galvo. Thus we take the boundary conditions at the base (*x*_{0}, *y*_{0}, and *θ*_{0}) to be known. We transform *Eqs. 1–6* to a form of a boundary-value problem (Hires et al. 2013). The equations are solved numerically with the shooting method (Press et al. 1992) implemented in a program written in C. Note that multiplying both the stiffness *T*(*s*)*EI*(*s*) and by the same factor does not modify the whisker shape. Therefore, the shape is not modified by varying *E* while keeping it uniform.

#### Parametrizing whisker-object contacts.

Unperturbed whiskers generally have a parabola-like shape within a plane (Towal et al. 2011). In most of the work reported here, the whisker is contained entirely within a plane perpendicular to the pole. For a whisker with intrinsic curvature, contact occurs in either the “concave backward” (CB) or “concave forward” (CF) direction (Hires et al. 2013; Quist and Hartmann 2012) (Fig. 1*B*). To quantify contact strength, we use the push angle θ_{p}, the angle through which the whisker is rotated into the object (Hires et al. 2013; Quist and Hartmann 2012). By convention, we define the sign of θ_{p} to be positive for CB and negative for CF. The arclength from (*x*_{0},*y*_{0}) to (*x*_{obj},*y*_{obj}) when the whisker barely contacts the object (θ_{p} = 0) is denoted by *s*_{obj,con}.

#### Tripartite stiffness function.

To model changes in stiffness along the whisker in a simple manner, we consider potential *T*(*s*) in the form of three-part linear functions that decrease with *s* along the proximal part of the whisker and increase with *s* along its distal part. The functions *T*(*s*) decrease linearly from *T*_{base} to 1 for 0 < *s* < *s*_{1}, are 1 for *s*_{1} < *s* < *s*_{2}, and increase linearly from 1 to *T*_{tip} for *s*_{2} <*s* < *L* (see Fig. 5*B*). The values of *T*_{base}, *T*_{tip}, *s*_{1}, and *s*_{2} are determined for each whisker such that the model best-fits experimental results (see results).

#### Tracking whisker movies.

Whiskers mounted to a slowly rotating (0.2 Hz) galvo were tracked with the Janelia Whisker Tracker (Clack et al. 2012) (https://openwiki.janelia.org/wiki/display/MyersLab/Whisker+Tracking). Pixel size was 0.0328 mm per side. The whisker medial axis is stored as an array of points (*x*_{i},*y*_{i}), *i* = 1,…, *N*, where *N* is on the order of several hundred. The arclength *s*_{i} is computed, and spline smoothing (de Boor 2001; Reinsch 1967) of the arrays *x*_{i} and *y*_{i} is computed as a function of *s*_{i}. The smoothing parameter *p* controls the trade-off between adherence to the data (large *p*) and smoothing, namely, reduction of the integral over the square of the second derivative of *x*(*s*) or *y*(*s*) (small *p*). We use the value *p* = 1 mm^{−3}, for which both good smoothing and adherence to the data are obtained. Choosing *p* = 10 mm^{−3} does not have a significant effect on the results.

To compute the effects of contact on whisker shape, we need to know that the intrinsic curvature is *κ*_{i}(*s*). This is computed from a parametric representation of the shape of an undeflected whisker as a function of the running arclength *s*, namely, (*x*,*y*) = (*x*(*s*),*y*(*s*)). We denote *d*/*ds* by ′. The intrinsic curvature is
(7)

#### Correcting when whisker plane not parallel to image plane.

A configuration in which the whisker plane before contact is parallel to the pole (when the whisker is undeflected) is called “concave down” (CD). Here, the whisker plane is perpendicular to the plane of video recording. Protraction results in rotation of the whisker plane while it is still parallel to the pole. Eventually, the whisker contacts the pole and bends, and its projection on the plane normal to the pole is no longer a straight line. For simplicity, we do not develop a full three-dimensional (3D) whisker model (Huet et al. 2015) but correct the 2D model for this configuration. The calculation is carried out by assuming that the 2D quasi-static is valid for the activity on the projection plane. This assumption is correct to the first order in θ_{p} (Quist and Hartmann 2012; Timoshenko and Gere 2009), and we examine whether it is valid for values that are not small. We need to compute the real value of *s* along the whisker for each point because the stiffness of the whisker depends on *s*. We do it by considering the shape of the undeflected whisker in the CB (or CF) configuration (see Fig. 8*A*). In the whisker plane, the whisker is translated and rotated such that the whisker base is located at the coordinate (0,0) and θ_{o} = 0. In this case, the coordinate *x* of the whisker in the CB configuration is equal to the video-recorded recorded arclength *s*_{proj} of the whisker in the CD configuration. To compute *s*(*s*_{proj}), we invert the function *x*(*s*) in the CB configuration.

#### Computing slopes of experimental curves.

Slopes of curves were computed by linear fitting using the Numerical Recipes function fit (Press et al. 1992). When computing the slopes of differences between theoretical and experimental values, we omit high-θ_{p} regimes in which stickiness is observed (see below).

#### Measuring whisker width.

Two methods were used for high-resolution measurement of whisker taper. For three galvo-mouted whiskers (*W1–W3*; see Fig. 5 and Fig. 6), after whisker deflections and dynamics were recorded the tube-mounted whiskers were affixed to glass coverslips with double-sided tape. High-resolution (0.75 μm/pixel) overlapping images of the whisker were taken every 250–500 μm along the whisker length with a macroscope. Each segment was independently focused. Each whole image segment was cropped, linearly translated, and rotated to digitally straighten the whisker. Dust on the tape provided fiducial marks for exact segment alignment.

For 28 whiskers that were not galvo mounted, the complete whisker from follicle base to tip was embedded in clear nail polish and sandwiched between coverslips. Images were acquired with a compound microscope under ×10 magnification (0.58 μm/pixel resolution) and translated to construct an overlapping set of ∼1.2 images/mm of whisker length. After assembly with either scheme, whisker edges were traced using an average of 75 points per whisker edge (range 25–143 points, dependent on whisker length) with the neuronJ plug-in for ImageJ. The edges were interpolated and smoothed to 1,000 points each in MATLAB. The radius was found as half the shortest distance between the two edge traces for each step along one edge's length.

#### Normalization of whisker width.

To compare the dependence of *r*_{w} of whiskers from different rows and arcs on *s*, we normalized their arclength *s* to their virtual length *L* and their radius *r*_{w}(*s*) to *r*_{base}. Then, we averaged the normalized *r*_{w}(*s*) for whiskers in each of the eight whisker groups. The stiffness factor *T*(*s*) caused by the deviation from the conical shape is (*r*_{w}/*r*_{base})^{4}. To compute this factor for all the whisker population, we averaged the values of *r*_{w}/*r*_{cone} over the eight whisker groups and computed the fourth power. Fit of a tripartite function is carried out for *s*/*L* between 0.075 and 0.85. Points along the fit [in the form of (*s*/*L*, (*r*_{w}/*r*_{base})^{4}] are (0.075, 1.33) (0.41, 0.72), (0.6, 0.72), and (0.85, 2.0) (see Fig. 7*F*). For the grand mean polynomial fit, each whisker's radius was normalized to twice its average radius from 0.075*L*_{w} to *L*_{w}. The best fit was a fourth-order polynomial (coefficients: −2.737, 5.014, −2.335, −0.9260, 1.0735; for decending powers; see Fig. 7*G*).

#### Tapering affects forces on whisker shaft.

The force , generated by contact, has two components in the coordinate frame defined by the whisker shaft: *F*_{ax} is the axial component, parallel to the whisker at the shaft, and *F*_{lat} is the perpendicular component. In polar coordinates, *F* = (*F*_{ax}^{2} + *F*_{lat}^{2}^{1/2}) is the force magnitude and θ_{F} = atan2(*F*_{ax}/*F*_{lat}), such that θ_{F} = 0 if the contact force has a lateral component only and θ_{F} = ±90° if it is oriented in the axial direction.

## RESULTS

We quantified differences between a standard model of whisker bending and experimental observations of whisker shape under controlled whisker deflections. We then identified and tested potential sources of these differences. The major cause of the differences was nonlinear taper of whiskers. This taper has a large effect on applied tactile forces and whisker dynamics during object contact.

#### The standard model of whisker bending.

In principle, the rodent whisker system provides an unparalleled system for quantifying sensory input during active sensation and linking it to neural activity. In practice, the following challenges have limited the characterization of tactile forces during behavior. Whiskers are tapered cones with intrinsic curvature through three dimensions. Whisker follicles translate, rotate, and twist during active touch. Even with multiple camera angles, occlusion of the whisker by the mouse body and lack of intrinsic fiducial marks along the whisker prevent total knowledge of the 3D shape of the whisker.

A simplified model of whisker bending has emerged (Birdwell et al. 2007; Hires et al. 2013; Pammer et al. 2013; Quist and Hartmann 2012) as an experimental tactic to make progress in investigating the neural basis of tactile sensation. This “standard” model makes several simplifying assumptions: *1*) whisker shape reflects a quasi-static model of elastic bending; *2*) the whisker has intrinsic curvature, fit as a parabola (Fig. 1*A*); *3*) the whisker radius is linearly tapered from follicle base (*L*_{0}) to a truncated tip (*L*_{w}) (Fig. 1*A*); *4*) Young's modulus is homogeneous throughout the whisker; *5*) friction is ignored. We sought to assess the validity of this model, with the goal of finding simple improvements with a positive impact on the study of the neurophysiology of touch.

#### A slowly moving whisker rotating and deflecting in the imaging plane.

We recreated a common tactile sensing task, whisking into a thin vertical pole (Bagdasarian et al. 2013; O'Connor et al. 2010a; Pammer et al. 2013), in a manner in which we could control whisker orientation and speed (Hires et al. 2013). Mouse whiskers from the C2 follicle were mounted to a computer-controlled galvo and slowly moved (*f*_{galvo} = 0.2 Hz) into and away from a smooth steel pole (radius *r*_{pole} = 0.25 mm) (Fig. 1*B*) oriented in the vertical (*z*) direction. The whisker was modeled as a linearly tapered, truncated cone of homogeneous Young's modulus and intrinsic curvature κ_{i}(*s*) (Fig. 1*A*). The intrinsic curvature was oriented in the plane of whisker movement (*x*,*y*) and perpendicular to the pole, either CF or CB (Quist and Hartmann 2012) (Fig. 1*B*). Contact-induced bending was maintained in the (*x*,*y*) plane, verified by a second camera angle (Hires et al. 2013). This eliminated possible effects of torsional rotation and whisker projection errors in measuring whisker deformation from contact. Whisker movement toward the pole is “protraction,” and movement away is “retraction” (Fig. 1*C*). The base of the whisker was located at the point (*x*_{0},*y*_{0}), with base whisker angle θ_{0}. As the galvo rotated, *x*_{0}, *y*_{0}, and θ_{0} evolved with time. Upon contact (θ_{p} = 0, *s*_{obj} = *s*_{obj,con}), the whisker began to bend. The whisker was protracted into and then retracted away from the pole without slipping past. The whisker shape at each time was determined by the quasi-static solution computed for the time-varying boundary conditions. The stable whisker shape is uniquely defined for a set of follicle (θ_{0,} *x*_{0}, *y*_{0}) and pole (*x*_{obj}, *y*_{obj}, *r*_{pole}) coordinates.

#### Effects of sliding friction.

Sliding friction might affect whisker shape during touch. We computed the theoretical whisker shape for a coefficient of friction μ = 0.3 during protraction and retraction contacts (Fig. 2). The magnitude of was weakly dependent on μ (values with and without friction differ by 8–13% for Fig. 2). The friction force (Pammer et al. 2013) was substantial ( for Fig. 2), which would alter axial force at the follicle and potentially affect the response of trigeminal ganglion neurons (Stuttgen et al. 2008). Despite this, whisker shapes with and without friction were almost indistinguishable for medial objects (maximum distance between whisker shapes <20 μm; Fig. 2*A*) and similar for distal objects (maximum distance between whisker shapes <100 μm; Fig. 2*B*). Since friction forces are parallel to the whisker shape, they have minimal effects on bending shape. Therefore they could not explain any potential discrepancies between measured and predicted whisker bending. Since our focus here is on the shape of whisker deformations, we ignored friction (i.e., μ = 0) for the following experiments.

#### Comparison between standard model and experimental observations.

Using model parameters measured from real mouse whiskers (length, base radius, and intrinsic curvature along the whisker), we examined whether experimental bending matched theoretical calculations assuming conical whisker shapes. Example snapshots of experimental measurements and theoretical results are shown in Fig. 3 for an object near the middle of the whisker (Fig. 3, *A* and *B*) and an object closer to the tip (Fig. 3, *C* and *D*) for the CB (Fig. 3, *A* and *C*) and CF (Fig. 3, *B* and *D*) configurations. For object contact near the whisker center (*s*_{obj} = 9.7 and 9 mm, *L* = 17.6 mm; Fig. 3, *A* and *B*), the whisker bent more than the model. For contact near the tip (*s*_{obj} = 14.6 and 14.2 mm; Fig. 3, *C* and *D*), the whisker bent less. We systematically addressed possible reasons for this discrepancy.

#### Linearity of stress-strain curve for whisker bending.

A nonlinear stress-strain curve of the whisker material could cause deviations from model during bending. If so, deviations between modeled object contact angle (θ_{obj,th}) and observed object contact angle (θ_{obj,exp}) would increase at least quadratically with the push angle θ_{p} rather than linearly. We measured the dependence of θ_{obj} on θ_{p} for medial (Fig. 4, *A* and *B*) and distal (Fig. 4, *D* and *E*) objects. The dependencies are linear for values of θ_{p} between −22° and 14° for medial contact and between −10° and 5° for distal contact. Observed θ_{obj} during protraction and retraction was indistinguishable in this range of θ_{p}, confirming that the effect of sliding friction on the whisker shape is small. The difference in whisker bending between model and experiment can be quantified by the slope (*S*_{T=1}) of the difference between model and experimental contact angle (θ_{obj,th} − θ_{obj,exp}) vs. push angle (Fig. 4, *C* and *F*). This slope is linear with push angle in a range of θ_{p}. Thus differences between the model and experimental results are not a result of nonlinearity of the stress-strain curve of the whisker material in this range of deflections.

#### Stick-slip during distal contacts.

During distal contacts at large θ_{p}, the whisker contact point sometimes transiently stuck to the object, accumulating tension, which was eventually abruptly released. An example of these stick-slip events is demonstrated in Fig. 4*F* during retraction for θ_{p} values around θ_{p} = 5°. As θ_{p} decreases, θ_{obj,th} − θ_{obj,exp} decreases to −8.8° and then slips abruptly to −1.2°. Snapshots of whisker shapes before and after the slip event are plotted in Fig. 4*F*, *inset*. Stick-slip mostly occurred during retraction, leading to differences in θ_{obj} trajectories between protraction and retraction at large θ_{p}. While dynamic friction exerts a force continuously proportional to the normal force, stick-slips were unpredictable and abrupt. We lack a good model for stickiness in the context of this work. Thus all comparisons with model and experiment exclude parameter regimes in which stickiness appears (i.e., some cases of motion reversal at large push angles and contact near tip).

#### Whisker stiffness variation accounts for model discrepancies.

Unaccounted-for variation in stiffness along the whisker is a third possible explanation for discrepancies between model and observation. To test this, we quantify the difference in whisker bending between model and experiment by the slope *S*_{T(s)} of the curve θ_{obj,th} − θ_{obj,exp} vs. θ_{p} (Fig. 4, *C* and *F*; Fig. 5*A*). Note that *S*_{T(s)} is a functional of the function *T*(*s*). For the standard model where whiskers are linearly tapered cones of homogeneous Young's modulus, *T* = 1 for all *s*, and therefore *S*_{T(s)} is *S*_{T=1}. The slope *S*_{T=1} varied with distance from object contact to whisker tip *L* − *s*_{obj,con}. Across five C2 whiskers (*W1–W5*) that underwent deflections at multiple object distances, this slope inverted sign at ∼5 mm from the whisker tip. Close to the tip slope was positive, indicating less bending than expected from the theory, while contacts in the whisker middle had negative slope, indicating excess bending (Fig. 5*A*).

This motivated a slightly more complex model, where the stiffness *T*(*s*) varied along the length of the whisker as a tripartite linear function (Fig. 5*B*). For each whisker, we found values for the boundaries of each line segment that minimized the error between expected slope [*S*_{T(s)}] and experimentally measured slope (Fig. 5*C*). This was the simplest piecewise function (3 segments) that minimized and effectively eliminated deviation between modeled and measured whisker bending (Fig. 5, *D* and *E*). This identified stiffness variation as a likely candidate for the deviation.

#### The source of stiffness variation is nonlinear taper.

There are two possible sources of the stiffness variation along the whisker. The tapering of rodent whiskers *r*_{w}(*s*) may deviate from that of a perfectly linear cone (Boubenec et al. 2014; Carl et al. 2012; Ibrahim and Wright 1975; Voges et al. 2012). Alternatively, Young's modulus [*E*(*s*)] could vary along the whisker. Differences between *E* for proximal and distal halves of rat whiskers have been observed (Quist et al. 2011), although the complete functional form of *E*(*s*) is experimentally unknown.

To test the first possibility, we examined whisker width as a function of *s* for three whiskers *W1–W3* from Fig. 5, *A–C* (note that deviations in taper past the point of object contact have no effect on the deflected whisker's shape). Each mouse whisker was thinner in the middle than expected from a linear taper (Fig. 6*A*). We transformed the observed thickness variation into a variable stiffness factor by computing the value *r*_{w}/[*r*_{base}(1 − *s*/*L*)], which is the ratio between the measured whisker radius *r*_{w} and that of a perfect cone. The stiffness of a homogeneous beam scales with the fourth power of its radius (Quist and Hartmann 2012; Timoshenko and Gere 2009). Reflecting this, the stiffness factor qualitatively matched the fourth root of the three-part function *T*(*s*) for each whisker (Fig. 6*B*), with one exception: the width in the most distal ∼4% of the whisker length rapidly narrowed. Whisker bending is not affected by whisker properties beyond the contact point. Therefore, this narrowing is irrelevant for objects at distances reported here. While Young's modulus may also vary along the whisker length (Quist et al. 2011), we conclude that the primary cause of the difference between modeled and measured whisker deflections stems from this thin-middle whisker taper.

#### Nonlinear whisker taper is general across the whisker pad.

To determine whether thin-middle taper was a general phenomenon, we measured whisker shape across multiple columns and rows. We examined the complete shape of one group of seven additional whiskers from C2 and seven groups of three whiskers each from A2, B2, D2, E2, C1, C3, and C4 follicles (Fig. 7*A*). Whiskers were consistently thinner in the middle and thicker in the distal region than linearly tapered cones of whisker length (*L*_{w}) across all follicles (Fig. 7, *B* and *C*). We assessed taper of C2 whiskers of water-restricted mice vs. mice given free access to water for at least one complete whisker growth cycle. There was no difference in taper between these groups from C2 (Fig. 7*D*). Across all whiskers (8 restricted, 24 unrestricted), normalized to virtual tip length, there was no significant difference in taper (Fig. 7*E*). The grand mean whisker stiffness (31 whiskers) normalized to virtual tip is well fit by a tripartite linear function (Fig. 7*F*; see methods). We also provide a polynomial fit of mean whisker radius normalized to a linear cone from follicle emergence to truncated tip (Fig. 7*G*; see methods). Since there is some variance in taper from whisker to whisker (Fig. 6*B*, Fig. 7), detailed shape information is needed for the most accurate computation of the fit and whisker bending properties. In the absence of high-resolution imaging, the whisker taper can be estimated from these functions by measuring the whisker length and the radius at follicle emergence and at a point near the tip.

#### Effects of orthogonal whisker orientation.

During natural exploration whiskers rarely push into objects in a purely CF or CB orientation. In many pole localization experiments (Hires et al. 2015; O'Connor et al. 2013), the orientation upon contact approximates a CD orientation (i.e., the plane of intrinsic curvature and the long axis of the pole are parallel and aligned at first contact; Fig. 8*A*). Upon contact, the pole bends the whisker out of its plane of intrinsic curvature, which could twist the whisker and cause additional model discrepancies. We assessed whether variable stiffness could account for differences observed in bending in the CD orientation.

We examined the difference between model and observed whisker shape for the case where stiffness varies as a linear cone (*T* = 1) and our variable stiffness fit [*T*(*s*); Fig. 8, *B* and *C*]. The same stiffness function from Fig. 5*B*, *top left*, strongly reduced model discrepancy in the CD configuration. This was true across multiple contact locations [*S*_{T(s)} range −0.17 to 0.06, *S*_{T=1} range −0.38 to 0.74; Fig. 5*D*]. A model accounting for droop and variable stiffness fits observed whisker shapes well in the CD orientation (Fig. 8, *E* and *F*). The remaining difference may be a result of twisting of the whisker from object forces, experimental error, or angular anisotropy of the mechanical properties of whisker.

#### Tapering affects forces at the whisker base.

Small deviations from linear taper can have a significant impact on how a whisker interacts with objects, because of the fourth-power dependence of stiffness on whisker radius. In particular, they change the relationships between the force vector (with magnitude *F* whisker angle at the object θ_{obj}) and the push angle θ_{p}. Rotating the coordinate frame such that the *y*-axis is parallel to the whisker at the base, we obtain the angle of the force with respect to the *x*-axis, θ_{F}. The relationship between and θ_{p} is behaviorally relevant, as it determines the angle of the force applied by the pole, components of which are used by mice to judge radial distance to objects (Pammer et al. 2013).

We modeled the effect of the variable stiffness function *T*(*s*) from our example whisker in Figs. 2–4 on this relationship. For contact near the middle of the whisker, the increased flexibility causes the angle of force applied by contact to pass through a larger range for a given push angle (Fig. 9*A*). In contrast, during contact near the whisker tip, the same push angle moves this force angle through a smaller range (Fig. 9*B*).

A second consequence of the thin-middle taper is related to force magnitudes. The force exerted by the pole for a given push angle increases for the tripartite function *T*(*s*) for both medial and distal contacts (Fig. 9, *C* and *D*). This is because the function *T*(*s*) of Fig. 5*B* is >1 by construction, and therefore the whisker stiffness is increased. The biologically relevant result is that, relative to conical whiskers, larger *F* values are obtained for distal objects than for medial objects because the whisker is stiffer near its tip.

A third consequence of the variable stiffness is a change in the push angle for which the whisker slips off the object. Tapered whiskers slip past objects at a critical push angle (Fig. 9, *A–D*, solid lines). The thin-middle whisker taper decreases the critical slipoff angle when contacting near the middle (Fig. 9*E*) and increases the slipoff angle when contacting near the tip (Fig. 9*F*). During natural exploration, rodents often preferentially investigate objects with their whisker tips (Grant et al. 2012; Mitchinson and Prescott 2013). Stiff tips substantially increase the push angle at which whiskers slip off the pole and increase the force magnitude that can be transmitted to the follicle, which could serve to increase the dynamic range of tactile exploration and improve sensory discrimination.

## DISCUSSION

#### Summary of results.

We assessed the accuracy of a standard model of whisker deflection by comparing empirical contacts with a pole to model predictions. Real deflections caused more bending than predicted when contact was in the middle of the whisker and less with contact near the tip (Fig. 3). Neither friction (Fig. 2) nor nonlinear stress-strain relationships (Fig. 4) accounted for the bending discrepancy. The differences were resolved if the real whisker stiffness is larger than expected from a linear cone near the whisker base and tip (Fig. 5), including for contacts in a CD orientation (Fig. 8). High-magnification inspection revealed that mouse whiskers have nonlinear taper, thicker than a linear cone at base and tip, which accounts for most of the deformation discrepancy (Fig. 6). This thin-in-middle taper is common across rows and columns of the whisker pad (Fig. 7). This taper increases the range of force angles sampled during middle contacts and increases the push angle reached before slipoff during distal contacts (Fig. 9).

#### Implications for quantifying tactile input.

Contact between an object and a whisker produces forces that are transmitted by the whisker to the follicle. These forces determine mechanosensory transduction in the follicle. The resulting neural response influences the motor plant (Deutsch et al. 2012; Pammer et al. 2013; Simony et al. 2010) and is decoded by neural circuits into representations of object position, properties, and identity (O'Connor et al. 2010a; Xu et al. 2012). Videography of whisker bending contact can provide experimentalists with a quantitative measurement of sensory input (Bagdasarian et al. 2013). This has inspired progressively more complex models of how whisker bending represents forces at the follicle.

Several groups have calculated tactile input forces from video of whisker bending using planar, quasi-static Euler-Bernoulli models with linear conical whiskers (Birdwell et al. 2007; Pammer et al. 2013; Quist and Hartmann 2012; Xu et al. 2012). Mismatch between these models and experimental deformation for deflections in the plane of whisker intrinsic curvature has been reported in rats (Birdwell et al. 2007; Quist and Hartmann 2012). These discrepancies could be due to a higher Young's modulus in the distal half of the whisker (Quist et al. 2011). Our measurements show that similar deviations in mice (Fig. 2, Fig. 4) can be accounted for by thin-middle whisker taper (Fig. 6, Fig. 7) modeled by a tripartite stiffness factor *T*(*s*) (Fig. 5). As other reports of rat whisker shape show a qualitatively similar “thinner in the middle” conical taper (Boubenec et al. 2014; Williams and Kramer 2010), we expect that thin-middle taper accounts for a large fraction of model discrepancies in rat as well.

Acquiring data in a manner that allows quantitative comparison across labs can increase the long-term impact and reproducibility of scientific findings. To enable this in whisker-based neurophysiology, the complete shape of every whisker at the time of experiment would ideally be known, to better relate sensory input to neural activity. However, timing experimental windows (e.g., the period of appropriate viral expression of a calcium indicator) to match the 2- to 4-wk window of growth quiescence at mature whisker length (Ibrahim and Wright 1975) is not always possible. Because of the rapid growth prior to whisker maturity [∼0.6 mm/day for mouse C2 (Ibrahim and Wright 1975)], a whisker plucked after neural recordings that span days or weeks may not be representative of the shape throughout the recording period. Conversely, after whiskers have reached a mature length, they can fall out unexpectedly between recording sessions, making high-resolution reconstruction impossible. The relationship between whisker width and distance to tip is consistent across cycles, during growth and at full length (Ibrahim and Wright 1975). Thus we recommend measurement of the whisker base diameter and total whisker length on each experimental day. Referencing this high-resolution whisker image after the last experimental day would allow accurate reconstruction of whisker shape across all days. If a final high-resolution image cannot be obtained, then mapping length and base diameter to our provided average whisker taper curves should provide a viable proxy for whisker shape.

Mice and rats twist their whiskers during whisking, leading to a range of intrinsic curvature orientations throughout a whisk cycle (Knutsen et al. 2008). Contacts during head-fixed radial distance discrimination are often near the CD configuration (Pammer et al. 2013). Yet many force estimates are based on models of bending in the plane of intrinsic curvature, which does not account for possible anisotropy of whisker bending. We show that the stiffness correction factor based on in-plane bending predicts the appropriate projected shape in the CD configuration (Fig. 8). Thus whisker bending is not directionally sensitive, and these bending models can be applied to a range of whisker orientations. However, in CD configurations, whisker droop substantially shortens the whisker projection. This can cause major errors in force calculations for distal contacts if not accounted for. We show that an adjusted area moment of inertia can be computed in the CD configuration from this projection of the whisker on the camera plane (Fig. 8).

A unique stable solution for whisker shape exists given the push angle, whisker dimensions, and object location (Birdwell et al. 2007; Hires et al. 2013; Williams and Kramer 2010). This shape determines θ_{obj}, which in turn determines the direction and magnitude of applied forces (θ_{F} and *F*). We show that thin-middle taper predicts values of θ_{F} and *F* significantly different from linear taper models. This difference is most pronounced for large push angles or distal contacts. This has a potentially large impact on the neural algorithms of object localization. Using the correct *T*(*s*) reduces the errors in estimating θ_{F} and *F*.

We show that sliding friction has a minimal effect on whisker shape, particularly for medial contacts, but does affect the magnitude and direction of forces at the base of the whisker (Fig. 2) (Pammer et al. 2013). Since friction force has opposite directions during protraction and retraction, accurate calculation of sensory input should consider direction of whisker motion. Furthermore, the force magnitude is dependent on the friction coefficient μ (Bhushan et al. 2005), which may vary with whisker oils, cleanliness of pole, humidity, and other factors. Estimates of μ may be possible by careful examination of the subtle dependence of whisker bending on direction of whisker motion for distal contacts (Fig. 2*B*; Fig. 4, *D–F*). However, this analysis is complicated by potential differences in static and kinetic friction coefficients. Despite using a smooth pole (specified surface roughness *R*_{a} < 0.25 μm), we observed sticks and slips during retraction on distal touches with large θ_{p}. These were most prominent at the moment of direction reversal and during high-frequency whisker sweeps. This stick-slip effect requires a more complex model, so its quantitative estimation is beyond the scope of this work.

Ultimately, a complete description of the sensory input forces requires 3D whisker tracking and a 3D model, but this may be excessively complex for many neuroscientific investigations. 3D models of whisker motion and deflection have recently been developed (Huet et al. 2015) and compared with a quasi-static 2D model. Huet et al. (2015) found that the axial and the 2D transverse forces on the whisker, as well as the 2D bending moment, match fairly closely with the corresponding 3D values. In this work, we examined contact of whiskers with a pole under the condition that the whisker moves perpendicular to the pole in all whisker configurations (CB, CF, and CD). The predicted 2D projected shape in each configuration matches experimental observations very well. Together, this shows that the 2D model is suitable for describing whisker kinematics under carefully constrained experimental conditions, such as head-fixed whisking into a vertical pole, provided the whisker does not have a significant velocity component parallel to the pole. A more complex 3D model may provide additional value in experimental paradigms where effects beyond the 2D representation need to be addressed, for example, when the whisker has a significant velocity component parallel to the pole.

#### Implications for behavior.

Accurate measurement of input forces is critical for understanding neural algorithms that underlie sensory processing and behavior. For example, mice can determine the radial distance of objects with a single touch of a single whisker (Pammer et al. 2013). Radial localization algorithms using a single force component (e.g., *F*_{ax} = *F* sin θ_{F} or *F*_{lat} = *F* cos θ_{F}) have degenerate solutions for radial distance, but those using relative amplitudes of force components provide unique solutions (Bagdasarian et al. 2013; Pammer et al. 2013). This is because tapered whiskers become much easier to bend as the contact point approaches the tip, allowing distal contacts to increase axial force more easily. A thin-middle taper steepens the stiffness gradient for contacts on the proximal half of the whisker and reduces this gradient for contacts on the distal half of the whisker (Fig. 7*G*). For example, with a linear taper, the ratio between the whisker stiffness at 0.2*L* and at 0.4*L* is 3.2, while the ratio between the stiffness at 0.6*L* and 0.8*L* is 16. With a thin-middle taper, the corresponding ratios are 4.7 and 6.6. This linearization of stiffness gradient along whisker length by thin-middle taper may simplify the neural algorithms and expand the range of single-whisker radial distance discrimination.

Whiskers are also thinner in the middle in rats (Ibrahim and Wright 1975). Since rat and mouse genera diverged an estimated 10 million years ago (Wu et al. 2012), thin-middle whisker taper is likely evolutionarily adaptive. During contacts in the middle of the whisker, the increased flexibility increases the range of force angles sampled for contacts with the same maximum push angle (Fig. 9*A*). This would increase the angles of interaction with an object, potentially providing greater information about object properties. Taper allows whiskers to more easily slip past proximal objects and rough surfaces, aiding maneuverability in tight or textured environments (Hires et al. 2013; Williams and Kramer 2010). Here we show that this slipoff occurs at greater push angles than predicted when contact is near the tip (Fig. 9). This is consistent with deviations between modeled and observed slipoff events in our previous work (Hires et al. 2013). Thus thin-middle whisker taper could be adaptive to the minimal impingement behavior of rodents during freely moving navigation and object exploration (Mitchinson et al. 2007). Since a stiff tip allows whiskers to hang on to objects over a greater range of push angles and a flexible middle increases the sampled object angles, we posit that thin-middle whiskers extract more tactile information per whisk than linear cones.

## GRANTS

The research was supported by United States-Israel Binational Science Foundation (BSF, Jerusalem, Israel) Grant 2013033 (S. A. Hires and D. Golomb), the Janelia Research Campus of the Howard Hughes Medical Institute (S. A. Hires and D. Golomb), Israel Science Foundation Grant 88/13 (D. Golomb), the Helmsley Charitable Trust through the Agricultural, Biological and Cognitive Robotics Center of Ben-Gurion University of the Negev (D. Golomb), the Dornsife College of Letters, Arts and Sciences at the University of Southern California (S. A. Hires), and the Whitehall Foundation (S. A. Hires).

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

S.A.H. and D.G. conception and design of research; S.A.H., A.S., J.S., V.H., I.W., and X.W. performed experiments; S.A.H. and D.G. analyzed data; S.A.H. and D.G. interpreted results of experiments; S.A.H. and D.G. prepared figures; S.A.H. and D.G. drafted manuscript; S.A.H., A.S., J.S., V.H., I.W., X.W., and D.G. edited and revised manuscript; S.A.H., A.S., J.S., V.H., I.W., X.W., and D.G. approved final version of manuscript.

## ACKNOWLEDGMENTS

The authors thank Mitra Hartmann, Brian Quist, Lucie Huet, Georges Debregeas, Daniel Shulz, and Rony Azouz for useful discussions. We also thank Karel Svoboda, who conceived and initiated the project together with D. Golomb and S. A. Hires, and in whose lab at Howard Hughes Medical Institute's Janelia Research Campus parts of the project were performed.

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