# Vibrissa mechanical properties

Mitra Hartmann (2015), Scholarpedia, 10(5):6636. | doi:10.4249/scholarpedia.6636 | revision #151934 [link to/cite this article] |

The vibrissal (whisker) array of the rodent has been an important model for the study of active touch and tactile perception for over a century (Richardson, 1909; Vincent, 1912). During exploratory behaviors many rodents brush and tap their whiskers against surfaces to tactually extract object features, similar in some ways to how humans use their fingers for tactual exploration.

One of the largest advantages of studying vibrissae is that they are relatively mechanically simple. The whisker can be modeled as a tapered cantilever beam that transmits mechanical information to mechanoreceptors in the follicle at the whisker base.

The relative mechanical simplicity of the whiskers offers a long-term vision in which we can compute the complete set of tactile (mechanical) inputs transmitted by the vibrissae during active tactile exploration. To realize this vision requires careful quantification of whisker mechanics under the full range of behavioral conditions associated with active touch.

The present article reviews whisker mechanics in a manner intended to provide physical intuition for the types of mechanical signals that characterize natural vibrissotactile exploratory behavior.

## Introduction: whisker geometry and material properties

Vibrissal mechanics is an exciting field of study because it allows us to begin to understand how other mammals perceive their world. It is not easy to gain intuition for the vibrissal experience, as humans count as one of very few mammals who do not use vibrissal-based sensing as part of their behavioral repertoire (Muchlinski, 2010; Prescott et al., 2011; Grant et al., 2013; Muchlinski et al., 2013). As a start, we can state with confidence that the mechanical behavior of whiskers is governed by their *geometry*, their *density*, and their *elastic moduli*.

To help gain an intuition for whisker geometry, the schematic of Figure 1 depicts a typical C3 rat whisker whose intrinsic curvature has been removed, and whose base diameter (120 μm) and length (2 cm) have been scaled up by a factor of ten. The whisker in the figure has been drawn with a base diameter of 1.2 mm and a length of 20 cm. At this magnified scale it is easier to gain an appreciation for how long the whisker is compared to its diameter. Whiskers are very thin.

The geometric features that characterize a rat whisker are illustrated in Figure 2a. Rat whiskers have an intrinsic curvature well-matched by a quadratic fit (Knutsen et al., 2008; Towal et al., 2011) and they taper approximately linearly from base to tip (Ibrahim and Wright, 1975; Williams and Kramer, 2010; Hires et al., 2013). The cross-section of a whisker contains three layers: the outermost layer is called the cuticle, followed by the cortex, and then the medulla, which is hollow (Quist et al., 2011; Voges et al., 2012; Adineh et al., 2015). The presence of the medulla changes the cross-sectional geometry of the whisker as well as local average density, but these effects are too detailed for the present article. This article neglects the medulla, and analyzes the whisker as though it were solid and had uniform density. Subsequent sections will show how both whisker taper and intrinsic curvature play important roles in shaping information transmission along the whisker to mechanoreceptors in the follicle at the whisker base.

Whiskers are made of alpha keratin (Fraser and Macrae, 1980), with an approximate average density between 1.1 mg/mm^{3} and 1.50 mg/mm^{3} (Mason, 1963; Neimark et al., 2003). With these density values, rat whiskers are expected to range in mass between ∼8 micrograms for the smallest, most rostral whiskers (e.g., C6) and ∼700 micrograms for the largest most caudal whiskers (e.g., beta, C1) (Hartmann et al., 2003; Neimark et al., 2003). If the whisker depicted at 10× scale in Figure 1 were made of keratin, it would weigh approximately 0.11 grams. This is approximately the same weight as if one were to use heavy-stock paper to construct a whisker of the same size.

The *elastic moduli* of the whisker describe how resistant it is to being deformed in different directions (Beer et al., 2008). All materials, including whiskers, are characterized by many different elastic moduli, but the two most relevant to whisker mechanics are Young’s modulus (*E*) and the shear modulus (*G*). The shear modulus is important when considering mechanics in three dimensions (3D), discussed only at the end of this article. In contrast, Young's modulus is essential to understanding basic whisker mechanics even in two dimensions (2D), so we describe it here in some detail.

Young's modulus is important because it helps determine the bending stiffness of the whisker, which in turn determines how the whisker will bend when it makes contact with an object.

To develop an understanding of how Young’s modulus contributes to stiffness, it is useful to perform a thought experiment. Imagine holding a length of stainless steel rod, say 2 cm in diameter, and trying to bend it in the middle. It is clear that you would have to exert considerable force in order for the rod to flex. Now imagine holding a length of nylon rod, also 2 cm in diameter and trying to bend it. Obviously it is much easier to imagine bending the nylon rod than the stainless steel one. Correspondingly, stainless steel has a Young's modulus of approximately 200 gigapascals (GPa) while nylon has a Young’s modulus between 2-4 GPa. The stainless steel rod will be about 100 times stiffer than the nylon rod.

But now consider a second thought experiment. Imagine that the stainless steel rod is only 1 mm in diameter. It is now easy to imagine that you could bend the stainless steel rod in the middle. Perhaps your intuition says that a 2 cm nylon rod would be stiffer than a 1 mm stainless steel rod, which is in fact, true. This thought experiment makes it clear that stiffness depends both on the geometry of the rod as well as the material properties of the rod.

It turns out that the bending stiffness of a beam with circular cross section can be written as a product of Young's modulus and the "area moment of inertia" *I*, defined here as $I = \pi r^4 / 4$:
\[
\begin{align}
\text{Stiffness} &= EI = \frac{E \pi r^4}{4}. \tag{1}
\end{align}
\]
An important feature of this equation is that stiffness depends **linearly** on Young's modulus, but depends on the fourth power of the radius (Beer et al., 2008). Notice that stiffness does not depend at all on the density of the whisker.

In the case of a cylindrical rod, the stiffness is equal at all locations along the length of the rod, because the radius does not change. But whiskers are not cylinders – they taper. This means that their stiffness decreases along their length, and it decreases **rapidly**, as the fourth power of the radius. This rapid stiffness drop off is illustrated in the bottom graph of Figure 2.

The rapid drop-off in stiffness explains why it is so difficult to see bending near the base of the whisker during experiments. The distal region of the whisker bends much more than the proximal region because it has a smaller radius – but, as we will see in section 5 – this does not mean that the forces and moments are larger in the distal regions.

How stiff are whiskers? Depending on experimental technique, studies typically estimate a range for Young's modulus between 0.33 GPa and 4.92 GPa (Hartmann et al., 2003; Herzog et al., 2005; Birdwell et al., 2007; Quist et al., 2011; Kan et al., 2013). Higher values have been found as outliers (Birdwell et al., 2007) and several studies that rely on bending tests or resonance tests have found significantly higher values, in the range of 7.5 GPa (Neimark et al., 2003; Carl et al., 2012). One recent nanoindentation study (Kan et al., 2013) found evidence that Young's modulus varies considerably from whisker to whisker within a single array, with an overall tendency to increase from caudal to rostral within a row. The most detailed study to date characterized Young's modulus across each of the component layers of the whisker (Adineh et al., 2015). Results showed that the cuticle region had the largest values for Young's modulus, with values gradually decreasing towards the interior, from cortex towards the medulla.

Regardless of the exact value, reporting Young's modulus alone does not provide good physical intuition for whisker stiffness. To improve intuition for whisker stiffness, imagine holding the 10× whisker at its base, recalling that it would weigh approximately as much as if it were constructed of heavy-weight paper. The whisker's own mass will cause it to deflect under the influence of gravity. If the 10× whisker had the same stiffness as a real whisker, its tip would deflect ("droop") under the influence of gravity by no more than approximately the diameter of the whisker base (1.2 mm). The tip of a real rat whisker will thus deflect under its own weight by no more than approximately 250 μm.

In the next three sections, we examine how these properties – geometry, density, and stiffness – influence the mechanical signals that the rat will obtain during whisking behavior.

## Non-contact whisking

Different species of animals exhibit "whisking" behavior to varying degrees (Muchlinski, 2010; Prescott et al., 2011; Grant et al., 2013; Muchlinski et al., 2013). Whisking involves an active sweeping motion of the whiskers forwards ("protraction") and backwards ("retraction"), and can occur independent of head movements. The term "non-contact" whisking is used to refer to situations in which the animal actively whisks in free air, that is to say, the whiskers do not touch any object. The term "contact" whisking refers to conditions in which the animal whisks so as to actively brush and tap its whiskers against a surface or object.

### The kinematics of whisking

The kinematics of whisking is complex (Knutsen, 2015). Here we describe those aspects of kinematics essential to understanding the mechanical analysis in section 4. Each of the ** bold, italicized** terms is defined further in Figure 3.

The most basic description of whisking kinematics is based on the simplifying assumption that each whisker undergoes a rigid two-dimensional (2D) angular rotation in a plane approximately parallel to the ground. Variables are defined within the 2D view that would be obtained from a top-down camera. The very proximal portion of each whisker (usually 0.5 – 1 cm) is taken to be linear, and the ** position** of the whisker is then defined as the angle that the linear portion of the whisker makes relative to a reference line, here chosen to be the midline of the rat (Figure 3a). The

**of the whisker is taken to be the time derivative of the angular position, and the**

*velocity***is the absolute value of the velocity.**

*speed*Actual whisking motions are considerably more complex than the 2D approximation would suggest. First, there is significant ** translation** of each whisker's basepoint, in addition to angular rotation. Second, during protraction, each whisker exhibits a small change in

**(Bermejo et al., 2002; Knutsen et al., 2008) and each whisker also**

*elevation***(Knutsen et al., 2008) about its own axis (Figure 3b). These effects are small when observed in 2D with a top-down camera, but highly significant when the 3D case is considered (Knutsen et al., 2008; Huet and Hartmann, 2014; Huet et al., 2015; Knutsen, 2015). The roll and elevation can be written as functions of the protraction angle (Knutsen et al., 2008; Knutsen, 2015), so that the entire kinematics of the whisk can be simulated (Huet and Hartmann, 2014; Hobbs et al., 2015). Together, these three effects – translation, elevation, and roll -- become particularly important when quantifying how much the rodent has deflected its whisker against an object (section 4).**

*rolls*### Developing an intuition for whisker kinetics

"Kinetics" is defined by different people in different fields in different ways. Here we use it simply as a broad term to indicate the forces and moments associated with the whisker, as distinct from the "kinematic" variables.

The study of forces and moments can be divided into studying ** quasi-static** and

**effects. This statement will be more useful if we are able to assign some intuitive meaning to these concepts. We will gain intuition by doing some experiments that impose an external force on an artificial whisker. Then we will apply the ideas of quasi-statics and dynamics to understand the forces and moments generated during non-contact whisking, collisions, and bending against an object.**

*dynamic*A simple artificial whisker can be constructed by cutting off the zipper portion of a quart-sized freezer bag, as shown in Figure 4a.

For the first experiment, hold the whisker horizontally and use your finger to push down on it *very slowly* at some point out along its length (Figure 4b). You will see the whisker gradually bend as you deflect it. Notice that all of the bending occurs proximal to your finger. There is no change in the whisker shape distal to your finger.

One of the most important points to notice is that the whisker will bend in the identical way, regardless of whether you push on it at a rate of 1 mm/minute, or 1 mm/hour, or 1 mm/year. The only parameter that matters to the bending is the *distance* that your finger has deflected the whisker. As long as you push "slowly enough", the rate at which bending occurs does not influence the shape of the bent whisker. This is an example of quasi-static bending. The bending depends only on the stiffness of the whisker (Young's modulus and geometry). The bending does not depend on the whisker's density or mass, and bending does not involve vibrations.

Real whiskers have an intrinsic curvature, so we need to determine how to incorporate the initial curvature into our analysis of bending. As it turns out, the solution is remarkably simple: the final shape of the whisker can be determined simply by adding the whisker's intrinsic curvature to the curvature induced by deflection. Importantly, however, this linear summation works only in the quasi-static regime – that is, if you push on the whisker "slowly enough".

What happens if you don't push "slowly enough" – suppose you push "too fast?" Try deflecting the whisker rapidly. Now you will see that the whisker bends, as before, but it also vibrates (Figure 4c). The vibrations are most visible in the distal region of the whisker, but the proximal region experiences important vibrations as well, even if you can't see them. This is an example of a dynamic effect. The shape of the whisker at every point in time depends on the stiffness of the whisker, the mass of the whisker, how the mass is distributed along the whisker, and how energy is dissipated in the whisker. Whisker dynamics also depend on the exact details of how your finger struck the whisker, and how you happened to choose to decelerate your finger after it struck the whisker. Clearly, whisker dynamics are much more complicated than quasi-statics.

These two experiments get at the heart of the distinction between quasi-statics and dynamics in the context of the vibrissal system. Dynamic effects depend on mass and acceleration (and stiffness, and other parameters); quasi-static effects depend only on stiffness. For completeness, we note that dynamic effects are also sometimes called "inertial effects". These two terms are often used somewhat interchangeably.

An excellent question to ask here is: would it be possible to describe the complete mechanical behavior of the whisker just by adding the dynamic effects on top of the quasistatic effects? This idea is called "linear superposition". The answer is no in general, but yes in the limit of small whisker deformations. This approach recently allowed researchers to correctly capture the behavior of the whisker, at least to within experimental resolution, just after a shock (Boubenec et al., 2012). This method could prove very useful in in other contexts but is expected to become incorrect in configurations in which the whisker experiences large deflections.

We conclude this section with an important final point. All forces and moments – regardless of whether they are generated by quasistatic or dynamic effects – can be computed at every point along the whisker length. In practice, however, it is most useful to compute these quantities at the whisker base, because those are the signals that will directly enter the follicle.

Thus one of the primary reasons that the rat vibrissal system is such an attractive model for the study of active touch is that we can make a very strong statement about the signals sent to the brain by the whisker: **all** mechanical information transmitted by a whisker to the nervous system can be represented by the time series of three forces and three moments at the whisker base.

The three forces describe how the whisker resists translation in three directions (e.g., *x*, *y*, and *z*). Two of the moments (the two "bending moments") will describe how the whisker tends to bend as it resists rotation. One of the moments (the "twisting moment", or the "torque") will describe how the whisker "twists" about its own axis. We now examine how these forces and moments vary during non-contact whisking.

### Kinetics during non-contact whisking

To gain intuition for the mechanical effects relevant to non-contact whisking it is useful to perform a few more experiments that highlight the difference between rigid body rotation and flexible body rotation.

During non-contact whisking, the whisker is driven at its base by both intrinsic and extrinsic muscles. To obtain a sense for what whisking would be like if the whisker were a rigid body, try rotating a pen as though it were a whisker, holding it at one end (Figure 5a). Notice that if you use a heavier pen, the mechanical signals you feel at the base during rotation are different than if you use a lighter pen. This is an immediate indication that we are dealing with a dynamic effect.

Next, try rotating the artificial whisker at about the same speed as you did the pen (Figure 5b). The whisker will start to bend during the rotation, and it will be difficult to avoid shaking the whisker at its resonance. As was the case for the pen, the mechanical signals generated during rotation of the artificial whisker are also dynamic effects, and depend on the whisker's mass distribution. You can test the effect of mass distribution by changing the location of the slider on the zipper as shown in Figure 5c. A cone has its center of mass at a distance L/4 from the base, so to gain as realistic impression as possible, place the slider about a quarter of the length out.

The key point here is that in both experiments – rotating either the pen or the artificial whisker – the forces and moments generated at the whisker base are a result of dynamic effects. A dynamic model is needed to compute mechanical signals at the whisker base during non-contact whisking. The open question now is – what does the rodent experience? Does the rodent sense something more similar to what you sensed when you rotated the pen (rigid body), or more similar to when you rotated the artificial whisker (flexible body)?

The experiment that involved rotating the pen (Figure 5a) approximates the whisker as a rigid body. In this case, the moment *M* at the whisker base can be computed directly as $M = \alpha I$, where α is the angular acceleration of the whisker and *I* is the mass moment of inertia, which depends on the whisker's length, *L*. Also notice that α is well defined *only* because we are treating the whisker as a rigid body. Once we allow the whisker to bend, the angular acceleration will be different at different points along the whisker, and it is not a particularly meaningful or useful quantity. Importantly, both experimental and modeling studies have found that the first 60 – 70% of the whisker does not appear to deform much during non-contact whisking (Knutsen et al., 2008; Quist et al., 2014), suggesting that the rigid body approximation is likely to be reasonable for much of the whisker.

The experiment that involved rotating the artificial whisker (Figure 5b) approximates whisker mechanics as a resonance phenomenon. Whiskers have first-mode resonance frequencies that range between 25 – 500 Hz for the fixed-free condition that describes non-contact whisking (Hartmann et al., 2003; Neimark et al., 2003; Boubenec et al., 2012; Yan et al., 2013). A whisker will oscillate with very large amplitude if it is driven near resonance, but the rat typically drives its whisker at much lower frequencies, between 8 and 15 Hz. This means that the whisker will primarily follow the driving frequency, and not much resonance activity will be observed.

With these factors in mind, we might predict that during non-contact whisking the rat would experience mechanical signals that *mostly resemble* those from rigid body rotation, but also contain some smaller components associated with flexible-body dynamics. A recent simulation study of the dynamics of non-contact whisking confirmed this prediction (Quist et al., 2014). With the important caveat that these simulations were limited to 2D, the study offered the following important conclusions about the mechanical signals generated during non-contact whisking behavior:

- During non-contact whisking, mechanical signals will be dominated by the rigid-body rotation associated with the driving frequency, but there will be small components of the mechanical signals near the resonance frequencies of the whisker.
- Dynamic effects will be larger and more significant for the caudal whiskers, because they are larger than the rostral whiskers and have more mass.
- During noncontact whisking, the time-varying bending moment closely follows the position of the whisker, while the axial force closely follows the whisking speed, which has two maxima per whisk. These findings suggest a basis for the neural coding of angular position (or spatial phase) as well as whisking speed.
- For all whiskers, the mechanical signals generated during non-contact whisking are extremely small. The rat could therefore regulate vibrissal motion by controlling the position of the vibrissa base; no force control is required.
- Because the mechanical signals generated during non-contact whisking are so small, associated neural responses (e.g., in the trigeminal ganglion) are expected to be largely stochastic. This prediction is consistent with the high degree of variability observed experimentally.

We emphasize that adding 3D effects and intrinsic whisker curvature may change some of these results. In addition, an intriguing earlier study (Yan et al., 2013) showed that the natural frequencies of the whisker during both non-contact and contact whisking can be very sensitive to change in the rotational constraint at the base; this could permit the animal to adjust the frequencies of mechanical signals during active behavior.

Overall, however, dynamic effects during non-contact whisking are expected to be well approximated by a rigid body model. It is of questionable value to develop a dynamic model of a flexible whisker just to study non-contact whisking, however, such a model is critical for describing collisions and ensuing vibrations, and for describing the signals the rat will obtain during exploration of a texture (Boubenec et al., 2012; Yan et al., 2013).

## Collision

Many readers may recall studying collisions in physics class by performing conservation of momentum calculations. These calculations tended to involve spheres (e.g., billiard balls) that collided and then either stuck together (a "perfectly inelastic" collision) or bounced off each other (an "elastic" collision), with or without some associated loss of energy. If energy losses were zero, the collision was said to be "perfectly elastic".

The reason that these physics problems always involved spheres is that spheres can be treated as rigid point objects. Conservation of momentum calculations become complicated once an object is allowed to contain more than one point, and even more complicated if the object can deform.

Applying conservation of momentum to whiskers is not easy. During and immediately after a collision, each point along the whisker will have different velocities and different accelerations. The energy of impact will be distributed and lost in different ways along the whisker depending on the configuration (shape) of the whisker, which in turn depends on its material properties, including density, stiffness parameters, and damping parameters. At each instant of time, the shape of the whisker changes, which in turn influences how it will lose energy at the next instant of time, which changes local velocities and accelerations, and so on. Numerical models are needed, but very few exist.

One recent study, by Boubenec et al., offered a solution to the collision problem based on decomposing the whisker's deformations into their quasistatic and resonant (dynamic) components, and then performing a linear superposition of these two deformations (Boubenec et al., 2012). This approach yielded a remarkably good fit to experimental data describing the process of whisker contact and detach. This study is also the first to report and quantify the presence of a "shock wave" that travels axially down the whisker. The authors found that the mechanical perturbation induced by the shock changes linearly with velocity measured near the whisker base at the time of collision.

A second recent study, by Quist et al., yielded several additional insights into the dynamics of vibrissal-object collisions (Quist et al., 2014). First, collisions are mostly inelastic, that is, the whiskers will not "bounce" significantly after collision. Thus when the rat explores a complex surface, vibrissae will tend to make contact only once, helping to ensure that tactile signals accurately reflect the object's spatial features.

Second, although most vibrissal-object collisions will generate impact forces that are well above the magnitude of those generated by non-contact whisking, some collisions will not. Specifically, collisions that occur on the distal regions of some of the larger, more caudal whiskers, may not generate signals much larger than those observed during non-contact whisking (Quist et al., 2014). This result may offer a mechanical explanation for "whisking" neurons of the trigeminal ganglion, which respond with equal magnitude to non-contact whisking and light touch (Szwed et al., 2003; Szwed et al., 2006; Leiser and Moxon, 2007).

The third finding of Quist et al. (2014) was that the mechanical effects of collision depend very little on the velocity of the whisker at the time of impact, and more on how the rat chooses to continue to whisk after the time of collision. This result may initially appear to be in contradiction to results of Boubenec et al. (2012), but the two studies are actually quite reconcilable; they simply emphasize different aspects of the data.

The study of Boubenec et al. (2012) carefully examined the shock wave that travels axially, along the length of the whisker, immediately following a collision. The shock wave is very small, but also very fast, and is likely to be the first indicator to the rat that a collision has occurred. The magnitude of this shock wave depends on the velocity of the whisker at its base at time of impact.

The study of Quist et al. (2014) also observes these shocks immediately following a collision, and notes in passing that their magnitude increases with collision velocity, in agreement with Boubenec et al. (2012). The reported results then focus on quantifying the maximum magnitude of the mechanical signals subsequent to the collision. These maxima are associated with much larger, but slower, components of the signal, and are dominated by the effects of whisker bending and deceleration. It is these larger, slower signals that are mostly independent of the velocity at time of collision.

Summarizing, the two studies agree that although the first shock after collision is small, its magnitude is governed by collision velocity, and its high frequency components suggest that it could be an important cue for the rat. Subsequent changes in mechanical signals (after the shock wave) are governed primarily by far and how fast the rat chooses to push its vibrissa against the object after initial contact. These effects are much larger, but also much slower, than the initial shock.

With these mechanics in mind, several recent studies have shown that rodents respond differently to the first whisk against an object (unexpected) compared to subsequent whisks (expected) (Grant et al., 2009; Deutsch et al., 2012; Grant et al., 2012). Specifically, the first whisk against an object appears to be associated with a fast velocity change immediately after collision. The timing of this large acceleration suggests that it is most likely to result from an involuntary reflex loop (Deutsch et al., 2012), possibly mediated by the fast-propagating shock wave. In contrast, the rodent's subsequent whisks involve longer latency changes in both ipsi- and contralateral contact profiles (Deutsch et al., 2012). The timing of these changes is most consistent with a voluntary sequence of motor actions, often resulting in a "double pump" against the object (Deutsch et al., 2012).

These voluntary contact-induced signals will have a tremendous effect on the signals received by the rat and are clearly an important area for future investigation. It is not easy to answer a question such as: "how does the C2 whisker respond differently if it collides with an object at 400°/s, compared with a collision that occurs at 700°/s?" After the initial shock wave, the forces and moments at the whisker base will depend almost entirely on the velocity profile that the rat chooses to continue after collision. In addition, if the rat is able to change the stiffness at the whisker base the resonant properties of the whisker may shift considerably (Yan et al., 2013).

## Post-collision: bending of the whisker against an object

After collision, the mechanical signals transmitted by the vibrissa are dominated by quasistatic bending. Quasistatic signals are significantly easier to compute than dynamic signals, though 3D effects make even the quasistatics challenging.

### Computing mechanical signals during contact whisking as the whisker bends

As discussed in section 2.1, the stiffness of the whisker in the proximal region is much larger than in the distal region, and it is therefore experimentally challenging to measure bending near the whisker base. Measurement of bending near the base will be dominated by errors in tracking the shape of the whisker. It is therefore essential to develop mechanical models to compute the signals that will enter the follicle.

Using the variable s to indicate the position along the arc length of the whisker, we can write an equation that relates the change (Δ) in bending moment *M* to the change in curvature *κ* at each point along the whisker. The equation will use a proportionality constant from section 1, namely the stiffness of the whisker. The bending moment is the product of the stiffness of the whisker (*EI*) and the curvature of the whisker (*κ*).
\[
\begin{align}
\Delta M\text{(s)} &= \text{(stiffness)} \Delta \kappa \text{(s)}\\
&= E I \Delta \kappa \text{(s)}\\
&= \frac{E \pi}{4} r\text{(s)}^4 \Delta \kappa \text{(s)}. \tag{2}
\end{align}
\]
Notice that Δ*M* and Δ*κ* are both functions of s, as is the radius, *r*. The bending moment and the curvature are not constant along the length of the whisker. At all locations along the whisker the two variables are related to each other by a constant of proportionality $E \pi / 4$ as well as by the fourth power of the radius at that location.

We emphasize that equation (2) illustrates why the curvature of a whisker cannot conceptually be simplified into a single value: it will vary along the whisker's length. Because the whisker is much stiffer at its base (by a power of four), changes in curvature will be much more evident distally than proximally. This is why models are essential to accurately determine changes in curvature, and thus the bending moment, at the whisker base, near s = 0.

Multiple studies have solved a two-dimensional version of equation (2) to obtain estimates of the signals that the rat will experience during whisker deflection (Kaneko et al., 1998; Ueno et al., 1998; Solomon and Hartmann, 2006; Solomon and Hartmann, 2008; O'Connor et al., 2010; Solomon and Hartmann, 2010; Solomon and Hartmann, 2011; Boubenec et al., 2012; Quist and Hartmann, 2012; Hires et al., 2013; Pammer et al., 2013). In two dimensions, these signals consist of the bending moment (*M*), as defined by equation (2), as well as the axial force (Fx), directed along the length of the whisker near the base, and the transverse force (*F _{y}*), perpendicular to the axial force. All of these variables (

*M*,

*F*, and

_{x}*F*) change continuously along the length of the whisker, but the signals sent into the follicle are the values of these variables measured at the vibrissal base.

_{y}Open source Matlab^{TM} code that implements equation (2) and permits an experimentalist to compute forces and moments at the whisker base by tracking the whisker's shape is available here: http://nxr.northwestern.edu/digital-rat

### Mappings between vibrissal mechanics and the two dimensional (2D) location of an object

One of the longest standing problems in the field of vibrissal research is how the rat might combine various mechanical cues from the vibrissa to determine the location of whisker-object contact. Here we address the most basic 2D version of the mapping problem, which assumes pure rotation of the whisker (no basepoint translation).

#### Coordinate systems

The 2D mapping problem can be expressed either in head-centered coordinates, resting-whisker coordinates, or whisker-centered coordinates.

The 2D head-centered coordinate system is illustrated in Figure 6a, with the origin (0,0) at the tip of the rat’s snout. The location of a point object can be expressed either in Cartesian coordinates (*x*, *y*), illustrated in blue, or in polar coordinates, (*r*, *θ*) illustrated in red. The point object is shaded purple to indicate that its location is equally well expressed in either coordinate system. The polar coordinates used here differ in one important respect from the standard geometrical definition: by convention, the angle *θ* is measured relative to the vector pointing caudal along the midline of the rat's head. Note that whiskers play no role in defining the head-centered coordinate system.

Some studies (Szwed et al, 2003; 2006; Bagdasarian et al., 2013) have employed a “resting-whisker” coordinate system that is very similar to head-centered coordinates. The only two differences are that the origin is placed at the whisker base instead of the rat’s snout and the azimuthal angle,*θ* is defined relative to the resting angle of the whisker. The resting-whisker coordinate system is illustrated in Figure 6b. In the resting-whisker coordinate system, each whisker has a unique, constant offset relative to the head-centered coordinate system.

The 2D whisker-centered coordinate system is illustrated in Figure 6c. In this coordinate system, the radial distance of the object is measured from the base of the whisker to the point of contact with the object. Whisker-centered coordinates rotate with the whisker by definition. The whisker base is the origin, and the positive *x*-axis points along the initial linear portion of the whisker as it emerges from the rat's face (c.f., Figure 3). The horizontal angle *θ* is measured relative to this *x*-axis.

An important difference between head-centered and whisker-centered coordinates is that as the rat whisks the location of the object does not change in head-centered coordinates, but it does change in whisker-centered coordinates.

The mapping problem becomes more complicated as we consider what happens as the whisker deflects against the object. As the whisker bends, two more angles become relevant to the mapping problem: *θ _{push}* and

*θ*, identified in Figure 6d. The angle

_{cp}*θ*, as its name suggests, is a measure of how far the whisker has pushed against the object since initial contact. The value of

_{push}*θ*is obtained by subtracting the angular position of the whisker at the instant of object contact from the current angular position of the whisker. The angle

_{push}*θ*is the angle of the contact point (subscript "cp") in standard polar coordinates in the reference frame of the whisker.

_{cp}If the whisker were perfectly straight, *θ _{cp}* and

*θ*would be exactly equal and opposite (one would be the negative of the other). Real whiskers, however, have an intrinsic curvature, so their relationship depends on the radial distance of contact.

_{push}#### Mappings for object location in two dimensions (2D)

A number of studies have focused on solving the 2D localization problem.
One recent study (Bagdasarian et al., 2013) has addressed how the rat might determine the (r, θ) position of an object, where the azimuthal angle θ is measured in resting-whisker coordinates (Figure 6b). The work identifies four morphological variables that can be involved in localization: Global Curvature (*G _{κ}*), Angle Absorption (

*θ*), Angle Protraction (

_{A}*θ*), and Base Curvature (

_{P}*B*).

_{κ}In this morphological coding scheme, the parameter *θ _{P}* is the angle through which the whisker protracts after making contact with the object, and the base curvature is the curvature near the whisker base. The parameter

*G*is defined as the maximal curvature along a spline fitted to the portion of the whisker between its base and the point of object contact. Angle absorption is defined as the difference between the angle through which the whisker would rotate during non-contact whisking and the angle through which the whisker actually protracted, having been obstructed by an object. The study experimentally finds that either the pair (

_{κ}*G*,

_{κ}*θ*) or the pair (

_{A}*θ*,

_{P}*B*) is sufficient to localize the (r, θ) position of an object. In other words, the rat’s knowledge of either of the two morphological pairs is sufficient to localize the object in the plane. The work further shows that changes in the angular velocity of the whisker upon contact with an object can code for the object’s radial distance.

_{κ}Some unpublished simulation results from our laboratory lead to significant reservations about the generality of these findings. Although preliminary, the simulations suggest that neither the pair (*G _{κ}*,

*θ*) nor the pair (

_{A}*θ*,

_{P}*B*) is sufficient to localize the (r, θ) position of an object if the whisker has intrinsic curvature. Results also suggest that – even if the whisker is straight – the azimuthal angle can be determined only if the rat knows the total amplitude of the whisk, and then subtracts off the protraction angle. Finally, results suggest that velocity, in the sense indicated in the paper, is related to the radial distance of contact only if the amplitude of the whisk is assumed to be constant.

_{κ}Other studies that address the 2D mapping problem have been done with artificial (robotic) whiskers within an engineering context. In 2D, two sets of mechanical variables have been shown to code uniquely for the radial distance of the object in whisker-centered coordinates. The first set is bending moment at the vibrissal base (*M*) combined with *θ _{push}*. The second set is bending moment at the vibrissal base (

*M*) and axial force at the vibrissal base (

*F*). The combination of

_{x}*M*and

*F*, both measured at the vibrissal base, has also been shown to code uniquely for the contact point

_{x}*θ*in whisker centered coordinates.

_{cp}Kaneko's work in the late 1990's first showed that both radial distance as well as object compliance could be determined by monitoring how the bending moment at the whisker base changed with *θ _{push}* (Kaneko et al., 1998). The algorithm required a cylindrical "antenna" (whisker) to be rotated so as to ensure a condition of no lateral slip (i.e., the whisker was adjusted in a manner that the problem remained fundamentally two-dimensional).

To generalize this work to rat vibrissae, the mechanical results shown by Kaneko's were extended to include tapered beams with large intrinsic curvature and to include the effect of lateral slip (Solomon and Hartmann, 2006). This work enabled development of a robotic whisker array that could push gently against an object to determine the radial distance to each contact point; splining the contact points together then enabled reconstruction of the object surface (Solomon and Hartmann, 2006).

The validity of the robotic model for real rat vibrissae was confirmed in a follow-up mechanical study (Birdwell et al., 2007), while on the neurobiological side it was independently demonstrated that neurons of the trigeminal ganglion represented more proximal radial distances with increases in firing rate (Szwed et al., 2003; Szwed et al., 2006).

Techniques have now been developed to accommodate for surface friction while determining radial distance(Solomon and Hartmann, 2008), and to permit distance extraction as the whisker continuously "sweeps" along the object (in contrast to the discrete pushes of the earlier work) (Solomon and Hartmann, 2010). In 2011 Solomon and Hartmann showed that by combining axial force and bending moment the rat could uniquely determine both radial distance as well as the horizontal angle of contact (*θ _{cp}*) in whisker centered coordinates. These results were shown to hold for a tapered, but not cylindrical, whisker (Solomon and Hartmann, 2011).

Experiments on mice subsequently provided behavioral confirmation that axial force and bending moment could be combined to determine radial distance (Pammer et al., 2013). This study did not specifically demonstrate that results were unique for horizontal angle of contact, but it provided the demonstration that the mouse could distinguish between a compliant object placed at a small radial distance and a non-compliant object placed more distally. Within the limits of a 2D analysis, this result almost completely rules out the possibility that rodents rely exclusively on combining *M* and *θ _{push}* to determine radial distance to an object (Pammer et al., 2013). The study suggests it is far more likely that rodents rely on combining information about

*M*and

*F*. Thus behavioral experiments come full circle to compare to the original compliance result of Kaneko (Kaneko et al., 1998).

_{x}Three painstaking behavioral experiments explored the accuracy with which rodents can determine the horizontal angle of contact with a peg (Knutsen et al., 2006; Mehta et al., 2007; O'Connor et al., 2010). These studies have shown that although rats can localize the relative location of two simultaneously-present poles at hyperacuity (Knutsen et al., 2006), their accuracy drops approximately to the level of inter-whisker spacing when pole location has to be memorized (Mehta et al., 2007). Head-fixed mice can perform these absolute (memorized) estimates of horizontal angle at higher levels of performance, achieving absolute object localizations to better than 0.95mm in the anterior–posterior dimension (< 6° of azimuthal angle) (O'Connor et al., 2010). One open question left by these studies is whether the rodents perform these tasks based on head-centered cues (e.g., time from whisk start) or whisker-centered cues (e.g., based on differential mechanical signals induced from the collision happening at different phases of the whisk), or a combination of both.

### Three dimensional (3D) mechanics and geometry

Neurons throughout the vibrissal-trigeminal system are exquisitely sensitive to deflections of the whisker in all three directions (Simons, 1978; Simons, 1985; Lichtenstein et al., 1990; Timofeeva et al., 2003; Jones et al., 2004; Furuta et al., 2006; Bellavance et al., 2010; Hemelt et al., 2010). It is therefore evident that 2D mechanical models will fail to capture significant information that the vibrissa transmits to the brain (Knutsen et al., 2008; Quist et al., 2012).

From a 3D kinematic perspective, an important feature of whisking is that the whisker rolls about its own axis, so its orientation changes continuously throughout both protraction and retraction (Knutsen et al., 2008; Knutsen et al., 2005, also see Figure 3). Notice that this orientation change is entirely a 3D effect; in all of the 2D models described above, the orientation of the whisker is assumed not to change during the trajectory.

From a 3D kinetic perspective, recent work has shown that the intrinsic curvature of the whisker will significantly affect the forces and moments at the vibrissal base because the vibrations and bending from collision depend strongly on the whisker's orientation (Boubenec et al., 2012; Quist and Hartmann, 2012; Yan et al., 2013; Quist et al., 2014).

Putting kinematic and kinetic effects together, it becomes clear that the mechanical effects of a whisker bending against an object will depend strongly on where the whisker contacts the object during its trajectory. In other words, the location of the collision in head-centered coordinates will influence the bending mechanics in whisker-centered coordinates.

This important point is depicted in Figure 7, which illustrates three whiskers colliding with a peg at two different phases of the whisk cycle. In Figure 7a, the three whiskers all collide near 50° relative to the rostral-caudal midline, while in Figure 7b, they collide near 120°. Because the whiskers are in different rows, they also collide at different heights on the peg relative to the rat's snout. Both of these coordinates (*θ*_{impact} and the height) clearly must be described in a head-centered coordinate system.

But notice also that each whisker makes contact with the peg at a different 3D location relative to its coordinate system. The mechanical signals at each whisker base – the information actually entering each follicle – must be calculated based on how each whisker deflects against the peg in whisker-centered coordinates. The transformation between head and whisker coordinate systems is not simply an offset, because the whisker has intrinsic curvature, and because each whisker translates, elevates, and rolls during both protraction and retraction.

The 3D whisker centered coordinate system is shown in Figure 7c. The origin is at the whisker base, and the *x*-axis lies along the linear portion of proximal part of the whisker. The *x*-*y* plane is defined to be the plane of the whisker's intrinsic curvature, and the *z*-axis is perpendicular to the *x*-*y* plane. This is the coordinate system more relevant to describing the signals experienced by mechanoreceptors within the follicle.

The ideas in this section highlight the critical importance of developing a 3D model of whisker mechanics. Our laboratory recently showed that during active whisking behavior the whisker will often bend out of its plane of rotation, generating sizeable 3D mechanical signals (Huet et al., 2015). In the same work, we developed a model of whisker bending that computes the 3D tactile signals – all six components of force and moment – at the vibrissal base during active whisking behavior of the awake rat (Huet et al., 2015).

A question that remains open is the 3D mapping problem: how might the rat infer the 3D contact point location based only on the tactile signals it receives at the vibrissal base? Our laboratory’s preliminary work in this area shows that many different combinations of force and moment components can uniquely determine the point of vibrissal-object contact ($r_{cp}$, $\theta_{cp}$, $\phi_{cp}$). Uniqueness of the mapping depends strongly on the exact shape of the whisker (i.e., taper and intrinsic curvature) and is an active topic of investigation.

### Mechanics during electrophysiological experiments in the anesthetized animal

This article has primarily addressed vibrissal mechanics in the awake animal. Two points are important when considering experiments in the anesthetized animal.

The first point is likely to be obvious to many readers, but it is worth stating for emphasis: when studies in the anesthetized animal report that "angular position", "velocity", or "speed" are coded by neurons of the trigeminal system, these variables are completely unrelated to the kinematic variables of the same names used to describe non-contact whisking (Section 2.1 and Figure 3).

In the anesthetized animal stimuli are generated by passively bending the whisker some distance out along its length. This bending most closely resembles the quasi-static bending described in Section 2.2 (Figure 4b). The term "position" in this context is fundamentally related to the magnitude and direction of bending, and the term "velocity" reflects the rate of change of bending in a given direction. In the context of active behavior, this type of bending will occur any time that the rat presses its whiskers against an object (Section 4.1), and forces and moments at the whisker base may be quite large.

In contrast, during non-contact whisking (Section 2.1 and Figure 3) the animal rotates the whisker at its base. The bending of the whisker is very small. The whisker behaves mostly like a rigid body, although it may resonate a bit, as described in Section 2.3 and illustrated in Figure 5b. The terms "position" and "velocity" in this context do not depend on bending. The forces and moments at the whisker base will be very small.

Second, if a whisker is trimmed and then fixed (e.g., with glue) to a piezoelectric stimulator (a "piezo") and the piezo is then moved linearly, the whisker will experience forces that tend to pull it out of the follicle. This is a condition that the rat will almost never experience. A more natural experimental paradigm is to let the whisker slip on the piezo a bit, or slip within a capillary tube that serves as an interface between the piezo and the whisker.

## Summary

The goal of this article was to provide physical intuition for the types of mechanical signals the rodent will obtain during natural vibrissotactile exploratory behavior. We now summarize some essential points that may help experimentalists identify the mechanical model most appropriate for computing signals during different behaviors.

In the case that the whisker does not make contact with any object (non-contact whisking), then the **relevant question** is to what extent the whisker can be treated as a rigid body versus a flexible body. In many cases, the mechanics of non-contact whisking might be simplified to an analysis of rigid body dynamics, while including resonance components. The rigid-body approximation may not always be appropriate to describe the behavior of the larger, more caudal whiskers, where the flexibility of the whisker will have a larger effect. During non-contact whisking forces and moments will tend to be small relative to those generated during object contact, but larger whiskers and very fast whisking motions will generate larger forces and moments.

In the case that the whisker does make contact with an object (presumed stiffer than the whisker) then the whisker must be treated as a flexible body. The **relevant question** is to what extent whisker mechanics can be described using quasi-static models, which include only stiffness and bending, versus the extent to which dynamic models are necessary. Mechanical descriptions of whisker collisions, vibrations, and interactions with surface texture will require dynamic models. In contrast, quasistatic models will provide good descriptions for most passive-displacement experiments in the anesthetized animal, and for how the rat deflects its whisker against a surface following a collision.

## Acknowledgments

We thank Lucie Anne Huet for helpful discussions and for running simulations to validate many of the ideas presented here. The work was sponsored by NSF awards CAREER-IOS-0846088; CRCNS-IIS-1208118; EFRI-BioSA-0938007; and IOS-0818414.

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