Comparative morphology of the whiskers and faces of mouse and rat (Mus musculus and Rattus norvegicus)

Understanding neural function requires quantification of the sensory signals that an animal’s brain evolved to interpret. These signals in turn depend on the morphology and mechanics of the animal’s sensory structures. Although the house mouse (Mus musculus) is one of the most common model species used in neuroscience, the spatial arrangement of its facial sensors has not yet been quantified. To address this gap, the present study quantifies the facial morphology of the mouse, with a particular focus on the geometry of its vibrissae (whiskers). The study develops equations that establish relationships between the three-dimensional (3D) locations of whisker basepoints, whisker geometry (arclength, curvature), and the 3D angles at which the whiskers emerge from the face. Additionally, the positions of facial sensory organs are quantified relative to bregma-lambda. Comparisons with the Norway rat (Rattus norvegicus) indicate that when normalized for head size, the whiskers of these two species have similar spacing density. The rostral-caudal distances between facial landmarks of the rat are a factor of ~2.1 greater than the mouse, while the scale of bilateral distances is larger and more variable. We interpret these data to suggest that the larger size of rats compared to mice is a derived (apomorphic) trait. As rodents are increasingly important models in behavioral neuroscience, the morphological model developed here will help researchers generate naturalistic, multimodal patterns of stimulation for neurophysiological experiments and allow the generation of synthetic datasets and simulations to close the loop between brain, body, and environment. Introduction Animal nervous systems evolve simultaneously with their peripheral sensory structures. To understand the signals that an animal’s brain evolved to process thus requires consideration of the relative positions and orientations of the animal’s sensors. The spatial arrangement of facial sensors is particularly important to the study of multimodal integration, as this arrangement directly shapes how visual, auditory, olfactory, and tactile signals are actively acquired by the animal and received by the brain. Given that house mice (Mus musculus) and Norway rats (Rattus norvegicus) are the most common model species used in basic neuroscience and biomedical research, it is surprising that the spatial arrangement of their sensory structures has not yet been quantified and compared. For both species (from here-on termed “mouse” and “rat”), one of the most important sensory organs is the whisker (vibrissal) array, which provides a set of rich tactile cues during multiple behaviors, ranging from locomotion and climbing, to texture discrimination, to social Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t interactions (Grant et al., 2018; Guić-Robles et al., 1989; Wolfe et al., 2011). The whiskers’ behavioral importance is paralleled by the hypertrophy of whisker-related structures within the rodent nervous system, particularly in the “barrel” region of primary somatosensory cortex (Ma and Woolsey, 1984; Rice et al., 1985; Van Der Loos, 1976; Woolsey and Van der Loos, 1970); the whisker system is thus ideal for electrophysiological study as well. The present study performs quantitative, morphological quantification and comparison of the facial features of mouse and rat with a particular focus on the vibrissae. The motivation for the work is three-fold, as the results fill scientific and technological gaps across multiple domains of research. One motivation is to develop a morphometric approach to compare the spatial arrangement of facial structures and whiskers across species, regardless of whether the whisker arrays are “ordered” into rows and columns, or “disordered,” with no apparent row/column arrangement (Muchlinski, 2010; Muchlinski et al., 2013). Although rats and mice happen to have similar row/column arrangements of whiskers, the present work compares their morphology without relying on this similarity. Another motivation is that a mouse morphological model would benefit behavioral and neurophysiological research in multiple ways. First, the model could be combined with videographic data to help quantify exploratory and sensory acquisition behaviors, as has been done for the rat (Hobbs et al., 2015; Hobbs et al., 2016). For example, overlaying the model on videos of a behaving mouse would allow researchers to estimate how the animal’s sensory structures are oriented relative to an object and how sensory signals arrive at each structure relative to each other; the latter is critical for analysis of multimodal integration. Second, combining the model with equations of motion for the whiskers (Knutsen et al., 2008) could improve whisker tracking, which remains as yet a challenging image-processing problem. Third, quantifying the arrangement of the whiskers is a necessary first step towards creating mechanical models to estimate the mechano-tactile input to the system, so that these input signals can be correlated with neurophysiological and behavioral recordings. Fourth, quantifying the arrangement of the whiskers could aid in the design of stimuli for neurophysiological experiments. For example, a researcher could construct a stimulus to move through the array of an anesthetized animal that deflected some whiskers but not others, or deflected the whiskers in a particular sequence or temporal pattern. Fifth, the morphological model could ultimately Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t enable development of simulations to examine the mechanosensory signals generated during natural tactile exploration, enabling the development of synthetic datasets that close the loop between the animal’s sensorimotor structures, the environment, and the nervous system. A final motivation for the present work is to quantitatively compare the facial and vibrissal morphologies of mouse and rat. As elaborated in the Discussion, there are several reasons to expect these morphologies to be similar, but equally good reasons to expect them to differ. The quantitative comparison performed in the present work allows us to suggest that the rat is “scaled up” from its most recent common ancestor with the mouse. To summarize, the present work quantifying the facial features of Mus musculus was undertaken in order to fill gaps across multiple fields. The work develops an important tool that we anticipate will be used widely in the study of morphometrics, animal behavior, multimodal integration, and neurophysiology. We anticipate the model will enable the creation of synthetic datasets of sensory acquisition behaviors that help close the loop between the brain, body, and environment (Bolaños et al., 2021; Hobbs et al., 2016; Ramalingasetty et al., 2021; Weiler et al., 2022; Zhuang et al., 2017; Zweifel et al., 2021) Methods The Animal Care and Use Committee of Northwestern University approved all procedures in advance. Data collection A total of 467 macrovibrissae from eight mice were collected for the present work. All animals were male C57BL/6 J mice between the ages of 6 and 8 weeks. Animals were housed communally (between 2 5 mice/cage) in standard rectangular laboratory caging on a 12/12 light cycle with wood chip bedding and a wire top. Data for the rat were obtained from a previous study (Belli et al., 2018), which developed equations based on a total of 518 macrovibrissae from nine female Long Evans rats between the ages of 5 and 36 months. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t A limitation of the present study is its exclusive use of male mice, while Belli et al. (2018) used only female rats. We expect that differences between individuals of the same species will likely be confined to size rather than the relative locations of facial structures or whisker geometry. We also expect that any potential differences between males and females of the same species and similar size will be smaller than the spatial resolution of the MicroscribeTM scanning approach. Future work should use animals of both sexes to confirm these expectations. Surgery and anesthesia: Animals were anesthetized with a mixture of ketamine/xylazine/acepromazine injected intraperitoneally (100 mg/kg ketamine hydrochloride, 10 mg/kg xylazine hydrochloride, and 3 mg/kg acepromazine maleate in a saline vehicle). Every fifteen minutes the toe pinch withdraw reflex was assessed and booster doses of anesthetic were given if needed to maintain deep anesthesia. The animal was secured with bite block and ear bars in a stereotaxic apparatus and placed on a heating pad. Next, to ensure that no vibrissae touched any surface, we performed a surgery to allow head fixation without the bite block and ear bars. An incision was made along the midline of the scalp, and skull screws were inserted to form a triangle: one screw was placed in the frontal bone, and one screw in each temporal bone. A bridge of methyl methacrylate (dental acrylic) was constructed between the skull screws and the arm of the stereotaxic unit. We were careful to ensure that the skull landmarks lambda and bregma remained exposed (Paxinos et al., 1985). The bite block and ear bars were removed after the acrylic cured to ensure that no object touched any vibrissa or any part of the animal’s face. Because all measurements were made in the anesthetized animal, the locations of all sensory organs including the whiskers were obtained when all muscles were relaxed. The measured angles at which the whiskers emerge from the face are thus by definition the “resting” angles. MicroscribeTM data acquisition: The MicroscribeTM is a passively movable mechanical arm with five degrees of freedom. At the start of an experiment, a probe of known length Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t is placed on the end of the arm and an origin (0,0,0) is defined. The user moves the probe tip to a chosen location, recorded as (x, y, z) coordinates relative to the origin. For these experiments we used a custom MicroscribeTM probe, consisting of a 500 micron diameter tungsten bipolar electrode held in a pin-vice that had been machined to thread into the MicroscribeTM. We followed manufacturer’s instructions to re-calibrate the custom tip before collecting data from each animal. The experimenter held the MicroscribeTM and sequentially placed the probe tip at a series of points on each of the anatomical structures of interest. The 3D coordinates (x, y, z) at the position of the MicroscribeTM tip were recorded by pressing a trigger button. We digitized multiple facial features for each of the eight mice, including the skull features lambda and bregma, points corresponding to the corners and contours of the nostrils, eyes, mouth, rostrum, incisors, and pinnae, the basepoint of each whisker, and 4 – 60 points along each whisker. On average, 30 whiskers could be measured on each mouse mystacial pad, compared to the 31 whiskers measured for rats in a previous study (Belli et al., 2018). This difference is due to the small size and delicate nature of rostral mouse whiskers, and does not necessarily reflect a difference in number of macrovibrissae between the two species. Because sling muscles were not identified in the present work, it was not possible to definitively determine the transition from macroto microvibrissae. 2D whisker scanning: After data collection with the MicroscribeTM was finished, animals were euthanized with an overdose of the ketamine-xylazine-acepromazine combination and subsequent decapitation. Whiskers from seven of the eight mice (mice 1-7) were cut at their base, using forceps and microscissors, and stored for one or two days in folded rectangles of aluminum foil, which were then stored in plastic bags. Extreme care was taken to cut the whiskers exactly at the base, pushing carefully and delicately on the skin to ensure the full arclength of the whisker was obtained. In initial experiments, we tried plucking each whisker, but found that the top of the follicle could not be consistently identified using shape, diameter, or color changes. For example, some Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t plucked whiskers tended to pull out more cleanly from the follicle than others, increasing their apparent length by a few mm. By using microscissors to cut carefully at the skin surface, we captured the full arclength of the whisker just as it emerged from the follicle. After cutting, whiskers were scanned on a flatbed scanner (Epson Perfection 4180 Photo) at a resolution of 2,400 dpi (10.6 microns per pixel), along with a ruler for calibration (1mm resolution). Whiskers from the eighth mouse were accidentally lost. Definition and quantification of whisker morphological parameters Following convention (Dörfl, 1982), approximately 30 large whiskers on each mystacial pad were taken to be macrovibrissae, including whiskers up to column 7 in rows that had whiskers in these positions. More rostral whiskers became increasingly short and fine so that visual identification and mechanical manipulation were not feasible. Whisker array morphology was quantified using a total of eight parameters, all of which have been defined in previous work (Belli et al., 2018; Knutsen et al., 2008; Luo and Hartmann, 2023; Towal et al., 2011). The arclength (S) and the intrinsic curvature coefficient (A) describe 2D whisker geometry. The term “arclength” is defined as the total, curved length of the whisker, to distinguish it from the chord length that describes the straight line between the whisker’s base and its tip. The 3D coordinates of the whisker basepoints are denoted as rbp, θbp, and φbp, and the orientation at which the whiskers emerge from the pad are described by the three angles θw, φw, ζw. Quantifying whisker arclength and the intrinsic curvature coefficient: The 2D images of whiskers from seven mice (mice 1-7) were imported into MATLABTM to measure arclength (S) and curvature coefficient (A). . Standard semi-automated image processing techniques were used to create a trace along the whisker length. This trace was then smoothed in the y-dimension using a 20-pixel moving average filter and whisker arclength S was calculated by summing the lengths of the segments between the data points. Measurement error was estimated as two pixels on each end of the whisker, or ~10.6 microns per pixel x 2 pixels per endpoint x 2 endpoints = ~42.4 microns. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Previous studies have shown that the 2D shape of a rat whisker can be approximated by the parabola y = Ax (Belli et al., 2018; Knutsen et al., 2008; Towal et al., 2011). The coefficient A is called the “intrinsic curvature coefficient,” or simply “curvature coefficient.” As will be shown in Results, we found that a quadratic fit was also a good approximation for whisker curvature in mice. We determined the value for A following the steps described in a previous study (Belli et al., 2018). The whisker was first oriented concave up, with its basepoint at the origin (0,0) and initial portion of the whisker aligned with the x-axis such that the majority of the whisker lay in the first quadrant. Because previous studies have indicated that only the proximal ~65% of the whisker remains planar (Knutsen et al., 2008; Towal et al., 2011), the arclength of the whisker was truncated to 65%. Although a detailed analysis of whisker intrinsic curvature lies outside the scope of the present work, a recent study reviewed various methods for its quantification (Luo and Hartmann, 2023). The study examines the whiskers of ten different species, including Mus musculus, and the analysis includes the full whisker arclength, instead of limiting to the proximal 65%. For full-length Mus musculus whiskers, a fractional exponent fit between quadratic and cubic was found to be the most accurate (Luo and Hartmann, 2023). An iterative optimization routine that minimized the difference between the smoothed traced whisker and the fit curve was used to determine the fraction of the whisker that should be aligned with the x-axis. We tested values between 1% and 30% of the whisker length. For each iteration, the curve y=Ax was fit to the truncated whisker, and the mean squared error (MSE) between the fit curve and whisker trace was minimized. On average across all whiskers, using 8% of the proximal whisker to align with the xaxis generated the smallest MSE. This percentage was therefore used for alignment for all whiskers. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Comparing rat and mouse base diameters: To compare the base diameters of the whiskers of mice and rats we used publicly available datasets from Hires et al. (2016) (Hires et al., 2016) and Belli et al., 2018. (Belli et al., 2018) The Hires dataset includes 28 whiskers: three each of the A2, B2, C1, C3, C4, D2, and E2 whiskers, and seven C2 whiskers. We selected whiskers from the Belli dataset to match those measured in Hires for a total of 146: 16 A2; 20 B2; 18 C1; 19 C2; 20 C3; 18 C4; 17 D2; and 18 E2 whiskers. Choice of coordinate system: Establishing the reference frame in which to define the whisker pad parameters (rbp, θbp, φbp, θw, φw, ζw) required us first to choose an origin and a horizontal plane in which to orient the mouse’s head. Following procedures previously established for the rat (Belli et al., 2018), the origin was chosen to be the mean position of all whisker basepoint locations on both left and right sides of the array. This choice placed the origin near the center of the muzzle, inside the animal’s head (Fig. 1A). Note that this procedure required “matched” basepoints between right and left sides, so if a particular whisker identity was present on only one side it was omitted from the calculation of the origin. Again following the approach of previous studies (Belli et al., 2018; Towal et al., 2011), we defined the “average whisker row plane” as the horizontal (x-y) plane. A plane was fit to the basepoints of each of the five whisker rows using least squares, and the mean of these five planes defined the average whisker row plane (Fig. 1B). With the x-y plane established as the average row plane, the x-axis was defined as the line connecting the centroids of the left and right arrays with the left side of the animal negative. The y-axis was defined to be orthogonal to the x-axis, with the positive y-axis pointing rostrally. Lastly, the axis perpendicular to the x-y plane was the z-axis (Fig. 1C). Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Quantifying 3D coordinates of the whisker basepoints: Having chosen an origin and horizontal plane, the 3D coordinates of all recorded points on the animal’s head, including the whisker basepoints, could be determined. We used an intuitive coordinate system that exploits bilateral symmetry of the animal by mirroring the left array across the midline. All equations and figures in the present work are defined in terms of right-sided whisker arrays. These “array-centered” axis conventions apply equally well to both sides of the head, with the caveat that the coordinates on the left side of the animal no longer follow the right-hand rule. The three spherical coordinates for basepoints (rbp, θbp, φbp) are illustrated in Fig. 1D. The radius (rbp) is defined as the straight-line distance between the basepoint and the origin, and is closely related to the size of the mouse’s mystacial pads. The coordinate θbp describes the rostro-caudal location of the whisker basepoint along the positive xaxis. If the whisker is rostral [caudal] to the x-z coronal plane, it has a positive [negative] value of θbp. The dorsoventral location of the basepoints with respect to the x-y horizontal plane is described by the coordinate φbp. If the basepoint location is dorsal [ventral] to this horizontal plane it has a positive [negative] value of φbp. Quantifying the 3D angles at which the whiskers emerge from the face: The “angles of emergence” describe the orientation of the whiskers as they emerge from the mystacial pad at their basepoint locations. In the present work, the angles are defined as Euler angles applicable to the right array. The angles θw and φw both describe the orientation of the approximately linear portion (proximal 8%) of the whisker, near its base; neither of these angles depends on the intrinsic curvature of the whisker. As shown in Fig. 1E, the angle θw is the angle of emergence in the horizontal plane, and ranges between 0° (whisker points caudally) and 180° (whisker points rostrally). Values of θw greater than 180° indicate that the whisker points across the rostro-caudal midline of the animal. The angle φw describes Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t the elevation of the whisker (Fig. 1F). Values range between ±90°, with 90° indicating that the whisker points dorsally and -90° ventrally. Because each whisker has an intrinsic curvature, a third angle, ζw, is required to describe the whisker’s twist about its own axis, as illustrated in Fig. 1G. If the whisker is oriented concave forward, it has an angle of ζw = 90°, while concave backwards is ζw = 90°, and concave dorsal and ventral are 180° and 0°, respectively. As indicated earlier, the angles of emergence are defined only for whiskers of the right array. All left side whisker arrays were mirrored across the y-z plane, and figures and equations in the present work are presented in terms of right-sided whisker arrays. To find the angles of emergence, we performed an optimization using the built-in MATLABTM function “fmincon.” The optimization parameters were θw, φw, ζw, S, and A. Starting with the whisker point cloud in standard position and orientation, the individual points were sorted by distance from the basepoint. The x, y, z coordinates of all points were smoothed with a moving average filter (window size of five points), and the whisker was then resampled into 500-micron segments and its base translated to the origin. The routine next varied θw, φw, ζw, S, and A to fit an idealized whisker model to the point cloud. The model whisker began with its base at the origin and its proximal portion aligned along the negative y-axis. The model was initially oriented concave down. To match the idealized model with the whisker point cloud, we performed an Euler rotation sequence in the order y-axis (roll, ζw), x-axis (pitch, φw), z-axis (yaw, θw). These rotations were extrinsic about the global y-x-z axes. The parameters S and A were also varied to best match the idealized model to the point cloud. The optimization minimized the mean sum-squared distance between the whisker point cloud and the points in the idealized whisker model. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Relationship between basepoint coordinates and row and column identity in the mouse The present work quantifies the morphology of the mouse vibrissal array using the 3D position of the whisker basepoints (θbp, φbp, rbp). To do so, it was first necessary to develop equations that related the basepoint coordinates for each whisker to its row and column position. In all equations, angles are measured in degrees and lengths in mm. As expected, θbp was linearly related to column (Col) and φbp was linearly related to row (Row): θbp = 11.7 Col – 41.2, Adj. R 2 = 0.84 (eq. 1) φbp = -16.8 Row + 56.3, Adj. R 2 = 0.92 (eq. 2) In equations 1 and 2, Col ranges between 1 and 7 and Row ranges between 1 (for the A row) and 5 (for the E row). Note that numbering the columns starting with 1 for the “Greek arc” is nontraditional, but follows the nomenclature used in Belli et al., 2018 (Belli et al., 2018), and accommodates comparisons with species that do not have an easily identifiable Greek arc. Equation 1 indicates that θbp increases from caudal to rostral across the mystacial pad, and equation 2 indicates that φbp decreases from dorsal to ventral; both of these relationships are shown in Fig. 2A. The quality of both fits is presented in Fig. 2B, while Fig. 2C visualizes the strong relationship between θbp and column, and between φbp and row. A third coordinate, rbp, is needed to fully describe basepoints in spherical coordinates. We first found rbp as a function of row and column position: rbp = 0.213Col + 0.372 Row + 3.97, Adj. R 2 = 0.612 (eq. 3a) Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t The coordinate rbp is more variable than θbp or φbp, reflected by the lower R , partially due to head size differences across individual mice. We next quantified the relationship between rbp and the basepoint parameters θbp and φbp, and found rbp to be linearly related to both variables (Fig. 2D): rbp = -0.0152 θbp 0.0165 φbp + 4.44, Adj. R 2 = 0.551 (eq. 3b) Although equation 3b has a slightly lower R value than equation 3a, it allows comparisons between species with different numbers of rows and columns. A sense for the quality of fit of equation 3b is given by the dispersion about the identity line in Fig. 2E. Fig. 2F illustrates the variation of rbp across the array, revealing the narrowing of the mouse’s snout rostrally and dorsally. Statistical analysis of morphological parameters and error analysis We used the method of “forward selection” to incrementally increase both model order and the number of independent variables included in a model, starting with simple firstorder linear regressions. We evaluated all models based on three metrics: first, the significance (as indicated by p<0.05) of the overall model and of individual covariates; second, the information lost by imposing a model on the dataset (as indicated by an Akaike Information Criterion (AIC) of <2 points compared to a competing model); and third, visual assessment of good correspondence between observed and predicted values, as indicated by a qualitatively “tight” clustering of points about the identity line on an observed vs predicted plot. Data for individual mice can be found at https://doi.org/10.5281/zenodo.7992354. Intrinsic curvature and arclength as functions of θbp and φbp: A total of 357 unique 2D scanned images of whiskers were obtained from seven of the eight mice. Values of whisker arclength were obtained from all 357 scans and values of the intrinsic curvature coefficient were obtained for 324 of the scans. To identify outliers, the mean and Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t standard deviation were calculated for whiskers grouped by their row and column identity. We eliminated whiskers greater than two standard deviations above or below the mean. Out of the dataset of 357 scans, 14 outliers (3.92%) for arclength (S) were removed, yielding a total of 343 whiskers. Out of the dataset of 324 scans for curvature coefficient, 12 outliers (3.70%) were eliminated, yielding a total of 312 whiskers for analysis. We constructed models for S and A as functions of θbp and φbp. To avoid overfitting, models were not fit directly to the entire dataset for each parameter. Instead, the best fit model was found for each mouse individually. The best fit models were selected using the following steps. First, we divided each dataset for S and A (with outliers removed) into seven subgroups by mouse identity. For each of the seven mice, histograms of S and A were found to not be quite normally distributed. We therefore analyzed the data both with and without logtransformations to the S and A coefficients to improve normality. Second, linear regression models were constructed for each mouse using both the original and the log-transformed data. We tested whether S and A were univariately associated with θbp or φbp. Specifically, we tested the null hypothesis that the regression coefficients for θbp and/or φbp were equal to zero. In accordance with standard procedures of forward selection model evaluation, if the p-value for the independent variable coefficients was less than or equal to 0.05, we rejected the null hypothesis for the lower order model and proceeded to evaluate a higher order model (square of θbp and/or square of φbp). The hypothesis that the regression coefficients for these higher order terms were equal to zero was then tested. For each quadratic order coefficient in the model, if the p-value was greater than 0.05, the model remained first order. If the pvalue was less than 0.05, then the second order model was selected. We also considered the Akaike Information Criterion (AIC) as an additional metric to avoid overfitting. The higher order model was selected only if the AIC for that model was more than two points lower than the lower order model, and the independent variable Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t coefficients were significant. We did not find any model for whiskers from an individual mouse to have independent variables greater than second-order. Third, if both basepoint parameters (θbp and φbp) were univariately associated to either first or second order with either the original data or log-transformed A or S, both parameters were included in a multivariable linear regression model for each individual mouse. The method of forward selection was again used to introduce the terms as independent variables. We tested the hypothesis that the regression coefficients for θbp or φbp were equal to zero. If the p-value for the independent variable coefficients was less than or equal to 0.05, a second order model (square of θbp and/or square of φbp) was fit and we again tested the hypothesis that the regression coefficients for these higher order terms were equal to zero. We then compared the best-fit simple or multivariable linear regression models across all seven mice, for the original data and log-transformed A or S. An overall model, based on data from all mice together, was fit only after considering the results of the models fit to each individual mouse. A parameter (i.e., θbp, φbp, or their squares) was included in the overall model only if it had first appeared as a significant independent variable in at least six of the seven individual mouse models. Each of these parameters was then retained or discarded in the overall model using the same forward selection methods described for the individual models. To avoid overfitting, the order of the combined model was not allowed to exceed the highest order of the six out of seven individual mouse models. Angles of emergence (θw, φw, and ζw) and basepoint radius (rbp) as functions of θbp and φbp: Of the 467 whiskers obtained from the eight mice, 461 were of sufficient quality to use in a 3D analysis of basepoints (rbp θbp φbp) and emergence angles (θw, φw, ζw). To identify outliers for each parameter, the mean and standard deviation were calculated for whiskers grouped by their row and column identity. As before, the value of a given parameter for a particular whisker was eliminated if it was greater than two standard deviations above or below the mean. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Out of the dataset of 461 whiskers, 21 outliers (4.56%) were removed for rbp, 14 outliers (3.04%) were removed for θbp, 19 outliers (4.12%) were removed for φbp, 19 outliers (4.12%) were removed for θw, 16 outliers (3.47%) were removed for φw, and 13 outliers (2.82%) were removed for ζw. Thus, after outlier removal, 440 values remained for rbp, 447 values remained for θbp, 442 values remained for φbp and θw, 445 values remained for φw, and 448 values remained for ζw. We next constructed models for rbp θw, φw, and ζw as functions of θbp and φbp. To avoid overfitting, the best fit model for the parameters rbp, θw, φw, and ζw, was found for each of the eight mice individually. We first determined that the parameters rbp, θw, φw, and ζw were all normally distributed for each of the eight mice. Therefore, a log-transformation was not necessary and linear regression models were then constructed for each mouse as follows. First, for each mouse, we tested whether rbp, θw, φw, and ζw were univariately associated with θbp and/or φbp using the forward selection procedure described previously. We did not find any models for whiskers from an individual mouse to have statistically significant independent variables greater than second-order. Second, if both basepoint parameters (θbp and φbp) were univariately associated to either first or second order with rbp, θw, φw, and ζw, then both parameters were included in a multivariable linear regression model for each individual mouse. Again, independent variables were added to the models using the method of “forward selection.” We tested the hypothesis that the regression coefficients for θbp and/or φbp were equal to zero. If the p-value for the independent variable coefficients was less than or equal to 0.05, a second order model (square of θbp and/or square of φbp) was fit and we retested the hypothesis that the regression coefficients for these higher order terms was equal to zero. We also considered the AIC to avoid overfitting. The higher order model was selected only if both the independent variable coefficients were significant and the AIC for the higher order model was more than two points lower than the lower order model. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Finally, for each parameter (rbp, θw, φw, and ζw), the best fit univariate or multivariable function of θbp and/or φbp was compared across all eight mice. An overall model, based on data from all mice together, was fit only after taking into account the results of the models fit to each individual mouse. This overall model was fit similarly to the individual models, except that a parameter was included in the final combined model only if it had first appeared as a significant independent variable in at least six of the eight individual mouse models. If a parameter met this criterion it was initially included in the model, and retained or discarded using the same forward selection methods described for the individual models. Error assessment: The MicroscribeTM origin was defined by making a small divot in a piece of laboratory labeling tape on the operating table near the animal. At the start of each experiment, we calibrated the MicroscribeTM by sampling this origin from five different orientations. The mean resolution over all scans and experiments was 0.5 mm. This error estimate includes any slight eccentricities of the tip, any deviation due to hand tremor of the user, and the intrinsic precision limitations of the device. Note also that the 0.5 mm error does not compound along the arclength of the whisker because each point along the whisker was sampled independently. As an additional estimate of measurement error, we performed simulations in which we randomly varied all points along the whisker arclength (including the basepoint) by +/0.5 mm in all three spatial dimensions. We then observed the effect on the 3D position and orientation of the whiskers. We performed these simulations for one rostral and one caudal whisker from each row, for a total of 10 whiskers. The maximum angular deviation of the whisker basepoint location was 0.015 mm in rbp, 1.07° in θbp, and 0.53° in φbp, and the maximum angular deviation of the whisker emergence angle was found to be 5.9° in θw, 4.1° in φw, and 28.7° in ζw. The maximum error for ζw was a true outlier, as the next largest error among the remaining nine whiskers checked was 0.74°. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Digitization of internal skull features To position the skull and facial features in standard orientation, the (x, y, z) coordinates of the skull and face collected using the MicroscribeTM were imported into MATLABTM. These coordinates were then rotated and translated to match the axis conventions shown in Fig. 1. The data used for the external auditory meatus and lateral canal coordinates came from serial computed tomography (CT) scans of Mus musculus skull, specimen TMM M3196, available through the digital morphology database DigiMorph.org (Demarest, 2001). The specimen was originally scanned along the coronal axis, for a total of 480 slices. Each 1024 x 1024 pixel slice is 0.02961 mm thick, with 0.02961 mm interslice spacing and a field of reconstruction of 28 mm. These images result in a resolution of 0.02734 mm in x and y (within each coronal plane slice) and an interslice resolution of 0.02961 mm in z. We used ReconstructTM to record the 3D coordinates of the bony labyrinth and other skull features in serial coronal CT images. Structures traced included the entire left and right bony labyrinths, the left and right external auditory meatuses, the lambdoid, sagittal, and coronal sutures, and the lateral corners of the left and right upper incisors. Data were imported into MATLABTM and we manually identified the (x, y, z) coordinates of the following skull features: (1) bregma and lambda; (2) the locations at which each of the two lateral canals terminated in a crista; (3) the lateral-most point of each of the two lateral canals; (4) five distinct points around the circumference of each auditory meatus; (5) the lateral corners of the incisors. Points from the CT scan data were brought into the same reference frame as the MicroscribeTM data by aligning a subset of corresponding points between the two datasets. These points included bregma, lambda, and the corners of the incisors. We found the translations and rotations that brought these points into register, and then applied these same translations and rotations to the points for all other features. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Results The first two sections of Results describe the morphology of the mouse vibrissal array using linear regression, while the last four sections compare the morphology of the mouse and rat. In all equations, angles are measured in degrees, lengths in mm, and curvature in mm. The model of the mouse is available at https://doi.org/10.5281/zenodo.7992354 Two-dimensional shape: arclength and curvature coefficient The most basic description of two-dimensional (2D) whisker geometry includes arclength (S) and the intrinsic curvature coefficient (A). We quantified these two parameters as functions of the basepoint coordinates θbp and φbp. Arclength was best described as an exponential function of θbp: S = e θbp + , Adj. R = 0.671 (eq. 4a) To provide a more intuitive sense for how whisker arclength varies across the array, equation 4a can be rewritten: S = 11.7e , Adj. R = 0.671 (eq. 4b) Equations 4ab indicate that S decays exponentially from caudal to rostral (Fig. 3A). Whisker arclength is ~11.7 mm near the rostro-caudal center of the array, where θbp = 0, and changes exponentially from the center. Experimental values for S vary between ~2 and ~30 mm. The quality of the fit for equation 4ab is shown in the plot of actual vs. predicted values for S (Fig. 3B), and Fig. 3C provides visual intuition for variations in S across the array. As described in Methods, whisker curvature was quantified using the 2D parameterization y = Ax, fit to the proximal 65% of the whisker. The curvature Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t coefficient, A, was best described as a linear function of θbp, a relationship shown in Fig. 3D: A = 0.00037 θbp + 0.017, Adj. R 2 = 0.233 (eq. 5) The R for equation 5 is low, due to the high variability in A for the rostral whiskers. However, equations 4ab and 5 depend only on θbp, indicating that mouse 2D whisker geometry is relatively consistent within a column. Fig. 3E illustrates the relationship between curvature coefficient and arclength. Although curvature is particularly variable for short whiskers, a strong upper bound characterizes the relationship between A and S. An equation for the upper bound (eq. 6) was generated by fitting to a series of maximum points using a moving average with a 4 mm window; this equation is visualized in Fig. 3F A = 0.294e + 0.012, Adj. R = 0.994 (eq. 6) The results presented in Figures 3D-F are subject to an important caveat. Although these figures show that the curvature coefficient is more variable for shorter whiskers, they do not imply that shorter whiskers have a more unpredictable tip location than longer whiskers. This statement is explained in the analysis of Fig. 3G-H. To create Fig. 3G-H, we first identified the full range of A values found experimentally. We then solved for S using eq. 6 for each value of A larger than the y-intercept (0.012). When A is smaller than the y-intercept, S is undefined, and for these whiskers S was fixed at the longest observed value. We then used these values for A and S to generate a set of “mock,” idealized whiskers that followed the curve y = Ax in standard Cartesian coordinates, where the whisker base was placed at the origin and its initial slope was aligned with the x-axis. These whiskers are idealized because their curvature is computed based only on their proximal 65%, and then extrapolated to the tip This approach accounts for the tendency of whisker tips to lie out of the primary plane of Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t intrinsic curvature (Knutsen et al., 2008; Luo and Hartmann, 2023; Towal et al., 2011); Luo and Hartmann, 2023 provides an analysis of curvature that includes the full whisker length(Luo and Hartmann, 2023). Eighteen example mock whiskers (six whiskers for each of three different values of S), are shown in Fig. 3G. The color of a whisker indicates its A value. Each thick black arc shows the range of possible tip locations for a whisker with that arclength, as the parameter A is varied across its observed values. The top of each black arc terminates when a whisker of arclength S reaches its maximal observed curvature. The six shorter whiskers have more variable colors than the longer whiskers because shorter whiskers have more variable curvatures. The analysis shown in Fig. 3G is extended across a more continuous distribution of whiskers in Fig 3H. In this figure, each mock whisker still has a single color (representing its value of A), but whiskers are plotted densely enough to form a continuous color field. The small, colorful left corner indicates that curvature is more variable for shorter whiskers, but that this variability does not cause large tip displacements. In contrast, the right side is mostly dark blue, indicating that changes in tip locations of the longer whiskers are not associated with large differences in A. The gray regions of Fig. 3H are those in which no whiskers are predicted to lie. This region is found by smoothly connecting the tops of the black arcs, bounding the curvatures for whiskers of given arclengths. However, note that, because A is calculated based only on the proximal 65% of the whisker, some of the actual whisker tips we sampled (31 of 312) would lie within the gray region. Given that the whiskers’ distal tips are often damaged, the shape of any given mouse whisker tip will be variable and unpredictable. Therefore, an advantage of the present model is that simulated whiskers more closely reflect a “typical” whisker instead of a particular individual mouse whisker. Angles of emergence of the whiskers as a function of basepoint coordinates As shown in Fig. 4A, the horizontal angle of emergence, θw, was best described as a linear function of θbp: Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t θw = 1.04 θbp + 82.5, Adj. R 2 = 0.544 (eq. 7a) Although equation 7a represents the best relationship found while avoiding overfitting, the R is relatively low. Moreover, plotting actual vs. predicted values (Fig. 4B) reveals moderately high dispersion about the identity line, limiting the predictive value of equation 7a. However, values for θw were particularly variable for whiskers in the A row (Fig. 4B, blue points). After omitting these, the equation for θw is: θw = 0.922 θbp + 85.1, Adj. R 2 = 0.739 (eq. 7b) The elevation angle, φw, was found to vary linearly only with φbp: φw = 1.00 φbp + 10.2, Adj. R 2 = 0.76 (eq. 8) This relationship is shown in Fig. 4C, and Fig. 4D shows the quality of the fits by comparing predicted versus actual values. The two colormaps of predicted θw and φw values across the array (Fig. 4E) confirm the visual intuition that more caudal whiskers emerge at more caudal angles than the rostral whiskers, while whiskers in more dorsal rows emerge at more elevated angles than whiskers in more ventral rows. The angle ζw, describes the twist of the whisker about its own axis. This angle can be computed accurately only if the whisker curves measurably away from its base. Therefore, whiskers shorter than 8 mm were excluded from the fits for ζw. Even after omitting these whiskers, the best fit for the angle ζw was poor: ζw = 0.71 φbp + 36.1, Adj. R 2 = 0.13 (eq. 9a) Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Fig. 4F shows values for ζw grouped by the row and column identity of the whisker and plotted as a function of φbp, with equation 9a shown in black. To explore the high variability in ζw we plotted it as a function of row and column (Fig. 4G), revealing that most variability comes from the A-row and the most rostral columns within each row. Models for each row individually were moderately to minimally predictive, as indicated by R values below 0.25. The p-value (>0.05) for the linear regression model for row A indicated that the values of ζw for these whiskers were not well-predicted by rostro-caudal position. In contrast, models ζw for whiskers in rows B E were well predicted by rostro-caudal position, as indicated by significant f-tests (p < 0.005). These results suggest that a substantially improved model for ζw might be obtained after excluding the A-row and the most rostral whiskers. We therefore excluded these whiskers and re-fit an overall model to the remaining 309 whiskers. As predicted, the fit improves significantly: ζw = 0.620θbp + 1.00φbp + 43.0, Adj. R 2 = 0.34 (eq. 9b) The reduced dataset, along with predicted values, is shown in Fig. 4H, and the quality of the model is illustrated in Fig. 4I. Predicted values from equation 9b are shown in the colormap in Fig. 4J. Our confidence in equation 9b is increased because its form and coefficients are similar to those for the rat (Belli et al., 2018) : ζw RAT = 0.876θbp RAT + 0.845φbp RAT + 37.9, Adj. R 2 = 0.42 (eq. 10) The rat and mouse models are not statistically different, as their 95% confidence intervals overlap for both predictors. It is also possible that a different choice for the horizontal plane would reveal a stronger relationship between ζw and θbp and/or φbp. Comparisons between the mouse and rat Table 1 compares equations for the mouse with equations for the rat obtained from Belli et al., 2018. Because mouse and rat have the same number of whisker rows and Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t columns, basepoint coordinates are easily compared. The first two rows of Table 1 and Fig. 5A show that rat and mouse arrays have similar relationships between the basepoint coordinates and row and column variables. One difference is that mouse φbp varies linearly with Row, while Belli et al., 2018 found that rat φbp varies quadratically with Row. However, when forced to a linear fit, the coefficients for the rat are not statistically different from those of the mouse, as indicated by overlap in the 95% confidence intervals of the two models (mouse interval [-16.28 -17.29], rat interval [16.81 -17.92]). These models result in predicted values with an average difference between species of less than 2°. An important goal of the present work was to develop equations for whisker morphology and arrangement that can be compared across species, regardless of whether the species has an “orderly” or “disorderly” whisker array. Rows 3 8 of Table 1 compare the morphology of mouse and rat whisker arrays using θbp and φbp as independent variables. These types of comparisons could be performed for any whiskered animal; the basepoint coordinates can be measured independently of row and column position (rows 3 – 8 do not depend on rows 1 and 2). The parameter rbp is compared in the third row of Table 1 and in Fig. 5B. Values for rbp depend strongly on head size, and are, on average, 56.1% larger in the rat compared to the mouse, with a range of 36.1% for the D5 whisker (the most similar) and 77.8% for the A1 whisker (the least similar). The larger rbp values for rat were statistically significant with p < 0.001 (independent samples t-test). This difference in size contrasts with the remarkable similarity between the two species for θbp and φbp by whisker identity (rows 1-2 of Table 1). To obtain further intuition for the differences in scale across species, Fig. 5C shows the whisker basepoint locations for the two animals in the y-z plane. We found that, for the mouse, the total area occupied by whisker basepoints was 50.90 ± 16.86 mm, for a density of 0.66 ± 0.44 whiskers/mm, while the rat has a whisker area of 139.35 ± 34.6 mm, yielding a density of 0.24 ± 0.05 whiskers/mm. When normalized for the obvious Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t size difference, however, mouse and rat have approximately the same number of whiskers in the same angular region (Fig. 5D). Comparisons of 2D parameters (arclength, curvature, base diameter) for mouse and rat We compared the relationships that describe 2D whisker geometry for mouse and rat (rows 4 and 5 of Table 1). Fig. 6A shows S as a function of θbp, along with the fits given by equations in the fourth and fifth rows of Table 1. Whisker arclength has a larger range for the rat than mouse, reflected by the steeper exponential decay. The longest rat whisker was approximately twice as long as the longest mouse whisker (60 mm vs. 30 mm). Fig. 6B shows curvature coefficient (A) as a function of θbp for both animals, again highlighting increased variability for more rostral (shorter) whiskers. The coefficient does not differ significantly between the two species, as reflected in the overlapping confidence intervals. Fig. 6C plots the curvature coefficient as a function of arclength, and the associated upper bounds, for mouse and rat. The upper bound on the rat is greater than that on the mouse for most whiskers, but there is considerable overlap. To complete the analysis of 2D parameters, we used datasets from Hires et al., 2016 (mouse) and Belli et al, 2018 (rat) to compare the base diameters of select mouse and rat whiskers (Fig. 6D). A total of 28 mouse whiskers and 146 rat whiskers were analyzed (see Methods). On average, rat whiskers have base diameters ~2.35x larger than mouse whiskers. Angles of emergence of the whiskers as a function of basepoint coordinates for mouse and rat We next compared the 3D angles of emergence (θw, φw, ζw) between mouse and rat (Table 1, rows 6, 7, and 8.) Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Horizontal Angle: As shown in Fig. 7A, mouse whiskers tend to emerge from the cheek pointing more rostrally than those of the rat, especially the rostral whiskers. The mean value for θw in columns 6 and 7 of the mouse is 120°, compared to 97. 0° in the rat. In contrast, θw is more similar between the mouse and rat for caudal whiskers: the mean value of θw for columns 1 and 2 (combined) is 59.3° for mouse and 51.0° for rat. Also, rat θw is a function of both θbp and φbp, while mouse θw depends only on θbp. These differences likely reflect the small size of mice, making it difficult to capture as much nuance in the model. Elevation angle: Comparing φw between rats and mice (Fig. 7B) shows that mouse whiskers lie at slightly higher elevation than rat whiskers, as indicated by the almost parallel lines offset by ~4°. Twist angle: Although the relationship for ζw in the mouse is not strong (equation 9b), it allows approximate comparison to the rat (Fig. 7C), with greatest differences found for the dorsal whiskers. In general, ζw is much more variable than the other angles for both species, in part due to limited measurement resolution. Using equations 1-9, the full 3D model of the average mouse vibrissal array is plotted in the left panel of Fig. 7D, and compared with the average rat array (overlaid) in the right panel of Fig.7D. Relationship of facial features to basepoint parameters (θbp and φbp) for mouse and rat Finally, we quantified the locations of mouse facial and skull features relative to whisker basepoints. For visual intuition, Fig. 8A shows two views of the facial features of an individual mouse and rat. To compare relative facial feature locations despite difference in absolute scale, Fig. 8B superimposes these facial features in 2D angular coordinates. The origin is the centroid of all vibrissal basepoints, in the middle of the head. To gain physical intuition for this figure, the reader should imagine looking into the page with their nose at the origin, and then wrapping the figure around their head so that bregma Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t and lambda are at the back. Only the width of the mouth appears to differ appreciably between the species. These data allow us to compare both rostral-caudal and bilateral distances between facial features of the rat and mouse. The top panel of Fig. 8C shows that in the rostralcaudal dimension, all rat facial features are nearly exactly 2.09 larger than those of the mouse. Bilateral distances between facial structures (Fig. 8C, bottom panel) are more variable and generally scale by less than the value of 2.09 observed for rostral-caudal distances. The straight-line distances between facial features of the mouse are shown in Fig. 9, highlighting clusters of rostral and caudal structures. In general, straight-line distances for mouse facial features are similar to those for the rat shown in Fig. 9 of Belli et al., 2018. However, after normalizing for facial size, some important differences became evident. On each side of the mouse’s face, the eye corners are slightly closer to the C3 whisker base points, and (again normalizing for facial size) the two ears of the mouse are further apart than the rat’s, consistent with the strong bilateral differences in Fig. 8C. Discussion Because different animal species have different numbers and arrangements of whiskers (Muchlinski, 2010; Muchlinski et al., 2013) row and column position do not lend themselves to developing equations that can be used to compare whisker geometry across species. To overcome this problem, we developed an approach that uses 3D whisker basepoint positions to quantify whisker properties including arclength, curvature, and emergence angles. Facial features can then be placed in this same coordinate system. Using this approach, we quantified and compared whisker and facial morphology across mice and rats. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Variations in whisker arclength and intrinsic curvature Consistent with previous studies on rats (Brecht et al., 1997; Pockock, 1914; Towal et al., 2011), the present work shows that whisker arclength increases exponentially from rostral to caudal across the mouse’s face. Rat whiskers, which range in arclength from 10 – 60 mm, are approximately twice as long as mouse whiskers (Fig. 6A), in line with the ~2x linear scaling of other facial features (Fig. 8). Data from previous studies (Belli et al., 2018; Hires et al., 2016) indicate that the base diameters of rat and mouse whiskers also differ by approximately a factor of two (Fig. 6D). The 2x linear scaling in both arclength and base diameter means that the volume (and hence mass) of rat whiskers is expected to be ~8x greater than that of the mouse, comparable to the ~10x body mass difference. The intrinsic curvature of rat and mouse whiskers is described by a quadratic function, with a strict upper bound on maximum curvature and thus tip position (Figs 3G and 6C). Rostral whiskers exhibit the largest variability in curvature coefficient, but that does not mean that their shapes are more variable. When a whisker is short, variations in its curvature coefficient have only a small effect on its tip position (Fig. 3G). Therefore, the coefficient of more rostral whiskers may not be under as strong selective pressure as the coefficient for longer whiskers. Whisker orientations at rest The present work measured the locations of sensory structures in the anesthetized animal, so whisker emergence angles are the “resting” angles. Both mice and rats move their whiskers through large angles during whisking behavior, with primary motion in the rostro-caudal direction. The horizontal angle θw can thus be considered an initialization parameter for simulations of whisking behavior, rather than a fixed parameter. For both mice and rats the elevation angle w is constant within a row (it varies only with bp), and the relationship between w and bp has one of the highest correlation coefficients (Table 1). We suggest that constant elevation angle (w) within each row simplifies the problem of localizing the height of an object in head-centered coordinates. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Whisking behavior continuously changes the azimuthal angle (rostral-caudal) but the elevation angle (dorsal-ventral) remains relatively fixed. We predict that the singlevariable, linear relationship between w and bp is a characteristic feature of whiskerbased tactile sensing that will generalize across all or most species. In contrast to the single-variable relationship between w and bp, the twist (ζw) of the whisker about its own axis varies diagonally, and depends on both θbp and bp. In addition, the fit quality is poor, mostly due to measurement error. However, it is also possible that choosing a different head pitch to define the horizontal plane could reveal a stronger linear relationship. The diagonal orientation gradient could be particularly useful for determining the direction of a stimulus moving across the array. Stimuli such as air or water, which interact with many stationary whiskers simultaneously, will encounter different gradients of whisker twist depending on the animal’s head pitch. These gradients could permit the animal to infer stimulus direction across a population of responsive neurons (Quist and Hartmann, 2012; Yu et al., 2016b). Reasons to expect that the facial and vibrissal morphologies of mice and rats will be similar, and reasons to expect that they will be different There are several reasons to expect rats and mice to have similar facial and vibrissal morphologies. Though rats and mice diverged earlier than did humans and chimps (~10mya compared to ~8mya (Patterson et al., 2006) (Steppan et al., 2004) they remain so physically similar that many non-experts confuse the two species. Both rats and mice are small, terrestrial, burrowing, cathemeral or crepuscular omnivores. Both species occupy a niche that relies on quick reproduction and on quantity rather than quality of offspring to maintain population (Berry and Bronson, 1992; Feng and Himsworth, 2013), though mice emphasize this strategy slightly more than rats (Berry and Bronson, 1992). The two species also retain many less obvious adaptations in common, such as longitudinal chewing with interlocking molar cusps, probably inherited from an insectivorous ancestor (Lazzari et al., 2008). Most importantly for the question of whisker morphology, both rats and mice actively tap and brush their whiskers against objects during tactile exploratory behaviors. These motions, called “whisking,” help the animal determine object location, size, orientation, Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t and texture (Carvell and Simons, 1990; Grant et al., 2009; Knutsen et al., 2006; Krupa et al., 2001; Mehta et al., 2007; Pammer et al., 2013; Polley et al., 2005; Vincent, 1912). Facial muscles are similar between the species as well, in addition to the numbering of the whiskers (Dörfl, 1982; Haidarliu et al., 2010). However, there are some distinct differences between mice and rats that could produce differing selective pressures on facial and vibrissal morphology. Rats weigh about ten times more than mice, conferring a survival advantage given that small predators are less willing to take larger prey (Parsons et al., 2018). Though both species whisk, rats move their whiskers at average frequencies between 5-15 Hz while mice whisk faster, between 10 and 25 Hz (Jin et al., 2004; Welker, 1964) Selective pressure for rat and mouse morphological divergence could also emerge from differences in diet, as rats lack a gallbladder while mice retain theirs (McMaster, 1922), possibly assisting the mouse’s relative emphasis on insectivory (Shiels et al., 2013). Though rats eat fewer insects, they are an occasional predator of mice (Yoshimura and Ueki, 1981). More evidence that the two species have specialized niches comes from their different colonization patterns. While the mouse spread across the Old World roughly following the expansion of human farming during the Bronze age (Auffray et al., 2008), the rat spread out of Asia much later, in the 18 century, after humans had already established urban environments. Finally, the whisker systems of mice and rats also contain notable differences in the barrel cortex – barrels in rat cortex have “cell-dense” centers compared to barrels with “cell-sparse” centers in mice, a difference the arises in rats during postnatal development (Rice et al., 1985). Additionally, inter-barrel septa are much less prominent in mice compared to rats (Rice et al., 1985). Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Rat facial sensory structures may be “scaled up” versions of mouse facial sensory structures The present results support the idea that the larger size of rats compared to mice is a derived (apomorphic) trait. After scaling, rat and mouse have extremely similar whisker array and facial feature morphology (Figs. 8, 9; Table 1). Both species exhibit strong linear relationships between θbp and column and between rbp and φbp. In addition, plotting whisker basepoint location as a function of φbp and θbp, which normalizes whisker location to head size, reveals that mouse and rat whisker basepoints occupy strikingly similar positions. One difference between species is that the relationships between φbp and row, and between rbp and θbp, are linear for mice but quadratic for rats. These results indicate that mice and rats have snouts with similar mediolateral curvature but that the rat’s snout curves slightly more dorsoventrally. Thus, although the rat has a slightly more elongated face, consistent with the tendency for larger animals to have longer faces (Cardini and Polly, 2013), the most prominent difference between the two species is simply scale. These results also agree with findings that the molar-to-mandible ratio scales linearly between mice and rats (Cai et al., 2007). Notably, the bilateral (but not rostral-caudal) distance between the ears is disproportionately large for the mouse, perhaps helping ensure larger interaural differences. At the same time, other studies indicate that the common ancestor of mice and rats is likely to have been the same scale as modern mice. For example, divergence in the mouse/rat lineage has been investigated using fossil molars from a probable common ancestor of mice and rats (Antemus chinjinensis), as well as later extinct murines (Kimura et al., 2015). These fossils indicate that Antemus molars are on the same scale as modern mouse molars. Given the essentially linear scaling between extant mice and rats found in the present work, we suggest that Antemus is likely to have had a similar facial feature arrangement, and that the facial morphology of the rat can be thought of as a “scaled up” version of the mouse. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Utility of the model for simulations and neurophysiological experiments Studying the nervous system using categorical stimulus sets (Bale et al., 2013; Jones et al., 2004; Lottem and Azouz, 2011; Shoykhet et al., 2000; Storchi et al., 2012; Stüttgen et al., 2008) can inadvertently mischaracterize the coding of primary sensory neurons (Bush et al., 2016a; Bush et al., 2016b; Campagner et al., 2016; Severson et al., 2019). We anticipate that the present work will allow simulation of the multimodal sensory data that the mouse acquires during natural exploratory behaviors, similar to other poseestimation or body-dynamics software(Bolaños et al., 2021; Ramalingasetty et al., 2021; Zweifel et al., 2021). After adding multiple sources of variability and noise, these simulations can help establish rich, naturalistic stimulus datasets for use in neurophysiological experiments. References Auffray, J.-C., Vanlerberghe, F. and Britton-Davidian, J. (2008). The house mouse progression in Eurasia: a palaeontological and archaeozoological approach. Biological Journal of the Linnean Society 41, 13-25. Bale, M. R., Davies, K., Freeman, O. J., Ince, R. A. and Petersen, R. S. (2013). Low-dimensional sensory feature representation by trigeminal primary afferents. J Neurosci 33, 12003-12. Belli, H. M., Bresee, C. S., Graff, M. M. and Hartmann, M. J. Z. (2018). 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A unifying framework underlying mechanotransduction in the somatosensory system. J Neurosci 31, 8520-32. Luo, Y. and Hartmann, M. J. (2023). On the intrinsic curvature of animal whiskers. PLoS One 18(1): , e0269210. Ma, P. M. and Woolsey, T. A. (1984). Cytoarchitectonic correlates of the vibrissae in the medullary trigeminal complex of the mouse. Brain Res 306, 374-9. McMaster, P. D. (1922). DO SPECIES LACKING A GALL BLADDER POSSESS ITS FUNCTIONAL EQUIVALENT? J Exp Med 35, 127-40. Mehta, S. B., Whitmer, D., Figueroa, R., Williams, B. A. and Kleinfeld, D. (2007). Active Spatial Perception in the Vibrissa Scanning Sensorimotor System. PLoSBIOLOGY 5, e15. doi:10.1371/journal.pbio.0050015. Muchlinski, M. N. (2010). A comparative analysis of vibrissa count and infraorbital foramen area in primates and other mammals. J Hum Evol 58, 447-73. Muchlinski, M. N., Durham, E. L., Smith, T. D. and Burrows, A. M. (2013). Comparative histomorphology of intrinsic vibrissa musculature among primates: implications for the evolution of sensory ecology and "face touch". Am J Phys Anthropol 150, 301-12. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Pammer, L., O'Connor, D. H., Hires, S. A., Clack, N. G., Huber, D., Myers, E. W. and Svoboda, K. (2013). The mechanical variables underlying object localization along the axis of the whisker. J Neurosci 33, 6726-41. Parsons, M. H., Banks, P. B., Deutsch, M. A. and Munshi-South, J. (2018). Temporal and SpaceUse Changes by Rats in Response to Predation by Feral Cats in an Urban Ecosystem. Frontiers in Ecology and Evolution 6. Patterson, N., Richter, D. J., Gnerre, S., Lander, E. S. and Reich, D. (2006). Genetic evidence for complex speciation of humans and chimpanzees. Nature 441, 1103-1108. Paxinos, G., Watson, C., Pennisi, M. and Topple, A. (1985). Bregma, lambda and the interaural midpoint in stereotaxic surgery with rats of different sex, strain and weight. Journal of Neuroscience Methods 13, 139-143. Pockock, R. I. (1914). On the Facial Vibrissae of Mammalia. Proceedings of the Zoological Society of London 60, 889-912. Polley, D. B., Rickert, J. L. and Frostig, R. D. (2005). Whisker-based discrimination of object orientation determined with a rapid training paradigm. Neurobiology of Learning and Memory 83, 134142. Quist, B. W. and Hartmann, M. J. (2012). Mechanical signals at the base of a rat vibrissa: the effect of intrinsic vibrissa curvature and implications for tactile exploration. J Neurophysiol 107, 2298312. Ramalingasetty, S. T., Danner, S. M., Arreguit, J., Markin, S. N., Rodarie, D., Kathe, C., Courtine, G., Rybak, I. A. and Ijspeert, A. J. (2021). A Whole-Body Musculoskeletal Model of the Mouse. IEEE Access 9, 163861-163881. Rice, F. L., Gomez, C., Barstow, C., Burnet, A. and Sands, P. (1985). AComparative analysis of the development of the primary somatosensory cortex: Interspecies similarities during barrel and laminar development. Journal of Comparative Neurology 236, 477-495. Severson, K. S., Xu, D., Yang, H. and O'Connor, D. H. (2019). Coding of whisker motion across the mouse face. Elife 8. Shiels, A. B., Flores, C. A., Khamsing, A., Krushelnycky, P. D., Mosher, S. M. and Drake, D. R. (2013). Dietary niche differentiation among three species of invasive rodents (Rattus rattus, R. exulans, Mus musculus). Biological Invasions 15, 1037-1048. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Shoykhet, M., Doherty, D. and Simons, D. J. (2000). Coding of deflection velocity and amplitude by whisker primary afferent neurons: implications for higher level processing. Somatosens Mot Res 17, 171-80. Steppan, S., Adkins, R. and Anderson, J. (2004). Phylogeny and divergence-date estimates of rapid radiations in muroid rodents based on multiple nuclear genes. Syst Biol 53, 533-53. Storchi, R., Bale, M. R., Biella, G. E. and Petersen, R. S. (2012). Comparison of latency and rate coding for the direction of whisker deflection in the subcortical somatosensory pathway. J Neurophysiol 108, 1810-21. Stüttgen, M. C., Kullmann, S. and Schwarz, C. (2008). Responses of rat trigeminal ganglion neurons to longitudinal whisker stimulation. J Neurophysiol 100, 1879-84. Towal, R. B., Quist, B. W., Gopal, V., Solomon, J. H. and Hartmann, M. J. Z. (2011). The Morphology of the Rat Vibrissal Array: A Model for Quantifying Spatiotemporal Patterns of WhiskerObject Contact. Plos Computational Biology 7. Van Der Loos, H. (1976). Barreloids in mouse somatosensory thalamus. Neurosci Lett 2, 1-6. Vincent, S. B. (1912). The Function of the Vibrissae In the Behavior of the White Rat. Animal Behavior Monographs 1, 84. Weiler, S., Rahmati, V., Isstas, M., Wutke, J., Stark, A. W., Franke, C., Geis, C., Witte, O. W., Hübener, M., Bolz, J. et al. (2022). A primary sensory cortical interareal feedforward inhibitory circuit for tacto-visual integration. bioRxiv, 2022.11.04.515161. Welker, W. I. (1964). Analysis of sniffing of the albino rat. Behav. 22, 223–244 Wolfe, J., Mende, C. and Brecht, M. (2011). Social facial touch in rats. Behav Neurosci 125, 90010. Woolsey, T. A. and Van der Loos, H. (1970). The structural organization of layer IV in the somatosensory region (SI) of mouse cerebral cortex. The description of a cortical field composed of discrete cytoarchitectonic units. Brain Res 17, 205-42. Yoshimura, H. and Ueki, S. A. (1981). Mouse-killing and hyperemotionality in rats induced by three different kinds of experimental manipulations: A comparative study. Physiological Psychology 9, 269-275. Yu, Y., Graff, M. and Hartmann, M. J. Z. (2016b). Mechanical responses of rat vibrissae to airflow. Journal of Experimental Biology 219, 937-948. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t Zhuang, C., Kubilius, J., Hartmann, M. J. and Yamins, D. L. (2017). Toward Goal-Driven Neural Network Models for the Rodent Whisker-Trigeminal System. In Advances in Neural Information Processing Systems, vol. 30 (ed. I. G. a. U. V. L. a. S. B. a. H. W. a. R. F. a. S. V. a. R. Garnett). San Diego, CA: . Associates, Inc. Zweifel, N. O., Bush, N. E., Abraham, I., Murphey, T. D. and Hartmann, M. J. Z. (2021). A dynamical model for generating synthetic data to active tactile behavior in the rat. Proceedings of the National Academy of Sciences of the United States of America 118, 10. Jo ur na l o f E xp er im en ta l B io lo gy • A cc ep te d m an us cr ip t

interactions (Grant et al., 2018;Guić-Robles et al., 1989;Wolfe et al., 2011). The whiskers' behavioral importance is paralleled by the hypertrophy of whisker-related structures within the rodent nervous system, particularly in the "barrel" region of primary somatosensory cortex (Ma and Woolsey, 1984;Rice et al., 1985;Van Der Loos, 1976;Woolsey and Van der Loos, 1970); the whisker system is thus ideal for electrophysiological study as well.
The present study performs quantitative, morphological quantification and comparison of the facial features of mouse and rat with a particular focus on the vibrissae. The motivation for the work is three-fold, as the results fill scientific and technological gaps across multiple domains of research.
One motivation is to develop a morphometric approach to compare the spatial arrangement of facial structures and whiskers across species, regardless of whether the whisker arrays are "ordered" into rows and columns, or "disordered," with no apparent row/column arrangement (Muchlinski, 2010;Muchlinski et al., 2013). Although rats and mice happen to have similar row/column arrangements of whiskers, the present work compares their morphology without relying on this similarity.
Another motivation is that a mouse morphological model would benefit behavioral and neurophysiological research in multiple ways. First, the model could be combined with videographic data to help quantify exploratory and sensory acquisition behaviors, as has been done for the rat (Hobbs et al., 2015;Hobbs et al., 2016). For example, overlaying the model on videos of a behaving mouse would allow researchers to estimate how the animal's sensory structures are oriented relative to an object and how sensory signals arrive at each structure relative to each other; the latter is critical for analysis of multimodal integration. Second, combining the model with equations of motion for the whiskers (Knutsen et al., 2008) could improve whisker tracking, which remains as yet a challenging image-processing problem. Third, quantifying the arrangement of the whiskers is a necessary first step towards creating mechanical models to estimate the mechano-tactile input to the system, so that these input signals can be correlated with neurophysiological and behavioral recordings. Fourth, quantifying the arrangement of the whiskers could aid in the design of stimuli for neurophysiological experiments. For example, a researcher could construct a stimulus to move through the array of an anesthetized animal that deflected some whiskers but not others, or deflected the whiskers in a particular sequence or temporal pattern. Fifth, the morphological model could ultimately Journal of Experimental Biology • Accepted manuscript enable development of simulations to examine the mechanosensory signals generated during natural tactile exploration, enabling the development of synthetic datasets that close the loop between the animal's sensorimotor structures, the environment, and the nervous system.
A final motivation for the present work is to quantitatively compare the facial and vibrissal morphologies of mouse and rat. As elaborated in the Discussion, there are several reasons to expect these morphologies to be similar, but equally good reasons to expect them to differ. The quantitative comparison performed in the present work allows us to suggest that the rat is "scaled up" from its most recent common ancestor with the mouse.
To summarize, the present work quantifying the facial features of Mus musculus was undertaken in order to fill gaps across multiple fields. The work develops an important tool that we anticipate will be used widely in the study of morphometrics, animal behavior, multimodal integration, and neurophysiology. We anticipate the model will enable the creation of synthetic datasets of sensory acquisition behaviors that help close the loop between the brain, body, and environment (Bolaños et al., 2021;Hobbs et al., 2016;Ramalingasetty et al., 2021;Weiler et al., 2022;Zhuang et al., 2017;Zweifel et al., 2021)

Methods
The Animal Care and Use Committee of Northwestern University approved all procedures in advance.
A limitation of the present study is its exclusive use of male mice, while Belli et al.
(2018) used only female rats. We expect that differences between individuals of the same species will likely be confined to size rather than the relative locations of facial structures or whisker geometry. We also expect that any potential differences between males and females of the same species and similar size will be smaller than the spatial resolution of the Microscribe™ scanning approach. Future work should use animals of both sexes to confirm these expectations.
Surgery and anesthesia: Animals were anesthetized with a mixture of ketamine/xylazine/acepromazine injected intraperitoneally (100 mg/kg ketamine hydrochloride, 10 mg/kg xylazine hydrochloride, and 3 mg/kg acepromazine maleate in a saline vehicle). Every fifteen minutes the toe pinch withdraw reflex was assessed and booster doses of anesthetic were given if needed to maintain deep anesthesia. The animal was secured with bite block and ear bars in a stereotaxic apparatus and placed on a heating pad. Next, to ensure that no vibrissae touched any surface, we performed a surgery to allow head fixation without the bite block and ear bars. An incision was made along the midline of the scalp, and skull screws were inserted to form a triangle: one screw was placed in the frontal bone, and one screw in each temporal bone. A bridge of methyl methacrylate (dental acrylic) was constructed between the skull screws and the arm of the stereotaxic unit. We were careful to ensure that the skull landmarks lambda and bregma remained exposed (Paxinos et al., 1985). The bite block and ear bars were removed after the acrylic cured to ensure that no object touched any vibrissa or any part of the animal's face.
Because all measurements were made in the anesthetized animal, the locations of all sensory organs including the whiskers were obtained when all muscles were relaxed.
The measured angles at which the whiskers emerge from the face are thus by definition the "resting" angles.
Microscribe™ data acquisition: The Microscribe™ is a passively movable mechanical arm with five degrees of freedom. At the start of an experiment, a probe of known length Journal of Experimental Biology • Accepted manuscript is placed on the end of the arm and an origin (0,0,0) is defined. The user moves the probe tip to a chosen location, recorded as (x, y, z) coordinates relative to the origin. For these experiments we used a custom Microscribe™ probe, consisting of a 500 micron diameter tungsten bipolar electrode held in a pin-vice that had been machined to thread into the Microscribe™. We followed manufacturer's instructions to re-calibrate the custom tip before collecting data from each animal.
The experimenter held the Microscribe™ and sequentially placed the probe tip at a series of points on each of the anatomical structures of interest. The 3D coordinates (x, y, z) at the position of the Microscribe™ tip were recorded by pressing a trigger button.
We digitized multiple facial features for each of the eight mice, including the skull features lambda and bregma, points corresponding to the corners and contours of the nostrils, eyes, mouth, rostrum, incisors, and pinnae, the basepoint of each whisker, and 4 -60 points along each whisker.
On average, 30 whiskers could be measured on each mouse mystacial pad, compared to the 31 whiskers measured for rats in a previous study (Belli et al., 2018). This difference is due to the small size and delicate nature of rostral mouse whiskers, and does not necessarily reflect a difference in number of macrovibrissae between the two species. Because sling muscles were not identified in the present work, it was not possible to definitively determine the transition from macro-to micro-vibrissae.
2D whisker scanning: After data collection with the Microscribe™ was finished, animals were euthanized with an overdose of the ketamine-xylazine-acepromazine combination and subsequent decapitation. Whiskers from seven of the eight mice (mice 1-7) were cut at their base, using forceps and microscissors, and stored for one or two days in folded rectangles of aluminum foil, which were then stored in plastic bags. Extreme care was taken to cut the whiskers exactly at the base, pushing carefully and delicately on the skin to ensure the full arclength of the whisker was obtained. In initial experiments, we tried plucking each whisker, but found that the top of the follicle could not be consistently identified using shape, diameter, or color changes. For example, some Journal of Experimental Biology • Accepted manuscript plucked whiskers tended to pull out more cleanly from the follicle than others, increasing their apparent length by a few mm. By using microscissors to cut carefully at the skin surface, we captured the full arclength of the whisker just as it emerged from the follicle.
After cutting, whiskers were scanned on a flatbed scanner (Epson Perfection 4180 Photo) at a resolution of 2,400 dpi (10.6 microns per pixel), along with a ruler for calibration (1mm resolution). Whiskers from the eighth mouse were accidentally lost.

Definition and quantification of whisker morphological parameters
Following convention (Dörfl, 1982), approximately 30 large whiskers on each mystacial pad were taken to be macrovibrissae, including whiskers up to column 7 in rows that had whiskers in these positions. More rostral whiskers became increasingly short and fine so that visual identification and mechanical manipulation were not feasible.
Whisker array morphology was quantified using a total of eight parameters, all of which have been defined in previous work (Belli et al., 2018;Knutsen et al., 2008;Luo and Hartmann, 2023;Towal et al., 2011). The arclength (S) and the intrinsic curvature coefficient (A) describe 2D whisker geometry. The term "arclength" is defined as the total, curved length of the whisker, to distinguish it from the chord length that describes the straight line between the whisker's base and its tip. The 3D coordinates of the whisker basepoints are denoted as r bp , θ bp , and φ bp , and the orientation at which the whiskers emerge from the pad are described by the three angles θ w , φ w , ζ w .
Quantifying whisker arclength and the intrinsic curvature coefficient: The 2D images of whiskers from seven mice (mice 1-7) were imported into MATLAB™ to measure arclength (S) and curvature coefficient (A). . Standard semi-automated image processing techniques were used to create a trace along the whisker length. This trace was then smoothed in the y-dimension using a 20-pixel moving average filter and whisker arclength S was calculated by summing the lengths of the segments between the data points. Measurement error was estimated as two pixels on each end of the whisker, or ~10.6 microns per pixel x 2 pixels per endpoint x 2 endpoints = ~42.4 microns.

Journal of Experimental Biology • Accepted manuscript
Previous studies have shown that the 2D shape of a rat whisker can be approximated by the parabola y = Ax 2 (Belli et al., 2018;Knutsen et al., 2008;Towal et al., 2011). The coefficient A is called the "intrinsic curvature coefficient," or simply "curvature coefficient." As will be shown in Results, we found that a quadratic fit was also a good approximation for whisker curvature in mice.
We determined the value for A following the steps described in a previous study (Belli et al., 2018). The whisker was first oriented concave up, with its basepoint at the origin (0,0) and initial portion of the whisker aligned with the x-axis such that the majority of the whisker lay in the first quadrant. Because previous studies have indicated that only the proximal ~65% of the whisker remains planar (Knutsen et al., 2008;Towal et al., 2011), the arclength of the whisker was truncated to 65%.
Although a detailed analysis of whisker intrinsic curvature lies outside the scope of the present work, a recent study reviewed various methods for its quantification (Luo and Hartmann, 2023). The study examines the whiskers of ten different species, including Mus musculus, and the analysis includes the full whisker arclength, instead of limiting to the proximal 65%. For full-length Mus musculus whiskers, a fractional exponent fit between quadratic and cubic was found to be the most accurate (Luo and Hartmann, 2023).
An iterative optimization routine that minimized the difference between the smoothed traced whisker and the fit curve was used to determine the fraction of the whisker that should be aligned with the x-axis. We tested values between 1% and 30% of the whisker length. For each iteration, the curve y=Ax 2 was fit to the truncated whisker, and the mean squared error (MSE) between the fit curve and whisker trace was minimized.
On average across all whiskers, using 8% of the proximal whisker to align with the xaxis generated the smallest MSE. This percentage was therefore used for alignment for all whiskers. Choice of coordinate system: Establishing the reference frame in which to define the whisker pad parameters (r bp , θ bp , φ bp , θ w , φ w , ζ w ) required us first to choose an origin and a horizontal plane in which to orient the mouse's head. we defined the "average whisker row plane" as the horizontal (x-y) plane. A plane was fit to the basepoints of each of the five whisker rows using least squares, and the mean of these five planes defined the average whisker row plane (Fig. 1B).

Journal of Experimental Biology • Accepted manuscript
With the x-y plane established as the average row plane, the x-axis was defined as the line connecting the centroids of the left and right arrays with the left side of the animal negative. The y-axis was defined to be orthogonal to the x-axis, with the positive y-axis pointing rostrally. Lastly, the axis perpendicular to the x-y plane was the z-axis (Fig. 1C).

Journal of Experimental Biology • Accepted manuscript
Quantifying 3D coordinates of the whisker basepoints: Having chosen an origin and horizontal plane, the 3D coordinates of all recorded points on the animal's head, including the whisker basepoints, could be determined.
We used an intuitive coordinate system that exploits bilateral symmetry of the animal by mirroring the left array across the midline. All equations and figures in the present work are defined in terms of right-sided whisker arrays. These "array-centered" axis conventions apply equally well to both sides of the head, with the caveat that the coordinates on the left side of the animal no longer follow the right-hand rule.
The three spherical coordinates for basepoints (r bp , θ bp , φ bp ) are illustrated in Fig. 1D.
The radius (r bp ) is defined as the straight-line distance between the basepoint and the origin, and is closely related to the size of the mouse's mystacial pads. The coordinate θ bp describes the rostro-caudal location of the whisker basepoint along the positive xaxis. If the whisker is rostral [caudal] to the x-z coronal plane, it has a positive [negative] value of θ bp . The dorsoventral location of the basepoints with respect to the x-y horizontal plane is described by the coordinate φ bp . If the basepoint location is dorsal [ventral] to this horizontal plane it has a positive [negative] value of φ bp .
Quantifying the 3D angles at which the whiskers emerge from the face: The "angles of emergence" describe the orientation of the whiskers as they emerge from the mystacial pad at their basepoint locations. In the present work, the angles are defined as Euler angles applicable to the right array.
The angles θ w and φ w both describe the orientation of the approximately linear portion (proximal 8%) of the whisker, near its base; neither of these angles depends on the intrinsic curvature of the whisker. As shown in Fig. 1E, the angle θ w is the angle of emergence in the horizontal plane, and ranges between 0° (whisker points caudally) and 180° (whisker points rostrally). Values of θ w greater than 180° indicate that the whisker points across the rostro-caudal midline of the animal. The angle φ w describes

Journal of Experimental Biology • Accepted manuscript
the elevation of the whisker (Fig. 1F). Values range between ±90°, with 90° indicating that the whisker points dorsally and -90° ventrally.
Because each whisker has an intrinsic curvature, a third angle, ζ w , is required to describe the whisker's twist about its own axis, as illustrated in Fig. 1G. If the whisker is oriented concave forward, it has an angle of ζ w = 90°, while concave backwards is ζ w = -90°, and concave dorsal and ventral are 180° and 0°, respectively.
As indicated earlier, the angles of emergence are defined only for whiskers of the right array. All left side whisker arrays were mirrored across the y-z plane, and figures and equations in the present work are presented in terms of right-sided whisker arrays.
To find the angles of emergence, we performed an optimization using the built-in MATLAB™ function "fmincon." The optimization parameters were θ w , φ w , ζ w , S, and A.
Starting with the whisker point cloud in standard position and orientation, the individual points were sorted by distance from the basepoint. The x, y, z coordinates of all points were smoothed with a moving average filter (window size of five points), and the whisker was then resampled into 500-micron segments and its base translated to the origin. The routine next varied θ w , φ w , ζ w , S, and A to fit an idealized whisker model to the point cloud. The model whisker began with its base at the origin and its proximal portion aligned along the negative y-axis. The model was initially oriented concave down. To match the idealized model with the whisker point cloud, we performed an Euler rotation sequence in the order y-axis (roll, ζ w ), x-axis (pitch, φ w ), z-axis (yaw, θ w ).
These rotations were extrinsic about the global y-x-z axes. The parameters S and A were also varied to best match the idealized model to the point cloud. The optimization minimized the mean sum-squared distance between the whisker point cloud and the points in the idealized whisker model.

Relationship between basepoint coordinates and row and column identity in the mouse
The present work quantifies the morphology of the mouse vibrissal array using the 3D position of the whisker basepoints (θ bp , φ bp , r bp ). To do so, it was first necessary to develop equations that related the basepoint coordinates for each whisker to its row and column position. In all equations, angles are measured in degrees and lengths in mm.
As expected, θ bp was linearly related to column (Col) and φ bp was linearly related to row (Row): In equations 1 and 2, Col ranges between 1 and 7 and Row ranges between 1 (for the A row) and 5 (for the E row). Note that numbering the columns starting with 1 for the "Greek arc" is nontraditional, but follows the nomenclature used in Belli et al., 2018(Belli et al., 2018, and accommodates comparisons with species that do not have an easily identifiable Greek arc.
Equation 1 indicates that θ bp increases from caudal to rostral across the mystacial pad, and equation 2 indicates that φ bp decreases from dorsal to ventral; both of these relationships are shown in Fig. 2A. The quality of both fits is presented in Fig. 2B, while A third coordinate, r bp , is needed to fully describe basepoints in spherical coordinates.

Journal of Experimental Biology • Accepted manuscript
The coordinate r bp is more variable than θ bp or φ bp , reflected by the lower R 2 , partially due to head size differences across individual mice.
We next quantified the relationship between r bp and the basepoint parameters θ bp and φ bp , and found r bp to be linearly related to both variables ( Fig. 2D): r bp = -0.0152 θ bp -0.0165 φ bp + 4.44, Adj. R 2 = 0.551 (eq. 3b) Although equation 3b has a slightly lower R 2 value than equation 3a, it allows comparisons between species with different numbers of rows and columns.
A sense for the quality of fit of equation 3b is given by the dispersion about the identity line in Fig. 2E. Fig. 2F illustrates the variation of r bp across the array, revealing the narrowing of the mouse's snout rostrally and dorsally.

Statistical analysis of morphological parameters and error analysis
We used the method of "forward selection" to incrementally increase both model order identity. We eliminated whiskers greater than two standard deviations above or below the mean. Out of the dataset of 357 scans, 14 outliers (3.92%) for arclength (S) were removed, yielding a total of 343 whiskers. Out of the dataset of 324 scans for curvature coefficient, 12 outliers (3.70%) were eliminated, yielding a total of 312 whiskers for analysis.
We constructed models for S and A as functions of θ bp and φ bp . To avoid overfitting, models were not fit directly to the entire dataset for each parameter. Instead, the best fit model was found for each mouse individually. The best fit models were selected using the following steps.
First, we divided each dataset for S and A (with outliers removed) into seven subgroups by mouse identity. For each of the seven mice, histograms of S and A were found to not be quite normally distributed. We therefore analyzed the data both with and without logtransformations to the S and A coefficients to improve normality.
Second, linear regression models were constructed for each mouse using both the original and the log-transformed data. We tested whether S and A were univariately associated with θ bp or φ bp . Specifically, we tested the null hypothesis that the regression coefficients for θ bp and/or φ bp were equal to zero. In accordance with standard procedures of forward selection model evaluation, if the p-value for the independent variable coefficients was less than or equal to 0.05, we rejected the null hypothesis for the lower order model and proceeded to evaluate a higher order model (square of θ bp and/or square of φ bp ). The hypothesis that the regression coefficients for these higher order terms were equal to zero was then tested. For each quadratic order coefficient in the model, if the p-value was greater than 0.05, the model remained first order. If the pvalue was less than 0.05, then the second order model was selected. We also considered the Akaike Information Criterion (AIC) as an additional metric to avoid overfitting. The higher order model was selected only if the AIC for that model was more than two points lower than the lower order model, and the independent variable Journal of Experimental Biology • Accepted manuscript coefficients were significant. We did not find any model for whiskers from an individual mouse to have independent variables greater than second-order.
Third, if both basepoint parameters (θ bp and φ bp ) were univariately associated to either first or second order with either the original data or log-transformed A or S, both parameters were included in a multivariable linear regression model for each individual mouse. The method of forward selection was again used to introduce the terms as independent variables. We tested the hypothesis that the regression coefficients for θ bp or φ bp were equal to zero. If the p-value for the independent variable coefficients was less than or equal to 0.05, a second order model (square of θ bp and/or square of φ bp ) was fit and we again tested the hypothesis that the regression coefficients for these higher order terms were equal to zero.
We then compared the best-fit simple or multivariable linear regression models across all seven mice, for the original data and log-transformed A or S. An overall model, based on data from all mice together, was fit only after considering the results of the models fit to each individual mouse. A parameter (i.e., θ bp , φ bp , or their squares) was included in the overall model only if it had first appeared as a significant independent variable in at least six of the seven individual mouse models. Each of these parameters was then retained or discarded in the overall model using the same forward selection methods described for the individual models. To avoid overfitting, the order of the combined model was not allowed to exceed the highest order of the six out of seven individual mouse models.
Angles of emergence (θ w , φ w , and ζ w ) and basepoint radius (r bp ) as functions of θ bp and φ bp : Of the 467 whiskers obtained from the eight mice, 461 were of sufficient quality to use in a 3D analysis of basepoints (r bp θ bp φ bp ) and emergence angles (θ w , φ w , ζ w ). To identify outliers for each parameter, the mean and standard deviation were calculated for whiskers grouped by their row and column identity. As before, the value of a given parameter for a particular whisker was eliminated if it was greater than two standard deviations above or below the mean.

Journal of Experimental Biology • Accepted manuscript
Out of the dataset of 461 whiskers, 21 outliers (4.56%) were removed for r bp , 14 outliers We next constructed models for r bp θ w , φ w , and ζ w as functions of θ bp and φ bp . To avoid overfitting, the best fit model for the parameters r bp , θ w , φ w , and ζ w , was found for each of the eight mice individually. We first determined that the parameters r bp , θ w , φ w , and ζ w were all normally distributed for each of the eight mice. Therefore, a log-transformation was not necessary and linear regression models were then constructed for each mouse as follows.
First, for each mouse, we tested whether r bp , θ w , φ w , and ζ w were univariately associated with θ bp and/or φ bp using the forward selection procedure described previously. We did not find any models for whiskers from an individual mouse to have statistically significant independent variables greater than second-order.
Second, if both basepoint parameters (θ bp and φ bp ) were univariately associated to either first or second order with r bp , θ w , φ w , and ζ w , then both parameters were included in a multivariable linear regression model for each individual mouse. Again, independent variables were added to the models using the method of "forward selection." We tested the hypothesis that the regression coefficients for θ bp and/or φ bp were equal to zero. If the p-value for the independent variable coefficients was less than or equal to 0.05, a second order model (square of θ bp and/or square of φ bp ) was fit and we retested the hypothesis that the regression coefficients for these higher order terms was equal to zero. We also considered the AIC to avoid overfitting. The higher order model was selected only if both the independent variable coefficients were significant and the AIC for the higher order model was more than two points lower than the lower order model.

Journal of Experimental Biology • Accepted manuscript
Finally, for each parameter (r bp , θ w , φ w , and ζ w ), the best fit univariate or multivariable function of θ bp and/or φ bp was compared across all eight mice. An overall model, based on data from all mice together, was fit only after taking into account the results of the models fit to each individual mouse. This overall model was fit similarly to the individual models, except that a parameter was included in the final combined model only if it had first appeared as a significant independent variable in at least six of the eight individual mouse models. If a parameter met this criterion it was initially included in the model, and retained or discarded using the same forward selection methods described for the individual models. This error estimate includes any slight eccentricities of the tip, any deviation due to hand tremor of the user, and the intrinsic precision limitations of the device. Note also that the 0.5 mm error does not compound along the arclength of the whisker because each point along the whisker was sampled independently.
As an additional estimate of measurement error, we performed simulations in which we randomly varied all points along the whisker arclength (including the basepoint) by +/-0.5 mm in all three spatial dimensions. We then observed the effect on the 3D position and orientation of the whiskers. We performed these simulations for one rostral and one caudal whisker from each row, for a total of 10 whiskers. The maximum angular deviation of the whisker basepoint location was 0.015 mm in r bp , 1.07° in θ bp , and 0.53° in φ bp , and the maximum angular deviation of the whisker emergence angle was found to be 5.9° in θ w , 4.1° in φ w , and 28.7° in ζ w . The maximum error for ζ w was a true outlier, as the next largest error among the remaining nine whiskers checked was 0.74°.

Digitization of internal skull features
To position the skull and facial features in standard orientation, the (x, y, z) coordinates of the skull and face collected using the Microscribe™ were imported into MATLAB™.
These coordinates were then rotated and translated to match the axis conventions shown in Fig. 1. (5) the lateral corners of the incisors.
Points from the CT scan data were brought into the same reference frame as the Microscribe™ data by aligning a subset of corresponding points between the two datasets. These points included bregma, lambda, and the corners of the incisors. We found the translations and rotations that brought these points into register, and then applied these same translations and rotations to the points for all other features.

Results
The first two sections of Results describe the morphology of the mouse vibrissal array using linear regression, while the last four sections compare the morphology of the mouse and rat. In all equations, angles are measured in degrees, lengths in mm, and curvature in mm -1 . The model of the mouse is available at https://doi.org/10.5281/zenodo.7992354

Two-dimensional shape: arclength and curvature coefficient
The most basic description of two-dimensional (2D) whisker geometry includes arclength (S) and the intrinsic curvature coefficient (A). We quantified these two parameters as functions of the basepoint coordinates θ bp and φ bp .
Whisker arclength is ~11.7 mm near the rostro-caudal center of the array, where θ bp = 0, and changes exponentially from the center. Experimental values for S vary between ~2 and ~30 mm. The quality of the fit for equation 4ab is shown in the plot of actual vs.
predicted values for S (Fig. 3B), and Fig. 3C provides visual intuition for variations in S across the array.
As described in Methods, whisker curvature was quantified using the 2D parameterization y = Ax 2 , fit to the proximal 65% of the whisker. The curvature Journal of Experimental Biology • Accepted manuscript coefficient, A, was best described as a linear function of θ bp, a relationship shown in Fig.   3D: A = 0.00037 θ bp + 0.017, Adj. R 2 = 0.233 (eq. 5) The R 2 for equation 5 is low, due to the high variability in A for the rostral whiskers.
However, equations 4ab and 5 depend only on θ bp , indicating that mouse 2D whisker geometry is relatively consistent within a column. To create Fig. 3G-H, we first identified the full range of A values found experimentally.
We then solved for S using eq. 6 for each value of A larger than the y-intercept (0.012).
When A is smaller than the y-intercept, S is undefined, and for these whiskers S was fixed at the longest observed value. We then used these values for A and S to generate a set of "mock," idealized whiskers that followed the curve y = Ax 2 in standard Cartesian coordinates, where the whisker base was placed at the origin and its initial slope was The analysis shown in Fig. 3G is extended across a more continuous distribution of whiskers in Fig 3H. In this figure, each mock whisker still has a single color (representing its value of A), but whiskers are plotted densely enough to form a continuous color field. The small, colorful left corner indicates that curvature is more variable for shorter whiskers, but that this variability does not cause large tip displacements. In contrast, the right side is mostly dark blue, indicating that changes in tip locations of the longer whiskers are not associated with large differences in A.
The gray regions of Fig. 3H are those in which no whiskers are predicted to lie. This region is found by smoothly connecting the tops of the black arcs, bounding the curvatures for whiskers of given arclengths. However, note that, because A is calculated based only on the proximal 65% of the whisker, some of the actual whisker tips we sampled (31 of 312) would lie within the gray region. Given that the whiskers' distal tips are often damaged, the shape of any given mouse whisker tip will be variable and unpredictable. Therefore, an advantage of the present model is that simulated whiskers more closely reflect a "typical" whisker instead of a particular individual mouse whisker.

Angles of emergence of the whiskers as a function of basepoint coordinates
As shown in Fig. 4A, the horizontal angle of emergence, θ w , was best described as a linear function of θ bp : Journal of Experimental Biology • Accepted manuscript θ w = 1.04 θ bp + 82.5, Adj. R 2 = 0.544 (eq. 7a) Although equation 7a represents the best relationship found while avoiding overfitting, the R 2 is relatively low. Moreover, plotting actual vs. predicted values (Fig. 4B) reveals moderately high dispersion about the identity line, limiting the predictive value of equation 7a.
However, values for θ w were particularly variable for whiskers in the A row (Fig. 4B, blue points). After omitting these, the equation for θ w is: θ w = 0.922 θ bp + 85.1, Adj. R 2 = 0.739 (eq. 7b) The elevation angle, φ w , was found to vary linearly only with φ bp : φ w = 1.00 φ bp + 10.2, Adj. R 2 = 0.76 (eq. 8) This relationship is shown in Fig. 4C, and Fig. 4D shows the quality of the fits by comparing predicted versus actual values.
The two colormaps of predicted θ w and φ w values across the array (Fig. 4E) confirm the visual intuition that more caudal whiskers emerge at more caudal angles than the rostral whiskers, while whiskers in more dorsal rows emerge at more elevated angles than whiskers in more ventral rows.
The angle ζ w , describes the twist of the whisker about its own axis. This angle can be computed accurately only if the whisker curves measurably away from its base.
Therefore, whiskers shorter than 8 mm were excluded from the fits for ζ w . Even after omitting these whiskers, the best fit for the angle ζ w was poor: ζ w = 0.71 φ bp + 36.1, Adj. R 2 = 0.13 (eq. 9a) Journal of Experimental Biology • Accepted manuscript To explore the high variability in ζ w we plotted it as a function of row and column (Fig.   4G), revealing that most variability comes from the A-row and the most rostral columns within each row. Models for each row individually were moderately to minimally predictive, as indicated by R 2 values below 0.25. The p-value (>0.05) for the linear regression model for row A indicated that the values of ζ w for these whiskers were not well-predicted by rostro-caudal position. In contrast, models ζ w for whiskers in rows B -E were well predicted by rostro-caudal position, as indicated by significant f-tests (p < 0.005). These results suggest that a substantially improved model for ζ w might be obtained after excluding the A-row and the most rostral whiskers. We therefore excluded these whiskers and re-fit an overall model to the remaining 309 whiskers. As predicted, the fit improves significantly: ζ w = 0.620θ bp + 1.00φ bp + 43.0, Adj. R 2 = 0.34 (eq. 9b) The reduced dataset, along with predicted values, is shown in Fig. 4H, and the quality of the model is illustrated in Fig. 4I. ζ w RAT = 0.876θ bp RAT + 0.845φ bp RAT + 37.9, Adj. R 2 = 0.42 (eq. 10) The rat and mouse models are not statistically different, as their 95% confidence intervals overlap for both predictors. It is also possible that a different choice for the horizontal plane would reveal a stronger relationship between ζ w and θ bp and/or φ bp . with Row. However, when forced to a linear fit, the coefficients for the rat are not statistically different from those of the mouse, as indicated by overlap in the 95% confidence intervals of the two models (mouse interval [-16.28 -17.29], rat interval [-16.81 -17.92]). These models result in predicted values with an average difference between species of less than 2°.

Comparisons between the mouse and rat
An important goal of the present work was to develop equations for whisker morphology and arrangement that can be compared across species, regardless of whether the species has an "orderly" or "disorderly" whisker array. Rows 3 -8 of Table 1 compare the morphology of mouse and rat whisker arrays using θ bp and φ bp as independent variables. These types of comparisons could be performed for any whiskered animal; the basepoint coordinates can be measured independently of row and column position (rows 3 -8 do not depend on rows 1 and 2).
The parameter r bp is compared in the third row of Table 1 and in Fig. 5B. Values for r bp depend strongly on head size, and are, on average, 56.1% larger in the rat compared to the mouse, with a range of 36.1% for the D5 whisker (the most similar) and 77.8% for the A1 whisker (the least similar). The larger r bp values for rat were statistically significant with p < 0.001 (independent samples t-test). This difference in size contrasts with the remarkable similarity between the two species for θ bp and φ bp by whisker identity (rows 1-2 of Table 1).
To obtain further intuition for the differences in scale across species, Fig. 5C shows the whisker basepoint locations for the two animals in the y-z plane. We found that, for the mouse, the total area occupied by whisker basepoints was 50.90 ± 16.86 mm 2 , for a density of 0.66 ± 0.44 whiskers/mm 2 , while the rat has a whisker area of 139.35 ± 34.6 mm 2 , yielding a density of 0.24 ± 0.05 whiskers/mm 2 . When normalized for the obvious Journal of Experimental Biology • Accepted manuscript size difference, however, mouse and rat have approximately the same number of whiskers in the same angular region (Fig. 5D).

Comparisons of 2D parameters (arclength, curvature, base diameter) for mouse and rat
We compared the relationships that describe 2D whisker geometry for mouse and rat (rows 4 and 5 of Table 1). Fig. 6A shows S as a function of θ bp , along with the fits given by equations in the fourth and fifth rows of Table 1. Whisker arclength has a larger range for the rat than mouse, reflected by the steeper exponential decay. The longest rat whisker was approximately twice as long as the longest mouse whisker (60 mm vs. 30 mm). highlighting increased variability for more rostral (shorter) whiskers. The coefficient does not differ significantly between the two species, as reflected in the overlapping confidence intervals. Fig. 6C plots the curvature coefficient as a function of arclength, and the associated upper bounds, for mouse and rat. The upper bound on the rat is greater than that on the mouse for most whiskers, but there is considerable overlap.
To complete the analysis of 2D parameters, we used datasets from Hires et al., 2016 (mouse) and Belli et al, 2018 (rat) to compare the base diameters of select mouse and rat whiskers (Fig. 6D). A total of 28 mouse whiskers and 146 rat whiskers were analyzed (see Methods). On average, rat whiskers have base diameters ~2.35x larger than mouse whiskers.

Angles of emergence of the whiskers as a function of basepoint coordinates for mouse and rat
We next compared the 3D angles of emergence (θ w , φ w , ζ w ) between mouse and rat (Table 1, rows 6, 7, and 8.)

Journal of Experimental Biology • Accepted manuscript
Horizontal Angle: As shown in Fig. 7A, mouse whiskers tend to emerge from the cheek pointing more rostrally than those of the rat, especially the rostral whiskers. The mean value for θ w in columns 6 and 7 of the mouse is 120°, compared to 97. 0° in the rat. In contrast, θ w is more similar between the mouse and rat for caudal whiskers: the mean value of θ w for columns 1 and 2 (combined) is 59.3° for mouse and 51.0° for rat. Also, rat θ w is a function of both θ bp and φ bp , while mouse θ w depends only on θ bp . These differences likely reflect the small size of mice, making it difficult to capture as much nuance in the model.
Elevation angle: Comparing φ w between rats and mice (Fig. 7B) shows that mouse whiskers lie at slightly higher elevation than rat whiskers, as indicated by the almost parallel lines offset by ~4°.
Twist angle: Although the relationship for ζ w in the mouse is not strong (equation 9b), it allows approximate comparison to the rat (Fig. 7C), with greatest differences found for the dorsal whiskers. In general, ζ w is much more variable than the other angles for both species, in part due to limited measurement resolution.
Using equations 1-9, the full 3D model of the average mouse vibrissal array is plotted in the left panel of Fig. 7D, and compared with the average rat array (overlaid) in the right panel of Fig.7D.

Discussion
Because different animal species have different numbers and arrangements of whiskers (Muchlinski, 2010;Muchlinski et al., 2013) row and column position do not lend themselves to developing equations that can be used to compare whisker geometry across species. To overcome this problem, we developed an approach that uses 3D whisker basepoint positions to quantify whisker properties including arclength, curvature, and emergence angles. Facial features can then be placed in this same coordinate system. Using this approach, we quantified and compared whisker and facial morphology across mice and rats.

Variations in whisker arclength and intrinsic curvature
Consistent with previous studies on rats (Brecht et al., 1997;Pockock, 1914;Towal et al., 2011), the present work shows that whisker arclength increases exponentially from rostral to caudal across the mouse's face. Rat whiskers, which range in arclength from 10 -60 mm, are approximately twice as long as mouse whiskers (Fig. 6A), in line with the ~2x linear scaling of other facial features (Fig. 8). Data from previous studies (Belli et al., 2018;Hires et al., 2016) indicate that the base diameters of rat and mouse whiskers also differ by approximately a factor of two (Fig. 6D). The 2x linear scaling in both arclength and base diameter means that the volume (and hence mass) of rat whiskers is expected to be ~8x greater than that of the mouse, comparable to the ~10x body mass difference.
The intrinsic curvature of rat and mouse whiskers is described by a quadratic function, with a strict upper bound on maximum curvature and thus tip position (Figs 3G and 6C).
Rostral whiskers exhibit the largest variability in curvature coefficient, but that does not mean that their shapes are more variable. When a whisker is short, variations in its curvature coefficient have only a small effect on its tip position (Fig. 3G). Therefore, the coefficient of more rostral whiskers may not be under as strong selective pressure as the coefficient for longer whiskers.

Whisker orientations at rest
The present work measured the locations of sensory structures in the anesthetized animal, so whisker emergence angles are the "resting" angles. Both mice and rats move their whiskers through large angles during whisking behavior, with primary motion in the rostro-caudal direction. The horizontal angle θ w can thus be considered an initialization parameter for simulations of whisking behavior, rather than a fixed parameter.
For both mice and rats the elevation angle  w is constant within a row (it varies only with  bp ), and the relationship between  w and  bp has one of the highest correlation coefficients (Table 1). We suggest that constant elevation angle ( w ) within each row simplifies the problem of localizing the height of an object in head-centered coordinates.

Journal of Experimental Biology • Accepted manuscript
Whisking behavior continuously changes the azimuthal angle (rostral-caudal) but the elevation angle (dorsal-ventral) remains relatively fixed. We predict that the singlevariable, linear relationship between  w and  bp is a characteristic feature of whiskerbased tactile sensing that will generalize across all or most species.
In contrast to the single-variable relationship between  w and  bp , the twist (ζ w ) of the whisker about its own axis varies diagonally, and depends on both θ bp and  bp . In addition, the fit quality is poor, mostly due to measurement error. However, it is also possible that choosing a different head pitch to define the horizontal plane could reveal a stronger linear relationship. The diagonal orientation gradient could be particularly useful for determining the direction of a stimulus moving across the array. Stimuli such as air or water, which interact with many stationary whiskers simultaneously, will encounter different gradients of whisker twist depending on the animal's head pitch.
These gradients could permit the animal to infer stimulus direction across a population of responsive neurons (Quist and Hartmann, 2012;Yu et al., 2016b).
Reasons to expect that the facial and vibrissal morphologies of mice and rats will be similar, and reasons to expect that they will be different There are several reasons to expect rats and mice to have similar facial and vibrissal morphologies. Though rats and mice diverged earlier than did humans and chimps (~10mya compared to ~8mya (Patterson et al., 2006) (Steppan et al., 2004 they remain so physically similar that many non-experts confuse the two species. Both rats and mice are small, terrestrial, burrowing, cathemeral or crepuscular omnivores. Both species occupy a niche that relies on quick reproduction and on quantity rather than quality of offspring to maintain population (Berry and Bronson, 1992;Feng and Himsworth, 2013), though mice emphasize this strategy slightly more than rats (Berry and Bronson, 1992).
The two species also retain many less obvious adaptations in common, such as longitudinal chewing with interlocking molar cusps, probably inherited from an insectivorous ancestor (Lazzari et al., 2008). Most importantly for the question of whisker morphology, both rats and mice actively tap and brush their whiskers against objects during tactile exploratory behaviors.
These motions, called "whisking," help the animal determine object location, size, orientation,
However, there are some distinct differences between mice and rats that could produce differing selective pressures on facial and vibrissal morphology. Rats weigh about ten times more than mice, conferring a survival advantage given that small predators are less willing to take larger prey (Parsons et al., 2018). Though both species whisk, rats move their whiskers at average frequencies between 5-15 Hz while mice whisk faster, between 10 and 25 Hz (Jin et al., 2004;Welker, 1964) Selective pressure for rat and mouse morphological divergence could also emerge from differences in diet, as rats lack a gallbladder while mice retain theirs (McMaster, 1922), possibly assisting the mouse's relative emphasis on insectivory (Shiels et al., 2013).
Though rats eat fewer insects, they are an occasional predator of mice (Yoshimura and Ueki, 1981). More evidence that the two species have specialized niches comes from their different colonization patterns. While the mouse spread across the Old World roughly following the expansion of human farming during the Bronze age (Auffray et al., 2008), the rat spread out of Asia much later, in the 18 th century, after humans had already established urban environments.
Finally, the whisker systems of mice and rats also contain notable differences in the barrel cortexbarrels in rat cortex have "cell-dense" centers compared to barrels with "cell-sparse" centers in mice, a difference the arises in rats during postnatal development (Rice et al., 1985). Additionally, inter-barrel septa are much less prominent in mice compared to rats (Rice et al., 1985).

Rat facial sensory structures may be "scaled up" versions of mouse facial sensory structures
The present results support the idea that the larger size of rats compared to mice is a derived (apomorphic) trait. After scaling, rat and mouse have extremely similar whisker array and facial feature morphology (Figs. 8, 9; Table 1). Both species exhibit strong linear relationships between θ bp and column and between r bp and φ bp . In addition, plotting whisker basepoint location as a function of φ bp and θ bp , which normalizes whisker location to head size, reveals that mouse and rat whisker basepoints occupy strikingly similar positions. One difference between species is that the relationships between φ bp and row, and between r bp and θ bp , are linear for mice but quadratic for rats.
These results indicate that mice and rats have snouts with similar mediolateral curvature but that the rat's snout curves slightly more dorsoventrally.
Thus, although the rat has a slightly more elongated face, consistent with the tendency for larger animals to have longer faces (Cardini and Polly, 2013), the most prominent difference between the two species is simply scale. These results also agree with findings that the molar-to-mandible ratio scales linearly between mice and rats (Cai et al., 2007). Notably, the bilateral (but not rostral-caudal) distance between the ears is disproportionately large for the mouse, perhaps helping ensure larger interaural differences.
At the same time, other studies indicate that the common ancestor of mice and rats is likely to have been the same scale as modern mice. For example, divergence in the mouse/rat lineage has been investigated using fossil molars from a probable common ancestor of mice and rats (Antemus chinjinensis), as well as later extinct murines (Kimura et al., 2015). These fossils indicate that Antemus molars are on the same scale as modern mouse molars. Given the essentially linear scaling between extant mice and rats found in the present work, we suggest that Antemus is likely to have had a similar facial feature arrangement, and that the facial morphology of the rat can be thought of as a "scaled up" version of the mouse.
We anticipate that the present work will allow simulation of the multimodal sensory data that the mouse acquires during natural exploratory behaviors, similar to other poseestimation or body-dynamics software (Bolaños et al., 2021;Ramalingasetty et al., 2021;Zweifel et al., 2021). After adding multiple sources of variability and noise, these simulations can help establish rich, naturalistic stimulus datasets for use in neurophysiological experiments.