Tapered whiskers are required for active tactile sensation
Identifieur interne : 002545 ( Pmc/Curation ); précédent : 002544; suivant : 002546Tapered whiskers are required for active tactile sensation
Auteurs : Samuel Andrew Hires [États-Unis] ; Lorenz Pammer [États-Unis, Allemagne] ; Karel Svoboda [États-Unis] ; David Golomb [États-Unis, Israël]Source :
- eLife [ 2050-084X ] ; 2013.
Abstract
Many mammals forage and burrow in dark constrained spaces. Touch through facial whiskers is important during these activities, but the close quarters makes whisker deployment challenging. The diverse shapes of facial whiskers reflect distinct ecological niches. Rodent whiskers are conical, often with a remarkably linear taper. Here we use theoretical and experimental methods to analyze interactions of mouse whiskers with objects. When pushed into objects, conical whiskers suddenly slip at a critical angle. In contrast, cylindrical whiskers do not slip for biologically plausible movements. Conical whiskers sweep across objects and textures in characteristic sequences of brief sticks and slips, which provide information about the tactile world. In contrast, cylindrical whiskers stick and remain stuck, even when sweeping across fine textures. Thus the conical whisker structure is adaptive for sensor mobility in constrained environments and in feature extraction during active haptic exploration of objects and surfaces.
Url:
DOI: 10.7554/eLife.01350
PubMed: 24252879
PubMed Central: 3828597
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uniqKey="Svoboda K" first="Karel" last="Svoboda">Karel Svoboda</name><affiliation wicri:level="1"><nlm:aff id="aff1"><institution>Janelia Farm Research Campus, Howard Hughes Medical Institute</institution>,<addr-line>Ashburn</addr-line>,<country>United States</country></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author><author><name sortKey="Golomb, David" sort="Golomb, David" uniqKey="Golomb D" first="David" last="Golomb">David Golomb</name><affiliation wicri:level="1"><nlm:aff id="aff1"><institution>Janelia Farm Research Campus, Howard Hughes Medical Institute</institution>,<addr-line>Ashburn</addr-line>,<country>United States</country></nlm:aff><country xml:lang="fr">États-Unis</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation><affiliation wicri:level="1"><nlm:aff id="aff3"><addr-line>Department of Physiology and Cell Biology</addr-line>,<institution>Ben Gurion University</institution>,<addr-line>Be’er-Sheva</addr-line>,<country>Israel</country></nlm:aff><country xml:lang="fr">Israël</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation><affiliation wicri:level="1"><nlm:aff id="aff4"><addr-line>Zlotowski Center for Neuroscience</addr-line>,<institution>Ben Gurion University</institution>,<addr-line>Be’er-Sheva</addr-line>,<country>Israel</country></nlm:aff><country xml:lang="fr">Israël</country><wicri:regionArea># see nlm:aff country strict</wicri:regionArea></affiliation></author></analytic><series><title level="j">eLife</title><idno type="eISSN">2050-084X</idno><imprint><date when="2013">2013</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>Many mammals forage and burrow in dark constrained spaces. Touch through facial whiskers is important during these activities, but the close quarters makes whisker deployment challenging. The diverse shapes of facial whiskers reflect distinct ecological niches. Rodent whiskers are conical, often with a remarkably linear taper. Here we use theoretical and experimental methods to analyze interactions of mouse whiskers with objects. When pushed into objects, conical whiskers suddenly slip at a critical angle. In contrast, cylindrical whiskers do not slip for biologically plausible movements. Conical whiskers sweep across objects and textures in characteristic sequences of brief sticks and slips, which provide information about the tactile world. In contrast, cylindrical whiskers stick and remain stuck, even when sweeping across fine textures. Thus the conical whisker structure is adaptive for sensor mobility in constrained environments and in feature extraction during active haptic exploration of objects and surfaces.</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.001">http://dx.doi.org/10.7554/eLife.01350.001</ext-link></p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Ahissar, E" uniqKey="Ahissar E">E Ahissar</name></author><author><name sortKey="Sosnik, R" uniqKey="Sosnik R">R Sosnik</name></author><author><name sortKey="Bagdasarian, K" uniqKey="Bagdasarian K">K Bagdasarian</name></author><author><name sortKey="Haidarliu, S" uniqKey="Haidarliu S">S Haidarliu</name></author></analytic></biblStruct><biblStruct><analytic><author><name sortKey="Anjum, F" uniqKey="Anjum F">F Anjum</name></author><author><name sortKey="Turni, H" uniqKey="Turni H">H Turni</name></author><author><name sortKey="Mulder, Pg" uniqKey="Mulder P">PG 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<front><journal-meta><journal-id journal-id-type="nlm-ta">eLife</journal-id><journal-id journal-id-type="iso-abbrev">Elife</journal-id><journal-id journal-id-type="hwp">eLife</journal-id><journal-id journal-id-type="publisher-id">eLife</journal-id><journal-title-group><journal-title>eLife</journal-title></journal-title-group><issn pub-type="epub">2050-084X</issn><publisher><publisher-name>eLife Sciences Publications, Ltd</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">24252879</article-id><article-id pub-id-type="pmc">3828597</article-id><article-id pub-id-type="publisher-id">01350</article-id><article-id pub-id-type="doi">10.7554/eLife.01350</article-id><article-categories><subj-group subj-group-type="display-channel"><subject>Research Article</subject></subj-group><subj-group subj-group-type="heading"><subject>Biophysics and Structural Biology</subject></subj-group><subj-group subj-group-type="heading"><subject>Neuroscience</subject></subj-group></article-categories><title-group><article-title>Tapered whiskers are required for active tactile sensation</article-title></title-group><contrib-group><contrib id="author-6964" contrib-type="author"><name><surname>Hires</surname><given-names>Samuel Andrew</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="other" rid="par-1"></xref><xref ref-type="fn" rid="con1"></xref><xref ref-type="fn" rid="conf1"></xref></contrib><contrib id="author-7120" contrib-type="author"><name><surname>Pammer</surname><given-names>Lorenz</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="aff" rid="aff2">2</xref><xref ref-type="other" rid="par-1"></xref><xref ref-type="fn" rid="con3"></xref><xref ref-type="fn" rid="conf1"></xref></contrib><contrib id="author-1328" contrib-type="author"><name><surname>Svoboda</surname><given-names>Karel</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="corresp" rid="cor1">*</xref><xref ref-type="other" rid="par-1"></xref><xref ref-type="fn" rid="con4"></xref><xref ref-type="fn" rid="conf1"></xref></contrib><contrib id="author-7121" contrib-type="author"><name><surname>Golomb</surname><given-names>David</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="aff" rid="aff3">3</xref><xref ref-type="aff" rid="aff4">4</xref><xref ref-type="corresp" rid="cor2">*</xref><xref ref-type="other" rid="par-1"></xref><xref ref-type="other" rid="par-2"></xref><xref ref-type="fn" rid="con2"></xref><xref ref-type="fn" rid="conf1"></xref></contrib><aff id="aff1"><label>1</label><institution>Janelia Farm Research Campus, Howard Hughes Medical Institute</institution>,<addr-line>Ashburn</addr-line>,<country>United States</country></aff><aff id="aff2"><label>2</label><institution>Max Planck Institute for Brain Research</institution>,<addr-line>Frankfurt am Main</addr-line>,<country>Germany</country></aff><aff id="aff3"><label>3</label><addr-line>Department of Physiology and Cell Biology</addr-line>,<institution>Ben Gurion University</institution>,<addr-line>Be’er-Sheva</addr-line>,<country>Israel</country></aff><aff id="aff4"><label>4</label><addr-line>Zlotowski Center for Neuroscience</addr-line>,<institution>Ben Gurion University</institution>,<addr-line>Be’er-Sheva</addr-line>,<country>Israel</country></aff></contrib-group><contrib-group><contrib contrib-type="editor"><name><surname>Tsodyks</surname><given-names>Misha</given-names></name><role>Reviewing editor</role><aff><institution>Weizmann Institute of Science</institution>,<country>Israel</country></aff></contrib></contrib-group><author-notes><corresp id="cor1"><label>*</label>For correspondence: <email>svobodak@janelia.hhmi.org</email> (KS);</corresp><corresp id="cor2"><label>*</label>For correspondence: <email>golomb@bgu.ac.il</email> (DG)</corresp></author-notes><pub-date pub-type="epub"><day>19</day><month>11</month><year>2013</year></pub-date><pub-date pub-type="collection"><year>2013</year></pub-date><volume>2</volume><elocation-id>e01350</elocation-id><history><date date-type="received"><day>08</day><month>8</month><year>2013</year></date><date date-type="accepted"><day>09</day><month>10</month><year>2013</year></date></history><permissions><copyright-statement>Copyright © 2013, Hires et al</copyright-statement><copyright-year>2013</copyright-year><copyright-holder>Hires et al</copyright-holder><license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by/3.0/"><license-p>This article is distributed under the terms of the <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/3.0/">Creative Commons Attribution License</ext-link>, which permits unrestricted use and redistribution provided that the original author and source are credited.</license-p></license></permissions><self-uri content-type="pdf" xlink:type="simple" xlink:href="elife01350.pdf"></self-uri><abstract><p>Many mammals forage and burrow in dark constrained spaces. Touch through facial whiskers is important during these activities, but the close quarters makes whisker deployment challenging. The diverse shapes of facial whiskers reflect distinct ecological niches. Rodent whiskers are conical, often with a remarkably linear taper. Here we use theoretical and experimental methods to analyze interactions of mouse whiskers with objects. When pushed into objects, conical whiskers suddenly slip at a critical angle. In contrast, cylindrical whiskers do not slip for biologically plausible movements. Conical whiskers sweep across objects and textures in characteristic sequences of brief sticks and slips, which provide information about the tactile world. In contrast, cylindrical whiskers stick and remain stuck, even when sweeping across fine textures. Thus the conical whisker structure is adaptive for sensor mobility in constrained environments and in feature extraction during active haptic exploration of objects and surfaces.</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.001">http://dx.doi.org/10.7554/eLife.01350.001</ext-link></p></abstract><abstract abstract-type="executive-summary"><title>eLife digest</title><p>When foraging in dark, confined spaces, mammals use the information gathered by their whiskers to ‘see’ the world around them. Mammalian whiskers come in a variety of shapes and sizes, most likely reflecting the way in which they are used. Rodent whiskers are conical and precisely tapered, whereas some harbor seals have flattened whiskers with wave-like undulations. Human hair is cylindrical.</p><p>Rodents sweep their whiskers back and forth over objects and surfaces without moving their head. They use this process, called whisking, to build up a three-dimensional picture of objects. Whisking allows the rodent to estimate where an object is located, how big it is, and what kind of surface texture it has. Information about surface texture can, for example, help the animal to distinguish a stone from a seed.</p><p>Hires et al. have used theoretical and experimental methods to analyze the interaction of mouse whiskers with objects. The conical shape of a mouse whisker makes the tip thousands of times more flexible than the base. Hires et al. show that this flexibility gradient allows the whiskers to slip past objects close to the face and to move freely across rough surfaces. Cylindrical whiskers, on the other hand, become stuck behind nearby objects and get caught on tiny features in an object’s surface texture.</p><p>Hires et al. conclude that conical whiskers are advantageous in the tight confines of the tunnels that mice live, forage and socialize in, because they are able to gather a more complete sensory picture of their surroundings. The maneuverability of the whiskers also allows the mouse to move their whiskers forwards or backwards when rough tunnel walls are close by. By contrast, the sticking experienced by cylindrical whiskers would lead to ‘blind spots’. In addition to providing insights into the ways that mice interact with their environment, this work could also lead to improvements in the design of the canes used by the visually impaired to navigate human environments.</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.002">http://dx.doi.org/10.7554/eLife.01350.002</ext-link></p></abstract><kwd-group kwd-group-type="author-keywords"><title>Author keywords</title><kwd>Whisker</kwd><kwd>Barrel Cortex</kwd><kwd>Somatosensation</kwd></kwd-group><kwd-group kwd-group-type="research-organism"><title>Research organism</title><kwd>Mouse</kwd></kwd-group><funding-group><award-group id="par-1"><funding-source>Howard Hughes Medical Institute</funding-source><principal-award-recipient><name><surname>Hires</surname><given-names>Samuel Andrew</given-names></name><name><surname>Pammer</surname><given-names>Lorenz</given-names></name><name><surname>Svoboda</surname><given-names>Karel</given-names></name><name><surname>Golomb</surname><given-names>David</given-names></name></principal-award-recipient></award-group><award-group id="par-2"><funding-source>Israel Science Foundation</funding-source><award-id>88/13</award-id><principal-award-recipient><name><surname>Golomb</surname><given-names>David</given-names></name></principal-award-recipient></award-group><funding-statement>The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.</funding-statement></funding-group><custom-meta-group><custom-meta><meta-name>elife-xml-version</meta-name><meta-value>0.7</meta-value></custom-meta><custom-meta specific-use="meta-only"><meta-name>Author impact statement</meta-name><meta-value>The tapered, conical shape of rodent whiskers is essential for sensor mobility in constrained spaces and feature extraction from textured surfaces.</meta-value></custom-meta></custom-meta-group></article-meta></front><body><sec id="s1"><title>Introduction</title><p>Many mammals use facial whiskers for navigation (<xref ref-type="bibr" rid="bib50">Vincent, 1912</xref>; <xref ref-type="bibr" rid="bib15">Dehnhardt et al., 2001</xref>), object localization (<xref ref-type="bibr" rid="bib26">Hutson and Masterton, 1986</xref>; <xref ref-type="bibr" rid="bib30">Knutsen et al., 2006</xref>; <xref ref-type="bibr" rid="bib31">Krupa et al., 2001</xref>; <xref ref-type="bibr" rid="bib33">Mehta et al., 2007</xref>; <xref ref-type="bibr" rid="bib35">O’Connor et al., 2010a</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>), texture discrimination (<xref ref-type="bibr" rid="bib10">Carvell and Simons, 1990</xref>; <xref ref-type="bibr" rid="bib54">Wolfe et al., 2008</xref>; <xref ref-type="bibr" rid="bib11">Chen et al., 2013</xref>), and object recognition (<xref ref-type="bibr" rid="bib2">Anjum et al., 2006</xref>). The shapes of mammalian whiskers are diverse. Rodent whiskers are conical (<xref ref-type="bibr" rid="bib7">Birdwell et al., 2007</xref>; <xref ref-type="bibr" rid="bib53">Williams and Kramer, 2010</xref>; <xref ref-type="bibr" rid="bib40">Quist et al., 2011</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>), whereas sea lion whiskers (<xref ref-type="bibr" rid="bib23">Hanke et al., 2010</xref>) and human hair are approximately cylindrical. Whiskers of harbor seals have elliptical cross-sections with an undulated structure (<xref ref-type="bibr" rid="bib23">Hanke et al., 2010</xref>). Differences in whisker shapes across different species likely reflect differences in how animals use their whiskers. For example, the undulating microstructure of harbor seal whiskers suppresses vibrations triggered by vortices and enhances the seal’s ability to analyze water movements (<xref ref-type="bibr" rid="bib23">Hanke et al., 2010</xref>).</p><p>What could be the advantages of the whisker taper seen in rodents? Rodents sense their surroundings by moving their whiskers over objects with large amplitudes (up to 50° peak–peak) in a rhythmic motion (<xref ref-type="bibr" rid="bib30">Knutsen et al., 2006</xref>; <xref ref-type="bibr" rid="bib35">O’Connor et al., 2010a</xref>; <xref ref-type="bibr" rid="bib36">O’Connor et al., 2013</xref>; <xref ref-type="bibr" rid="bib51">Voigts et al., 2008</xref>). Rodents can localize and recognize objects in three dimensions (<xref ref-type="bibr" rid="bib30">Knutsen et al., 2006</xref>; <xref ref-type="bibr" rid="bib31">Krupa et al., 2001</xref>; <xref ref-type="bibr" rid="bib35">O’Connor et al., 2010a</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>; <xref ref-type="bibr" rid="bib51">Voigts et al., 2008</xref>) and also discriminate subtle differences in surface textures (<xref ref-type="bibr" rid="bib10">Carvell and Simons, 1990</xref>; <xref ref-type="bibr" rid="bib54">Wolfe et al., 2008</xref>) (reviewed in <xref ref-type="bibr" rid="bib17">Diamond, 2010</xref>). These behaviors are based on collisions between whiskers and objects, which cause time-varying forces at the whisker base and excitation of sensory neurons in the follicles (<xref ref-type="bibr" rid="bib56">Zucker and Welker, 1969</xref>; <xref ref-type="bibr" rid="bib46">Szwed et al., 2006</xref>). Whisker mechanics thus couples the tactile world to forces at the whisker base (<xref ref-type="bibr" rid="bib44">Solomon and Hartmann, 2006</xref>; <xref ref-type="bibr" rid="bib7">Birdwell et al., 2007</xref>; <xref ref-type="bibr" rid="bib5">Bagdasarian et al., 2013</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>).</p><p>Rodent whiskers are thin, approximately linear and homogenous elastic cones (<xref ref-type="bibr" rid="bib44">Solomon and Hartmann, 2006</xref>; <xref ref-type="bibr" rid="bib7">Birdwell et al., 2007</xref>; <xref ref-type="bibr" rid="bib53">Williams and Kramer, 2010</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>). As a result of the linear taper, whisker bending stiffness decreases with distance from the face over five orders of magnitude. Behavioral measurements have shown that mice use distance-dependent whisker mechanics as a ruler to estimate object location along the length of the whisker (<xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>).</p><p>Here we used theoretical and experimental methods to analyze the interactions of whiskers with objects. We uncover additional decisive advantages of conical whiskers compared to cylindrical whiskers for tactile exploration. Conical whiskers sweep across textures with informative micromotions, whereas cylindrical whiskers get stuck. The steep increase in flexibility from base to tip of conical whiskers allow rodents to maneuver their sensors past objects with relative ease. Conical whisker shape is thus critical for tactile exploration in confined spaces.</p></sec><sec sec-type="results" id="s2"><title>Results</title><sec id="s2-1"><title>Mechanical model of whiskers interacting with an object</title><p>We modeled rodent whiskers as truncated cones with a cylindrical cross section, base radius <italic>r</italic><sub>base</sub>, tip radius <italic>r</italic><sub>tip</sub>, and length <italic>L</italic><sub>W</sub> (<xref ref-type="bibr" rid="bib27">Ibrahim and Wright, 1975</xref>; <xref ref-type="bibr" rid="bib8">Boubenec et al., 2012</xref>) (<xref ref-type="fig" rid="fig1">Figure 1A</xref>). Whiskers have intrinsic curvature (<xref ref-type="bibr" rid="bib41">Quist and Hartmann, 2012</xref>) and are further deflected by forces that are caused by interactions with objects (<xref ref-type="bibr" rid="bib35">O’Connor et al., 2010a</xref>; <xref ref-type="bibr" rid="bib5">Bagdasarian et al., 2013</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>) (<xref ref-type="fig" rid="fig1">Figure 1B</xref>). In our model, contacts occurred either in the ‘concave backward’ (CB) or ‘concave forward’ (CF) directions (<xref ref-type="fig" rid="fig1">Figure 1C</xref>) (<xref ref-type="bibr" rid="bib41">Quist and Hartmann, 2012</xref>). We quantified contact strength using the push angle <italic>θ</italic><sub>p</sub> (<xref ref-type="bibr" rid="bib41">Quist and Hartmann, 2012</xref>), the angle through which the whisker is rotated into the object (<xref ref-type="fig" rid="fig1">Figure 1D</xref>). By convention, contacts for the CB configuration correspond to <italic>θ</italic><sub>p</sub> > 0, and for the CF configuration to <italic>θ</italic><sub>p</sub> < 0; <italic>θ</italic><sub>p</sub> = 0 defines the angle of initial touch. In all cases whisker movement and bending were limited to the <italic>x</italic>-<italic>y</italic> plane. We computed whisker shape by solving the Euler–Bernoulli beam equation in the quasi-static regime (<xref ref-type="bibr" rid="bib21">Euler, 1744</xref>; <xref ref-type="bibr" rid="bib7">Birdwell et al., 2007</xref>; <xref ref-type="bibr" rid="bib44">Solomon and Hartmann, 2006</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>). The beam equation describing whisker shape was converted to a boundary-value problem formulation (‘Materials and methods’; <xref ref-type="bibr" rid="bib39">Press et al., 1992</xref>), a set of differential equations with defined boundary conditions at the whisker base and at the point of contact with the object. The object was assumed to be a cylindrical pole perpendicular to the plane of motion, as is typically used in object localization experiments (<xref ref-type="bibr" rid="bib30">Knutsen et al., 2006</xref>; <xref ref-type="bibr" rid="bib33">Mehta et al., 2007</xref>; <xref ref-type="bibr" rid="bib35">O’Connor et al., 2010a</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>) (<xref ref-type="fig" rid="fig1">Figure 1B</xref>). The whisker shape at each time was determined by the static solution computed for the time-varying boundary conditions. Using identical methods we also modeled hypothetical cylindrical whiskers.<fig id="fig1" orientation="portrait" position="float"><object-id pub-id-type="doi">10.7554/eLife.01350.003</object-id><label>Figure 1.</label><caption><title>Schematic of the whisker in two dimensions.</title><p>(<bold>A</bold>) The whisker is modeled as a truncated cone of length <italic>L</italic><sub>W</sub>, virtually extended to length <italic>L</italic>. (<bold>B</bold>) The base of the whisker (in the follicle, or attached to a galvo, <xref ref-type="fig" rid="fig4 fig6">Figures 4–6</xref>) is at point (<italic>x</italic><sub>0</sub>, <italic>y</italic><sub>0</sub>) and angle <italic>θ</italic><sub>0</sub>, measured clockwise. The position of a point along the whisker is (<italic>x</italic>(<italic>s</italic>), <italic>y</italic>(<italic>s</italic>)) and its angle with the x-axis is <italic>θ</italic>(<italic>s</italic>). The contacted object is a cylindrical pole with radius <italic>r</italic><sub>pole</sub> centered at (<italic>x</italic><sub>cen</sub>, <italic>y</italic><sub>cen</sub>); the pole and whisker are shown at a magnified scale in the inset on the left. The whisker contacts the object at the point (<italic>x</italic><sub>obj</sub>, <italic>y</italic><sub>obj</sub>) at an angle <italic>θ</italic><sub>obj</sub>. The object distance, <italic>d</italic>, is the distance between (<italic>x</italic><sub>obj</sub>, <italic>y</italic><sub>obj</sub>) and the whisker base. The pole applies a force <inline-formula><mml:math id="inf1"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>F</mml:mi><mml:mo>→</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:math></inline-formula> on the whisker. (<bold>C</bold>) The concave forward (CF, left) and concave backward (CB, right) whisker configurations. Thick black lines, whiskers; solid circle, poles; gray arrows, movement directions. (<bold>D</bold>), Definition of the push angle, <italic>θ</italic><sub>p</sub>, which measures the strain on the whisker imposed by the object. The deflected and undeflected whiskers are shown as black and gray lines respectively. The pole is a dark gray circle. The undeflected whisker is translated and rotated in the plane such that it has the same <italic>x</italic><sub>0</sub>, <italic>y</italic><sub>0</sub> and <italic>θ</italic><sub>0</sub> as the deflected whisker. This generates a virtual undeflected whisker (dashed gray line). A virtual pole (light gray circle) is generated by shifting the real pole such that it will be tangent to the virtual undeflected whisker. In addition, the distance from the contact point of the virtual unbent whisker and the virtual pole, (<italic>x</italic><sub>virtual</sub>, <italic>y</italic><sub>virtual</sub>) and the base (<italic>x</italic><sub>0</sub>, <italic>y</italic><sub>0</sub>) is equal to <italic>d</italic>, the distance between (<italic>x</italic><sub>obj</sub>, <italic>y</italic><sub>obj</sub>) and the base. The angle between the two line segments connecting the base with the real and virtual contact points is <italic>θ</italic><sub>p</sub>.</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.003">http://dx.doi.org/10.7554/eLife.01350.003</ext-link></p></caption><graphic xlink:href="elife01350f001"></graphic></fig></p><p>The boundary-value problem for whisker shape generally has two solutions, one stable and the other unstable (<xref ref-type="fig" rid="fig2">Figure 2A</xref>). During object contact, the whisker shape matches the stable solution since small perturbations from it will decay back to the stable solution (<xref ref-type="bibr" rid="bib45">Strogatz, 1994</xref>). The bending of the stable solution is weaker compared to the unstable solution. As the whisker pushes further into the object it becomes increasingly deflected (<xref ref-type="fig" rid="fig2">Figure 2B</xref>). At the same time the whisker slides along the object and the arclength from the whisker base to the point of contact, <italic>s</italic><sub>obj</sub>, increases until the whisker detaches from the object. We note two qualitatively different types of detachment. First, under some conditions detachment occurs suddenly before the end of the whisker has reached the object, <italic>s</italic><sub>obj</sub> < <italic>L</italic><sub>W</sub>; we refer to this type of detachment as ‘slip-off’ (<xref ref-type="fig" rid="fig2">Figure 2B</xref>). Second, detachment has to occur when the tip reaches the end of the object, <italic>s</italic><sub>obj</sub> = <italic>L</italic><sub>W</sub>; we refer to this type of detachment as ‘pull-off’ (<xref ref-type="fig" rid="fig2">Figure 2C,D</xref>).<fig id="fig2" orientation="portrait" position="float"><object-id pub-id-type="doi">10.7554/eLife.01350.004</object-id><label>Figure 2.</label><caption><title>Interactions between whiskers and an object.</title><p>Solutions of the quasi-static model (<xref ref-type="disp-formula" rid="equ8">Equations 8</xref>–<xref ref-type="disp-formula" rid="equ14">14</xref>) for a conical whisker (<bold>A</bold> and <bold>B</bold>) and a cylindrical whisker (<bold>C</bold> and <bold>D</bold>). The pole is denoted by a gray circle. The resting shape of the whisker is <italic>y</italic> = Ax<sup>2</sup>, where A = 0.02 mm<sup>−1</sup> (<xref ref-type="bibr" rid="bib41">Quist and Hartmann, 2012</xref>), and the whisker touches the pole in the concave backward configuration. (<bold>A</bold>) Two solutions for a conical whisker. For <italic>θ</italic><sub>p</sub> = 10°, there are two solutions for whisker shape, one is stable (solid line) and one is unstable (dashed line). (<bold>B</bold>) Whisker shape at the saddle-node bifurcation (<italic>θ</italic><sub>p</sub> = 15.6°). There is only one solution as the stable and unstable solutions coalesce. The object touches the whisker not at the tip. For any larger <italic>θ</italic><sub>p</sub>, static solutions cease to exist and the whisker slips off the pole. (<bold>C</bold>) Whisker shape for a cylindrical ‘whisker’ and <italic>θ</italic><sub>p</sub> = 10°. Only one solution (stable) exists; the unstable solution is not physical because its computed arclength is longer than <italic>L</italic><sub>w</sub> = 20 mm. (<bold>D</bold>) For <italic>θ</italic><sub>p</sub> = 62.7°, the tip of the cylindrical ‘whisker’ reaches the object. Beyond this value of <italic>θ</italic><sub>p</sub>, the ‘whisker’ is pulled off the object. Parameters for all panels: <italic>L</italic><sub>w</sub> = 20 mm, <italic>r</italic><sub>base</sub>= 30 µm, <italic>r</italic><sub>tip</sub> = 1.5 µm for the conical whisker and 30 µm for the cylindrical ‘whisker’, <italic>d</italic>=15.7 mm, <italic>E</italic> = 3 GPa, <italic>x</italic><sub>0</sub> = 0, <italic>y</italic><sub>0</sub> = 0. The pole has <italic>r</italic><sub>pole</sub>=0.25 mm and its center is located at <italic>x</italic><sub>cen</sub> = 15.13 mm, <italic>y</italic><sub>cen</sub> = 4.29 mm.</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.004">http://dx.doi.org/10.7554/eLife.01350.004</ext-link></p></caption><graphic xlink:href="elife01350f002"></graphic></fig></p><p>Bending of the whisker can be characterized by the angle of the whisker at the point of object contact, <italic>θ</italic><sub>obj</sub> (<xref ref-type="fig" rid="fig1">Figure 1B</xref>). The whisker first touches the pole at <italic>θ</italic><sub>p</sub> = 0 (<xref ref-type="fig" rid="fig3">Figure 3A–D</xref>, open circles). As the whisker pushes into the pole (|<italic>θ</italic><sub>p</sub>| > 0), <italic>θ</italic><sub>obj</sub> changes monotonically (<xref ref-type="fig" rid="fig3">Figure 3B</xref>). In the CB configuration, the whisker bends and its shape becomes more ‘concave backwards’. The force <italic>F</italic> acting on the whisker increases as more elastic energy is stored in the whisker (<xref ref-type="fig" rid="fig3">Figure 3C</xref>); <italic>s</italic><sub>obj</sub>, also increases (<xref ref-type="fig" rid="fig3">Figure 3D</xref>). At a critical <italic>θ</italic><sub>p</sub>, the two solutions (solid lines, dashed lines) coalesce and disappear at a saddle-node bifurcation (SNB) (<xref ref-type="bibr" rid="bib45">Strogatz, 1994</xref>) (<xref ref-type="fig" rid="fig3">Figure 3B–D</xref>, solid circles). No solution exists above this critical <italic>θ</italic><sub>p</sub> value, which also corresponds to a critical <italic>s</italic><sub>obj</sub> < <italic>L</italic><sub>W</sub>. The whisker slips suddenly and rapidly past the pole. The saddle-node bifurcation corresponds to slip-off. In the CF configuration, <italic>s</italic><sub>obj</sub> first decreases as |<italic>θ</italic><sub>p</sub>| increases from 0, because touch forces straighten the whisker (<xref ref-type="fig" rid="fig3">Figure 3D</xref>). With further increases in <italic>θ</italic><sub>p</sub>, the whisker bends in the other direction and <italic>s</italic><sub>obj</sub> increases.<fig id="fig3" orientation="portrait" position="float"><object-id pub-id-type="doi">10.7554/eLife.01350.005</object-id><label>Figure 3.</label><caption><title>Analysis of conical (<bold>A</bold>–<bold>E</bold>) and cylindrical whiskers (<bold>F</bold>–<bold>J</bold>) pushing into a pole.</title><p>(<bold>A</bold>) Schematic of a conical whisker. Parameters for panels (<bold>B</bold>–<bold>E</bold>): <italic>L</italic><sub>w</sub> = 20 mm, <italic>r</italic><sub>base</sub>= 30 µm, <italic>r</italic><sub>tip</sub> = 1.5 µm, <italic>x</italic><sub>0</sub> = 0, <italic>y</italic><sub>0</sub> = 0, <italic>r</italic><sub>pole</sub> = 0.25 mm, <italic>E</italic> = 3 GPa. The equation of the undeflected whisker is <italic>y</italic> = Ax<sup>2</sup> where A = 0.02 mm<sup>−1</sup> (<xref ref-type="bibr" rid="bib41">Quist and Hartmann, 2012</xref>). For CB configurations, <italic>x</italic><sub>cen</sub> = 15.13 mm, <italic>y</italic><sub>cen</sub> = 4.29 mm; for CF configurations, <italic>x</italic><sub>cen</sub> = 14.87 mm, <italic>y</italic><sub>cen</sub> = 4.71 mm. Positive and negative values of <italic>θ</italic><sub>p</sub> correspond to CB and CF configurations respectively. (<bold>B</bold>) <italic>θ</italic><sub>obj</sub> as a function of <italic>θ</italic><sub>p</sub>. Left, concave forward (CF); right, concave backward (CB). Solid lines, stable solutions; dashed lines, unstable solutions (<xref ref-type="disp-formula" rid="equ8">Equations 8</xref>–<xref ref-type="disp-formula" rid="equ14">14</xref>). Solid circles denote saddle-node bifurcations (SNB). (<bold>C</bold>) Force <italic>F</italic> as a function of <italic>θ</italic><sub>p</sub>. (<bold>D</bold>) Location of object contact along the whisker arc, <italic>s</italic><sub>obj</sub>, as a function of <italic>θ</italic><sub>p</sub>. Arrows correspond to <xref ref-type="fig" rid="fig2">Figure 2A</xref> (a) and <xref ref-type="fig" rid="fig2">Figure 2B</xref> (b, SNB). (<bold>E</bold>) The detachment curve in the <italic>θ</italic><sub>p</sub>−<italic>d</italic> plane bounds the parameter regime with a stable solution for a whisker contacting an object. Black lines represent the points when the stable solution coalesces with an unstable solution and disappears via a saddle-node bifurcation (slip-offs). Blue line represents the points where the whisker is pulled off because the tip has reached the object, <italic>s</italic><sub>obj</sub> = <italic>L</italic><sub>W</sub> (pull-offs). (<bold>F</bold>) Schematic of a cylindrical whisker. Parameters as for conical whisker, except that <italic>L</italic><sub>w</sub> = 20 mm, <italic>r</italic><sub>base</sub>= <italic>r</italic><sub>tip</sub> = 30 µm. Panels (<bold>G</bold>–<bold>J</bold>) correspond to panels (<bold>B</bold>–<bold>E</bold>). Two object distances are considered in panels (<bold>G</bold>–<bold>I</bold>). Arrows in panel <bold>i</bold> correspond to <xref ref-type="fig" rid="fig2">Figure 2C,D</xref>. <italic>d</italic> = 15.7 mm (blue lines) corresponds to the pole location used in (<bold>B</bold>–<bold>D</bold>). The ends of the blue lines correspond to pull-offs. Additionally, an object distance <italic>d</italic> = 10 mm is shown (black lines). The black solid circles correspond to slip-offs (SNBs).</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.005">http://dx.doi.org/10.7554/eLife.01350.005</ext-link></p></caption><graphic xlink:href="elife01350f003"></graphic></fig></p><p>The regime in which a stable static solution exists for whisker shape can be visualized by plotting a ‘detachment’ curve in the <italic>θ</italic><sub>p</sub>−<italic>d</italic> plane, where <italic>d</italic> is the distance of the object from the base of the whisker (<xref ref-type="fig" rid="fig3">Figure 3E</xref>). The detachment curve is the set of points in the <italic>θ</italic><sub>p</sub>−<italic>d</italic> plane where detachments occur. It encloses an area where stable contacting solutions exist. When the object is close to the face (small <italic>d</italic>), the whisker contacts the object near its base and a static solution exists for most practical <italic>θ</italic><sub>p</sub> values (peak-to-peak amplitude of whisker movements, 50° [<xref ref-type="bibr" rid="bib51">Voigts et al., 2008</xref>; <xref ref-type="bibr" rid="bib14">Curtis and Kleinfeld, 2009</xref>; <xref ref-type="bibr" rid="bib36">O’Connor et al., 2013</xref>]). For larger <italic>d</italic>, the <italic>θ</italic><sub>p</sub> regime with a stable solution decreases approximately linearly. Detachments correspond to slip-offs. The range of <italic>θ</italic><sub>p</sub> with a stable solution is larger for the CF than the CB configuration. This is consistent with experimental observations (<xref ref-type="bibr" rid="bib41">Quist and Hartmann, 2012</xref>) and intuition: in the CB configuration the intrinsic curvature aids slip-off. When the object touches near the whisker tip (large <italic>d</italic>), the saddle-node bifurcation ceases to exist and the whisker is ‘pulled off’ the object. (<xref ref-type="fig" rid="fig3">Figure 3E</xref>, blue line). The pull-offs are the result of whisker truncation, and would not occur for a perfect cone.</p><p>Identical analyses were performed for hypothetical cylindrical whiskers (<xref ref-type="fig" rid="fig3">Figure 3F–J</xref>). Although the bifurcation diagrams were superficially similar for conical and cylindrical whiskers (c.f. <xref ref-type="fig" rid="fig3">Figure 3B–E,G–J</xref>), cylindrical whiskers exhibit stable solutions at much larger <italic>θ</italic><sub>p</sub>. The SNB occurs for <italic>θ</italic><sub>p</sub> > 90° (<xref ref-type="fig" rid="fig3">Figure 3G–I</xref>, black lines and solid circles; <italic>d</italic> = 10 mm), which is beyond plausible ranges of whisking since whiskers cannot move into the face. When cylindrical whiskers touch the object close to their end, they are pulled off at moderate <italic>θ</italic><sub>p</sub>, because the whisker tip reaches the object (<xref ref-type="fig" rid="fig3">Figure 3I</xref>, blue line) <italic>s</italic><sub>obj</sub> = <italic>L</italic><sub>W</sub>. Therefore, cylindrical whiskers do not slip-off the pole. For a homogenous cylinder this effect is independent of the cylinder’s bending stiffness and thus its thickness. Our model thus predicts that conical and cylindrical whiskers interact with objects in a fundamentally different way. For a large range of object distances conical whiskers slip past objects, whereas cylindrical whiskers get stuck. This difference is expected to have profound consequences for object-whisker interactions during haptic sensation.</p></sec><sec id="s2-2"><title>Whisker-object interactions</title><p>We compared our model with measurements made on mouse whiskers (conical) and human hair (cylindrical) (<xref ref-type="fig" rid="fig4">Figure 4</xref>). Mouse and human hair have similar Young’s Modulus (<xref ref-type="bibr" rid="bib25">Hu et al., 2010</xref>; <xref ref-type="bibr" rid="bib40">Quist et al., 2011</xref>). A C2 mouse whisker was mounted on a galvanometer scanner so that its intrinsic curvature was in the plane of whisker movement (<xref ref-type="fig" rid="fig4">Figure 4A</xref>). The whisker was then moved slowly (<italic>f</italic><sub>galvo</sub> = 0.2 Hz, peak-to-peak amplitude 30°) against a pole. As the whisker rotated into the pole it was deformed until, at a critical <italic>θ</italic><sub>p</sub>, it suddenly slipped off the pole (<xref ref-type="fig" rid="fig4">Figure 4B</xref>, <xref ref-type="other" rid="video1">Video 1</xref>). The red line in <xref ref-type="fig" rid="fig4">Figure 4B</xref> shows the whisker immediately before detachment. Whisker slip-offs occurred before the tip of the whisker had reached the point of contact. In contrast, for the cylindrical hair, slip-offs did not occur. Detachments always coincided with the whisker tip reaching the point of contact and were thus pull-offs (<xref ref-type="fig" rid="fig4">Figure 4C</xref>, red line, <xref ref-type="other" rid="video2">Video 2</xref>).<fig id="fig4" orientation="portrait" position="float"><object-id pub-id-type="doi">10.7554/eLife.01350.006</object-id><label>Figure 4.</label><caption><title>Isolated whiskers interacting with cylindrical poles.</title><p>(<bold>A</bold>) Top-down view of a mouse C2 whisker mounted on a galvanometer scanner. The scanner rotates the whisker into a vertical pole. The distance of the pole from the base of the whisker, <italic>d</italic>, is varied across experiments. (<bold>B</bold>) Snapshots of the whisker at 32 Hz as it is smoothly rotated (0.2 Hz, counter clockwise) into and past the pole. Red line, whisker shape immediately (<32 ms) before slip-off. Note that the end of the whisker had not reached the point of object contact. (<bold>C</bold>) Snapshots of a near-cylindrical hair. Red line, hair shape immediately before pull-off. Note that the end of the hair had reached the point of object contact. (<bold>D</bold>) The detachment curve in the <italic>θ</italic><sub>p</sub>−<italic>d</italic> plane. Solid line, theoretical prediction for conical whisker; open circles, experimental measurements for conical whisker. Dashed line, theoretical prediction for cylindrical hair; solid circles, experimental measurements for cylindrical hair. Blue, pull-offs occur because whisker tip has reached the object. Black, slip-offs occurs because of saddle-node bifurcation. Parameters of the conical whisker: <italic>L</italic><sub>w</sub> = 15.25 mm, <italic>r</italic><sub>base</sub>= 32.5 µm, <italic>r</italic><sub>tip</sub> = 2 µm, A = 0.02 mm<sup>−1</sup>. Parameters of the approximately cylindrical hair: <italic>L</italic><sub>w</sub> = 15.0 mm, <italic>r</italic><sub>base</sub>= 30 µm, <italic>r</italic><sub>tip</sub> = 26.5 µm, A = 0.017 mm<sup>−1</sup>. Pole radius, <italic>r</italic><sub>pole</sub> = 0.25 mm.</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.006">http://dx.doi.org/10.7554/eLife.01350.006</ext-link></p></caption><graphic xlink:href="elife01350f004"></graphic></fig><fig id="video1" position="anchor"><label>Video 1.</label><caption><p>Example video of a conical whisker mounted on the galvo (<xref ref-type="fig" rid="fig4">Figure 4</xref>) slipping off a pole. Speed 16fps, 0.5x real-time.</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.007">http://dx.doi.org/10.7554/eLife.01350.007</ext-link></p></caption><media xlink:href="elife01350v001.avi"><object-id pub-id-type="doi">10.7554/eLife.01350.007</object-id></media></fig><fig id="video2" position="anchor"><label>Video 2.</label><caption><p>Example video of a cylindrical hair mounted on the galvo (<xref ref-type="fig" rid="fig4">Figure 4</xref>) pulling off a pole. Speed 16fps, 0.5x real-time.</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.008">http://dx.doi.org/10.7554/eLife.01350.008</ext-link></p></caption><media xlink:href="elife01350v002.avi"><object-id pub-id-type="doi">10.7554/eLife.01350.008</object-id></media></fig></p><p>We performed the same type of measurement for multiple object locations along the whisker (<italic>d</italic>, <xref ref-type="fig" rid="fig4">Figure 4A</xref>). The regime of stable interactions between whisker and pole, bounded by the detachment curve, can be visualized in the <italic>θ</italic><sub>p</sub>−<italic>d</italic> plane (<xref ref-type="fig" rid="fig4">Figure 4D</xref>). The experimental results were in agreement with the model. For conical whiskers, slip-off occurred before the whisker tip reached the object, and the critical <italic>θ</italic><sub>p</sub> decreased rapidly with object distance (<xref ref-type="fig" rid="fig4">Figure 4D</xref>, black circles). The observed deviations between the idealized conical model and actual whisker are expected because the whisker is not a perfect cone (<xref ref-type="bibr" rid="bib27">Ibrahim and Wright, 1975</xref>) and because the whisker’s Young’s modulus may vary slightly along its length (<xref ref-type="bibr" rid="bib40">Quist et al., 2011</xref>). In contrast, the cylindrical hair only pulled off when the whisker tip reached the pole (<xref ref-type="fig" rid="fig4">Figure 4C</xref>), with a close fit between experimental and theoretical results (<xref ref-type="fig" rid="fig4">Figure 4D</xref>, blue circles).</p><p>We next tested if slip-offs occur normally during whisker-dependent behavior (<xref ref-type="fig" rid="fig5">Figure 5</xref>). We analyzed data from head-fixed mice trained in a vibrissa-based object location discrimination task (<xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>). Mice reported the presence of a pole at a target position (the ‘Go stimulus’; proximal) or in a distracter position (the ‘No Go stimulus’; distal) (<xref ref-type="fig" rid="fig5">Figure 5A</xref>) by either licking (Go response) or withholding licking (No Go response). In each trial, the pole was presented at a single location. Whiskers were trimmed so that mice performed the task with a single whisker (C2). For the trials analyzed here the pole distance from the face was randomly chosen from the range d = 7–13 mm (measured from the follicle; the No Go stimuli). We used high-speed (500 Hz) videography and automated whisker tracking to measure the position and shape of the whisker in two mice (<xref ref-type="bibr" rid="bib12">Clack et al., 2012</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>) (140 slip events).<fig id="fig5" orientation="portrait" position="float"><object-id pub-id-type="doi">10.7554/eLife.01350.009</object-id><label>Figure 5.</label><caption><title>Slip-offs during object location discrimination behavior.</title><p>(<bold>A</bold>) Schematic of a mouse whisking to touch a pole (experiments from <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>). (<bold>B</bold>) Time series (250 Hz) of whisker shape around example protraction slip event. Frame of slip-off is highlighted in red. (<bold>C</bold>) Detachment curves in the <italic>θ</italic><sub>p</sub>−<italic>d</italic> plane for two whiskers. Solid line, theoretical predictions for conical whisker; open circles, experimental measurements for conical whiskers. Dashed line, theoretical predictions for cylindrical hair. Blue, pull-offs. Black, slip-offs. Left, truncated whisker with parameters: <italic>L</italic><sub>w</sub> = 12.5 mm, <italic>r</italic><sub>base</sub>= 35 µm, <italic>r</italic><sub>tip</sub> = 8.5 µm. Right, whisker parameters: <italic>L</italic><sub>w</sub> = 15.3 mm, <italic>r</italic><sub>base</sub> = 33.5 µm, <italic>r</italic><sub>tip</sub> = 2 µm. For both whiskers, intrinsic curvature was <italic>y</italic> = A(x−2.2 mm)<sup>2</sup> where A = 0.02 mm<sup>−1</sup>.</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.009">http://dx.doi.org/10.7554/eLife.01350.009</ext-link></p></caption><graphic xlink:href="elife01350f005"></graphic></fig></p><p>In behaving mice, the intrinsic whisker curvature is not parallel to the plane of whisking and imaging (<xref ref-type="bibr" rid="bib49">Towal et al., 2011</xref>). We corrected for the curvature out of the imaging plane using a simple procedure (‘Materials and methods’). Furthermore, whiskers exhibit torsion during movement, rotating from a partially concave backward orientation, thru concave down to partially concave forward during protraction (<xref ref-type="bibr" rid="bib29">Knutsen et al., 2008</xref>). We thus define positive and negative <italic>θ</italic><sub>p</sub> to denote whisker movement in the protraction and retraction directions respectively, independent of intrinsic curvature.</p><p>Slip-offs were more likely for more distant object locations (<xref ref-type="fig" rid="fig5">Figure 5B</xref>) and occurred at larger <italic>θ</italic><sub>p</sub> for smaller object distances (<xref ref-type="fig" rid="fig5">Figure 5C</xref>). Overall, slip-offs occurred in approximately 15% of behavioral trials. We again compared model and experiment in the <italic>θ</italic><sub>p</sub>−<italic>d</italic> plane. One of the two whiskers was truncated (<xref ref-type="fig" rid="fig5">Figure 5C</xref>, left). For objects touching the whisker near the tip, detachments occurred for small <italic>θ</italic><sub>p</sub> (<20°), with the whisker tip reaching the pole (i.e., pull-offs). For smaller object distances slip-off occurred at larger <italic>θ</italic><sub>p</sub> (>20°), consistent with a saddle-node bifurcation (i.e., slip-offs). The second whisker was less truncated (<xref ref-type="fig" rid="fig5">Figure 5C</xref>, right). For the object distances tested we observed slip-offs only along the whisker. These results are consistent with our model. The critical <italic>θ</italic><sub>p</sub> values for slip-off varied significantly across trials even for identical object distances. This variability is likely caused by differences in whisker movement and whisker elevation across trials.</p><p>Rodents move their whiskers over objects to explore surfaces. For example, mice can discriminate surface roughness over a few whisking cycles (<xref ref-type="bibr" rid="bib11">Chen et al., 2013</xref>). Texture is likely inferred from the statistics of whisker micromotions produced by the interactions between whiskers and objects (<xref ref-type="bibr" rid="bib17">Diamond, 2010</xref>). In particular, as whiskers move over objects whiskers occasionally get stuck, followed by high-velocity slips. The pattern of stick-slip events is highly informative about surface texture (<xref ref-type="bibr" rid="bib3">Arabzadeh et al., 2005</xref>; <xref ref-type="bibr" rid="bib54">Wolfe et al., 2008</xref>).</p><p>We wondered whether whisker shape determines the nature of the stick-slip events underlying texture exploration. We moved a C2 mouse whisker over extra fine (600 grit) sandpaper using a galvanometer scanner (<italic>f</italic><sub>galvo</sub> = 0.2 Hz; peak-to-peak amplitude, 30<bold>°</bold>) while tracking whisker shape in three dimensions using dual view videography (<xref ref-type="fig" rid="fig6">Figure 6A</xref>, <xref ref-type="other" rid="video3">Video 3</xref>). The tips of mouse whiskers moved along the surface in an irregular manner, during protractions and retractions. Whisker tips were transiently trapped (<xref ref-type="fig" rid="fig6">Figure 6B</xref>, red) followed by small, high-velocity slips (<xref ref-type="fig" rid="fig6">Figure 6B,C</xref>). The pattern of stick-slip events differed for different wall distances (<xref ref-type="fig" rid="fig6">Figure 6C</xref>).<fig id="fig6" orientation="portrait" position="float"><object-id pub-id-type="doi">10.7554/eLife.01350.010</object-id><label>Figure 6.</label><caption><title>The whisker taper is necessary for slips across textures.</title><p>(<bold>A</bold>) Dual-perspective imaging of a conical whisker, mounted on a galvo, sweeping across a texture (600 grit sandpaper). ‘Top’, Side view; ‘Bottom’, Top view. (<bold>B</bold>) Conical whisker swept past the texture at four distances: Free air, push distance <italic>d</italic><sub>p</sub> = 0.33 mm, 1.5 mm and 4.5 mm. <italic>d</italic><sub>p</sub> = ||x(L<sub>w</sub>)−x(0), y(L<sub>w</sub>)−y(0)||−<italic>d</italic>, where <italic>d</italic> is the nearest distance from the base of the whisker to the surface. In other words, <italic>d</italic><sub>p</sub> is the distance the surface is moved radially into the whisker beyond just touching. Red traces indicate frames where the whisker tip is stuck, gray traces where the tip is slipping along the surface. Surface texture is schematic and exaggerated. (<bold>C</bold>) Black lines, histograms of tip position over time. Gray lines, trajectories of the whisker tip over the first three whisking periods. Traces are aligned to peak of theta at base. (<bold>D</bold>–<bold>F</bold>), as (<bold>A</bold>–<bold>C</bold>), but using a cylindrical hair of similar length. Free air, push distance <italic>d</italic><sub>p</sub> = 0.33 mm, 2 mm and 3.3 mm. Whisker parameters <italic>L</italic><sub>w</sub> = 16.4 mm, <italic>r</italic><sub>base</sub>= 33.5 µm, <italic>r</italic><sub>tip</sub> = 2 µm. Hair parameters as in <xref ref-type="fig" rid="fig4">Figure 4</xref>.</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.010">http://dx.doi.org/10.7554/eLife.01350.010</ext-link></p></caption><graphic xlink:href="elife01350f006"></graphic></fig><fig id="video3" position="anchor"><label>Video 3.</label><caption><p>Composite video of the conical whisker mounted on a galvo slipping across the textured surface in <xref ref-type="fig" rid="fig6">Figure 6B</xref> (4.5 mm push distance). Upper video is the side view, lower video is the top view. Speed 32fps, 1x real-time.</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.011">http://dx.doi.org/10.7554/eLife.01350.011</ext-link></p></caption><media xlink:href="elife01350v003.avi"><object-id pub-id-type="doi">10.7554/eLife.01350.011</object-id></media></fig></p><p>In contrast, as cylindrical hairs swept across the surface they were trapped during initial protraction and remained trapped for the remainder of the experiment lasting multiple whisking cycles (<xref ref-type="fig" rid="fig6">Figure 6D–F</xref>, <xref ref-type="other" rid="video4">Video 4</xref>). When the distance between follicle and the site of trapping was shorter than the hair length, the hair buckled out of the plane of movement (<xref ref-type="fig" rid="fig6">Figure 6E</xref>, top). The tips of hairs escaped the traps only when the distance to the tip along the path of an undeflected hair exceeded the actual hair arclength (<xref ref-type="fig" rid="fig6">Figure 6E,F</xref>, pane 2). The whisker tip was thus pulled out of the trap (<xref ref-type="fig" rid="fig3">Figure 3J</xref>, blue line). These measurements show that the conical whisker shape is critical for the sweeping motions of whisker tips across objects and surfaces, which supports feature extraction via stick-slip events. More generally, conical whiskers can move past walls and objects, which may be critical for positioning of whiskers in confined spaces, such as tunnels, during directed tactile exploration.<fig id="video4" position="anchor"><label>Video 4.</label><caption><p>Composite video of the cylindrical hair mounted on a galvo getting stuck on the textured surface in <xref ref-type="fig" rid="fig6">Figure 6E</xref> (3.3 mm push distance). Upper video is the side view, lower video is the top view. Speed 32fps, 1x real-time.</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.012">http://dx.doi.org/10.7554/eLife.01350.012</ext-link></p></caption><media xlink:href="elife01350v004.avi"><object-id pub-id-type="doi">10.7554/eLife.01350.012</object-id></media></fig></p><p>We investigated whether slip-offs convey specific sensory information to cortex. Silicon probes were inserted into the C2 barrel column (<xref ref-type="bibr" rid="bib36">O’Connor et al., 2013</xref>) (<xref ref-type="fig" rid="fig7">Figure 7A</xref>). We recorded multi-unit activity across cortical layers 2–5 while mice performed an object location discrimination task with the C2 whisker (<xref ref-type="bibr" rid="bib36">O’Connor et al., 2013</xref>). Mice touched the pole multiple times during a trial (<xref ref-type="fig" rid="fig7">Figure 7B</xref>). The first touch within a trial caused a large peak in activity with a rapid rise (<xref ref-type="fig" rid="fig7">Figure 7B,C</xref>), consistent with previous work (<xref ref-type="bibr" rid="bib43">Simons, 1978</xref>; <xref ref-type="bibr" rid="bib4">Armstrong-James et al., 1992</xref>; <xref ref-type="bibr" rid="bib16">de Kock et al., 2007</xref>; <xref ref-type="bibr" rid="bib37">O’Connor et al., 2010b</xref>; <xref ref-type="bibr" rid="bib36">O’Connor et al., 2013</xref>). Later touches within a series, during which slip-offs were more commonly seen, produced smaller responses (<xref ref-type="fig" rid="fig7">Figure 7D</xref>) (<xref ref-type="bibr" rid="bib1">Ahissar et al., 2001</xref>). When slip-off did not occur, the detach-related signals were almost undetectable. In contrast, when slip-off did occur, the detach-related signals were large, comparable to the first touch (<xref ref-type="fig" rid="fig7">Figure 7F,G</xref>).<fig id="fig7" orientation="portrait" position="float"><object-id pub-id-type="doi">10.7554/eLife.01350.013</object-id><label>Figure 7.</label><caption><title>Neural signals of slip-off in the barrel cortex.</title><p>(<bold>A</bold>) Silicon probe recording during a pole localization task (experiments from <xref ref-type="bibr" rid="bib36">O’Connor et al., 2013</xref>). (<bold>B</bold>) Spikes and whisker forces, one behavioral trial. ‘Top’, multi-unit activity. Arrow, slip-off event. ‘Bottom’<italic>,</italic> contact induced forces. Solid circles, time points with non-zero contact-mediated forces calculated from changes in whisker curvature. A 2 ms period during which the whisker was slipping off was removed as the quasi-static model is invalid for such highly dynamic events. (<bold>C</bold>–<bold>F</bold>) Multi-unit spike responses in the barrel cortex (shank < 300 μm from the center of the C2 barrel). (<bold>C</bold>) Activity aligned to the first touch in a trial (720 responses; two animals, three sessions, six electrode shanks). (<bold>D</bold>) Same as (<bold>C</bold>), but aligned to the last touch before a behavioral response (i.e., lick) (720 responses). (<bold>E</bold>) Same as (<bold>C</bold>), but aligned to the moment of detach on the last touch before a behavioral response in trials without slip-offs (392 responses). To prevent contamination by touch-onset this analysis was restricted to touches that were longer than 50 ms. (<bold>F</bold>) Aligned to slip-off (34 responses). (<bold>G</bold>) Change in spike rate triggered by the event (activity 10–30 ms post event, minus activity −50 to 0 ms pre event). Error bars, SEM. Pairwise comparison showed no significant difference in evoked spikes between first touch and slip-off groups (p=0.20), a significant difference between last touch and slip-off (p=0.037) and significant differences between all other groups (every pair, p<10<sup>−12</sup>).</p><p><bold>DOI:</bold><ext-link ext-link-type="doi" xlink:href="10.7554/eLife.01350.013">http://dx.doi.org/10.7554/eLife.01350.013</ext-link></p></caption><graphic xlink:href="elife01350f007"></graphic></fig></p></sec></sec><sec sec-type="discussion" id="s3"><title>Discussion</title><p>We have developed a mathematical framework for whisker deflection in the context of dynamical systems theory (<xref ref-type="fig" rid="fig1 fig2 fig3">Figures 1–3</xref>) to explore the functional consequences of whisker taper. Recent findings have shown that whisker taper is used as a ruler by mice to gauge the distance to objects with a single whisker (<xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>). Tapered whiskers have resonance frequencies that are robust to wear and truncation damage to their tips (<xref ref-type="bibr" rid="bib53">Williams and Kramer, 2010</xref>). Tapered whiskers also detach from objects at shallower push angles than cylindrical whisker substitutes.</p><p>Here we go beyond prior observations and uncover fundamental differences in how tapered and untapered hairs interact with objects; tapered whiskers slip-off when the contact point is along the whisker body (s<sub>obj</sub> < L<sub>w</sub>), whereas cylindrical hairs require the tip to be pulled off (s<sub>obj</sub> = L<sub>w</sub>) for biologically plausible push angles (<xref ref-type="fig" rid="fig3">Figure 3E,J</xref>, <xref ref-type="fig" rid="fig4">Figure 4</xref>). Thus, tapered whiskers have greater freedom of movement past obstacles compared to cylindrical hairs. This mobility is seen in reduced preparations (<xref ref-type="fig" rid="fig4">Figure 4</xref>) and also during active behavior, as whisker slip-off occurs in a variety of object location discrimination tasks (<xref ref-type="fig" rid="fig5 fig7">Figures 5, 7</xref>). The intrinsic difference in detachment also produces qualitatively different interaction patterns during palpation of textured surfaces (<xref ref-type="fig" rid="fig6">Figure 6</xref>). Tapered whiskers sweep past with stick-slip micromotions, whereas cylindrical whiskers become immobilized on surface imperfections.</p><p>Theoretical treatment of whisker mechanics is a necessary foundation for understanding how sensory input shapes neural representations of the tactile world. Previous work has computed whisker deflections based on the quasi-static solution of the Euler-Bernoulli equation (<xref ref-type="bibr" rid="bib7">Birdwell et al., 2007</xref>; <xref ref-type="bibr" rid="bib53">Williams and Kramer, 2010</xref>; <xref ref-type="bibr" rid="bib41">Quist and Hartmann, 2012</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>). Aspects of whisker vibrations have also been treated, including resonant frequencies (<xref ref-type="bibr" rid="bib24">Hartmann et al., 2003</xref>; <xref ref-type="bibr" rid="bib34">Neimark et al., 2003</xref>) and wave propagation following contact-induced impulses (<xref ref-type="bibr" rid="bib8">Boubenec et al., 2012</xref>).</p><p>We framed whisker-object interactions in the language of boundary-value problems. This allowed us to carry out bifurcation analysis and distinguish stable from unstable shapes. We demonstrate that for conical whiskers there are only two possible solutions for whisker shape for a given object distance and push angle, one stable, one unstable. We identify how the saddle-node bifurcations separating the two branches of solutions vary as a function of parameters, such as <italic>θ</italic><sub>p</sub> (<xref ref-type="fig" rid="fig3">Figure 3</xref>). This is not possible using previously developed numerical approaches (<xref ref-type="bibr" rid="bib7">Birdwell et al., 2007</xref>; <xref ref-type="bibr" rid="bib41">Quist and Hartmann, 2012</xref>). Slip-offs occur suddenly at a critical push angle, <italic>θ</italic><sub>p</sub>, corresponding to the angle where these saddle-node bifurcations occur. Curves of saddle-node bifurcations in a two-parameter plane define the regime in which stable solutions can be obtained (<xref ref-type="fig" rid="fig3">Figure 3E,J</xref>). For conical whiskers the critical push angles are within the normal range of whisking (<xref ref-type="fig" rid="fig4">Figure 4</xref>). For cylindrical hairs the critical angles fall outside the range of whisking. Thus our theory predicts that conical and cylindrical whiskers will interact with objects in a fundamentally different manner.</p><p>Our theory addresses the effects of whisker truncations (<xref ref-type="fig" rid="fig3 fig4 fig5">Figures 3–5</xref>). Truncations of conical whiskers make whisker behavior more ‘cylindrical-like’ (<xref ref-type="fig" rid="fig5">Figure 5</xref>). The intuition obtained from our analysis led us to distinguish between the dynamics of conical whiskers and cylindrical hairs during sweeping across textures (<xref ref-type="fig" rid="fig6">Figure 6</xref>).</p><p>We compared theoretical predictions with videos of whiskers and cylindrical hair rotated into a steel pole. In situations where the whisker curvature was contained within a plane, the agreement between theory and experiment was very good (<xref ref-type="fig" rid="fig4">Figure 4</xref>), despite our model ignoring frictional forces. The small remnant differences between theory and experiment are due to deviations of whisker geometry from perfect conical shape (AH, KS, DG, unpublished) and possible inhomogeneities in the Young’s modulus (<xref ref-type="bibr" rid="bib40">Quist et al., 2011</xref>) (but also see <xref ref-type="bibr" rid="bib9">Carl et al., 2012</xref>). In behaving mice the whisker droops out of the plane of whisking (<xref ref-type="bibr" rid="bib49">Towal et al., 2011</xref>). Rigorous treatment of whisker deflection by an object would thus require a three-dimensional model. We developed a phenomenological model to predict slip-offs even for behaving mice (‘Materials and methods’), which produced qualitative agreement with experiments (<xref ref-type="fig" rid="fig5">Figure 5</xref>).</p><p>Our results suggest several functions for which the conical shape of rodent whiskers is evolutionarily adaptive. Within their natural habitat, many rodents, including house mice (<xref ref-type="bibr" rid="bib6">Berry, 1968</xref>) and African pouched mice (<xref ref-type="bibr" rid="bib19">Ellison, 1993</xref>), live in group nests consisting of chambers connected by long, body-width tunnels. During running, whiskers point forward to touch unanticipated objects. When a new object is encountered rodents foveate their whiskers on the object for fine-scale exploration (<xref ref-type="bibr" rid="bib22">Grant et al., 2009</xref>). Within these dark, radially constrained tunnels, navigation (<xref ref-type="bibr" rid="bib50">Vincent, 1912</xref>; <xref ref-type="bibr" rid="bib15">Dehnhardt et al., 2001</xref>), localization of objects (<xref ref-type="bibr" rid="bib26">Hutson and Masterton, 1986</xref>; <xref ref-type="bibr" rid="bib31">Krupa et al., 2001</xref>), social touch (<xref ref-type="bibr" rid="bib55">Wolfe et al., 2011</xref>), and determination of friend or foe (<xref ref-type="bibr" rid="bib2">Anjum et al., 2006</xref>) demands freedom of whisker motion. Whiskers have to be moved past the rough walls of the tunnel. Without the flexibility provided by whisker taper, the whiskers could be trapped in a far protracted or retracted orientation, causing a tactile ‘blind-spot’.</p><p>Whisker taper is also desirable outside of constrained spaces. A major sensory avenue for the localization and identification of objects and their properties is via directed sweeping of whiskers across object surfaces (<xref ref-type="bibr" rid="bib10">Carvell and Simons, 1990</xref>; <xref ref-type="bibr" rid="bib42">Ritt et al., 2008</xref>; <xref ref-type="bibr" rid="bib52">von Heimendahl et al., 2007</xref>; <xref ref-type="bibr" rid="bib54">Wolfe et al., 2008</xref>). During artificial periodic palpation of fine-grained textures, conical whiskers traversed the surface with complex micromotions, whereas cylindrical hair became trapped against the surface (<xref ref-type="fig" rid="fig6">Figure 6</xref>). Although a precise understanding of the interaction between a tapered whisker and textured surface during a stick-slip event has not been treated mathematically, it is likely that forces at the tip build up until they bend the whisker tip sufficiently to free it from traps. In the cylindrical case, the constant bending stiffness of the body and tip render the whisker incapable of transmitting sufficient lateral force to buckle the much stiffer tip and release it from the surface.</p><p>Beyond mechanical maneuverability, do slips contribute to the neural representation of tactile sensation? During active whisking, stick-slip micromotions on textured surfaces drive sparse, precisely timed spikes in barrel cortex that provide a sensory cue for surface texture (<xref ref-type="bibr" rid="bib28">Jadhav et al., 2009</xref>). Neural responses in barrel cortex to repeated contacts between whiskers and objects show strong adaptation during active touch (<xref ref-type="bibr" rid="bib1">Ahissar et al., 2001</xref>; <xref ref-type="bibr" rid="bib13">Crochet et al., 2011</xref>) and object location discrimination (<xref ref-type="fig" rid="fig7">Figure 7C,D,G</xref>). Despite occurring when the circuit is adapted to touch, slip-offs produce strong volleys of cortical activity, of comparable magnitude to pre-adapted touch (<xref ref-type="fig" rid="fig7">Figure 7F</xref>). Thus, slip-related excitation can overcome cortical touch adaptation and likely contributes to sensation and perception in a variety of tactile behaviors (<xref ref-type="bibr" rid="bib3">Arabzadeh et al., 2005</xref>).</p></sec><sec sec-type="materials|methods" id="s4"><title>Materials and methods</title><sec id="s4-1"><title>Model of a whisker deflected by a cylindrical pole</title><p>We model whiskers as truncated cones with length <italic>L</italic><sub>W</sub>, base radius <italic>r</italic><sub>base</sub>, and tip radius <italic>r</italic><sub>tip</sub> (<xref ref-type="fig" rid="fig1">Figure 1A</xref>). The conical shape is virtually extended to a perfect cone of length <italic>L</italic>. The whisker is located in the <italic>x</italic>-<italic>y</italic> plane. The arclength along the whisker, <italic>s</italic>, is <italic>s</italic> = 0 at the base, <italic>s</italic> = <italic>s</italic><sub>obj</sub> at the point of object contact, <italic>s</italic> = <italic>L</italic><sub>W</sub> at the tip, and <italic>s</italic> = <italic>L</italic> at the virtual tip (<xref ref-type="fig" rid="fig1">Figure 1B</xref>). The whisker base is located at point (<italic>x</italic><sub>0</sub>, <italic>y</italic><sub>0</sub>), and the positions of a point along the whisker is (<italic>x</italic>(<italic>s</italic>), <italic>y</italic>(<italic>s</italic>)), 0≤<italic>s</italic>≤<italic>L</italic><sub>W</sub>. The running angle between the whisker and the <italic>x</italic>-axis is <italic>θ</italic>(<italic>s</italic>), and <italic>θ</italic>(0) = <italic>θ</italic><sub>0</sub>. The whisker radius is <italic>r</italic><sub>w</sub> = (<italic>L−s</italic>) <italic>r</italic><sub>base</sub>/<italic>L</italic> and the area moment of inertia is <inline-formula><mml:math id="inf2"><mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>π</mml:mi><mml:msubsup><mml:mi>r</mml:mi><mml:mi>w</mml:mi><mml:mn>4</mml:mn></mml:msubsup></mml:mrow><mml:mn>4</mml:mn></mml:mfrac></mml:mrow></mml:math></inline-formula>. The Young’s modulus is <italic>E</italic> = 3 GPa (<xref ref-type="bibr" rid="bib7">Birdwell et al., 2007</xref>; <xref ref-type="bibr" rid="bib40">Quist et al., 2011</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>). Similar calculations were carried out for cylindrical hair with <italic>r</italic><sub>w</sub> = <italic>r</italic><sub>base</sub>.</p><p>The bending stiffness of the whisker is the product <italic>EI</italic>(<italic>s</italic>). In the absence of contact with an object, the intrinsic curvature of the whisker is <italic>κ</italic><sub>i</sub>(s). The object is a cylindrical pole oriented perpendicular to the <italic>x</italic>-<italic>y</italic> plane with radius <italic>r</italic><sub>pole</sub>, centered at (<italic>x</italic><sub>cen</sub>, <italic>y</italic><sub>cen</sub>). Upon contact, the whisker touches the object at (<italic>x</italic><sub>obj</sub>, <italic>y</italic><sub>obj</sub>) with angle <italic>θ</italic><sub>obj</sub> (<xref ref-type="fig" rid="fig1">Figure 1B</xref>), where<disp-formula id="equ1"><label>(1)</label><mml:math id="m1"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi mathvariant="normal">cen</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi mathvariant="normal">pole</mml:mi></mml:mrow></mml:msub><mml:mi>sin</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi mathvariant="normal">cen</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi mathvariant="normal">pole</mml:mi></mml:mrow></mml:msub><mml:mi>cos</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></disp-formula></p><p>The Euclidian distance between the whisker base and the contact point is <italic>d</italic>. The object applies force <inline-formula><mml:math id="inf3"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>F</mml:mi><mml:mo>→</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:math></inline-formula> on the whisker:<disp-formula id="equ2"><label>(2)</label><mml:math id="m2"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>F</mml:mi><mml:mo>→</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mo>−</mml:mo><mml:mi>F</mml:mi><mml:mi>sin</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>F</mml:mi><mml:mi>cos</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></disp-formula></p><p>At steady state, the shape of the whisker is determined by the solution of the static Euler-Bernoulli equation (<xref ref-type="bibr" rid="bib32">Landau and Lifshitz, 1986</xref>; <xref ref-type="bibr" rid="bib7">Birdwell et al., 2007</xref>; <xref ref-type="bibr" rid="bib53">Williams and Kramer, 2010</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>)<disp-formula id="equ3"><label>(3)</label><mml:math id="m3"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi>z</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>E</mml:mi><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math></disp-formula>where <italic>M</italic><sub>z</sub> is the component of the bending moment <inline-formula><mml:math id="inf4"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>M</mml:mi><mml:mo>→</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover accent="true"><mml:mi>r</mml:mi><mml:mo>→</mml:mo></mml:mover></mml:mrow><mml:mo>×</mml:mo><mml:mrow><mml:mover accent="true"><mml:mi>F</mml:mi><mml:mo>→</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:math></inline-formula> perpendicular to the <italic>x-y</italic> plane and <inline-formula><mml:math id="inf5"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>r</mml:mi><mml:mo>→</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>y</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula>, together with the equations<disp-formula id="equ4"><label>(4)</label><mml:math id="m4"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>cos</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:math></disp-formula><disp-formula id="equ5"><label>(5)</label><mml:math id="m5"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>sin</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:math></disp-formula></p><p>Substituting <xref ref-type="disp-formula" rid="equ1">Equations 1,2</xref> in <xref ref-type="disp-formula" rid="equ3">Equation 3</xref>, we obtain<disp-formula id="equ6"><label>(6)</label><mml:math id="m6"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>κ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mi>F</mml:mi><mml:mrow><mml:mi>E</mml:mi><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi mathvariant="normal">cen</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi mathvariant="normal">pole</mml:mi></mml:mrow></mml:msub><mml:mi>sin</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>cos</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi mathvariant="normal">cen</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi mathvariant="normal">pole</mml:mi></mml:mrow></mml:msub><mml:mi>cos</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>sin</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></disp-formula></p><p>We seek a solution for <xref ref-type="disp-formula" rid="equ4">Equations 4</xref>–<xref ref-type="disp-formula" rid="equ6">6</xref> given the boundary conditions at the base (<italic>x</italic><sub>0</sub>, <italic>y</italic><sub>0</sub> and <italic>θ</italic><sub>0</sub>), and that the whisker contacts the pole at an (initially unknown) arclength <italic>s</italic><sub>obj</sub>.</p><p>Given the shape of an undeflected whisker as a function of the running arclength <italic>s</italic>, namely <italic>(x,y)=(g(s),h(s))</italic>, the intrinsic curvature is<disp-formula id="equ7"><label>(7)</label><mml:math id="m7"><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo> </mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi>g</mml:mi><mml:mo>'</mml:mo></mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mo>'</mml:mo><mml:mo>'</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>−</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>'</mml:mo></mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:msup><mml:mi>g</mml:mi><mml:mrow><mml:mo>'</mml:mo><mml:mo>'</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msup><mml:mi>g</mml:mi><mml:mo>'</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>'</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math></disp-formula>where <italic>d</italic>/<italic>ds</italic> is denoted by <italic>‘</italic>. The shape of the undeflected whisker is considered to be parabolic, <italic>y = Ax</italic><sup><italic>2</italic></sup> (<xref ref-type="bibr" rid="bib41">Quist and Hartmann, 2012</xref>).</p><p>To compute the whisker shape <xref ref-type="disp-formula" rid="equ4">Equations 4</xref>–<xref ref-type="disp-formula" rid="equ7">7</xref> are transformed to a form of a boundary-value problem (BVP) by introducing a variable <italic>σ</italic> = <italic>s</italic>/<italic>s</italic><sub>obj</sub><disp-formula id="equ8"><label>(8)</label><mml:math id="m8"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>cos</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:math></disp-formula><disp-formula id="equ9"><label>(9)</label><mml:math id="m9"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>sin</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:math></disp-formula><disp-formula id="equ10"><label>(10)</label><mml:math id="m10"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:msup><mml:mi>g</mml:mi><mml:mo>'</mml:mo></mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>σ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mo>'</mml:mo><mml:mo>'</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>σ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>−</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>'</mml:mo></mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>σ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:msup><mml:mi>g</mml:mi><mml:mrow><mml:mo>'</mml:mo><mml:mo>'</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>σ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msup><mml:mi>g</mml:mi><mml:mtext>'</mml:mtext></mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>σ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mtext>'</mml:mtext></mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>σ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>E</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>σ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>σ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mfrac><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi mathvariant="normal">cen</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi mathvariant="normal">pole</mml:mi></mml:mrow></mml:msub><mml:mi>sin</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>cos</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi mathvariant="normal">cen</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi mathvariant="normal">pole</mml:mi></mml:mrow></mml:msub><mml:mi>cos</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>sin</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></disp-formula><disp-formula id="equ11"><label>(11)</label><mml:math id="m11"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></disp-formula><disp-formula id="equ12"><label>(12)</label><mml:math id="m12"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></disp-formula><disp-formula id="equ13"><label>(13)</label><mml:math id="m13"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></disp-formula></p><p>The differential <xref ref-type="disp-formula" rid="equ8">Equations 8</xref>–<xref ref-type="disp-formula" rid="equ13">13</xref> are solved on the interval 0 <italic>≤ σ ≤</italic> 1 together with the equations<disp-formula id="equ14"><label>(14)</label><mml:math id="m14"><mml:mrow><mml:mi>I</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>σ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mi>π</mml:mi><mml:mn>4</mml:mn></mml:mfrac><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>L</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mi mathvariant="normal">obj</mml:mi></mml:mrow></mml:msub><mml:mi>σ</mml:mi></mml:mrow><mml:mi>L</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:math></disp-formula></p><p>The boundary-value conditions for σ = 0 are: <italic>x</italic>(0) = <italic>x</italic><sub>0</sub>, <italic>y</italic>(0) = <italic>y</italic><sub>0</sub>, <italic>θ</italic>(0) = <italic>θ</italic><sub>0</sub>. The conditions for σ = 1 are: <italic>x</italic>(1) = <italic>x</italic><sub>cen</sub> − <italic>r</italic><sub>pole</sub> sin <italic>θ</italic><sub>obj</sub>, <italic>y</italic>(1) = <italic>y</italic><sub>cen</sub> + <italic>r</italic><sub>pole</sub> cos <italic>θ</italic><sub>obj</sub>, <italic>θ</italic>(1) = <italic>θ</italic><sub>obj</sub>. Solutions to <xref ref-type="disp-formula" rid="equ8">Equations 8</xref>–<xref ref-type="disp-formula" rid="equ14">14</xref> have physical meaning if <italic>s</italic><sub>obj</sub> < <italic>L</italic><sub>W</sub>. If the whisker tip reaches the contact point (<italic>s</italic><sub>obj</sub> = <italic>L</italic><sub>W</sub>) the whisker detaches because it is pulled off the pole.</p><p>We solved six differential <xref ref-type="disp-formula" rid="equ8">Equations 8</xref>–<xref ref-type="disp-formula" rid="equ13">13</xref> together with their boundary conditions to find six unknown variables (<italic>x</italic>, <italic>y</italic>, <italic>θ</italic>, <italic>F</italic>, <italic>θ</italic><sub>obj</sub>, <italic>s</italic><sub>obj</sub>) as functions <italic>σ</italic> on the interval 0 <italic>≤ σ ≤</italic> 1. The variable <italic>θ</italic><sub>obj</sub> is treated as a separate variable from <italic>θ</italic>, but the boundary condition <italic>θ</italic>(1) = <italic>θ</italic><sub>obj</sub> guarantees that the solution is self-consistent. The equations were solved numerically using the iterative shooting method (<xref ref-type="bibr" rid="bib39">Press et al., 1992</xref>). We start with guessed initial values for the unknown variables for <italic>σ</italic> = 0 and integrate the differential equation until <italic>σ</italic> = 1. The initial conditions are then varied to reduce the difference between the given boundary conditions and those that are obtained by the most recent integration. The method converges if the initial conditions are sufficiently close to the solution. We begin by solving the boundary-value problem with <italic>θ</italic><sub>obj</sub> corresponding to <italic>θ</italic><sub>p</sub> = 0 (i.e., the whisker is barely touching the pole). We then vary <italic>θ</italic><sub>obj</sub> slightly, compute the whisker shape, and repeat the process until the desired <italic>θ</italic><sub>obj</sub> is reached. We used the boundary-value problem solver software package XPPAUT (<xref ref-type="bibr" rid="bib20">Ermentrout, 2002</xref>). The software package AUTO (<xref ref-type="bibr" rid="bib18">Doedel, 1981</xref>), which is incorporated into XPPAUT, was used to compute bifurcation diagrams (<xref ref-type="fig" rid="fig3">Figure 3</xref>), by following the solutions of the boundary-value problem as parameters, such as <italic>θ</italic><sub>obj</sub>, vary.</p><p>The static solution of <xref ref-type="disp-formula" rid="equ8">Equations 8</xref>–<xref ref-type="disp-formula" rid="equ14">14</xref> is a fixed point of a spatiotemporal dynamical system representing whisker movement. The full dynamical system can be formulated as a partial differential equation only for small <italic>θ</italic><sub>p</sub> and straight beams (<xref ref-type="bibr" rid="bib47">Timoshenko, 1961</xref>; <xref ref-type="bibr" rid="bib8">Boubenec et al., 2012</xref>). Since the full dynamical system for all <italic>θ</italic><sub>p</sub> and beams with intrinsic curvature is not known we cannot linearize a dynamic equation. However, the static solution for small <italic>θ</italic><sub>p</sub> must be stable. In addition, bifurcation theory implies that if we increase <italic>θ</italic><sub>p</sub> the solution will coalesce with an unstable solution and they both disappear, via a saddle-node bifurcation (<xref ref-type="bibr" rid="bib45">Strogatz, 1994</xref>). In principle, a branch of stable solutions can lose stability via a Hopf bifurcation before the saddle-node bifurcation. Slip-off will occur at <italic>θ</italic><sub>p</sub> values smaller than predicted by our quasi-static theory. The good correspondence between the computed saddle-node bifurcation and the experimentally measured value for <italic>θ</italic><sub>p</sub> at slip-off shows that the static solution disappears via a saddle-node bifurcation (<xref ref-type="fig" rid="fig4">Figure 4</xref>).</p><p>Undeflected whiskers can be modeled as parabolas within a plane (<xref ref-type="bibr" rid="bib49">Towal et al., 2011</xref>). In the work reported here the whisker is contained entirely within a plane perpendicular to the pole. For a whisker with intrinsic curvature, contact occurs in either the ‘concave backward’ (CB) or ‘concave forward’ (CF) directions (<xref ref-type="fig" rid="fig1">Figure 1C</xref>) (<xref ref-type="bibr" rid="bib41">Quist and Hartmann, 2012</xref>). To quantify contact strength, we use the push angle <italic>θ</italic><sub>p</sub> (<xref ref-type="fig" rid="fig1">Figure 1D</xref>) (<xref ref-type="bibr" rid="bib41">Quist and Hartmann, 2012</xref>). Suppose a whisker originates at (<italic>x</italic><sub>0</sub>, <italic>y</italic><sub>0</sub>, <italic>θ</italic><sub>0</sub>) and touches a pole at (<italic>x</italic><sub>obj</sub>, <italic>y</italic><sub>obj</sub>). We plot an undeflected whisker with the same (<italic>x</italic><sub>0</sub>, <italic>y</italic><sub>0</sub>, <italic>θ</italic><sub>0</sub>), and find a point along the whisker with the same Euclidian distance <italic>d</italic> from (<italic>x</italic><sub>0</sub>, <italic>y</italic><sub>0</sub>) as (<italic>x</italic><sub>obj</sub>, <italic>y</italic><sub>obj</sub>), defined as (<italic>x</italic><sub>virtual</sub>, <italic>y</italic><sub>virtual</sub>) The angle between the two rays starting at (<italic>x</italic><sub>0</sub>, <italic>y</italic><sub>0</sub>) towards (<italic>x</italic><sub>obj</sub>, <italic>y</italic><sub>obj</sub>) and (<italic>x</italic><sub>virtual</sub>, <italic>y</italic><sub>virtual</sub>) is defined as <italic>θ</italic><sub>p</sub>. By convention, we define the sign of <italic>θ</italic><sub>p</sub> to be positive for CB and negative for CF.</p><p>Exact treatment of whisker deflection in behaving rodents demands a three-dimensional model that is outside the scope of this work. Instead, we developed a phenomenological two-dimensional model (<xref ref-type="fig" rid="fig5">Figure 5</xref>). First we assume that the whisker touches the object in a concave-down configuration. Second, the deflection of the whisker is described by the two-dimensional model (<xref ref-type="disp-formula" rid="equ8">Equation 8</xref>–<xref ref-type="disp-formula" rid="equ14">14</xref>) when the projection of the whisker on that plane is treated as a two-dimensional whisker. The area moment of inertia (<italic>I</italic>) is computed by estimating the arclength <italic>s</italic> along the real whisker from the whisker projection and using this value in <xref ref-type="disp-formula" rid="equ14">Equation 14</xref>. This correction in <italic>s</italic> was on the order of 3%. We measured the length of the isolated whisker. Estimating the whisker base is inaccurate because of the fur on the face. We therefore determined the effective whisker length from the estimated whisker base to the tip based on the video recordings. If the whisker slips off at its tip, we find the maximal projected length during events of slip-off at the tip. If there are no such events, we compute the projected length that yields the theoretically-obtained slip-off at the largest <italic>d</italic> for which slip-off is obtained. For all cases, this estimated value is less than 1 mm smaller than the length measured directly.</p></sec><sec id="s4-2"><title>Whisker measurements</title><p>For galvo experiments (<xref ref-type="fig" rid="fig4 fig6">Figures 4, 6</xref>), we used plucked, full-grown mouse C2 whiskers. The shapes of these whiskers were measured under a light microscope at high magnification (<xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>). The follicle ends of the whiskers were embedded in the barrel of a cut 21 gauge needle filled with Kwik-Cast silicon sealant (World Precision Instruments, Sarasota, FL). Needles were mounted on the top edge of a galvo scan mirror (6800HP; Cambridge Technology, Bedford, MA). Whiskers were then rotated into a cylindrical object (steel Wiretrol II plunger; Drummond Scientific, Broomall, PA) at 0.2 Hz, 30° peak-to-peak amplitude (<xref ref-type="fig" rid="fig4">Figure 4</xref>). Dual-perspective imaging confirmed that whiskers remained in the concave forward or concave backward orientation during the interaction with the pole (data not shown). The same whiskers were used for imaging whisker motion across textured surfaces (<xref ref-type="fig" rid="fig6">Figure 6</xref>). The surface was fine sandpaper (600 grit) rigidly mounted on a glass slide and positioned perpendicular to both planes of imaging. A variety of human hair was characterized. Hair from an Asian female closely matched the whisker diameter close to the base and was used as a cylindrical hair. The hair dimensions were: base diameter, 60 μm; tip diameter, 53 μm; length, 15.0 mm.</p><p>High-speed videography was used to measure the position and shape of mouse whiskers during galvo experiments (<xref ref-type="fig" rid="fig4 fig6">Figures 4, 6</xref>) (X-PRI camera, 32 fps, 0.6 ms exposure, 8-bit depth, AOS Technologies, Switzerland) and behavior (<xref ref-type="fig" rid="fig5 fig7">Figures 5, 7</xref>) (1000 fps, 0.2 ms exposure, 8-bit depth, Basler 504 k, Germany). Pixel size was 0.07 mm (<xref ref-type="fig" rid="fig5">Figure 5</xref>), or 0.031 mm (<xref ref-type="fig" rid="fig7">Figure 7</xref>), or 0.032 mm (<xref ref-type="fig" rid="fig4 fig6">Figures 4, 6</xref>). Illumination was with a 940 nm infrared LED delivered through a diffuser and condenser lens and projected directly into the camera. A silver mirror (PFSQ10-03-P01, Thor Labs, Newton, NJ) reflected an orthogonal side view projection onto the same camera (<xref ref-type="bibr" rid="bib29">Knutsen et al., 2008</xref>). Videos were split and cropped prior to whisker tracking.</p><p>Whiskers were tracked with the Janelia Whisker Tracker (<xref ref-type="bibr" rid="bib12">Clack et al., 2012</xref>) (<ext-link ext-link-type="uri" xlink:href="https://openwiki.janelia.org/wiki/display/MyersLab/Whisker+Tracking">https://openwiki.janelia.org/wiki/display/MyersLab/Whisker+Tracking</ext-link>). The whisker medial axis is stored as an array of points (<italic>x</italic><sub><italic>i</italic></sub>,<italic>y</italic><sub><italic>i</italic></sub>), i = 1,…,<italic>N</italic>, where <italic>N</italic> is on the order of several hundreds. To remove artifacts associated with tracking variation at the base when calculating <italic>θ</italic><sub>0</sub>, the angle of the whisker base was determined by a linear fit of the fifth through tenth points closest to the base. Forces acting on the whisker (<xref ref-type="fig" rid="fig7">Figure 7B</xref>) were calculated using published methods (<xref ref-type="bibr" rid="bib35">O’Connor et al., 2010a</xref>; <xref ref-type="bibr" rid="bib12">Clack et al., 2012</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>).</p><p>The behavioral task and apparatus have been described in detail elsewhere (<xref ref-type="bibr" rid="bib35">O’Connor et al., 2010a</xref>; <xref ref-type="bibr" rid="bib36">O’Connor et al., 2013</xref>; <xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>). Briefly, head-fixed mice judged the distance to a metal pole that was presented at a range of positions along the whisker in the radial dimension (<xref ref-type="fig" rid="fig5">Figure 5</xref>) or horizontal dimension (<xref ref-type="fig" rid="fig7">Figure 7</xref>). For radial discrimination, a proximal position (5 mm radially from follicle) was defined as the Go position, distal positions (7–13 mm) were defined as No Go positions. For horizontal discrimination, Go and No go positions were separated by 4.5 mm along a parallel to the anteroposterior axis of the mouse at a radial distance of 7–11 mm. Mice performed object location discrimination with a single C2 whisker. Within two days of the behavioral experiments we plucked whiskers and measured their shape and material properties using a macroscope and microgram balance (Mx5; Mettler Toledo, Columbus, OH) (<xref ref-type="bibr" rid="bib38">Pammer et al., 2013</xref>).</p></sec><sec id="s4-3"><title>Electrophysiology</title><p>Parts of the electrophysiology dataset is a reanalysis of previously acquired data (see <xref ref-type="bibr" rid="bib36">O’Connor et al., 2013</xref> for detailed methods). During a head-fixed pole location discrimination task, a 32 channel, four shank silicon probe (Buz32, Neuronexus, Ann Arbor, MI) was lowered into barrel cortex, with an estimated tip depth of 375–720 μm from the pia. Prior to insertion, probes were painted with DiI. Following recordings, mouse brains were fixed, stained for cytochrome oxidase and tangentially sectioned to determine the location of the shanks within the barrel field. Shanks within 300 μm of the center of C2 were included for analysis for slip-off responses (two animals, three behavioral sessions, six shanks). Following common signal subtraction, bandpass filtering between 300 and 6,000 Hz, spike extraction of 4 s.d. threshold crossing, and spike merging, multiunit responses were aligned to whisker behavioral events. Spikes with peaks <307.5 μs jitter on the same shank were considered a single spike. Each multiunit was the sum of activity on all eight electrodes on a single shank (six total multiunit recordings). Slip-off events were rare (17 total in three sessions) compared to detach without slip-off. Significance was calculated as unpaired two-tailed t-tests on the difference between the number of spikes in the period 10–30 ms post event and the 50 ms prior to event normalized to the respective period lengths followed by Bonferroni–Holm correction for multiple comparisons.</p></sec></sec></body><back><sec sec-type="funding-information"><title>Funding Information</title><p>This paper was supported by the following grants:</p><list list-type="bullet"><list-item><p><funding-source>Howard Hughes Medical Institute</funding-source> to Samuel Andrew Hires, Lorenz Pammer, Karel Svoboda, David Golomb.</p></list-item><list-item><p><funding-source>Israel Science Foundation</funding-source><award-id>88/13</award-id> to David Golomb.</p></list-item></list></sec><ack id="ack"><title>Acknowledgements</title><p>The authors thank Mitra Hartmann, Brian Quist, Lucie Huet and Tansu Celikel for useful discussions, Diego Gutnisky, Nick Sofroniew and Daniel O’Connor for comments on our manuscript.</p></ack><sec sec-type="additional-information"><title>Additional information</title><fn-group content-type="competing-interest"><title><bold>Competing interests</bold></title><fn fn-type="conflict" id="conf1"><p>The authors declare that no competing interests exist.</p></fn></fn-group><fn-group content-type="author-contribution"><title><bold>Author contributions</bold></title><fn fn-type="con" id="con1"><p>SAH, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article.</p></fn><fn fn-type="con" id="con2"><p>DG, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article.</p></fn><fn fn-type="con" id="con3"><p>LP, Acquisition of data, Analysis and interpretation of data.</p></fn><fn fn-type="con" id="con4"><p>KS, Conception and design, Analysis and interpretation of data, Drafting or revising the article.</p></fn></fn-group><fn-group content-type="ethics-information"><title><bold>Ethics</bold></title><fn fn-type="other"><p>Animal experimentation: This study was performed in strict accordance with the recommendations in the Guide for the Care and Use of Laboratory Animals of the National Institutes of Health. 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pub-id-type="doi">10.1016/0006-8993(69)90061-4</pub-id><pub-id pub-id-type="pmid">5802473</pub-id></mixed-citation></ref></ref-list></back><sub-article id="SA1" article-type="article-commentary"><front-stub><article-id pub-id-type="doi">10.7554/eLife.01350.014</article-id><title-group><article-title>Decision letter</article-title></title-group><contrib-group><contrib contrib-type="editor"><name><surname>Tsodyks</surname><given-names>Misha</given-names></name><role>Reviewing editor</role><aff><institution>Weizmann Institute of Science</institution>,<country>Israel</country></aff></contrib></contrib-group></front-stub><body><boxed-text position="float" orientation="portrait"><p>eLife posts the editorial decision letter and author response on a selection of the published articles (subject to the approval of the authors). An edited version of the letter sent to the authors after peer review is shown, indicating the substantive concerns or comments; minor concerns are not usually shown. Reviewers have the opportunity to discuss the decision before the letter is sent (see <ext-link ext-link-type="uri" xlink:href="http://elife.elifesciences.org/review-process">review process</ext-link>). Similarly, the author response typically shows only responses to the major concerns raised by the reviewers.</p></boxed-text><p>Thank you for sending your work entitled “Tapered whiskers are required for active tactile sensation” for consideration at <italic>eLife</italic>. Your article has been favorably evaluated by a Senior editor and 2 reviewers, one of whom is a member of our Board of Reviewing Editors. As described below, the requested revisions are quite minor and we should be able to make a final decision in a very short time after receipt of your revision.</p><p>The Reviewing editor and the other reviewer discussed their comments before we reached this decision, and the Reviewing editor has assembled the following comments to help you prepare a revised submission.</p><p>It is well established that rodent and cat whiskers are conical in contrast to human hairs and seal whiskers. A fundamental question, unanswered for many years, is whether the conical shape of the formers has a functional significance.</p><p>This paper provides an important contribution in the answer to that question. This work is the most comprehensive study of the relationship between the shape of a whisker and the profile it takes upon touch to a small object or a texture. Combining analytical, numerical, and experimental approaches, the authors demonstrate that the profile depends crucially on the whisker's tapering. A central result here is that when a conically tapered whisker moves and contact an object, it can slip off the object well before the latter has touched the tip of the whisker. To get some insight on the possible functional relevance of this effect, the authors complete their study by performing electrophysiological recordings in S1 of behaving mice while monitoring the whiskers' motion. They confirm previous studies, which showed that S1 neurons respond strongly to the first touch of whiskers and adapts to subsequent touch. They also report that there is no positive response to detach. Remarkably, they also found that S1 neurons respond to slip-off almost as strongly as to the first touch. This is a completely new result that suggests that information on slip off is encoded in cortex. Thus, tapering of the whiskers may have functional significance.</p><p>Minor issues to address in the revision:</p><p>1) The authors should provide more explanations about how they solved <xref ref-type="disp-formula" rid="equ8">equations 8</xref>uations –<xref ref-type="disp-formula" rid="equ13">equations 13</xref>. These equations contain unknown free parameters F, \theta_{obj} and s_{obj}. Presumably these are obtained in the shooting solution? How is stability of solutions obtained with the shooting method?</p><p>2) The paper is well written and easy to follow. Clearly, the authors have made a lot of effort in order to make the modeling part accessible to a large audience. Yet, this can be further improved. For instance a sentence like the one opening the second paragraph of the Results (“Generically, the boundary value problem…“) may be very cryptic to a reader without an appropriate mathematical background. Elaborating, somewhat more on the qualitative meaning of a “boundary value problem” or on the meaning of an “unstable solution” should be straightforward and will be of significant help to the reader.</p><p>3) The novelty of the theoretical results must be emphasized more in the Discussion. In particular, the authors can elaborate a bit more on the comparison of their approach and results with those of <xref ref-type="bibr" rid="bib7">Birdwell et al. (2007)</xref>.</p></body></sub-article><sub-article id="SA2" article-type="reply"><front-stub><article-id pub-id-type="doi">10.7554/eLife.01350.015</article-id><title-group><article-title>Author response</article-title></title-group></front-stub><body><p>We have made textual revisions and a few tweaks to the manuscript with the goal to:</p><p>1) Explain in more detail how we solved the boundary-value problem.</p><p>2) Explain the mathematical terms for non-mathematicians.</p><p>3) Further compare our results to published results in this area based mainly on numerical methods.</p></body></sub-article></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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