Conveyance of texture signals along a rat whisker

Neuronal activities underlying a percept are constrained by the physics of sensory signals. In the tactile sense such constraints are frictional stick–slip events, occurring, amongst other vibrotactile features, when tactile sensors are in contact with objects. We reveal new biomechanical phenomena about the transmission of these microNewton forces at the tip of a rat’s whisker, where they occur, to the base where they engage primary afferents. Using high resolution videography and accurate measurement of axial and normal forces at the follicle, we show that the conical and curved rat whisker acts as a sign-converting amplification filter for moment to robustly engage primary afferents. Furthermore, we present a model based on geometrically nonlinear Cosserat rod theory and a friction model that recreates the observed whole-beam whisker dynamics. The model quantifies the relation between kinematics (positions and velocities) and dynamic variables (forces and moments). Thus, only videographic assessment of acceleration is required to estimate forces and moments measured by the primary afferents. Our study highlights how sensory systems deal with complex physical constraints of perceptual targets and sensors.

. Speed of transmission along the whisker beam. Moving contact at a distance 7 − 7 mm (A) and -1 mm (B). In both panels the contact with the rough p80 sandpaper 8 is shown at the top, contact with smooth p1200 sandpaper at the bottom. In the left 9 columns acceleration traces measured at the tip (s=24mm) and base (s=3mm) are 10 shown. The right columns show cross-correlograms of acceleration traces at tip vs. 11 base, calculated from all trials measured in this configuration. The central negative 12 peaks (vertical grey line) are located at time delay 0, up to measurement precision. 13 are formed on to measure multi-axis forces applied to the tip of this cantilever type 20 structure; two of them are formed on the surface of the beams as R1 and R4, and the 21 others are formed on the side walls as R2 and R3. x and y directional forces applied to 22 the tip of the sensor can be detected by comparing the resistance changes of these 23 piezo-resistors formed on the spring type beams. The measured directional forces are 24 then the and latitudinal (normal, x) and longitudinal (axial, y) forces acting on the whisker 25 base (see figure 4 in the main text for orientation of the whisker mounted on the sensor). 26  Supplementary text related to figure S5.

Numerical implementation.
Here we present a full and condensed description of our Cosserat rod model of whiskers in dimensionless form, together with a finite difference semi-discretization scheme allowing its translation into a set of first order differential algebraic equations (DAEs) at the heart of the method of lines that we use to solve the equations of motion of the whisker.
The method of lines is a general technique for solving partial differential equations (e.g. see [48] for a didactic description of the method). It relies on the discretization of the spatial domain and derivatives which yields a large set of differential equations whose time integration can be performed using numerical routines developed for the numerical integration of ODEs and DAEs. Complete details are given to make the numerical implementation our method into a computer code in any computing language relatively straightforward.
We point out that our dimensionless formulation, explained below, renders our approach very generic in the sense that any whiskers with the same dimensionless characteristics will behave in exactly the same fashion. Another important feature of our formulation relies on the shape function σ(s), which describes the rod geometry and its mechanical characteristics (like inertia and bending stiffness). The whisker geometry in this study is based on a linear cone, but any other shape could be assumed in principle from modifying σ to any other relevant and realistic function of the arclength s. With this view in mind, our formulation could be particularly useful for the study of vibrissae across the animal kingdom and for the design of synthetic whiskers to explore their mechanical response to dynamic contacts. Finally note that the present formulation is not restricted to a particular point contact model either.
All in all only slight modifications of the dimensionless formulation and its numerical implementation presented below would be required for the study of natural or synthetic whiskers with different shapes and with more complex inner structure.

Non-dimensionalisation and notations.
To non-dimensionalise system (7)-(13) in the main document and to resolve short wavelengths and high frequencies, we use b/2, (b/2)/c and πEb 2 as characteristic scales of length (s) and displacement (x, y ), time (t) and force (f , g ), respectively. Note that the rod longitudinal wavespeed is denoted c = E /ρ and that moments are measured in units of πEb 2 .
With abuse of notation, denoting the rod aspect ratio ε = b/(2l), the dimensionless dynamics of the rod is governed by the partial differential-algebraic system, The associated boundary conditions are straightforward to write down. The whisker shape function σ(s) (i.e. dimensionless whisker radius) is defined as for a truncated linear cone with a dimensionless truncation length λ c = c / and a dimensionless arclength s ∈ [0, 1/ε]. Note that different whisker geometries could be considered from assuming different functions σ(s).
The semi-discrete formulation of system (1) needed for the method of lines is obtained as follows. We have adapted the spatial finite difference discretization scheme of McMillen and Holmes (2006) [40] consisting in decomposing the rod over the discrete arc-length grid s i := ih (i ∈ {0, 1, ... , n}), h = 1/(nε), as a chain of small rigid segments (labelled by index i) of length h, mass µ i and moment of inertia ν i defined by See figure S5. Correspondingly, the discrete version of the equations of motion reads where the discrete shear force and moment are given by The discretization of the constraint of inextensibility yields while the quadrature formula that approximates integral conditions (1) 6 reads A discrete Differential Algebraic Equations (DAE) scheme.
We now express the system of equations above as a first order system of ODEs.
In the following, we denote ∆ + = E + − I the matrix corresponding to the difference opera- The vector e i is the Euclidian basis vector that has unity in its i-th position and zeros elsewhere.
Writing the vector of unknown variables z := (x, y, θ,ẋ,ẏ,θ, f, g, p) T , denoting for instance x = (x 1 , ... , x n ) T := (x(s 1 ), ... , x(s n )) T ((•) T being the transpose of a vector), we then construct from (8)-(10) the DAE system to be solved by defining the mass matrix, the 'body' force and 'contact' vectors as We denote I n the n-dimensional identity matrix, 0 n representing either the zero matrix or vector of the relevant size.
When rate-and-state friction is assumed (see Eqs. (13) in the main document), this DAE system is augmented with the state evolution law equation (appropriately rescaled), the tip velocity relative to the driving surface (speed V ) being evaluated with Note that a change in the tip contact model would be implemented by modifying the non-zero components in the contact force F 0 .
In practice, the first order system of ODEs (12)