Electromechanical model for object roughness perception during finger sliding

Abstract

Touch allows us to gather abundant information in the world around us. However, how sensory cells embedded in the fingers convey texture information into their firing patterns is still poorly understood. Here, we develop an electromechanical model for roughness perception by incorporating main ingredients such as voltage-gated ion channels, active ion pumps, mechanosensitive channels, and cell deformation. The model reveals that sensory cells can convey texture wavelengths into the period of their firing patterns as the finger slides across object surfaces, but they can only convey a limited range of texture wavelengths. We also show that an increase in sliding speed broadens the decoding wavelength range at the cost of reduction of lower perception limits. Thus, a smaller sliding speed and a bigger contact force may be needed to successfully discern a smooth surface, consistent with previous psychophysical observations. Moreover, we show that cells with slowly adapting mechanosensitive channels can still fire action potentials under static loadings, indicating that slowly adapting mechanosensitive channels may contribute to the perception of coarse textures under static touch. Our work thus provides a new theoretical framework to study roughness perception and may have important implications for the design of electronic skin, artificial touch, and haptic interfaces.

Significance

Touch is essential for environmental exploration, social interaction, tactile discrimination, and other tasks in life. Thus, it is of great significance to understand the mechanisms responsible for touch. However, how sensory cells embedded in our skin convey texture information into their spiking patterns is still poorly understood. In this work, we develop a theoretical framework for roughness perception at cellular level. We show that living sensory cells function as frequency filter and can only convey a limited range of texture wavelengths into the period of their firing patterns. Bigger sliding speeds and contact forces broaden the perceived range. Our findings thus have a potential application in the design of tactile sensors and haptic interfaces for humanoid robots.

Introduction

Aristotle classified touch, along with hearing, sight, taste, and smell, as one of the five main senses of humans, which allows us to perceive a wealth of information from the physical world (1). For example, touch allows us to recognize the magnitudes and types of mechanical stimuli (2) and to discern physical properties of objects and subtly manipulate objects (3,4). Moreover, it is also involved in prosocial comforting behaviors, like fondling and caressing (5).

Among various kinds of tactile perceptions, the exquisite texture perception is the most versatile (6). Natural textures vary in a wide range from the order of micrometers to the order of millimeters. Previous experiments suggest that the perceptions of coarse and fine textures are conveyed by two independent coding mechanisms (7,8,9). Coarse textures (feature size larger than 200 μm) can be detected by spatial variations of the finger under static touch and are mediated primarily by slowly adapting (SA) type 1 afferent fibers (7,10). In contrast, the perception of fine textures requires sliding the finger across the surface (dynamic touch) to elicit temporal vibrations in the skin. These vibrations are encoded primarily by rapidly adapting (RA) and Pacinian afferent fibers (7,11).

An afferent fiber usually links with plenty of sensory neurons (1,12), indicating that the afferent fiber receives abundant information from various sensory cells. Thus, the prerequisite of texture perception may involve the activation of sensory cells embedded in the finger. Although the coding mechanisms at the scale of afferent fiber are well studied (7,13), at the cellular scale, how sensory cells convey information about textured features into their firing patterns is still elusive. Besides, as we slide the finger across the textured surface, how the sliding speed and the contact force between the finger and the surface affect roughness perceiving performance is also poorly understood.

To address these questions, we propose a generic theoretical framework for roughness perception at the cellular level, without restricting ourselves to certain specialized cell types. We find that sensory cells containing RA mechanosensitive channels can convey the textured wavelength into the period of firing action potentials as the finger slides across the textured surface. Moreover, the perceiving range increases with the sliding speed of the finger and the contact force between the finger and the textured surface, which agrees well with previous psychophysical observations. In contrast, cells containing SA mechanosensitive channels can still generate action potentials under static loading and thus may play critical roles in the perception of coarse texture under static touch.

Materials and methods

To recognize the texture, we actively slide our fingers across the textured surface (Fig. 1 a). Here, we focus on rigid wavy surface that is characterized by a roughness profile $h\left(x\right)=A_0\left[1-\text{cos}\left(2\pi x/\lambda \right)\right]$ with a texture wavelength λ and amplitude $A_0$ (Fig. 1 b). When the finger slides from valleys to crests (finger displacement is denoted by $x=vt$, where v is sliding speed and t is time), sensory cells embedded in the finger will be compressed (Fig. 1 b, blue oval), then they return to an uncompressed state in the next valley (Fig. 1 b, blue circle). Therefore, sensory cells suffer from periodic compression during finger sliding (Fig. 1 c and d). Under compression, the cell height H is given as $H=H_0-d$, where $H_0$ is initial height and d is compression depth (Fig. 1 c). We assume that d takes the form of $d=A\left(F_c\right)h\left(x\right)$ (Fig. 1 d), where $A\left(F_c\right)$ is a function of the contact force between the finger and wavy surface $F_c$. Based on contact mechanics models of a rigid wavy surface with elastic surfaces (14,15), we have $d=0$ when $F_c=0$, thus $A=0$. However, if $F_c$ is very big, the finger will fully contact with the wavy surface (16,17), thus there is no gap between the finger and the surface. In this case, we assume $d\left(x\right)=h\left(x\right)$ and that is $A=1$. Based on these limit cases, we assume $A=1-\text{exp}\left(-F_c/F_0\right)$, where $F_0$ is a constant. Note that in addition to periodic patterns, our model can expand to more complex patterns as long as we can connect the indentation depth of cell $d\left(x\right)$ to the roughness profile of the object $h\left(x\right)$.

Figure 1.

Schematic diagram of texture perception. (a) A finger slides across a surface with a speed v and a contact force $F_c$. (b) Wavy surfaces can be characterized by a roughness profile $h\left(x\right)=A_0\left[1-\text{cos}\left(2\pi x/\lambda \right)\right]$ with a texture wavelength λ and amplitude $A_0$. Sensory cells (blue) embedded in the finger are compressed periodically during sliding. Finger displacement is $x=vt$, where t is time. (c) Cell shapes before (dash) and after (solid) compression, where d is compression depth, and $H_0$ and H are the cell heights before and after compression, respectively. (d) Compression depth d takes a form of $A\left(F_c\right)h\left(x\right)$, where $A\left(F_c\right)$ is a function of $F_c$ and ranges from 0 to 1. (e) Ion channels and mechanosensitive (MS) channels are embedded in cell membrane. (f) MS channels have three states: closed, open, and inactivated states. (g) A step compression is applied. (h) Ion fluxes generated by rapidly adapting (RA) and slowly adapting (SA) MS channels under the loading in (g). Relaxation times of RA and SA channels are $\tau =4.4$ ms and $\tau =259.4$ ms, respectively. To see this figure in color, go online.

Similar to previous studies (18,19,20), we consider four charged species, Na⁺, K⁺, Cl^–, and negatively charged macromolecules $A^{-}$. The first three ions can flow across the membrane by various ion channels (Fig. 1 e), whereas $A^{-}$ is not permeable through the membrane. Here, we consider four kinds of ion channels: 1) Na⁺/K⁺ pumps, which transport three K⁺ into cells and two Na⁺ out of cells (21); 2) Na⁺-K⁺-Cl^– cotransporters, which transport one Na⁺, one K⁺, and two Cl^– in the same direction (22,23); 3) voltage-gated Na⁺, K⁺, and Cl^– channels; and 4) mechanosensitive (MS) channels. Experiments have shown that if Na⁺ in extracellular solution is removed, the ion current generated by MS channels is dramatically reduced (24,25). Furthermore, the extracellular concentration of Na⁺ is much bigger than that of K⁺. Thus, we will first assume that the ion transported by MS channels is mostly Na⁺. Previously, we have shown that vesicle release from sensory cells can also affect ion transport (26). For simplicity, here we neglect the ion transport mediated by vesicles kinetics. Hence, the dynamic equations of each ion are

$$\frac{d{n}_{N{a}^{+}}}{dt}=S_{eff}\left(-3J_{pump}+J_{KCC2}+J_{Na}+J_{ms}\right),$$ (1)

$$\frac{d{n}_{K^{+}}}{dt}=S_{eff}\left(2J_{pump}+J_{KCC2}+J_K\right),\text{and}$$ (2)

$$\frac{d{n}_{C{l}^{-}}}{dt}=S_{eff}\left(2J_{KCC2}+J_{Cl}\right),$$ (3)

where $n_{N{a}^{+}}$, $n_{K^{+}}$, and $n_{C{l}^{-}}$ are intracellular Na⁺, K⁺, and Cl^– molar number, respectively. $S_{eff}$ is the effective membrane surface area for ion transport. $J_{pump}$, $J_{KCC2}$, $J_{Na}$, $J_K$, $J_{Cl}$, and $J_{ms}$ are ion fluxes of Na⁺/K⁺ pumps, Na⁺-K⁺-Cl^– cotransporters, voltage-gated Na⁺ channels, voltage-gated K⁺ channels, voltage-gated Cl^– channels, and MS channels, respectively (see supporting material for details).

MS channels are the most important participators that translate mechanical stimuli into electrical signals and play critical roles in touch sensation (27). We use three states, i.e., closed, open, and inactivated states (Fig. 1 f), to describe the behaviors of MS channels (28,29). The dynamic equations of these states are

$$\frac{dC}{dt}=-k_1\left(\sigma \right)C-k_4\left(\sigma \right)C+k_6\left(\sigma \right)O+k_3\left(\sigma \right)I,$$ (4)

$$\frac{dO}{dt}=-k_{2}O-k_6\left(\sigma \right)O+k_1\left(\sigma \right)C+k_{5}I,\text{and}$$ (5)

$$\frac{dI}{dt}=-k_{5}I-k_3\left(\sigma \right)I+k_4\left(\sigma \right)C+k_{2}O,$$ (6)

where C, O, and I are the proportions of MS in the closed, open, and inactivated states respectively, $k_i$ (i = 1, 2, …, 6) are transform rates between these states (see supporting material for details). MS channels can be activated by cortical tension (30) or hydrostatic pressure (31,32). According to the “force-from-lipids” and “force-from-tether” mechanisms for Piezo channels activation (33), we assume that the transform rates $k_i$ are related to the cellular surface stress σ. This stress includes both the passive stress resulting from membrane deformation and the active stress resulting from myosin contraction.

Following previous works (34,35), we treat the membrane-cortex-combined layer as an active contractile elastic layer

$$\sigma =\frac{K}{2}\left(\frac{S}{S_0}-1\right)-{\sigma }_a,$$ (7)

where K is the elastic modulus of the combined layer, S is cell surface area, $S_0$ is reference cell surface area, and ${\sigma }_a$ is the active stress. Note that this constitutive equation does not account for skin roughness, the interaction of different cells, and the physics of MS channel activation.

Cell membrane voltage $V_m$ is determined by total ion currents as (36)

$$C_m\frac{d{V}_m}{dt}=S_{eff}F\left(-J_{pump}+J_{Na}+J_K-J_{Cl}+J_{ms}\right),$$ (8)

where $C_m$ is membrane capacitance and F is the Faraday constant.

Cell compression will also lead to the change in cell volume V, which is given as (37,38)

$$\frac{dV}{dt}=S_{eff}{J}_{water},$$ (9)

where $J_{water}$ denotes the water flux (see supporting material for details).

Two typical ion fluxes generated by RA (big inactivated rate $k_2$) and SA (small $k_2$) MS channels after a step compression (Fig. 1 g) are shown in Fig. 1 h. The relaxation times of ion fluxes of RA and SA MS channels are $\tau =4.4$ ms and $\tau =259.4$ ms, respectively (Fig. 1 h), which are consistent with previous experimental observations (24,39).

Results

Four distinct firing patterns for various texture wavelengths

We first study how sensory cells containing RA MS channels (such as Piezo1 and Piezo2 (40)) convert information about textured features into their firing patterns during sliding the finger across the textured surface. We find that, for a small λ, the cell only generates one action potential (Figs. 2 a and S3). Notably, small perturbations of $V_m$ in Fig. 2 a are not action potentials, since the depolarized threshold potential is not reached, and there is no depolarization phase and overshoot phase ($V_m>0$) (41). As λ increases to 100 μm, the cell periodically fires action potentials, generating one action potential in every two texture wavelengths (Figs. 2 b and S4). Surprisingly, the cell generates one action potential per texture wavelength when $\lambda =200$ μm (Figs. 2 c and S5), indicating that the firing pattern carries information about the texture wavelength. However, for a very big λ, the cell does not generate any action potential (Figs. 2 d and S6). These findings suggest that sensory cells can only periodically generate action potentials during sliding across surfaces with intermediate texture wavelengths.

Figure 2.

Four distinct action potential patterns during sliding across surfaces with various texture wavelengths. (a) Only one action potential ($\lambda =50$μm). (b) One action potential every two texture wavelengths ($\lambda =100$μm). (c) One action potential per texture wavelength ($\lambda =200$μm). (d) No action potential ($\lambda ={10}^4$μm). $v=10$ mm/s, $F_c=15$ nN, and $A_0=9$μm unless otherwise specified in this work. Other parameters are given in Table S1. To see this figure in color, go online.

During the sliding, the loading frequency of the cell is set by the ratio of the sliding speed to the texture wavelength $v/\lambda$. Hence, for a small λ, the loading is too fast, thus the response of the cell cannot catch up with the periodic loading (Fig. S3 g–i). In contrast, for a big λ, the loading is too slow (quasi-static loading), so MS channels will easily become inactivated (Fig. S6 d–f), leading to zero action potential. Hence, only when the loading frequency is comparable to the cell response timescale can the cell fire in a periodic fashion.

Cells can only convey a limited wavelength range into their firing periods

We next slide the finger across various textures at a fixed speed. For a given speed v, the oscillation period of the skin is $\lambda /v$. To successfully capture texture features, the period of firing pattern needs to reflect that of skin oscillation during finger sliding (6,7). That is when the cell fires one action potential per texture wavelength (Fig. 2 c). In this case, the spatial interval between two adjacent action potentials $\text{Δ}x$ (Fig. 2 c) (for nonperiodic firing patterns, we let $\text{Δ}x=\infty$) is equal to λ. As shown in Fig. 3 a, there is a lower and an upper limit of wavelength, ${\lambda }_l$ and ${\lambda }_u$, only between which $\text{Δ}x=\lambda$. Thus, the firing patterns of sensory cells can only convey sufficient information to identify texture wavelengths in the limited range of $\left[{\lambda }_l\text{,}{\lambda }_u\right]$. We denote this range as the “perception range” of sensory cells (Fig. 3 a). This result also suggests that cells can only encode a limited oscillations frequency, consistent with recent experimental findings (29,42).

Figure 3.

Perception range of sensory cells. (a) For a fixed sliding speed, sensory cells can only recognize textures with wavelengths in a limited range of $\left[{\lambda }_l,{\lambda }_u\right]$. $\text{Δ}x$ is the interval between two adjacent action potentials (see Fig. 2b). (b) ${\lambda }_l$ and ${\lambda }_u$ increase with v. (c) Increase of $F_c$ broadens perception range. (d) Threshold contact force $F_{c,t}$ (green dot in c) decreases with texture amplitude $A_0$. (e) Only in $\left[v_l\text{,}{v}_u\right]$ can sensory cells discriminate a specific texture. $\lambda =100$μm in (d) and (e). (f) Both $v_l$ and $v_u$ increase with λ. To see this figure in color, go online.

Moreover, we find that cells containing SA MS channels usually generate multiple action potentials during the finger sliding across one texture wavelength (Fig. S7 a and b). Thus, the perception range of texture wavelength, i.e., the range of $\lambda /\text{Δ}x=1$, is much smaller than that of cells containing RA MS channels (Fig. S7 c and d). Thus, these results may indicate that the roughness perception under dynamic sliding is mediated primarily by RA MS channels.

Effects of sliding speed and contact force

In reality, we can explore the surface with various sliding speeds or contact forces (6). Thus, an interesting question is how the sliding speed v and the contact force $F_c$ affect the roughness perception. We find that for a constant $F_c$, the perception range increases with v (Fig. 3 b, orange region). Likewise, the lower wavelength limit ${\lambda }_l$ also increases with v (Fig. 3 b, red line). These results demonstrate that a faster sliding speed broadens the perceiving wavelength range at the cost of the reduction of lower perception limit. That may be why humans will use a lower scanning velocity for smooth texture perception (43,44).

For a constant v, the perception range increases with $F_c$ (Fig. 3 c). However, if $F_c$ is smaller than a threshold $F_{c,t}$ (Fig. 3 c, green dot), the cell fails to convey any texture wavelength into the period of firing pattern ($\text{Δ}x/\lambda \ne 1$). Moreover, $F_{c,t}$ decreases as $A_0$ becomes bigger (Fig. 3 d), which is consistent with recent psychophysical experiments where the perception of rougher textures needs smaller contact forces (43,44).

We next study how the sliding speed affects the detection of the texture roughness with a specific wavelength. Similar to Fig. 3 a, there is a lower and an upper limit of sliding speed, $v_l$ and $v_u$, between which $\text{Δ}x=\lambda$ (Fig. 3 e). Thus, only in a limited range of v can sensory cells convey a given texture wavelength λ into their firing periods. Furthermore, both $v_l$ and $v_u$ increase with λ (Fig. 3 f), indicating that the perception of a rougher texture permits a wider range of sliding speed and requires a biggersliding speed. Together, these results demonstrate that we need to optimize v and $F_c$ for the perception of various texture wavelengths.

These results also indicate that cells can only encode oscillations in a limited frequency bandwidth. Let $\left[f_l\text{,}{f}_u\right]$ be the frequency bandwidth. The oscillation frequency of cells is set by the ratio of the sliding speed to the texture wavelength $v/\lambda$. Hence, the lower and upper thresholds of the wavelength are given as ${\lambda }_l=v/f_u$ and ${\lambda }_u=v/f_l$, respectively. On the contrary, the lower and upper thresholds of speed are given as $v_l=f_l\lambda$ and $v_u=f_u\lambda$, respectively. That is why ${\lambda }_l$ and ${\lambda }_u$ increase linearly with v (Fig. 3 b) and why $v_l$ and $v_u$ increase linearly with λ (Fig. 3 f).

Cells actively regulate their perception ranges by expressing various kinds of MS channels

There are various kinds of MS channels with different mechanical properties. For example, Piezo channels open at a low mechanical stimulus (40), while a possible novel channel TACAN opens at high mechanical stimulus (45,46). Therefore, we next study how the opening stress threshold of MS channels, ${\sigma }_s$, affects roughness perception. We find the perception range of texture wavelength, $\text{Δ}\lambda ={\lambda }_u-{\lambda }_l$, is a biphasic function of ${\sigma }_s$ (Figs. 4 a and S8). For a given contact force, if ${\sigma }_s$ is very big, the contact force will be too small to activate MS channels, so the cell cannot fire any action potential. Thus, $\text{Δ}\lambda =0$ (Fig. 4 a). For a given speed, as ${\sigma }_s$ increases, the contact force threshold $F_{c,t}$, above which cells can successfully convey the texture wavelength into the firing period also becomes bigger (Fig. 4 b, white dashed line). These results indicate that the perception range is related to the mechanical properties of MS channels at the molecular level.

Figure 4.

Cells actively regulate their perception ranges by expressing various kinds of MS channels. (a) Perception range $\text{Δ}\lambda ={\lambda }_u-{\lambda }_l$ as a function of v and ${\sigma }_s$ for $F_c=25$ nN. (b) Perception range $\text{Δ}\lambda$ as a function of $F_c$ and ${\sigma }_s$. The white dashed line indicates when $\text{Δ}\lambda =0$. (c) Dependence of $\text{Δ}\lambda$ with v. (d) Dependence of $\text{Δ}\lambda$ with $F_c$. Blue lines represent cells containing only RA channels. Auburn lines represent cells containing both RA and SA channels. Yellow lines represent cells containing both RA channels and SA channels with negative ion efflux. To see this figure in color, go online.

Cells can express multiple kinds of MS channels (45,47), leading to the question of how these channels coordinate to perform roughness perception. For simplicity, we consider sensory cells containing two kinds of MS channels. Previous experiments show that mechanical loading can also elicit an “intermediate” current (47,48), which is considered as the combined effects of RA current and SA current. Hence, we first consider cells containing both RA and SA MS channels. We find that, compared with cells containing only RA MS channels, cells containing these two kinds of MS channels show a broader perception range (Fig. 4 c and d, auburn line) and a smaller lower perception limit (Fig. S9 c and d).

Generally, most SA MS channels show an inward ion current (49,50). However, recent experiments reveal that there are also SA MS channels with an outward ion current acting as mechanical brake in the senses of touch, which are termed as negative channels (51,52,53). Hence, we next study how the negative ion efflux influences roughness perception. We find that the negative SA MS channels will reduce the perception range (Fig. 4 c and d, yellow line) and the lower perception limit (Fig. S9 c and d). Together, these findings show that sensory cells can actively regulate their perception ranges by expressing various kinds of MS channels, which could help us to achieve a more elaborate texture perception.

SA, rather than RA, MS channels contribute to the perception of coarse textures under static touch

Previous experiments show that under static touch, the perception of coarse texture is primarily mediated by SA, rather than RA, mechanoreceptors (7,10). To reveal the underlying mechanism, we next study cell responses under step loadings. Specifically, the compression depth d first increases linearly with a speed k (linear stage) and then keeps constant at $d=d_0$ (static stage) (Fig. 5 a). Note that sufficient ion influx is required to initiate an action potential, and a big ion influx can trigger multiple action potentials (54,55). We find if k or $d_0$ is small, the ion influx generated by MS channels is too small to depolarize the membrane potential (Fig. S10). Thus, there is no action potential (Fig. 5 b, c, e, and f). However, for big k and $d_0$, the ion influx resulting from the mechanical loading is big enough to trigger action potentials (Fig. S10). Moreover, we find that cells with SA MS channels generate multiple action potentials (Figs. 5 b, e, and f and S11 a and c), due to slow attenuation of $J_{ms}$ (Figs. 5 d and S11 b and d). This is consistent with the firing pattern of slow low-threshold mechanoreceptors (1). Moreover, when the cell generates an action potential, the membrane potential changes from negative to positive. Then cations flow out of the cell, leading to a negative ion flux of MS channels (Fig. 5 d).

Figure 5.

SA, rather than RA, MS channels contribute to the perception of coarse textures under static touch. (a) The loading process. (b and c) Firing patterns of cells with SA ($k_2=0.3$ ms⁻¹) or RA ($k_2=0.002$ ms⁻¹) channels. (d) Ion flux of MS channels ($J_{ms}$) for cells containing RA (auburn line) or SA (blue line) channels. (e and f) Action potential (AP) number as a function of k or $d_0$. (g) AP number decreases with inactivated rate $k_2$. (h and i) Phase diagrams of MS channels. White and blue curves indicate where AP number equals 0 and 1, respectively. To see this figure in color, go online.

In contrast, due to fast attenuation of $J_{ms}$ (Fig. S11 f and h), cells containing only RA MS channels generate one action potential during the linear loading stage but zero action potential during the static loading stage (Figs. 5 c, e, and f and S11 e and g). Under static touch, the texture wavelength is mainly sensed by the distance between adjacent sensory cells firing action potentials. Therefore, the multiple action potentials generated by SA MS channels become very important under static touch.

Since the inactivated rate $k_2$ determines the major difference between RA and SA channels (Fig. 1 h), the above results also indicate that the action potential number should decrease dramatically with increasing $k_2$ (Fig. 5 g). Biologically, the classifications of RA and SA channels are primarily based on the relaxation time constant of the decaying current (Fig. 1 h) (24,39). Here, we find there is a critical value of $k_2$ (denoted by $k_{2,c}$ in Fig. 5 g), above which cells can only generate one action potential under static loadings. Therefore, mathematically, we can rigorously classify RA and SA channels according to $k_2$ (Fig. 5 h and i).

Discussion

Although the mechanism of texture perception at the level of afferent fibers has been identified, little is known about how sensory cells convey texture roughness into their firing patterns. Here, we develop a generic theoretical model for roughness perception at the cellular level. Expect the cells presenting in the proximity of fingertips, this generic model may also be extended to multiple cell types that exist in glabrous skin (1,2), such as Merkel cells, Pacinian corpuscles, and Meissner’s corpuscles. Our work thus goes beyond previous studies of texture perception at the afferent level (7,8,9) and further demonstrates the electro-mechanical coupling mechanism in living cells (56,57).

Our work shows that sensory cells convey the texture wavelength into the period of action potentials as the finger slides across the textured surface, which is mediated primarily by RA MS channels. However, we also find that the sensor cells function as a frequency flitter and can only convey a limited range of texture wavelengths into their firing periods, which is denoted by “perception range.” Previous studies have shown that there is a sensitive frequency range of RA and Pacinian afferent fibers (42,58). Thus, our findings indicate that the frequency filter feature of RA and Pacinian afferent fibers may originate from the MS channels expressing in the sensory cells that link to these fibers.

We also show that the perception range strongly relies on the sliding speed of the finger and the contact force between the finger and the textured surface. A bigger sliding speed broadens the perception range; however, the lower perception limit increases with the sliding speed. Thus, a fast sliding speed is adverse to the perception of smooth textures. That may explain why humans need a lower scanning velocity for the perception of fine, smooth surfaces (43,44). Moreover, we find that as the amplitude of texture increases, the smallest contact force needed to successfully convey the texture wavelength into the firing period becomes smaller. This is consistent with the psychophysical observations that humans use a smaller contact force to perceive rough surfaces (43,44,59). Our results thus open up new avenues of roughness perception and may have a potential application in the design of tactile sensors and haptic interfaces for humanoid robots.

There are some limitations in our study. First, we focus on the roughness detection of rigid surface. However, the mechanical properties of the object are also important in texture perception. Especially when the object very soft, the object surface may lose its own textural features due to large deformation induced by finger compression, such that the cells fail to discern the intrinsic texture of the object. One easy way to consider the stiffness of the object is by replacing one of the rigid surface in Fig. 1 c with a flexible surface.

Second, we neglect some other ion species, such as Mg₂⁺ and Ca₂⁺, in our model. Moreover, we assume a sodium-specific flow for MS channels. However, most MS channels are nonselective cation channels, thus other ion species can also flow across the MS channels. Thus, how the transports of other charged species affect the roughness perception is an important open question for future studies.

Third, the local force on the cell is different from that on the tissue surface. However, since we do not consider the skin mechanics, our model cannot connect the tissue compression force to the local cell compression force. Future studies can include the skin mechanics into our theoretical framework, for example, based on continuum mechanics (60). The cells are also embedded within complicated tissue structures in vivo. However, these structures and the interactions between cells and the surrounding tissue are not considered in our model. How the tissue structures affect the roughness detection deserves more studies.

Fourth, the model does not incorporate the effects of fingerprints. However, the fingerprints may filter vibration frequency during finger sliding (6,42). Thus, there are at least two principal vibration frequencies during finder sliding: one is set by the ratio of the sliding speed to the texture wavelength, and the other is set by the ratio of the sliding speed to the wavelength of the fingerprint. How these two oscillation frequencies cooperate and compete with each other in the coding of surface roughness may need further studies.

Author contributions

H.J. and Y.Y. initiated and supervised the project. F.M. performed the simulations. All authors analyzed the data and wrote the manuscript.