Mechanical memory of excitable cells induced by mechanosensitive channel inactivation

Abstract

Recent experiments have unveiled a fascinating phenomenon: excitable cells exhibit mechanical memory, whereby their present excitation behaviors strongly depend on their past mechanical experiences. However, the underlying mechanism of this phenomenon remains elusive. Here, we introduce an electromechanical framework that integrates mechanical cell deformation, state transformations of mechanosensitive (MS) channels (such as Piezo channels), and transmembrane ion fluxes. We reveal that MS channel inactivation yields history-dependent excitation dynamics, characterized by a progressive decline in subsequent activated currents with increasing amplitude, speed, and duration of prior mechanical stimuli. Moreover, MS channel inactivation in preceding stimulation results in a refractory period during which cells cannot elicit new action potentials upon subsequent mechanical stimuli. Finally, we show that cells can adapt to preceding mechanical stimulation due to inactivation of MS channels, resulting in a higher activated threshold stimulation. Thus, MS channel inactivation favors the reduction of firing activities in response to prolonged and repeated mechanical stimuli (“neural adaptation”), which may protect neurons against overactivation and damage. We then conduct two virtual experiments to predict how changes in mechanical properties of neurons modify their excitation behaviors. These findings together emphasize a critical role of MS channel inactivation in governing the mechanical memory and neural adaptation of excitable cells, shedding new light on the intricate interplay between mechanical forces and cellular electrical responses.

Significance

This study introduces an electromechanical framework to elucidate the mechanical memory and neural adaptation in excitable cells, mediated by mechanosensitive (MS) channel inactivation. We demonstrate that MS channel inactivation induces history-dependent excitation dynamics, characterized by reduced activated currents and a refractory period post stimulation. This adaptation elevates the threshold for subsequent activations, protecting cells from overexcitation and damage. Our findings highlight the pivotal role of MS channel inactivation in modulating cellular responses to mechanical stimuli, offering new insights into the interplay between mechanical forces and electrical signaling in excitable cells.

Introduction

Memories prevail in both living and non-living systems. In physical systems, the remarkable property of ferroelectric materials switching between two stable polarization states can be harnessed to create non-volatile memories (1,2). Similarly, chemical systems utilize switchable oscillating reactions, such as the Belousov-Zhabotinsky reaction, to develop chemical computers capable of retaining memory (3,4). In living biological systems, stem cells can retain information about past mechanical environments, and this mechanical history significantly influences future cell-fate decisions (5,6). Moreover, excitable cells such as neurons and cardiomyocytes exhibit memory (7,8,9), as their activation dynamics show multiple timescales (10,11).

A remarkable feature of systems exhibiting memory is that their present dynamic behaviors depend on past history. For instance, the mechanical properties of living cells are closely linked to their loading history (12), as living cells are open systems capable of exchanging water and ions with their external environments (13,14). Experimental investigations have also shown that the activated currents of excitable cells, when subjected to mechanical forces, exhibit a strong dependence on their loading history. Consequently, even when the same mechanical stimulation is applied to the same nerve cell, the activated currents vary significantly for different loading histories (15,16,17). Changes in neuronal responses induced by prior cell stimulation are also known as neural adaptation in neuroscience (18,19,20). However, the mechanism underlying the mechanical memories of excitable cells remains poorly understood.

Traditional conductance-based models (21,22,23) have served as the prevailing theoretical framework for elucidating the excitation mechanism of excitable cells. Unfortunately, these models are purely electrical, as they treat the cell as a current circuit consisting of capacitors and conductors. However, excitable cells, to be excitable in response to mechanical stimuli, must convert the force-induced deformation into electrical signals. Thus, existing conductance-based models (21,22,23,24) fail to describe the activation dynamics of cells triggered by mechanical forces, as they neglect cellular mechanical deformation and lack explicit incorporation of mechanical transducer.

Prime candidates serving as force sensors may include mechanosensitive (MS) channels embedded in the cell membrane, such as Piezo1 (25), Piezo2 (26,27), and TACAN (28,29). The protein conformations of these channels change reversibly with membrane tension (deformation) (30,31). Thus, mechanical cellular deformation can affect the activation dynamics of cells by altering the conformation of MS channels (32,33). For example, a step force can open the MS channel and activate ion current. The activated currents then decay to zero under constant forces, a phenomenon known as inactivation (15,17,26,28,34,35). Based on the time constant of decaying current during the inactivated phase, MS channels can be broadly classified into rapidly adapting (RA, <10 ms), intermediately adapting (IA, 10$–$100 ms), and slowly adapting (SA, >100 ms) channels (15,17,26,28,34,35). Lewis et al. demonstrated that channel inactivation governs frequency filtering in repetitive mechanical stimuli (24). Similarly, Azevedo and Wilson revealed that voltage-gated conductances shape frequency-dependent responses in Drosophila mechanosensory circuits (36). These studies collectively highlight the importance of both channel-intrinsic properties and cellular-scale mechanisms in mechanotransduction. However, a theoretical model accounting for the coupling between mechanical cell deformation, conformation changes of MS channels, and the electrical dynamics of excitable cells is still lacking.

In this work, we develop an electromechanical framework that integrates the conformation changes of MS channels, mechanical cellular deformation, and electrical ion fluxes. We find that inactivation of MS channels leads to history-dependent responses of cells and results in a mechanical refractory period, suggesting that MS channel inactivation plays a critical role in the mechanical memory of excitable cells. Cells containing rapidly adapting MS channels, such Piezo1 and Piezo2, exhibit greater sensitivity to loading history compared to cells containing slowly adapting MS channels. These findings are all quantitatively consistent with previous experimental observations. Furthermore, we find that inactivation of MS channels favors the reduction in firing activities in response to prolonged and repeated mechanical stimuli, which are also referred to as “neural adaptation.” This mechanism may enables neurons to conserve energy and become more sensitive to new or changing stimuli while ignoring constant, insignificant background stimuli, thereby helping in optimizing sensory perception, decision making, and learning.

Materials and methods

Mechanical stimuli

In most electrophysiological experiments, a glass pipette cemented to a piezo-electric actuator is usually used to mechanically compress the cell (15,17,26,37) (Fig. 1 a, top). Thus, we also use a pipette to apply mechanical stimuli to the cell (Fig. 1 a, bottom). For a glass pipette with an indenter half-conical angle $\alpha$ and a indenter tip radius $r_p$ (Fig. 1 b, right), the total cellular indentation depth $d$ (Fig. 1 b) is given by

$$d=d_1+d_2+d_3,$$ (1)

$$d_1=H_0-r_a\left[E_1\left(\pi \right)-E_1\left({\theta }_0\right)-A{E}_2\left(\pi \right)+A{E}_2\left({\theta }_0\right)\right]+r_{a}B\cos {\theta }_0+r_{a}B,$$ (2)

$$d_2=r_a\left[E_1\left(\pi \right)-E_1\left(\frac{3}{2}\pi -\alpha \right)-A{E}_2\left(\pi \right)+A{E}_2\left(\frac{3}{2}\pi -\alpha \right)\right]+r_{a}B-r_{a}B\sin \alpha ,$$ (3)

$$d_3=\left(r_d-r_p\right)\cot \alpha ,$$ (4)

where $H_0$ is the initial cell height before cell indentation (Fig. 1 a),

$$E_1\left(\theta \right)={\int }_0^{\theta }\sqrt{A+B^2{\sin }^2\psi }d\psi ,E_2\left(\theta \right)={\int }_0^{\theta }1/\sqrt{A+B^2{\sin }^2\psi }d\psi ,B=\sigma h_c/\left(\Delta P{r}_a\right),A=1-2B\sin {\theta }_0,\sigma$$

is the membrane stress, $h_c$ is the membrane thickness, $\Delta P$ is the hydrostatic pressure difference across the cell membrane, $r_a$ and ${\theta }_0$ are the contact radius and angle between the cell and the substrate, and $r_d$ is the biggest contact radius between the pipette and the cell (Fig. 1 b). More details regarding the cell morphology under mechanical indentation are given in the supporting material.

Figure 1.

Electromechanical model for excitable cells. (a) Typical experimental setups (top, adapted with permission from (38)) and the schematic of our model (bottom). Cell indentation depth is denoted by $d$, with the cell’s height before and after indentation indicated as $H_0$ and $H$, respectively. (b) Cross section of the cell shape under an indentation force $F_c$ exerted by a glass pipette. The half-conical angle of the pipette is $\alpha$, its tip radius is $r_p$, and $r_d$ is the biggest contact radius between the pipette and the cell (enlarged view on the right). The cell indentation depth $d$ consists of global deformation $d_1$, local deformation $d_2$, and contact height $d_3$. The contact radius and angle between the cell and substrate are denoted by $r_a$ and ${\theta }_0$, respectively. The dashed green line represents the cell shape before indentation. (c) Cellular deformation induced by mechanical stimuli alters the conformations of MS channels, which then trigger subsequent electrical responses. (d) MS channels have three conformations: closed, open, and inactivated. Increased membrane tension promotes both the opening and inactivation of MS channels, whereas deceased membrane tension favors MS channel closure. (e) Schematic of the energy landscape of MS channel. The proportions of channels in closed, open, and inactivated states are denoted by $C,O$, and $I$, respectively. Transition rates between these states, $k_i$ (i = CO, OI, IC, CI, IO, OC), are partly dependent on the membrane stress $\sigma$.

As the cell is indented, the cell deforms, leading to change in cellular surface area and membrane tension, which then induces a cascade of changes in electric signals, including ion flux through MS channels and generation of action potential (Fig. 1 c). The cellular surface area under indentation, $S$, is given as (see supporting material for more details)

$$\begin{array}{cc}S& =2\pi {\left(r_{a}B\right)}^2\left[\frac{2}{B}\left(E_1\left(\frac{3\pi }{2}-\alpha \right)-E_1\left({\theta }_0\right)\right)-\frac{A}{B}\left(E_2\left(\frac{3\pi }{2}-\alpha \right)-E_2\left({\theta }_0\right)\right)+2\cos {\theta }_0+2\sin \alpha \right]\\ & +\pi r_a^2+\pi r_p^2+\frac{\pi }{\sin \alpha }\left(r_d^2-r_p^2\right)\text{.}\end{array}$$ (5)

Under mechanical indentation, the cell volume may also change. Previous studies have shown that changes in cell volume can affect the adhesive and detachment dynamics of single cells (12,39), as well as the wave generation and stress field of cell monolayers (14,40). However, the timescale of activation dynamics of excitable cells is on the order of milliseconds, which is much shorter than the timescale of cell volume regulation (on the order of 10 min). Thus, we assume that the cell volume is constant in this study. While volume remains constant during subsecond activation dynamics, slower volume regulation processes (minutes to hours) may modulate membrane tension and MS channel states through mechanochemical feedback. This emergent timescale hierarchy warrants further investigation.

Membrane mechanics

Motivated by previous works (13,39,41), we consider the membrane and cortex as a unified layer. The apparent stress of this combined layer is expressed as $\sigma ={\sigma }_a+{\sigma }_p$ (13,39,41), where ${\sigma }_a$ is the active stress generated by myosin contraction, and ${\sigma }_p$ is the passive stress primarily resulting from deformation of the actin cortex, as the cortical tension is significantly larger than the membrane tension (42,43). Generally, the actin cortex exhibits viscoelastic properties, where its viscosity results from the turnover of actin network (44,45). However, the activation timescale of excitable cells (∼milliseconds) is much shorter than that of actin turnover (∼10 s) (44,45). Consequently, on timescales shorter than its turnover, the actin cortex behaves akin to an elastic shell (44,45). Moreover, we deliberately exclude the relaxation of membrane tension to study whether conformation changes of MS channel alone is sufficient to explain previous experimental observations. Thus, we treat the combined layer of cell membrane and actin cortex as an active contractile elastic layer. The constitutive equation of this combined layer can be described by (13,14,39,41)

$$\sigma =\frac{E}{2}\left(\frac{S}{S_0}-1\right)+{\sigma }_a,$$ (6)

where $E$ is the elastic modulus, ${\sigma }_a$ is the active stress generated by myosin contraction, and $S_0$ and $S$ (Eq. 5) are the reference and deformed cell surface area, respectively.

Kinetics of MS channels

Previous experiments have convincingly demonstrated that membrane tension can modify the conformations of MS channels (46,47,48,49) (see also Fig. 1, c and d). Thus, mechanical tension can activate electrical ion currents by opening MS channels (32,33). Additionally, following the transition from a closed to an open conformation, the activated currents of MS channels can decay to zero under constant stimuli, which is referred to as inactivation (15,17,26,50,51). For example, globular C-terminal extracellular domain confers kinetics of inactivation of Piezo ion channels (50,51), and this channel inactivation is the molecular mechanism underlying frequency filtering under repetitive mechanical stimuli (24). Hence, we use three states, i.e., closed, open, and inactivated, to describe the behaviors of MS channels (Fig. 1 d).

As the open probability of MS channels increases with membrane tension (26,27), we assume that an increase in membrane stress promotes transitions from the closed state to both open and inactivated states (Fig. 1 d). Conversely, a decrease in membrane tension promotes transitions from open and inactivated states back to the closed state. Thus, the membrane stress $\sigma$ can modulate the energy barriers between various states and affect the transition rates among different states (Fig. 1 e). A recent study shows that Piezo1 density has no effect on its mechanical sensitivity or open probability, indicating that Piezo channels behave as independent mechanotransducers (52). Thus, the total open probability for all the MS channels in the membrane can be summed up and described by a single open probability $O$. Taken together, the dynamic equations governing the state transformations of MS channels are given as

$$\frac{dC}{dt}=-k_{CO}\left(\sigma \right)\beta C-k_{CI}\left(\sigma \right)C+k_{OC}\left(\sigma \right)O+k_{IC}\left(\sigma \right)I,$$ (7)

$$\frac{dO}{dt}=-k_{OI}O-k_{OC}\left(\sigma \right)O+k_{CO}\left(\sigma \right)\beta C+k_{IO}I,$$ (8)

$$\frac{dI}{dt}=-k_{IO}I-k_{IC}\left(\sigma \right)I+k_{CI}\left(\sigma \right)C+k_{OI}O,$$ (9)

where $C,O$, and $I$ are the proportions of MS in closed, open, and inactivated states, respectively. Transformation rate coefficients $k_i$ (where i = CO, OI, IC, CI, IO, and OC) govern the transitions between these states and are partly dependent on the membrane stress (see supporting material for more details). Previous studies show that there are two recovery-time constants of the activated current after removing the mechanical stimuli (15,17,24). The short recovery time accounts for the transition from inactivated state to closed state, while the existence of a long recovery time indicates that there is a second inactivated state in the long timescale. To account for this long-time recovery, we introduce a variable $\beta$, representing the percentage of closed MS channels available for transformation to the open state (see supporting material for details).

Ion flux through MS channels

Ions can only flow across MS channels when the channels are in open states (26). After being open, ions flow through MS channels driven by the electrochemical potential gradient of ions across the membrane. The electrochemical potential of an ion $\Delta {\mu }_{\text{ion}}$ is the sum of chemical potential and electrical potential $\Delta {\mu }_{\text{ion}}=RT\ln {\mathrm{\Gamma }}_{\text{ion}}+zF{V}_m$ (53,54), where ${\mathrm{\Gamma }}_{\text{ion}}={\left[\text{ion}\right]}_{\text{in}}/{\left[\text{ion}\right]}_{\text{ex}}$ is the ratio of intracellular to extracellular ion concentrations, $R$ is gas constant, $T$ is temperature, $z$ is the valence of ion, $F$ is the Faraday constant, and $V_m$ is the membrane potential. It has been shown that most MS channels are non-selective cation channels (26,27,55). In addition, ${\text{Na}}^{+}$ and ${\mathrm{K}}^{+}$ are the most frequent cations inside nerve cells (21,56). Therefore, we consider both the fluxes of ${\text{Na}}^{+}$ and ${\mathrm{K}}^{+}$ through MS channels (denoted by $J_{ms,\text{Na}}$ and $J_{ms,\mathrm{K}}$), which are given as

$$J_{ms,\text{Na}}=-g_{\text{Na}}O\left(RT\ln {\mathrm{\Gamma }}_{Na}+F{V}_m\right),$$ (10)

$$J_{ms,\mathrm{K}}=-g_{\mathrm{K}}O\left(RT\ln {\mathrm{\Gamma }}_K+F{V}_m\right),$$ (11)

where $g_{\text{Na}}$ and $g_{\mathrm{K}}$ are permeation rates of ${\text{Na}}^{+}$ and ${\mathrm{K}}^{+}$, respectively. It has been shown that the ionic permeations of ${\text{Na}}^{+}$ and ${\mathrm{K}}^{+}$ across MS channels are nearly identical (57), thus we assume $g_{\text{Na}}=g_{\mathrm{K}}=g_{ms}$ for simplicity. The total ion flux through MS channels, $J_{ms}$, is the sum of $J_{ms,\text{Na}}$ and $J_{ms,\mathrm{K}}$, and is given by

$$J_{ms}=-g_{ms}O\left(2F{V}_m+RT\ln {\mathrm{\Gamma }}_{Na}+RT\ln {\mathrm{\Gamma }}_K\right)\text{.}$$ (12)

Kinetics of membrane potential

To account for the dynamic of membrane potential, we use the well-known Hodgkin-Huxley (H-H) model. According to the H-H model (21), the generation of action potential relies on voltage-gated ${\text{Na}}^{+},{\mathrm{K}}^{+}$, and ${\text{Cl}}^{-}$ channels on the membrane. The ion fluxes across these voltage-gated channels, $J_i$ ($i=$ $\text{Na},\mathrm{K}$, and $\text{Cl}$), depend on the membrane potential $V_m$, and can be written as

$$J_{Na}=-\frac{1}{F}{g}_{Na}{m}^{3}h\left(V_m-E_{Na}\right),$$ (13)

$$J_K=-\frac{1}{F}{g}_K{n}^4\left(V_m-E_K\right),$$ (14)

$$J_{Cl}=\frac{1}{F}{g}_{Cl}\left(V_m-E_{Cl}\right),$$ (15)

where $g_i$ ($i=$ $\text{Na},\mathrm{K}$, and $\text{Cl}$) are the specific membrane conductances, $E_i$ are the equilibrium (or Nernst) potentials, and $n,m$, and $h$ are gating variables that range from 0 to 1 (21) (see supporting material for details). The flux of ion across the membrane may change the ionic concentration inside the cell. However, the timescale of action potential is on the order of milliseconds, which is much shorter than that of ion concentration regulation (∼minutes). As a result, the ionic concentrations are almost constant (Fig. S2). Thus, we do not consider the changes in ion concentrations during the activated dynamics of cells.

The membrane potential $V_m$ is regulated by ion fluxes across all ion channels. Thus, the dynamic equation of $V_m$ is given as (21,58)

$$C_m\frac{d{V}_m}{dt}=F\left(J_{Na}+J_K-J_{Cl}+J_{ms}\right),$$ (16)

where $C_m$ is the cell membrane capacitance per area.

Numerical methods

Calculations were performed in two steps. First, we use the MATLAB function ode15s to solve the steady states of the coupled differential-algebraic equations (Eqs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16). Using these steady states as initial conditions, we then simulate the activated dynamic of the cell under various mechanical stimuli.

Results

Activated currents of MS channels under step mechanical indentations

We first study the activated properties of MS channels under step mechanical stimuli. To this end, mechanically activated currents of RA and SA MS channels are quantified by subjecting the cell to incremental step indentations while clamping the membrane voltage at $V_m=-65.7$ mV (Fig. 2 a). We find that, under step indentation, the membrane stress $\sigma$ initially rises with increasing indentation depth $d$ and then remains constant after $d$ is kept constant (Fig. 2 b). However, for small $d$, the membrane stress is too small to open the MS channel, resulting in zero activated current (Fig. 2, c and d, blue curves). Thus, there exists a critical depth above which the mechanical indentation can open MS channels, resulting in non-zero activated currents (Fig. 2 g, blue region), which is consistent with previous experimental findings (15,17). In this case, the activated currents initially rise to their peaks (denoted by $I_p$) before gradually diminishing to zero (Fig. 2, c and d, red and yellow lines).

Figure 2.

Decaying timescales of the mechanically activated current of RA and SA channels are different, whereas their activated behaviors are similar. (a) Cells undergo ramp-and-hold indentation with an amplitude of $d$, while the membrane voltage is clamped at $V_m=-65.7\text{mV}$. The loading speed is 1 $\mu$m/ms, consistent with previous experiments (15). (b) Dynamic changes of the membrane stress $\sigma$. (c and d) Mechanically activated currents generated by RA and SA mechanosensitive (MS) channels for different $d$. The peak of activated current is denoted by $I_p$, with $I_{p,\max }$ representing the maximum peak current as $d$ increases. The decaying timescale $\tau$ of RA channel is around 3.5 s (estimated from single-exponential fits, dashed line), whereas the decaying time of SA channel is around 309 s. (e and f) Fraction of MS channels in closed (C), open (O), and inactivated (I) states. (g) Normalized current-depth activation curve. For small $d$, the activated current is zero, indicating no MS channels are closed (purple region). As $d$ further increases, the MS channel will open (blue region), after which the activated current reaches saturation (gray region). Solid lines are our simulation (Sim.) results, whereas circles are experimental (Exp.) data of dorsal root ganglion (DRG) neurons (15).

The decline in activated current may be attributed to either the relaxation of membrane stress acting on the MS channels or the transition of MS channels into inactivated states. We find that the membrane stress $\sigma$ remains constant after $d$ is kept constant (Fig. 2 b), indicating that the current decline observed here is not due to the relaxation of membrane stress. As shown in Fig. 2, e and f, the fractions of MS channel in the open and closed state decrease (Fig. 2, e and f, purple and green lines), while the inactivated fraction of MS channels increases after $d$ is kept constant (Fig. 2, e and f, blue lines). To further explore the contribution of stress dissipation to the current decline, we then use a viscoelastic constitutive law to describe the membrane stress (an elastic spring connected in series with a viscous dashpot). We find that the stress dissipation only contributes slightly to the decline of activated current under constant compression depth (Fig. S5), which further justifies the assumption of an elastic constitutive law. These results indicate that MS channels transform into inactivated states under constant stimuli, consequently leading to the decline in activated current.

The inactivation dynamics of RA and SA channels are distinct. Specifically, the activated currents of RA channels rapidly decay to zero under constant $d$ (Fig. 2 c). The relaxation time constant derived from exponential fits (dashed line in Fig. 2 c) of the decaying phase of currents ranges from 3.5 to 8 ms as $d$ varies (Figs. 2 c and S6 a). In contrast, the activated currents of SA channels exhibit a much slower decay (Fig. 2 d), with a significantly larger relaxation-time constant ranging from 309 to 317 ms (Figs. 2 d and S6 b), consistent with prior experimental data (15,17).

Despite strong differences in their inactivated behaviors, the activated curves (current-depth curves) of RA and SA channels overlap (Fig. 2 g), suggesting that their activated behaviors are indistinguishable. Their peak currents $\left(I_p\right)$ both increase with $d$ as more MS channels open, and approaches a saturation value $I_{p,\max }$ above an upper critical $d$ value when all channels are open (Fig. 2 g, gray region). Moreover, we find that our simulation results (solid lines in Fig. 2 g) are in good agreement with previous experimental data (15) (circles and squares in Fig. 2 g). By fitting our simulation to previous experimental data, we can determine both the transition rates of RA and SA MS channels (listed in Table S1), which then remain constant throughout all the following simulations. Taken together, these findings reveal that even the inactivated timescales of RA and SA channels are distinct, and their activated dynamics are similar.

Inactivation of MS channels in prior loadings leads to smaller activated currents in subsequent loadings

Motivated by recent experiments (15,17), we next study how MS channel inactivation during prior stimuli affects subsequent activation dynamics of cells by employing a two-step stimulation protocol. This protocol involves a first conditioning loading followed by a test loading, while the membrane potential is clamped at $V_m=-65.7\text{mV}$ . The first loading, referred to as the historical loading, is a step compression with an amplitude $d_h$, a loading speed $v$, and a duration of $T_h$ (light-green region in Fig. 3 a). After removing the first loading, a recovery time $T_r$ (gray region in Fig. 3 a) is included to allow cells to erase the effects of first loading before applying the second stimulus. The amplitude and duration of the second loading are denoted by $d_r$ and $T_2$, respectively (purple region in Fig. 3 a).

Figure 3.

Inactivations of MS channels during prior loadings results in reduced subsequent activated currents. (a) Two-step loading protocol while clamping the membrane voltage at −65.7 mV. Initially, a first loading with a duration of $T_h$ and an amplitude $d_h$ is applied, then cells undergo a recovery period of time $T_r$ before a second stimulus with a duration of $T_2$ and an amplitude $d_r$ is applied. The loading and unloading speeds are equal and denoted by $v$. Unless specified otherwise, $d_h=5.5\mu \mathrm{m},T_h=10\text{ms},v=1\mu \mathrm{m}/\text{ms},T_r=20\text{ms},d_r=8.5\mu \mathrm{m}$, and $T_2=10\text{ms}$. (b) Activated currents generated by RA channels under the loads depicted in (a). Here, $I_{p,w}$ represents the peak current with first loading, while $I_{p,wo}$ represents the peak current without first loading. (c and e) Decaying degree in the peak current $1-I_{p,w}/I_{p,wo}$ as a function of $d_h$ and $v$. (d and f) Fractions of MS channels in the inactivated state before applying the second loading (green point in a) as a function of $d_h$ and $v$. (g) Decaying degree $1-I_{p,w}/I_{p,wo}$ as a function of $T_h$. (h and i) Decaying degree $1-I_{p,w}/I_{p,wo}$ as a function of $T_r$ for RA and SA channels, respectively. Circles in (c) and (h) are experimental data from (15).

As shown in Fig. 3 b, the peak of activated current with first loading, $I_{p,w}$, is smaller than that without first loading, $I_{p,wo}$, indicating reduced sensitivity of the cell to subsequent mechanical stimuli after experiencing a prior loading, a phenomenon termed “desensitization” in previous experiments (15,17). To further investigate the degree of desensitization induced by the first loading, we quantify the decaying degree in the peak current with $1-I_{p,w}/I_{p,wo}$. A value of $1-I_{p,w}/I_{p,wo}=1$ implies that MS channels are fully desensitized, while $1-I_{p,w}/I_{p,wo}=0$ indicates that the first loading has no effect on the activation dynamics of cells during the second loading.

We first focus on cells containing RA MS channels. For small $d_h$, MS channels exhibit no desensitization and $1-I_{p,w}/I_{p,wo}\approx 0$ (Fig. 3 c, red curve). Before applying the second loading, the ratio of MS channels in inactivated state is nearly 0 for small $d_h$ (Fig. 3 d, purple region), indicating that no MS channel becomes inactivated during the first loading. This is because MS channels cannot transition from the closed state to open state under small $d$ (i.e., the activated current is zero, see Fig. 2, c and g), thereby no channels become inactivated (Fig. 3 d, purple region). Thus, the first loading cannot affect the activated currents triggered by the following loading. However, as $d_h$ increases, the desensitization effect becomes more pronounced, leading to an increase in $1-I_{p,w}/I_{p,wo}$ (Fig. 3 c, red curve). This is attributed to more MS channels transitioning into inactivated states during the first loading as $d_h$ increases (Fig. 3 d, blue region), thereby reducing the availability of MS channels for subsequent stimulus. Thus, these results suggest that a stronger first loading results in more significant reduction of the activated currents during the second loading. In addition to $d_h$, the speed of first loading $v$ also affects the desensitization degree. As $v$ increases, the decaying degree in peak current $1-I_{p,w}/I_{p,wo}$ significantly increases (Fig. 3 e, red curve), as a faster first loading promotes the inactivation of MS channels (Fig. 3 f, red curve).

In contrast to RA channels, various values of $d_h$ and $v$ only have slight effects on the desensitization degree of SA channels (blue curves in Fig. 3, c and e), as SA channels hardly become inactivated during the first loading (Fig. 3, d and f, blue curves). These results suggest that the mechanically activated current and inactivation of SA MS channels are barely influenced by short-term loading history (on the scale of 10 ms).

It is worth noting that our simulation results are in good agreement with previous experimental findings (15) (circles in Fig. 3 c). To further examine whether our simulation results rely on the specific values of fitting parameters, we perform parametric sensitivity analysis for the activated current ratio $I_{p,w}/I_{p,wo}$ when the amplitude of first loading is $d_h=9\mu \mathrm{m}$. We find that the local sensitivities of various parameters are all on the order of 1 (Figs. S3 and S4), indicating that our simulation results are insensitive to the changes in fitting parameters.

We then study how the duration of the first loading, $T_h$, affects the activated current elicited by the second loading. We find that for both classes of MS channels, the decaying degree in the peak current increases as $T_h$ increases (Fig. 3 g), as the inactivated fraction of MS channels increases with longer durations $T_h$ (Fig. S7, a and b). Moreover, we find that the degree of decay $1-I_{p,w}/I_{p,wo}$ decreases as $T_r$ increases (Fig. 3, h and i), indicating that more MS channels recover from the inactivated state to the closed state with longer recovery times (Fig. S7, c and d). However, a clear distinction was observed between RA and SA channels: RA channels require a longer recovery time for the peak current $I_{p,w}$ to return to the unloaded value $I_{p,wo}$ (i.e., when $1-I_{p,w}/I_{p,wo}\approx 0$) (Fig. 3 h), primarily due to significant inactivation during the initial loading phase. The recovery time for RA channels is typically around 1000 ms (Fig. 3 h), while for SA channels it is approximately 100 ms (Fig. 3 i). These findings indicate that, following the removal of mechanical stimuli, the recovery time for RA channels to achieve full availability for reopening (i.e., the regulation time of $\beta$) exceeds that of SA channels. Our simulation results regarding the effects of $T_r$ also align well with previous experimental data (15) (Fig. 3 h), further validating the robustness of our model.

To further assess the role of MS channel inactivation in generating history-dependent responses of activated currents, we investigated the current responses after disabling MS channel inactivation. Remarkably, when MS channels were unable to undergo inactivation (i.e., with the ratio of MS channels in the inactivated state set to 0), the mechanically activated currents during subsequent stimuli were independent of the speed, amplitude, and duration of prior loadings (Fig. S8). These findings strongly suggest that MS channel inactivation during prior loadings is essential for the mechanical memory of excitable cells.

Taken together, these findings suggest that MS channel inactivation during prior loadings leads to history-dependent activation dynamics, characterized by a decline in the mechanically activated current in subsequent stimuli with increasing amplitude, speed, and duration of prior loading. This decline occurs because MS channel inactivation reduces the number of available channels for activation during subsequent stimuli. Additionally, due to the rapid inactivation of RA channels, cells possessing RA channels exhibit greater sensitivity to short-term loading history compared to those with SA channels.

Mechanical refractory period results from MS channel inactivation in prior loadings

Next, we investigate how the decline in mechanically activated currents affects the firing dynamics of cells. We propose that this decline may hinder the generation of action potential in subsequent stimuli, as a sufficient ion influx is necessary to trigger action potentials. To test this hypothesis, we investigate the firing dynamics in a two-step loading protocol without clamping the membrane voltage (Fig. 4 a).

Figure 4.

Mechanical refractory period results from MS channel inactivation in preceding stimuli renders cells non-excitable. (a) Two-step loading protocol without clamping the membrane voltage. First, an initial loading with an amplitude $d_h$ and a duration of $T_h$ is applied, followed by a recovery period of $T_r$ before the subsequent stimulus. Unless specified otherwise, $v=1\mu \mathrm{m}/\text{ms},T_h=4000$ ms, $T_r=5$ ms, and $d_r=8.5\mu \mathrm{m}$. (b) Firing curves of cells under various $d_h$. The cell is either excitable or non-excitable during the second stimulus (dashed gray box). (c–e) The refractory period, during which the cell is non-excitable, lengthens with increasing amplitude $d_h$, duration $T_h$, and speed $v$ of the first loading. (f–h) The critical loading threshold, above which the cell can generate action potential, increases with the amplitude $d_h$, duration $T_h$, and speed $v$ of first loading.

We first focus on cells with RA MS channels. We observe that cells can still fire action potentials in the subsequent mechanical stimulus for a small amplitude of first loading $d_h$, i.e., cells are excitable (red line in Fig. 4 b, dashed gray box). However, for a large $d_h$, cells fail to fire action potentials in the subsequent stimulus, i.e., cells are non-excitable (blue line in Fig. 4 b, dashed gray box), as the decaying degree of activated current increases significantly with increasing $d_h$ (Fig. 3 c).

In previous sections, we have demonstrated that the mechanically activated current can recover its initial value without first loading if the recovery time $T_r$ is sufficiently long (Fig. 3 h). Thus, with a prolonged $T_r$, the cell may regain excitability even when subjected to a large $d_h$. We observe a significant increase in the recovery time required for the cell to regain excitability as $d_h$ increases (Fig. 4 c). Therefore, there is a “refractory period,” during which a second stimulus fails to elicit new action potentials, rendering the cells non-excitable. This refractory period may protect nerve cells against overactivation and damage. Moreover, this refractory period differs from the absolute refractory period observed during the depolarization phase of action potentials (59,60). While the refractory period identified here arises from MS channel inactivation, the absolute refractory period during the depolarization phase of action potential is caused by the inactivation of voltage-gated ${\text{Na}}^{+}$ channels (59,60).

The critical recovery time, beyond which cells are excitable, increases with the duration time $T_h$ and the speed $v$ of the first loading (Fig. 4, d and e). This trend suggests that cells require a prolonged recovery period to regain excitability after undergoing extended or rapid mechanical loading in the past. Thus, these findings imply the potential to modulate the excitability of cells by subjecting them to diverse forms of prior loading. Future experiments could further validate these predictions by quantifying the impact of prior loading on the firing patterns of excitable cells.

Due to the attenuation of activated current induced by past loadings, the critical loading amplitude required to trigger an action potential in subsequent stimuli increases with increasing $d_h,T_h$, and $v$ (Fig. 4, f–h). Note that we find similar results for cells containing SA MS channels as well (Fig. S9). Altogether, these results suggest that cells become insensitive after their MS channels become inactivated in prior stimuli. Thus, a stronger stimulus is needed to trigger action potentials, which may be detrimental to nociceptive sensation.

Adaptation of cellular mechanically activated behaviors results from MS channel inactivation

Changes in neuronal responses stemming from prior cell stimulation are sometimes referred to as “neural adaptation.” To further elucidate how cell adaptation arises from MS channel inactivation, we subject the cell to a range of preloads with varying durations $T_c$ and amplitudes $d_c$ while clamping the membrane voltage at −65.7 mV (Fig. 5 a). In contrast to the mechanical stimuli used in Figs. 3 and 4, the prior loading is not removed before compressing the cell to the target amplitude $d_r$ (Fig. 5 a). Similar to diminished currents activated by subsequent loadings shown in Fig. 3, we also observe a reduction in the peak of mechanically activated current with preload (Fig. 5 b), indicating MS channel inactivation induced by the preload. These findings suggest the presence of history-dependent activation behaviors across various forms of prior mechanical stimuli.

Figure 5.

Adaptation of cellular mechanically activated behaviors induced by MS channel inactivation. (a) Blue line: the cell is preloaded with an amplitude $d_c$ and a duration $T_c$ before being compressed to the target amplitude $d_r$. Red dashed line: the cell is compressed directly to $d_r$ without preload. The membrane voltage is clamped at −65.7 mV. Loading and unloading speeds are fixed at $v=1\mu \mathrm{m}/\text{ms}$, and the duration of target load is 200 ms. Unless specified otherwise, $d_c=5.5\mu \mathrm{m}$ and $d_r=9\mu \mathrm{m}$. (b) Activated currents generated by RA channels under the loads depicted in (a), where $I_{p,wo}$ represents the peak current without preload and $I_{p,w}$ represents the peak current with preload. (c and d) Current ratio $I_{p,w}/I_{p,wo}$ as a function of $d_c$ under various $d_r$ for RA channels (c) and SA channels (d). Pillars are experimental data with $d_r=6\mu \mathrm{m}$ (17) and $d_r=9\mu \mathrm{m}$ (15). (e) $I_{p,w}/I_{p,wo}$ as a function of $T_c$. Note different timescales between the upper x axis for RA channels and the lower x axis for SA channels. (f and g) Normalized activated peak currents ($I_{p,w}^{\max }$ is the maximum of $I_{p,w}$ as $d_r$ increases) as a function of $d_r$ under various $T_c$ for RA channels (f) and SA channels (g). The half-activation midpoint (red dots) is denoted by $d_{50\%}$. (h) Half-activation midpoint $d_{50\%}$ as a function of $T_c$. The saturation amplitude of $d_{50\%}$ is defined as “adaptive shift.” Note different timescales between the upper x axis for RA channels and the lower x axis for SA channels. (i) The adaptive shift increases with $d_c$ both for RA (yellow) and SA (blue) channels. Circles in (e), (h), and (i) are experimental data from (15).

The reduction in the activated current can be quantified by the ratio of peak current $I_{p,w}/I_{p,wo}$, where $I_{p,w}$ represents the peak current with preload and $I_{p,wo}$ represents the peak current without preload (Fig. 5 b). As shown in Fig. 5, c and d, the current ratio $I_{p,w}/I_{p,wo}$ of RA and SA channels both significantly decreases with increasing $d_c$, as a stronger preload leads to greater inactivated fraction of MS channels (Fig. S10, a and c). Note that our simulation results are in excellent agreement with previous experimental data (pillars in Fig. 5, c and d), without any adjustments to the model parameters.

Lengthening the duration of preload $T_c$ results in smaller activated current in the subsequent test compression for cells containing either RA or SA MS channels (Fig. 5 e), similar to the findings shown in Fig. 3 g. This is because increasing $T_c$ results in more inactivation of MS channels (Fig. S10, b and d). However, RA channels display higher sensitivity to $T_c$ compared to SA channels, as evidenced by the decrease in $I_{p,w}/I_{p,wo}$ to approximately 0.25 for $T_c>50$ ms (Fig. 5 e, top x axis), indicating substantial inactivation of RA channels. In contrast, the timescale for significant inactivation of SA channels, represented by the value of $T_c$, is on the order of 1000 ms (Fig. 5 e, bottom x axis), which is much longer than that of RA channels. These results further confirm much slower inactivated dynamics of SA channels. Increasing $T_c$ also leads to a rightward shift in the activated curves (Fig. 5, f and g). These findings suggest that cells become less sensitive as MS channels undergo inactivation in the preload necessitating a stronger mechanical stimulus to achieve activation, similar to the findings shown in Fig. 4, f–h.

To further examine the effects of preload on the rightward shift of the activated curve of MS channels, we quantify how the activated midpoint (denoted by $d_{50\%}$ in Fig. 5, f and g) changes with the duration of preload $T_c$. As shown in Fig. 5 h, $d_{50\%}$ initially increases with $T_c$ and then eventually approaches a platform, consistent with previous findings (15). We use a single exponential function to fit the curve of $d_{50\%}$ versus $T_r$ shown in Fig. 5 h, which allows us to determine the characteristic time of the rightward shift. For RA channels, the characteristic time of shift is approximately 4.5 ms, whereas for SA channels it extends to around 200 ms. These results suggest that SA channels require a longer preload duration to adapt their activation behavior in subsequent stimuli.

The extent of rightward shift is termed “adaptive shift” (Fig. 5 h). We find that the adaptive shift increases significantly with the amplitude of the preload $d_c$ both for RA and SA channels (curves in Fig. 5 i), consistent with previous experimental data (15) (circles in Fig. 5 i). These findings further suggest that MS channel inactivation alters the activated currents of nerve cells, prompting adaptation in cellular mechanically activated behaviors.

MS channel inactivation reduces the firing activities in response to prolonged mechanical stimuli

Another remarkable feature of neural adaptation is the decay in firing activities in response to prolonged or repeated stimulation. To further investigate the effects of MS channel inactivation on neural adaptation, we study the firing patterns of cells in response to prolonged and repeated mechanical stimulation.

We first study the firing behaviors of cells in response to step mechanical loading with a prolonged duration when the membrane voltage is not clamped (Fig. 6 a). As demonstrated in previous studies (61,62), the input current must surpass a critical threshold to trigger action potentials. In other words, the amplitude $d$ of the step loading must exceed a critical threshold to evoke action potentials, given that the activated current increases with $d$ (Fig. 2 g). Therefore, for small $d$, the mechanical stimulus fails to trigger action potentials in cells containing RA channels (Fig. 6 b, top, blue curves), as the activated current of RA channels rapidly decreases to zero before the action potentials can be generated. Furthermore, cells containing RA channels only fires a single action potential, even when $d$ is very large (Fig. 6 b, top, red curves). Note that a loading amplitude of $d=8.5\mu \mathrm{m}$ can activate the maximum current (Fig. 2 g). In contrast, cells containing SA channels fire multiple action potentials under sustained loading, even for small $d$ (Fig. 6 b, bottom, blue curves), owing to the slow decline in the activated current.

Figure 6.

Inactivation of MS channels is beneficial to avoid overexcitation under prolonged step mechanical stimuli. (a) Cells undergo step mechanical stimuli with different amplitude $d$, while the loading speed $v$ is $2.5\mu \mathrm{m}/\text{ms}$ (without clamping membrane voltage $V_m$). (b) Corresponding firing patterns for the loads shown in (a). Left: experimental results; right: simulation results. Top: RA channels; bottom: SA channels. (c) Number of action potentials (APs) as a function of $d$. Blue and red colors represent the results of RA and SA channels, respectively. Solid lines are our simulation results, while squares are experimental data from (15). (d) Cells undergo step mechanical stimuli with different speed $v$, while the loading depth is $d=8.5\mu \mathrm{m}$ (without clamping membrane voltage $V_m$). (e) Corresponding firing patterns for the loads shown in (d). (f) Number of action potentials (APs) as a function of $v$. (g) Cells undergo step mechanical stimuli with different speed $v$, while the loading depth is $d=10\mu \mathrm{m}$ and $V_m$ is clamped at −65.7 mV. (h) Corresponding activated currents for the loads shown in (g). The peak currents are denoted by $I_p$, while $I_{p,\max }$ is the maximum peak current. (i) Normalized peak current $I_p$ (normalized to its maximum $I_{p,\max }$) as a function of $v$. Squares in (f) and (i) are experimental data from (17).

We then quantify the number of action potentials under various values of $d$ (Fig. 6 c). For RA channels, the number of action potentials is zero when $d$ is small, while it remains at 1 for large $d$, indicating an all-or-none response of RA channels (Fig. 6 c, blue curves). In contrast, for cells containing SA channels, the number of action potentials increases significantly with $d$ (Fig. 6 c, red curves), in excellent agreement with experimental data (15). These findings thus suggest that rapid inactivation of MS channel favors the reduction of firing activities of cells in response to strong mechanical stimuli with high amplitudes.

To further delineate the differences in the firing behaviors between RA and SA channels, we subject the cells to step compression with various loading speeds $v$ (Fig. 6 d). Cells containing RA channels fails to fire action potentials at low $v$ (Fig. 6 e, top, red curves), whereas fast loadings can trigger action potentials (Fig. 6 e, top, blue curves). This is attributed to the significant decrease in the activated current of RA channels as $v$ decreases (Fig. 6, h (top) and i (blue line)), consistent with previous observations (17,63).

In contrast, cells containing SA channels can still fire action potentials even under slow mechanical loads (Fig. 6 e, bottom), as the activated current of SA channels only decreases slightly as $v$ decreases (Fig. 6, h (bottom) and i (red curve)). As $v$ increases, the number of action potentials for cells containing SA channels increases significantly (Fig. 6 f, red curve), whereas the number of action potentials for cells containing RA channels remains at 1 (Fig. 6 f, blue curve). Furthermore, cells containing RA channels require faster critical loading to trigger action potentials (Fig. 6 f, blue curve).

Together, these results suggest that rapid inactivation of MS channels favors the reduction of firing activities in response to fast and intense mechanical stimuli, potentially serving as a protective mechanism against overexcitation.

MS channel inactivation reduces the firing activities in response to repeated mechanical stimuli

Next, we study the firing behaviors of cells in response to repeated stimuli, where cells undergo multiple prior loadings (Fig. 7, a and b). We find that, for a periodic load of 12 Hz, cells with RA channels generate one action potential in each loading period and faithfully follow the mechanical stimuli (Fig. 7 c). To evoke an action potential, the inward currents must be sufficiently large (59,60). Due to fast inactivation, the activated currents of RA channels under high-frequency loading drop quickly blow the threshold value required to trigger action potentials. Therefore, cells containing RA channels will only fire two action potentials for a periodic loading of 21 Hz and cannot follow the mechanical stimuli (Fig. 7 d), consistent with previous experimental findings on Piezo2 ion channels (with an inactivation time course of <10 ms (24)). In contrast, if we decrease the inactivated rate of the channel, i.e., ultra-slowly (US) inactivated ($\tau \approx 1000$ ms), the MS channels exhibit a very slow decrease in activated current; thus, the activated current is almost constant during the repeated stimuli, allowing them to follow periodic loading with a frequency of 21 Hz (Fig. 7 e). These results are all consistent with previous experimental findings (15).

Figure 7.

Inactivation of MS channels is beneficial to avoid overexcitation under high-frequency repeated mechanical stimuli. (a and b) Cell indentation depth changes periodically with a frequency of 12 Hz (a) or 21 Hz (b), while the indentation amplitude is 9 $\mu \mathrm{m}$ and the membrane voltage is not clamped. (c and d) Firing patterns of cells containing RA channels under 12-Hz or 21-Hz stimuli. Left: experiments (15); right: simulations. (e) Firing patterns of cells containing ultra-slowly (US) adaptive MS channels under 21-Hz stimulus. Left: experiments (15); right: simulations. (f) Number of action potentials (Aps) in response to six loading pulses as a function of the loading frequency. Lines are our simulation (Sim.) results, whereas squares are previous experimental (Exp.) data (15).

To further elucidate how the loading frequency affects the firing behaviors of cells, we quantify the number of action potentials as a function of loading frequency (Fig. 7 f). Compared to cells possessing US channels, cells containing RA channels only fire one action potential under high-frequency repeated mechanical stimuli, which is in accordance with prior experimental results (15). Thus, these results indicate that rapid inactivation of MS channels will prevent cells from overexcitation under high-frequency mechanical stimuli.

Predictions of mechanical modulation of firing adaptation

While previous models neglect neuronal mechanical deformation, our framework explicitly integrates three coupled biophysical processes: 1) cell deformation under mechanical stimuli, 2) MS channel conformational changes driven by membrane tension, and 3) MS channel state transformations inducing subsequent changes in electric signals, including activated ion currents and action potential generation. This coupling enables quantitative predictions of how neuronal mechanics regulate electrophysiological outputs, including activated currents and action potential dynamics.

We find that increased cortical stiffness $\left(E\right)$ and active contraction $\left({\sigma }_a\right)$ elevates baseline membrane tension, thereby lowering the mechanical stimulus threshold required to activate MS currents (Fig. 8, a–d). Even though elevated $E$ and ${\sigma }_a$ render the neurons easier to be activated, the activated peak currents under a ramp-and-hold stimuli depend non-monotonically on $E$ and ${\sigma }_a$ (Fig. 8, e–h). At low $E$ and ${\sigma }_a$ regimes, $I_p$ increases with with $E$ and ${\sigma }_a$, as increasing $E$ and ${\sigma }_a$ promotes the opening of MS channel by enhancing membrane tension. However, beyond critical $E$ and ${\sigma }_a$ values, excessive tension induces MS channel inactivation (Fig. S11), thus suppressing $I_p$. This biphasic relationship (activation vs. inactivation dominance) may provide a unified framework for interpreting 1) mechanical silencing under pathological stiffening (e.g., fibrosis and scarring) and 2) activity-dependent plasticity via contractility modulation.

Figure 8.

Enhanced cortical stiffness and active contraction reduce activation thresholds and lead to biphasic changes in peak activated current. (a and b) Normalized current-depth activation curve for RA channels under various cortical stiffness $E$ (a) and active contraction ${\sigma }_a$ (b). Cells undergo ramp-and-hold indentation with various amplitudes $d$ while the membrane voltage is clamped at $V_m=-65.7\text{mV}$. (c and d) Normalized current-depth activation curve for SA channels under various $E$ (c) and ${\sigma }_a$ (d). (e and f) Biphasic dependence of peak currents (activated by a ramp-and-hold indentation with $d=6\mu \mathrm{m}$) on $E$ (e) and ${\sigma }_a$ (f) for RA channels. (g and h) Biphasic dependence of peak currents on $E$ (g) and ${\sigma }_a$ (h) for SA channels.

Beyond modulating activated peak currents under step loading, our model demonstrates that elevated $E$ and ${\sigma }_a$ accelerate ion current decay under two-step loading protocol (Fig. 9, c and d), as more MS channels become inactivated as $E$ and ${\sigma }_a$ increase (Fig. 9, e and f). This enhanced current decay in subsequent stimuli further drives neurons toward non-excitable states as $E$ and ${\sigma }_a$ increase (Fig. 9, g–j).

Figure 9.

Increased cortical stiffness and active contraction both lead to higher decaying degree of current and render the cell non-excitable. (a) Two-step loading protocol while clamping the membrane voltage at −65.7 mV. Initially, a first loading with a duration of $T_h$ and an amplitude $d_h$ is applied, then cells undergo a recovery period of time $T_r$ before a second stimulus with a duration of $T_2$ and an amplitude $d_r$ is applied. The loading and unloading speeds are equal and denoted by $v$. Unless specified otherwise, $d_h=5.5\mu \mathrm{m},T_h=10\text{ms},v=1\mu \mathrm{m}/\text{ms},T_r=20\text{ms},d_r=8.5\mu \mathrm{m}$, and $T_2=10\text{ms}$. (b) Activated currents generated by RA channels under the loads depicted in (a). Here, $I_{p,w}$ represents the peak current with first loading, while $I_{p,wo}$ represents the peak current without first loading. (c and d) Decaying degree in the peak current $1-I_{p,w}/I_{p,wo}$ as a function of $d_h$ for different cortical stiffness $E$ (c) or active contraction ${\sigma }_a$ (d). (e and f) Fractions of MS channels in the inactivated state before applying the second loading (green point in a) as a function of $d_h$ for different cortical stiffness $E$ (e) or active contraction ${\sigma }_a$ (f). (g–j) Increased cortical stiffness $E$ (g and h) and active contraction ${\sigma }_a$ (i and j) render the cell non-excitable.

Together, these predictions position neuronal mechanics as a central regulatory mechanism in mechanoelectric responses, which can be further verified via various perturbations to manipulate cortical stiffness (e.g., cytochalasin D, latrunculin A, and jasplakinolide) or myosin II contraction (e.g., blebbistatin and Y-27632).

Discussion

Excitable cells exhibit mechanical memory, whereby their present excitation behaviors strongly depend on their past mechanical experiences (7,8,9). In this study, we introduce an electromechanical theory to elucidate the observed mechanical memory in excitable cells. In contrast to previous electrical conductance-based models (21,22,23), our model provides detailed descriptions of cell deformation induced by mechanical stimuli and how this mechanical deformation alters the conformations of MS channels, leading to subsequent changes in electric signals, including activated ion current and action potential generation. Previously, a four-state model (closed, open, and two inactivation states) was proposed to explain the frequency filtering character and the mechanotransduction mechanical energy of the Piezo ion channel (24,64). Compared to these multiple states’ models of MS channels, we also incorporate the dynamics of membrane voltage in our model. This critical extension enables joint simulation of both mechanoactivated currents and action potential dynamics, a capability absent in previous multi-state channel models. Moreover, unlike prior neuronal models (21,22,23,24) that neglect cellular mechanics, we explicitly couple cellular mechanical deformation to electrophysiological outputs. This integration permits quantitative predictions of how mechanical factors, including cortical stiffness and active contraction, modulate neuronal excitability. Our model faithfully reproduces numerous experimental observations using consistent parameters (15,17), highlighting its robustness and reliability. Consequently, the validated model serves as a potent tool for investigating and predicting electromechanical behaviors of excitable cells subjected to diverse mechanical stimuli.

Eastwood et al. previously attributed declining mechanoactivated currents in sensory neurons to extracellular viscoelasticity within surrounding tissues (63). Our findings reveal that mechanosensitive channel inactivation constitutes an autonomous cellular mechanism driving current decay. By focusing on isolated cells, our model intentionally excludes extracellular viscoelastic components to isolate the intrinsic contribution of channel inactivation. This reductionist approach establishes cellular activation adaptation as a fundamental mechanism that operates independently of—yet potentially synergizes with—tissue-scale viscoelastic adaptation. To validate these distinct mechanisms, future studies that quantitatively assess the forces acting on the channels and the activated currents simultaneously are needed, which is still very experimentally challenging.

It has been shown that inactivation of MS channel plays a critical role in frequency filtering under repetitive mechanical stimuli (24). Here, we highlight that MS channel inactivation in past loadings yields history-dependent excitation, which further indicate the importance of MS channel inactivation in shaping neuronal firing responses. The degree of MS channel inactivation increases with the depth, duration, and loading speed of preceding stimuli, leading to smaller activated currents in the subsequent stimuli, consistent with previous experimental observations (15,17). On the other hand, subjecting cells to a preload induces a rightward shift in the activation curve as MS channels become inactivated, indicating that cells become less sensitive to mechanical stimuli (adaptation). These adaptations may help neurons to filter out constant or unchanging stimuli in the environment and conserve energy. Importantly, all of these history-dependent behaviors vanish when MS channels cannot undergo inactivation. Traditional neural adaptation to electrical/chemical stimuli (e.g., voltage-gated sodium channel inactivation in spike-frequency adaptation; neurotransmitter depletion at synapses) primarily enables temporal filtering of neural codes. Our findings extends this concept to mechanical stimuli by revealing that MS channel inactivation in past mechanical stimuli impedes subsequent excitability, a feature absent in purely electrical adaptation. Collectively, these results demonstrate that MS channel inactivation is responsible for the mechanical memory of excitable cells, providing theoretical support for previous experimental findings (15,17).

We demonstrate that due to the inactivation of MS channels in prior loadings, there is a mechanical refractory period during which cells are unable to generate new action potentials in response to subsequent mechanical stimuli. This refractory period differs from the absolute refractory period observed during the depolarization phase of action potentials (59,60). This observation highlights an important aspect to consider: the electromechanical responses of cells from various experiments may not be directly comparable if an adequate waiting time is not allowed between sequential stimuli. Therefore, our findings emphasize the significance of considering differences in loading history when comparing the electromechanical behaviors of cells. Taking into account this temporal aspect will enhance the accuracy and reliability of experimental comparisons and contribute to a more comprehensive understanding of the electromechanical dynamics of excitable cells.

Insensitivity (neural adaption) that results from MS channel inactivation may have important physiological implications. In some instances it can act as a defense mechanism, while in others it may lead to functional disorders or diseases. For example, neuronal insensitivity to mechanical stimuli helps prevent overactivation, protecting nerve cells against damage caused by continuous or intense stimulation. In dealing with pain or other discomforts, neuronal desensitization can alleviate chronic pain symptoms (65,66). However, prolonged desensitization may lead to a decline in sensory functions, such as reduced sensitivity in vision, hearing, and touch (67). Moreover, the reactivity of neurons to stimuli is crucial for the formation of learning and memory formation. Desensitization may impact these cognitive functions. Thus, long-term neuronal desensitization may be associated with the development of neurodegenerative diseases, such as Alzheimer’s disease (68).

There are some limitations to our study. First, our model focuses on RA and SA channels without considering IA channels with inactivation timescale between 10 ms and 100 ms. Our model can easily accommodate IA channels by adjusting transition rates ($k_{OI}$ and $k_{IC}$) to reflect intermediate inactivation timescales (10–100 ms). Second, our model focuses on the effects of MS channel inactivation in mechanical memory, consistent with experimental observations of rapid adaptation. However, cytoskeletal remodeling and $\mathrm{C}{\mathrm{a}}^{2+}$ signaling may also affect the activation dynamics. While MS channels inactivate on rapid timescales (milliseconds to seconds), cytoskeletal remodeling and $\mathrm{C}{\mathrm{a}}^{2+}$-mediated changes operate over longer timescales (minutes to hours). Thus, MS channel inactivation may provide short-term desensitization, while cytoskeletal remodeling and $\mathrm{C}{\mathrm{a}}^{2+}$ signaling may enforce long-term adaptation. In the future, a hierarchical model integrating both fast (MS inactivation) and slow (cytoskeletal remodeling or $\mathrm{C}{\mathrm{a}}^{2+}$ signaling) processes could help to explain multi-timescale mechanical memory.

Conclusion

In summary, we reveal that MS channel inactivation serves as a central mechanism underpinning mechanical memory and neural adaptation in excitable cells. By reducing neuronal firing under sustained mechanical stress, this adaptive mechanism likely safeguards excitable cells from overactivation and mechanical damage, ensuring cellular homeostasis. The widespread presence of mechanosensitive channels in various cell types (37,69) suggests that this mechanism may have broader implications across cell biology. The inactivation of these channels could similarly lead to mechanical memory effects, where the cell’s response to subsequent mechanical stimuli is influenced by its past experiences. Understanding how MS channel inactivation mediates mechanical memory in these different cell types could provide new insights into mechanotransduction mechanisms in various physiological and pathological contexts. More importantly, our findings may have implications for the development of novel therapeutic strategies targeting conditions associated with mechanosensitive channel inactivation dysfunction, such as hereditary xerocytosis (70), muscle fatigue (71,72), and chronic pain (73). By unraveling the mechanism of mechanical memory, our research opens new avenues for exploring potential interventions and treatment approaches for these diseases.

Data and code availability

All data needed to evaluate the conclusions in the paper are present in the paper and/or the supporting material. The codes used to generate results reported in this study are available via GitHub at https://github.com/yuehua92/mechanical-memory.git.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant nos. 12372319, 12025207), USTC Research Funds of the Double First-Class Initiative (YD2090002024). This work was partially carried out at the University of Science and Technology of China Center for Micro and Nanoscale Research and Fabrication.

Author contributions

Supervision, H.J. and Y.Y.; modeling and simulations, J.X., F.M., and Y.Y.; writing, J.X., F.M., Y.Y., and H.J.

Declaration of interests

The authors declare no competing interests.

Editor: Yun Lyna Luo.