Complex dynamics of hair bundle of auditory nervous system (II): forced oscillations related to two cases of steady state

Abstract

The forced oscillations of hair bundle of inner hair cells of auditory nervous system evoked by external force from steady state are related to the fast adaption of hair cells, which are very important for auditory amplification. In the present paper, comprehensive and deep understandings to nonlinear dynamics of forced oscillations are acquired in four aspects. Firstly, the complex dynamics underlying the twitch (fast recoil of displacement X which is fast variable) induced from Case-1 and Case-2 steady states by external pulse force are obtained. With help of vector fields and nullclines, the phase trajectory of forced oscillations is identified to be an evolution process between two equilibrium points corresponding to zero force and pulse force, respectively, and then the twitch is obtained as the behavior running along the nonlinear part of X-nullcline. Especially, twitch observed in experiment are classified into 6 types, which are induced by negative change of force, negative and positive changes of force, and positive change of force, respectively, and further build relationships to three subcases of Case-2 steady state with N-shaped X-nullcline (equilibrium point locates on the left, middle, and right branches of X-nullcline, respectively). Secondly, the experimental observation of fatigue of twitch induced by continual two pulse forces, i.e. the reduced amplitude of the latter twitch when interval between two forces is short, is also explained as a nonlinear behavior beginning from an initial value different from that of the former one. Thirdly, the experimental observation of transition between sustained oscillations and steady state induced by pulse force can be simulated for Case-1 steady state with Z-shaped X-nullcline instead of Case-2, due to that there exists bifurcations with respect to external force for Case-1 while no bifurcations for Case-2. Last, the threshold phenomenon induced by simple pulse stimulation exists for Case-1 steady state rather than Case-2, due to that the upper and lower branches of Z-shaped X-nullcline close to the middle branch exhibit coexisting behaviors of variable X while N-shaped X-nullcline does not. The nonlinear dynamics of forced oscillations are helpful for explanations to the complex experimental observations, which presents potential measures to modulate the functions of twitch such as the fast adaption.

Introduction

The complex phenomena of spontaneous oscillations or forced oscillations in nature, including mechanical system, nervous system, and auditory system (Clausznitzer et al. 2008; Roongthumskul et al. 2021), can be explained theoretically by using nonlinear dynamics such as threshold, bifurcation, chaos as well as the nullclines and phase trajectories. As an important sensory nervous system to communicate with the outside world, the auditory system can realize a variety of auditory functions, containing signal amplification, frequency selection, compression nonlinearity, and otoacoustic emission (Belousov et al. 2020; Hudspeth 2008; Maoiléidigh et al. 2019). The complex auditory functions are realized by the nonlinear oscillations including the mechanical and electrical oscillations. The nonlinear mechanical oscillations produced by the hair bundles in the cochlea, which locates in the inner ear, are important to the auditory system for receiving and processing sound signals from the outside world (Barral and Martin 2012; Fettiplace 2020). Hair bundles can receive and process sound signals and produce nonlinear mechanical oscillations. Subsequently, the mechanical oscillation signal can be converted into electrical signal through mechanical-electrical transition (MET) channels (Caprara et al. 2020; Fettiplace 2017). This process has been well simulated by recent studies on the acoustic nerve circuits with piezoelectric components (Guo et al. 2021; Zhou et al. 2021). The acoustic nerve element circuit with piezoelectric elements provides support for the development of cochlear implant (Guo et al. 2021), and the biological model in the present paper can better simulate the effect of calcium ion on MET process. The nonlinear oscillation of hair bundles include the spontaneous oscillations and the forced oscillations under the action of external forces (Dinis et al. 2012). The spontaneous oscillations of hair bundles are considered to be the basis of otoacoustic emission function (Roongthumskul et al. 2019). The forced oscillation under the action of external pulse forces, also called twitch that is an important manifestation of fast adaptation, has been widely studied in biological experiments (Benser et al. 1996; Strimbu et al. 2012; Tinevez et al. 2007). The study to nonlinear oscillations of hair bundles is of great significance for understanding the important roles of hair bundles in auditory function (Fredrickson-Hemsing et al. 2012; Ji et al. 2018; Sheth et al. 2018).

The hair bundle is a biological system with nonlinear biological regulation, which contains the calcium ion related driving force(driving force related to the calcium ion), which is provided by the myosin motors (Li et al. 2020; Maoiléidigh et al. 2012). The force provided by myosin motors is a form of adaptation which is produced by active biological regulation. In addition, adaptation also affects the open probability of MET channels, and further influences the displacement of hair bundles (Hudspeth 1989; Maoiléidigh et al. 2019). Adaptation is identified to modulate the nonlinear oscillations through regulation to negative or nonlinear stiffness of hair bundles (Le Goff et al. 2005; Peng et al. 2011). In order to simulate the adaptation force and the impact of adaptation on opening probabilities of MET channel, a two-dimensional mathematical model considering two key parameters are widely used (Amro and Neiman 2014; Clausznitzer et al. 2008; Tinevez et al. 2007; Willareth et al. 2017). One parameter is labeled with S (the degree to which the adaptation force affects the system), and the other parameter is D (the swing generated by the opening of the MET channel represents the degree to which the adaptation generated by the opening of the MET channel affects the displacement). In addition, many sophisticated models with open probability of MET channel as independent variables are proposed (Barral et al. 2018). Complex oscillation modes of hair bundles including the periodic oscillations of spike, periodic oscillations of burst, and chaotic oscillations are simulated (Cao et al. 2021).

The forced vibration of hair bundles induced by external pulse forces, twitch, has been extensively studied in biological experiments (Benser et al. 1996; Strimbu et al. 2012; Tinevez et al. 2007). Twitch of hair bundle refers to the rapid increase or decrease of the displacement followed by a rapid recoil, which is induced by the action of a step force (Tinevez et al. 2007) or the cancellation of a force (Benser et al. 1996). Twitch is an important manifestation of the fast adaptation, which represents the fast nonlinear regulation of hair bundle movement by organisms under external force stimulation (Maoiléidigh et al. 2019). Fast adaptation is considered as a potential mechanism of signal amplification (Bozovic 2019) for non-mammalian, which plays an important role in frequency selection (Fettiplace et al. 2001). For example, in biological experiments of drosophila, the effect of fast adaptation on the enhancement of sound encoding ability has been reported (Clemens et al. 2018). Fast adaptation also regulates the dynamical behavior of hair bundles by influencing the MET channels. For instance, fast adaptation helps to filter the response signal of the MET channel (Stepanyan and Frolenkov 2009), and control the amount of energy transferred by hair bundle movement by controlling the sensitivity of the MET channel (Azimzadeh et al. 2018). In experiments, different types of twitch can be produced by hair bundles stimulated by different force stimulation such as step force and pulse force with a relatively long duration. In Ref (Tinevez et al. 2007), two types of twitch induced by force step are reported. One is that a positive twitch is induced by a positive step force, and the other is that a negative twitch is induced by a negative step force in other species of hair bundles (different hair bundles). In another experiment, the force pulse which begins from a force step, remains at a constant force for a relatively long duration, and terminates by the cancellation of the force is applied to the hair bundles. The constant force can be either positive or negative, correspondingly, the force pulse is called positive pulse or negative pulse for convenience in the present paper, respectively. For the positive pulse stimulation, three types of twitch appear for some species of hair bundles. First, only a positive twitch appears at the phase corresponding to the positive step force. Second, only a negative twitch appears at the phase corresponding to the cancellation of the force. Last, a positive twitch appears at the phase corresponding to the step force and a negative twitch at the phase corresponding to the cancellation of the force. Similarly, three types of twitch appear for the negative pulse stimulation. First, only a negative twitch appears at the phase corresponding to the negative step force. Second, only a positive twitch appears at the phase corresponding to the cancellation of the force. Last, a negative twitch appears at the phase corresponding to the negative step force and a positive twitch at the phase corresponding to the cancellation of the force (Benser et al. 1996). In summary, 6 types of twitch have been observed in the experiment on hair bundles. Except for the auditory system, twitch exists in other biological systems such as muscle, which is related to muscle contractions that affect the ability to generate explosive force of muscle (Dideriksen et al. 2020).

Therefore, it is very important to simulate, explain, and even present measure to modulate the twitch in the theoretical studies. However, compared with many studies of forced vibration (twitch) in experiments (Benser et al. 1996), the simulations of forced vibration (twitch) in theoretical models are much less, and the nonlinear dynamical explanations are more rarely. In the previous study (Tinevez et al. 2007), two types of twitch for the step force and one example for the positive force pulse (a positive twitch and a negative twitch simultaneously appearing at the initiation and termination of the force pulse) are simulated. However, another type for the positive pulse force and two types of twitch for the negative pulse observed in the experiment have not been simulated. Although the force-displacement curve is used to explain the dynamical mechanism of twitch in biophysics (Tinevez et al. 2007), the cause for the fast changes of displacement for the twitch has not been acquired. For example, the relationships to the fast change of displacement to the variables and the derivatives of the variables of mathematical model have not been acquired. Although the phase trajectory as well as the nullclines of variables and vector filed have been verified to be effective measures to explained dynamics of action potential or subthreshold oscillations in the neurons (Cao et al. 2021; Zhao et al. 2020), which has not been used to explain the dynamics of twitch of hair bundles. Therefore, the diverse types of the twitch observed in the biological experiment are lack of explanations and theoretical classifications, which is the first question to be answered in the present paper.

Furthermore, twitch fatigue induced by continuous two force pulse stimulation with an interval has been observed in the biological experiment (Benser et al. 1996). With the decrease of the time interval between the two force pulses, the amplitude of latter twitch becomes lower than the former twitch, which is called the fatigue of twitch. Fatigue of twitch is an important dynamical manifestation modulated by changes of external force, which can be easily acquired in experiments. However, such a dynamical behavior of twitch has not been simulated or explained in theoretical models. This is the second question to be answered in the present paper.

Except for the twitch and the fatigue of twitch induced by long square wave of external force stimulation, a long square wave of external force stimulation induces the hair bundle in a stable steady state transited to sustained oscillations in experiment (Benser et al. 1996). Such a transition behavior is different from the twitch, which is simulated in the previous study (Maoiléidigh et al. 2019). According to the nonlinear theory, it can be inferred that the transition maybe related to a bifurcation from resting state to oscillations while the twitch maybe not related to a bifurcation. However, no bifurcation mechanism for the transition has been provided, and no non-bifurcation mechanism for the twitch has been provided. Therefore, for either the twitch or the transition from resting state to sustained oscillations induced by external force with a relatively long duration, no bifurcation mechanism related to the resting state has been provided. The bifurcation mechanism for the transition or non-bifurcation mechanism for the twitch is the third question to answered in the present paper.

In addition, as the forced oscillations, the twitch of the hair bundle exhibits characteristic different from the forced oscillation (such as action potential induced by external stimulation near the threshold) of a neuron (Guan et al. 2019; Wang et al. 2021). Near the threshold of a neuron, it is well-known that an action potential can be evoked by a simple suprathreshold stimulation. The amplitude of action potential is large enough to approximate to that of the spontaneous oscillations, i.e. firing. However, for the hair bundle, the amplitude of twitch is far smaller than that of the spontaneous oscillations. Therefore, it can be inferred that the twitch phenomenon maybe not a dynamical behavior near the threshold. However, up to now, little is known about the threshold phenomenon of oscillations of hair bundle, either experimentally or theoretically. Therefore, to identify the existence of the threshold in the mathematical model to describe oscillations of hair bundle, and to reveal the relationship of the twitch or the spontaneous oscillations to the threshold, are very important to present comprehensive understanding to the oscillations of the hair bundle. This is the last question to be answered in the present paper.

In the present paper, based on the two cases (five subcases) of steady state acquired in Ref (Cao et al. 2021b, the four questions mentioned above are answered with bifurcations, nonlinear characteristic of nullclines, vector field, and fast-slow dynamics of variables. The comprehensive and deep understanding to nonlinear dynamics of forced vibration evoked from two cases of steady state by force, such as the twitch, fatigue of twitch, transition behavior between resting state and sustained oscillations, and threshold phenomenon, are acquired. The studies are performed on the classical hair bundle vibration model (Tinevez et al. 2007) considering the nonlinear modulation related to parameter D and S.

The rest of the present paper is organized as follows. Section "Model and methods" presents models of a single hair bundle and methods. The results are reported in Section "Results". Discussion and conclusion are provided in Section "Discussion and conclusion".

Model and methods

Models of hair bundle

The physical, mechanical, oscillation models of a hair bundle are introduced in Ref Cao et al. (2021b). In the present paper, the oscillation model is considered and described as flows:

$$\begin{array}{cc}\hfill \lambda \frac{\mathit{dX}}{\mathit{dt}}=& -K_{\mathit{GS}}(X-X_a-D{P}_0)-K_{\mathit{SP}}X+F\hfill \end{array}$$ 1

$$\begin{array}{cc}\hfill {\lambda }_a\frac{d{X}_a}{\mathit{dt}}=& K_{\mathit{GS}}(X-X_a-D{P}_0)\hfill \\ \hfill & -K_{\mathit{ES}}(X_a-X_{\mathit{ES}})-F_{\mathit{max}}(1-S{P}_0)\hfill \end{array}$$ 2

where the variable X is the displacement of the hair bundle, which is the fast variable to describe the mechanical oscillation of the hair bundle model, and $X_a$ is the displacement of the myosin motor (a protein that is active in movement), which is the slow variable. Four forces related to hair bundle are the external force F, damping force $-\lambda \frac{\mathit{dX}}{\mathit{dt}}$, restoring force $-K_{\mathit{SP}}X$, and biological regulation force $F_{\mathit{GS}}=-K_{\mathit{GS}}(X-X_a-D{P}_0)$. For the myosin motor, the four forces are the damping force $-{\lambda }_a\frac{d{X}_a}{\mathit{dt}}$, restoring force $-K_{\mathit{ES}}(X_a-X_{\mathit{ES}})$, biological regulation forces $F_{\mathit{GS}}=-K_{\mathit{GS}}(X-X_a-D{P}_0)$ related to fast adaptation, and the slow adaptation force $-F_a=-F_{\mathit{max}}(1-S{P}_0)$. $P_0=1/(1+Aexp(\frac{-Z(X-X_a)}{k_{b}T}))$ is nonlinear function of the variables X and $X_a$. More details of the model refer to Ref Cao et al. (2021b).

The items related to the biological regulation, such as $D{P}_0$ and $S{P}_0$, which are nonlinear functions due to the nonlinearity of $P_0$, induce hair bundle to produce complex vibrations different from those of model without biological regulation (i.e. the item $D{P}_0$ and $S{P}_0$ are ignored) (Cao et al. 2021b). Therefore, the parameters D and S, which are the coefficients before the nonlinear term $P_0$ and determine the strength of biological regulation, are chosen as the important parameters discussed in the present paper. In terms of biological regulation, D and S respectively determine the intensity of fast adaptation and slow adaptation of hair cells. In addition, to study the forced oscillations such as twitch induced by external force F, the parameter F is chosen as another control parameter in the present paper.

The values of other parameters are as follows: $\lambda$ = 0.28 $\mu$Ns/m, ${\lambda }_a$ = 10 $\mu$Ns/m, $K_{\mathit{GS}}$ = 0.75 mN/m, $K_{\mathit{SP}}$ = 1 mN/m, $K_{\mathit{ES}}$ = 0.25 mN/m, $X_{\mathit{ES}}$ = 0 nm, $F_{\mathit{max}}$ = 90 pN, N = 50, T = 295.15 K.

Methods

The equations of the model are integrated using the fourth-order Runge-Kutta method and the integration time step is chosen as 0.01 ms. The bifurcations and nullclines are acquired with software of XPPAUT (Bard 2002). The existence condition of the solution of the equation to describe the border between two cases of steady states is obtained by the software of Mathematica 8.0.

Results

Two cases (5 subcases) of the steady state at F = 0

The distribution of steady state on (S, D) plane

In the plane (S, D), there are three dynamical behaviors, the spontaneous oscillations (red region), the coexistence of double steady states (blue region), and the monostable steady state (blanket area), as shown in Fig. 1 (reproduced from Fig. 12 of Ref (Cao et al. 2021b) to ensure the self-contain of the present paper). The border of the red region (spontaneous oscillations) mainly corresponds to codimension-1 curve of Hopf bifurcations (black solid (dotted) curve: Supercritical (Subcritical) Hopf bifurcation). The border of the coexisting steady states (blue area) corresponds to codimension-1 curve of subcritical Hopf bifurcations (black dotted curve). For the coexisting steady states, one exhibits a higher X value and the other manifests a lower X value. The point P1 is a codimension-2 bifurcation point (Cao et al. 2021b).

Fig. 1.

Different dynamical behaviors in (S, D) plane. The blanket, blue, and red regions represent the monostable steady state, the coexistence of double steady states (surrounded by the ${\text{H}}_{\mathit{sub}}$ bifurcation curves (black dotted curve)), and spontaneous oscillation (mainly surrounded by the supercritical (black solid) and subcritical (black dotted) Hopf bifurcation curves). Blanket region upper and lower to $D\approx$ 50.3 nm represents Case-1 and Case-2 steady states, respectively. Case-1 steady state left and right to the spontaneous oscillations represents Case-1-1 and Case-1-2 steady states, respectively. The Case-2 steady state lower than $D\approx$ 50.3 nm is divided into three subcases, Case-2-1, Case-2-2, and Case-2-3, which corresponds to the left (horizontal lines), middle (slash lines), and right (grids) regions. The magenta curves represent the borders between three subcases of Case-2 steady state

In the plane (S, D), the distribution of spontaneous oscillation, Case-1 steady state, and Case-2 steady state are shown in Fig. 1. The Case-1 steady state locates upper to the blue dashed line ($D\approx$ 50.3 nm) and is distinguished into two subcases, called Case-1-1 and Case-1-2, which locates left and right to the spontaneous oscillations, respectively. The Case-2 steady state is lower to the blue dashed line ($D\approx$ 50.3 nm) and is divided into three subcases Case-2-1, Case-2-2, and Case-2-3 from left to right. The blue dashed line ($D\approx$ 50.3 nm) is the border between Case-1 and Case-2 steady states and is also the border between the Z-shaped and N-shaped X-nullclines.

Different nullclines, equilibrium points, and vector fields of the steady state

The Case-1 steady state exhibits a Z-shaped X-nullcline (black curve), which is composed of upper, middle, and lower branches. The branches are separated by the local maximum (orange solid square) and minimal (orange hollow square) values of variable $X_a$, as shown in Fig. 2a, c. The X-nullcline exhibits nonlinear characteristic around the maximal or minimal values of variable $X_a$. For Case-1-1 and Case-1-2 steady states, the intersection points of Z-shaped X-nullcline and $X_a$-nullcline (magenta curve) are located on the upper and lower branches of X-nullcline, respectively, as depicted by solid green circle in Fig. 2a, c. It should be noticed that the upper and lower branches of the X-nullcline near the middle branch exhibit coexisting behaviors of the fast variable X. Such a characteristic is the basis of threshold for Case-1 steady state, which is introduced in Section "Threshold for case-1 steady state". Figure 2b represents the dynamics of spontaneous oscillations (green curve).

Fig. 2.

Nullclines, equilibrium point, and vector fields for 5 subcases of steady state and spontaneous oscillations. a Case-1-1 steady state for D = 70 nm and S = 0.55; b Spontaneous oscillations (green curve) for D = 70 nm and S = 0.8; c Case-1-2 steady state for D = 70 nm and S = 5; d Case-2-1 steady state for D = 50 nm and S = 0.2; e Case-2-2 steady state for D = 50 nm and S = 0.45; f Case-2-3 steady state for D = 50 nm and S = 5. The X-nullcline and $X_a$-nullcline is represented by black curve and magenta curve, respectively, and the vector fields are denoted by black arrows. For the Z-shaped X-nullcline, the orange solid and hollow squares represent the maximum and minimum values of variable $X_a$, respectively. For the N-shaped X-nullcline, the orange solid and hollow triangles represent the maximum and minimum values of X value in the X-nullcline, respectively. The parameter plane is divided into four regions by both nullclines: region I ($\dot{X}<$ 0 and ${\dot{X}}_a>$ 0, labeled by I), region II ($\dot{X}<$ 0 and ${\dot{X}}_a<$ 0, labeled by II), region III ($\dot{X}>$ 0 and ${\dot{X}}_a<$ 0, labeled by III), region IV ($\dot{X}>$ 0 and ${\dot{X}}_a>$ 0, labeled by IV). (Color figure online)

The Case-2 steady state manifests a N-shaped X-nullcline (black curve), which is composed of left, middle, and right branches. The branches are separated by the local maximum (orange solid triangle) and minimal (orange hollow triangle) points of variable X, as shown in Fig. 2d, e, f. Around the maximal or minimal values of variable X, the X-nullcline manifests nonlinear characteristic. For Case-2-1, Case-2-2, and Case-2-3 steady states, the intersection points (solid green circle) of Z-shaped X-nullcline and $X_a$-nullcline (magenta curve) are located on the left, middle, and right branches of X-nullcline, respectively, as depicted in Fig. 2d, e, f. Different from Case-1 steady state with Z-shaped X-nullcline, the N-shaped X-nullcline does not exhibit coexisting behaviors of X variables. Such a characteristic determines no threshold phenomenon for Case-2 steady state, which is introduced in Section "Threshold mechanism: coexistence of fast variable X on Z-shaped X-nullcline for case-1 steady state".

For each subcase of steady state shown in Fig. 2, the phase plane is divided into four regions by both nullclines, which are labeled as region I, II, III, and IV, respectively. The four regions exhibit different combinations of signs for $\dot{X}$ and ${\dot{X}}_a$. In region I, $\dot{X}<$ 0 and ${\dot{X}}_a>$ 0; in region II, $\dot{X}<$ 0 and ${\dot{X}}_a<$ 0; in region III, $\dot{X}>$ 0 and ${\dot{X}}_a<$ 0; and in region IV, $\dot{X}>$ 0 and ${\dot{X}}_a>$ 0. The vector fields determined by the signs of $\dot{X}$ and ${\dot{X}}_a$ are depicted by the black arrows in Fig. 2, which is helpful for understanding the running direction of the phase trajectory, for example, the phase trajectory of spontaneous oscillations in Fig. 2b. Of course, the vector fields can be used to identify the running direction of the phase trajectory of forced oscillations induced by external force F.

Trajectory running along nonlinear part of X-nullcline for Case-2 steady state

Case-2-1

For Case-2-1, the equilibrium point is on the left branch of the X-nullcline, which is prior to the nonlinear part of the X-nullcline. A trajectory beginning from a point within region II exhibits a complex process, as shown by the blue curve in Fig. 3a1. The trajectory (blue curve) drops to right branch of X-nullcline, and then runs along the X-nullcline from right to left to reduce X value, runs to minimal value of X, jumps along the middle branch to the left branch of X-nullcline to enhance X value, runs along the left branch from left to right, evolves to the equilibrium point, and remains at the point. Therefore, the trajectory runs along the nonlinear part of the X-nullcline, resulting in a decrease and a recoil of X value, as shown in Fig. 3a1, a2. The recoil corresponds to the jump from right branch to left branch (i.e. nearly along middle branch) of X-nullcline,, which is related to the nonlinear characteristic of the X-nullcline. Such a characteristic is related to twitch induced by negative change of external force, which is introduced in Section "Twitch induced by pulse force for case-2 steady state".

Fig. 3.

Transient behavior beginning from a phase point in region II (blue) and IV (gray) for Case-2 steady state. Left: Both phase trajectories (blue and gray) in phase plane with vector fields (black arrows) and nullclines (X-nullcline represented by black curve, and $X_a$-nullcline by magenta curve); Right: The changes of X with respect to time for both phase trajectories (blue and gray), and black line is the steady state. a1 and a2 Case-2-1 steady state for D = 50 nm and S = 0.2. The blue and gray curves correspond to the initial values of (−100, −37) and (−151, −46), respectively; b1 and b2 Case-2-2 steady state for D = 50 nm and S = 0.45. The blue and gray curves correspond to the initial values of (−91, −36) and (−144, −45), respectively; c1 and c2 Case-2-3 steady state for D = 50 nm and S = 5. The blue and gray curves correspond to the initial values of (−92, −37) and (−157, −50), respectively. (Color figure online)

However, a trajectory (gray curve) beginning from a point in region IV jumps to left branch of X-nullcline, and then runs along the X-nullcline from left to right and evolves to the equilibrium point and remains at the point, as shown in Fig. 3a1. The temporal process of the trajectory is shown in Fig. 3a2. Such a characteristic is related to no twitch induced by positive change of external force, which is introduced in Section "Twitch induced by pulse force for Case-2 steady state".

Case-2-2

Different from Case-2-1, the equilibrium point for Case-2-2 steady state is on the middle branch of the X-nullcline. Then, as shown in Fig. 3b1, the trajectory (blue curve) beginning from a point within region II runs along the nonlinear part of X-nullcline from right branch to middle branch, and beginning from a point within region IV (gray curve) runs along the nonlinear part of X-nullcline from left branch to middle branch. The trajectories from region II and region IV exhibit negative and positive recoils, respectively, due to running along the nonlinear part of X-nullcline. Such results are associated with twitch induced by positive and negative changes of external force, which is introduced in Section "Twitch induced by pulse force for Case-2 steady state".

Case-2-3

Different from above two cases, the equilibrium point for Case-2-3 steady state is on the right branch of X-nullcline. The vector fields pass through the left and middle branches of the X-nullcline. Then, a trajectory (gray curve) beginning from a point within region IV runs in a direction paralleling to the left branch of X-nullcline, resulting in increases of the X value, then runs across the middle branch and turns to a direction paralleling to the middle branch, resulting in decrease of X value, i.e. a recoil, as shown in Fig. 3c1, and approaches to the right branch and runs along X-nullcline to reach to the equilibrium point. The temporal process of the trajectory (gray curve) is shown in Fig. 3c2. Such a result corresponds to twitch induced by positive change of external force, which is introduced in Section "Twitch induced by pulse force for Case-2 steady state".

However, a trajectory (blue curve) beginning from a point in region II jumps down to left branch of X-nullcline, and then runs along the X-nullcline from left to right and evolves to the equilibrium point, at last remains at the point, as shown in Fig. 3c1. The temporal process of X is shown in Fig. 3c2. Such a result corresponds to no twitch induced by negative change of external force, which is introduced in Section "Twitch induced by pulse force for Case-2 steady state".

For each of five subcases of steady state, the trajectories beginning from a point within region I or III have no relationships to the twitch, which are not addressed in the present paper.

The dynamics of twitch induced by external force

The influence of static F on X-nullcline and equilibrium point

The influence of static F on X -nullcline

When static force F is applied, the X-nullcline is acquired from $\lambda \frac{\mathit{dX}}{\mathit{dt}}=-K_{\mathit{GS}}(X-X_a-D{P}_0)-K_{\mathit{SP}}X+F=0$, which is described as follows:

$$\begin{array}{cc}\hfill X=& \frac{K_{\mathit{GS}}}{K_{\mathit{GS}}+K_{\mathit{SP}}}{X}_a+\frac{K_{\mathit{GS}}}{K_{\mathit{GS}}+K_{\mathit{SP}}}D{P}_0\hfill \\ \hfill & +\frac{F}{K_{\mathit{GS}}+K_{\mathit{SP}}}\hfill \end{array}$$ 3

A representative of X-nullcline for D = 50 nm is shown in Fig. 4a. For F = 0 pN, the X-nullcline is shown by the black curve in Fig. 4a. With increasing F to 10, 35, and 50 pN, the X-nullclines are depicted by the red, magenta, and gray curves, respectively, as depicted in Fig. 4a. As the positive static force F increases, the X-nullcline moves up-right, which is consistent with Eq. (3). With decreasing F to −30 pN and to −40 pN, the X-nullclines are shown by the blue and cyan curves, respectively. As the absolute value of negative external force F increases, the X-nullcline moves down-left, which is consistent with Eq. (3). The change of X-nullcline with respect to force F is the basis to identify the dynamics of twitch induced by force.

Fig. 4.

The influence of static F on X-nullcline and equilibrium point for Case-2 steady state with D = 50 nm and S = 0.2. a The X-nullclines (solid curves) at different values of force F; b The X-nullclines (solid curves) intersect with $X_a$-nullcline (black dashed curve) at different values of force F to form the equilibrium points (hollow triangles). The numbers represent the value of F

The influence of static F on equilibrium point

The $X_a$-nullcline is independent of external force F due to that the equation of $X_a$-nullcline does not contain the parameter F, which is described as follows:

$$\begin{array}{c}\hfill {\lambda }_a\frac{d{X}_a}{\mathit{dt}}=K_{\mathit{GS}}(X-X_a-D{P}_0)-K_{\mathit{ES}}(X_a-X_{\mathit{ES}})-F_{\mathit{max}}(1-S{P}_0)=0\end{array}$$ 4

Then, the $X_a$-nullcline is acquired as follows:

$$\begin{array}{cc}\hfill X=& \frac{K_{\mathit{GS}}+K_{\mathit{ES}}}{K_{\mathit{GS}}}{X}_a+\frac{K_{\mathit{GS}}D-F_{\mathit{max}}S}{K_{\mathit{GS}}}{P}_0\hfill \\ \hfill & +\frac{F_{\mathit{max}}-K_{\mathit{ES}}{X}_{\mathit{ES}}}{K_{\mathit{GS}}}\hfill \end{array}$$ 5

From Eq. (5), $X_a$-nullcline is independent of static force F, as shown by the dashed black line of Fig. 4b. The intersection points of $X_a$-nullcline (dashed black line) and X-nullclines (solid colorful curves) at different values of F are the equilibrium points (hollow triangles), as shown in Fig. 4b. The different intersection points (hollow triangles) show the influence of F on the equilibrium point. With the increase of F, both X value and $X_a$ value of the equilibrium point increase, i.e. the equilibrium point moves to up-right along the $X_a$-nullcline. In addition, the change of F also affects the position of the equilibrium point on the X-nullcline. As F increases, the position of equilibrium point on lower branch of X-nullcline moves to down-left and approaches the middle branch from right to left. As F decreases, the position of equilibrium point on lower branch moves up-right, i.e. moves away from the middle branch from left to right. The different positions of equilibrium point at different values of external force F are the basis to explain the twitch.

Twitch induced by pulse force for Case-2 steady state

The transient behaviors induced by pulse force: during and after the pulse force

A representative of the dynamical behavior induced by impulse force from the Case-2-1 steady state is shown in Fig. 5. In the present Section, the force is changed from 0 to a nonzero value $F_0$ at phase A and remains unchanged in a duration 100 ms, and then the force recovers to 0 at phase C, as shown by the blue curve in Fig. 5a1. The dynamical behavior induced by pulse force can be divided into 2 durations. One is the duration within the pulse force (from phase A to phase C), and the other is the duration after the termination timing of the pulse (after phase C). The duration of force 100 ms (from 9000 ms to 9100 ms in Fig. 5a1) is long enough to ensure that the dynamical behavior reaches the steady state. In the present paper, to be consistent with experiment wherein 100 ms is used (Benser et al. 1996), the duration of force 100ms is chosen as a representative. In fact, the results are similar when force duration with value long enough is chosen.

Fig. 5.

The negative twitch induced by pulse force for Case-2-1 steady state with D = 50 nm and S = 0.2. Positive force with $F_0$ = 50 pN: a1 The twitch (black) appearing after the termination time (phase C at 9100 ms) of the pulse force (blue); a2 The relationship between the trajectory of twitch (black curve), X-nullclines for F = 0 pN (red curve) and F = 50 pN (blue curve), $X_a$-nullcline (green line), and equilibrium points for F = 0 pN (red star) and F = 50 pN (blue star). Negative force F = −30 pN: b1 The twitch (black) appearing at the beginning phase (phase A at 9000 ms) of the pulse force (blue); b2 The relationship between the trajectory of twitch (black curve), X-nullclines for F = 0 pN (red curve) and F = −30 pN (blue curve), $X_a$-nullcline (green line), and equilibrium points for F = 0 pN (red star) and F = −30 pN (blue star). (Color figure online)

The behaviors within two durations are as follows:

1. Before the application of pulse force $F_0$ ($t<$ 9000 ms), i.e. before phase A, the behavior remains at the equilibrium point (red star) for F = 0 pN. Therefore, within the pulse force (9000 ms $<t<$ 9100 ms), i.e. between phase A and C, the dynamical behavior corresponds to a transient behavior for F = $F_0$, which begins from the equilibrium point (red star) for F = 0 pN and evolves to the equilibrium point (blue star) for F = $F_0$.

2. Similarly, for the duration after the termination time of the pulse force ($t>$ 9100 ms), i.e. after phase C, the dynamical behavior corresponds to a transient behavior for F = 0 pN, which begins from the initial value corresponding to steady state of F= $F_0$ and at last approaches to the steady state of F = 0 pN, that is, a dynamical behavior runs from the equilibrium point of F = $F_0$ to equilibrium point of F = 0 pN.

Based on the recognitions on two dynamical behaviors and the transient behavior of 3 subcases of Case-2 steady state shown in Fig. 3, the diverse twitches induced by the change of force are explained in the following paragraphs.

Negative change of F induces negative twitch: Case-2-1 steady state

For D = 50 nm and S = 0.2, the equilibrium point for F = 0 pN is on left branch of the X-nullcline, as shown in Fig. 5a2, which belongs to Case-2-1 steady state. The results for the positive pulse force with $F_0$ = 50 pN are shown in Fig. 5a1, a2.

Within the duration of the pulse force (from phase A to C), the dynamical behavior corresponds to the transient behavior for $F_0$ = 50 pN, which begins from the equilibrium point (phase A) of F = 0 pN to equilibrium point (phase C) of F = 50 pN, as shown in Fig. 5a1. In phase plane, the initial value (phase A) of the transient behavior is within region IV divided by the X-nullcline (blue curve) of F = 50 pN, as shown in Fig. 5a2. Therefore, the transient behavior from point A to C (black curve) resembles the gray curve shown in Fig. 3a1, a2. The trajectory from point A to C (black) does not run across the nonlinear part of X-nullcline (blue curve) for F = 50 pN, resulting in no twitch.

After the termination time of the force pulse (after phase C), the dynamical behavior corresponds to the transient behavior for F = 0 pN, which begins from the equilibrium point (phase C) of F = 50 pN to equilibrium point (phase G) of F = 0 pN, as shown in Fig. 5a1. The initial value (phase C) of the transient behavior is within region II divided by the X-nullcline (red curve) of F = 0 pN, as shown in Fig. 5a2. Then, in phase plane, the transient behavior (black curve) from point C (blue star) to G (red star, same as phase A) resembles the blue curve shown in Fig. 3a1, a2. The behavior from phase D to E and to F runs along the nonlinear part of X-nullcline, resulting in decrease of X and then increase of X to form the negative twitch, which is induced by the negative change of the force.

The result for the negative pulse force is illustrated in Fig. 5b1 and b2, with $F_0$ = −30 pN chosen as representative. Within the duration of the pulse force (phase A to E), the dynamical behavior is transient behavior beginning from the equilibrium point (phase A, red star) for F = 0 pN to the equilibrium point (phase E, blue star) for F = −30 pN. The initial value, i.e. the phase A, locates within region II for F = −30 pN. Then, the transient behavior resembles the blue curve in Fig. 3a1, a2 and the twitch shown in Fig. 5a1, a2. The behavior from phase B, to C, and to D runs along the nonlinear part of X-nullcline for F = −30 pN, which corresponds to the twitch induced by negative change of force.

After the termination time of the force pulse (form phase E to F, to G, G is the same as phase A in phase plane), as shown in Fig. 5b1, b2, the dynamical behavior is transient behavior beginning from equilibrium point (phase E, blue star) for F = −30 pN to equilibrium point (phase G, red star, same as phase A in phase plane) for F = 0 pN. The initial value of the dynamical behavior, i.e. point E, is within region IV for X-nullcline for F = 0 pN. Then, no twitch appears, which resemble to the gray curve in Fig. 3a1, a2 and the non-twitch in Fig. 5a1, a2 induced by positive change of external force.

In summary, for Case-2-1 steady state, the equilibrium point is on the left branch of X-nullcline. Therefore, only trajectory beginning from the initial value within region II runs along the nonlinear part of X-nullcline to form a negative twitch, which corresponds to the decreases of X along the lower branch first and then recovery to the upper branch. The negative change of force F plays a role to modulate the initial value to be within region II. Therefore, the negative change of force F induces the negative twitch.

Case-2-2 steady state: Positive and negative changes of F induce positive and negative twitches, respectively

A representative of the twitch induced by impulse force from the Case-2-2 steady state (D = 50 nm and S = 0.45) is shown in Fig. 6. Different from Case-2-1, positive change of F induces positive twitch, as shown in Fig. 6a1, a2, and negative change of F induces negative twitch.

Fig. 6.

The positive and negative twitch induced by pulse force for Case-2-2 steady state with D = 50 nm and S = 0.45. Positive force with $F_0$ = 10 pN: a1 The positive and negative twitches (black) appearing at the beginning phase and after the termination time of the pulse force (blue), respectively; a2 The relationship between the trajectory of twitch (black curve), X-nullclines for F = 0 pN (red curve) and F = 10 pN (blue curve), $X_a$-nullcline (green line), and equilibrium points for F = 0 pN (red star) and F = 10 pN (blue star). Negative force F = −2 pN: b1 The negative and positive twitches (black) appearing at the beginning phase and after the termination time of the pulse force (blue), respectively; b2 The relationship between the trajectory of twitch (black curve), X-nullclines for F = 0 pN (red curve) and F = -2 pN (blue curve), $X_a$-nullcline (green line), and equilibrium points for F = 0 pN(red star) and F = −2 pN (blue star). (Color figure online)

The results for the positive pulse force with $F_0$ = 10 pN are shown in Fig. 6a1, a2. Within the pulse force (from phase A to E), the dynamical behavior corresponds to the transient behavior for F = 10 pN with initial value corresponding to equilibrium point (phase A) for F = 0 pN, which is within region IV for F = 10 pN. Then, the behavior from phase B, to D, and to E runs along the nonlinear part of X-nullcline of F = 10 pN, which resembles the behavior the gray curve in Fig. 3b1, b2, resulting in the positive twitch induced by the positive change of force. Before reaching phase E, damping oscillations appear due to the focus of the equilibrium point for F = 10 pN. After the termination time of the force pulse (from phase E to I), the dynamical behavior corresponds to the transient behavior for F = 0 pN and the initial value corresponds to equilibrium point (phase E) for F = 10 pN, which is within region IV for F = 0 pN. Therefore, the behavior from phase E to I runs along the nonlinear part of X-nullcline for F = 0 pN and forms the negative twitch induced by negative change of the external force, resembling the blue curve in Fig. 3b1, b2. Similarly to the positive twitch, damping oscillations appear before reaching the equilibrium point for F = 0 pN (phase I, same as phase A in phase plane).

The results for the negative pulse force with $F_0$ = −2 pN are shown in Fig. 6b1, b2. Within the force pulse (phase A to D), the dynamical behavior is transient behavior beginning from the initial value (phase A, equilibrium point for F = 0 pN) within region II for F = −2 pN and evolving to equilibrium point (phase D) for F = −2 pN, as shown in Fig. 6b2, which corresponds to the blue curve in Fig. 3b1, b2. The negative twitch is induced as the behavior runs near the nonlinear part of X-nullcline for F = −2 pN (from phase A, to B, to C, and to D). After the termination time of the force pulse (from phase A to D), the dynamical behavior is the transient behavior beginning from the initial value (phase D, blue star, equilibrium point for F = −2 pN) within region IV for F = 0 pN and evolving to the equilibrium point (phase G, red star) for F = 0 pN, as shown in Fig. 6b2, which corresponds to the gray curve in Fig. 3b1, b2. The positive twitch is induced as the behavior runs near the nonlinear part (from phase D, to E, and to F) of X-nullcline for F = 0 pN(red).

In summary, for Case-2-2 steady state, the equilibrium point is on the middle branch of X-nullcline. The trajectories beginning from the initial values within region II and region IV can run near the nonlinear part of X-nullcline to form the twitch. The positive and negative changes of external force F can induce positive and negative twitches, respectively.

Case-2-3 steady state: Positive changes of F induces positive twitch

A representative of the twitch induced by impulse force from the Case-2-3 steady state for D = 50 nm and S = 5 is shown in Fig. 7. The results for the positive pulse force with $F_0$ = 50 pN are shown in Fig. 7a1, a2. For the duration within the pulse force (from phase A to D), the dynamical behavior corresponds to the transient behavior for F = 50 pN and initial value corresponds to steady state with F = 0 pN (phase A), as shown in Fig. 7a1. The transient behavior beginns from the initial value (phase A) within region IV for F = 50 pN, as shown in Fig. 7a2, which corresponds to the gray curve in Fig. 3c1, c2. Since the trajectory pass through the X-nullcline near the nonlinear region to reduce the X value, the positive twitch is induced.

Fig. 7.

The positive twitch induced by pulse force for Case-2-3 steady state with D = 50 nm and S = 5. Positive force with $F_0$ = 50 pN: a1 The twitch (black) appearing at the beginning phase of the pulse force (blue); a2 The relationship between the trajectory of twitch (black curve), X-nullcline for F = 0 (red curve) and F = 50 pN (blue curve), $X_a$-nullcline (green line), and equilibrium point for F = 0 pN (red star) and F = 50 pN (blue star). Negative force F = −40 pN: b1 The twitch (black) appearing after the termination time of the pulse force (blue); b2 The relationship between the trajectory of twitch (black curve), X-nullcline for F = 0 pN (red curve) and F = −40 pN (blue curve), $X_a$-nullcline (green line), and equilibrium point for F = 0 pN (red star) and F = −40 pN (blue star). (Color figure online)

After the termination time of the force pulse (after phase D), the dynamical behavior corresponds to the transient behavior for F = 0 pN, which begins from the equilibrium point (phase D) of F = 50 pN to equilibrium point (phase F) of F = 0 pN, as shown in Fig. 7a1. The initial value (phase D) is within region II for F = 0 pN, as shown in Fig. 7a2. Then, the transient behavior (black curve) from point D (blue star) to point F (red star, same as phase A) resembles the blue curve shown in Fig. 3c1, c2. The behavior from phase D to E and to F runs along the linear part of X-nullcline, resulting in no twitch.

The result for the negative pulse force is illustrated in Fig. 7b1 and b2, with $F_0$ = −40 pN chosen as representative. Within the duration of the pulse force (from phase A to C), the dynamical behavior is transient behavior beginning from the equilibrium point (phase A, red star) for F = 0 pN to the equilibrium point (phase C, blue star) for F = −40 pN. The initial value, i.e. the phase A, locates within region II for F = −40 pN. Then, the transient behavior resembles the blue curve in Fig. 3c1, c2. The behavior from phase A, to B, and to C runs along right branch of X-nullcline for F = −40 pN, resulting in no twitch.

After the termination time of the force pulse (from phase C, to D, to E, to F, and F is the same as phase A in the phase plane), the dynamical behavior is transient behavior beginning from equilibrium point (phase C blue star) for F = −40 pN to equilibrium point (phase F, red star) for F = 0 pN. The initial value, i.e. point C, is within region IV for F = 0 pN. Then, positive twitch appears, which resembles the gray curve in Fig. 3c1, c2. The behavior from phase D to E and to F runs along the nonlinear part of X-nullcline.

In summary, for Case-2-3 steady state, the equilibrium point is on the right branch of X-nullcline. Therefore, only the trajectory beginning from the initial value within region IV runs across the nonlinear part of X-nullcline to form a positive twitch. The positive change of force F plays a role to regulate the initial value to be within region IV. Therefore, the positive change of force F induces the negative twitch.

Twitch exhibits fast-slow dynamics

The twitch is related to fast-slow dynamics of two variables (X and $X_a$), which is characterized by the changes of derivatives of the two variables with respect to time. A representative of positive and negative twitches (upper panel) induced by pulse force with $F_0$ = 10 pN for D = 50 nm and S = 0.45 and the corresponding derivatives (lower panel) are shown in Fig. 8. Two characteristics can be found. One is that dX/dt (black, left y-axe) is significantly larger than $d{X}_a/dt$ (red, right y-axe), and the other is that dX/dt changes much faster than $d{X}_a/dt$. Such results indicate that X is the fast variable and $X_a$ is the slow variable, as shown in the lower panel of Fig. 8.

Fig. 8.

Fast-slow dynamics of twitch for Case-2-2 steady state with D = 50 nm and S = 0.45. Upper panel: The twitch (black) appearing at the beginning phase of pulse force with $F_0$ = 10 pN (blue) and after the termination time of the pulse force. Lower panel: dX/dt (black) and $d{X}_a/dt$ (red). Insertin lower panel: Partial enlargement. (Color figure online)

Within a short duration (from phase A to phase B) after the application of the force at t = 9000 ms, dX/dt exhibits a large value (black in lower panel), and $d{X}_a/dt$ (red) can be negligible, compared with dX/dt, which causes the displacement X (black in upper pannel) to rise to phase B quickly. Subsequently (before phase D), the value of dX/dt drastically decreases to nearly zero (the left insert figure of lower pannel), while the slowly changing $d{X}_a/dt$ (red) is still non-zero, which plays a dominant role and induces the recoil of displacement X, resulting in the positive twitch. After phase D, both deviations approach to zero, the behavior tends to the equilibrium point for F = 10 pN. Similarly, negative twitch corresponds to a process from phase E to F (negative spike of dX/dt), to phase G and H (negative pulse of $d{X}_a/dt$). The rapid decrease of displacement X is dominated by dX/dt, and the recoil of displacement X is mainly determined by $d{X}_a/dt$. Other twitches exhibit similar fast-slow dynamics, which are not addressed to avoid repetitions.

Twitch induced by pulse force for Case-1 steady state

For two subcases of Case-1 steady state, the twitch can be induced by the pulse force, as depicted in Figs. 9 and 10, respectively. Compared with twitch for Case-2, the relationship between the phase trajectory of twitch for Case-1 and the X-nullcline exhibits local distinction near the middle branch of X-nullcline, as illustrated in Figs. 9 and  10, due to Z-shaped X-nullcline for Case-1 and N-shaped X-nullcline for Case-2.

Fig. 9.

The negative twitch induced by pulse force for Case-1-1 steady state with D = 80 nm and S = 0.5. Positive force with $F_0$ = 70 pN: a1 The twitch (black) appearing after the termination time of the pulse force (blue); a2 The relationship between the trajectory of twitch (black curve), X-nullclines for F = 0 pN (red curve) and F = 70 pN (blue curve), $X_a$-nullcline (green line), and equilibrium points for F = 0 pN (red star) and F = 70 pN (blue star). Negative force F = −40 pN: b1 The twitch (black) appearing at the beginning phase of the pulse force (blue); b2 The relationship between the trajectory of twitch (black curve), X-nullclines for F = 0 pN (red curve) and F = -40 pN (blue curve), $X_a$-nullcline (green line), and equilibrium points for F = 0 pN (red star) and F = −40 pN (blue star). (Color figure online)

Fig. 10.

The positive twitch induced by pulse force for Case-1-2 steady state with D = 80 nm and S = 3. Positive force with $F_0$ = 40 pN: a1 The twitch (black) appearing at the beginning phase of the pulse force (blue); a2 The relationship between the trajectory of twitch (black curve), X-nullclines for F = 0 pN (red curve) and F = 40 pN (blue curve), $X_a$-nullcline (green line), and equilibrium points for F = 0 pN (red star) and F = 40 pN (blue star). Negative force F = −50 pN: b1 The twitch (black) appearing after the termination time of the pulse force (blue); b2 The relationship between the trajectory of twitch (black curve), X-nullclines for F = 0 pN (red curve) and F = −50 pN (blue curve), $X_a$-nullcline (green line), and equilibrium points for F = 0 pN (red star) and F = −50 pN (blue star). (Color figure online)

Case-1-1: negative twitch induced by negative change of force

For D = 80 nm and S = 0.5, the equilibrium point for F = 0 pN is on left branch of the X-nullcline, as shown in Fig. 9a2, which belongs to Case-1-1 steady state. After the termination time of positive pulse force $F_0$ = 70 pN, the negative twitch (from phase C to F) can be induced, as shown in Fig. 9a1, which resembles Case-2-1 (Fig. 5a1). However, the relationship between the phase trajectory of twitch and X-nullcline exhibits local distinction to that of Case-2-1, as shown in Fig. 9a2, due to the Z-shaped X-nullcline for Case-1-1 and N-shaped X-nullcline for Case-2-1. From phase C, to D, and to E, the phase trajectory resembles that of Case-2-1. The relationship between trajectory of twitch and X-nullclines from phase E to F for Case-1-1 is different from that of Case-2-1. The trajectory does not run along middle branch of X-nullcline for Case-1-1 (Fig. 9a2), while does for Case-2-1 (Fig. 5a2), due to different vector fields around the middle branch. From phase F to G, phase trajectory for Case-1-1 steady state exhibits characteristic similar to that of Case-2-1. In addition, the phase trajectory during the force pulse (from phase A, to B , and to C) manifests similar dynamics for both Case-1-1 and Case-2-1, as shown in Figs. 9a2 and 5a2.

For the negative pulse force with $F_0$ = −40 pN, the negative twitch can be induced at the beginning time of pulse, as shown in Fig. 9b1, which is similar to Case-2-1 (Fig. 5b1). The relationship between phase trajectory of twitch and X-nullcline manifests distinction to that of Case-2-1 from phase C to D, as shown in Fig. 9b2. The phase trajectory of twitch for Case-1-1 does not run along the middle branch of X-nullcline while for Case-2-1 does. Other parts of the phase trajectory for Case-1-1 and Case-2-1 have similar relationships to X-nullcline. After the termination time of the pulse force, the relationship of phase trajectory to X-nullcline for Case-1-1 and Case-2-1 exhibits similar characteristics.

Case-1-2: positive twitch induced by positive change of force

For D = 80 nm and S = 3, the equilibrium point for F = 0 is on right branch of the X-nullcline, as shown in Fig. 10a2, b2, which belongs to Case-1-2 steady state.

The positive twitch can be induced at the beginning phase of positive pulse force $F_0$ = 40 pN, as shown in Fig. 10a1, resembling that of Case-2-3 (Fig. 7a1). In phase plane, the distinction between phase trajectory of Case-1-2 and Case-2-3 happens from phase C to D. The phase trajectory of twitch for Case-1-2 does not run along the middle branch of X-nullcline while for Case-2-3 does, as shown in Figs. 10a2 and 7a2. Other parts of both trajectories exhibit similar characteristics.

For the negative pulse force with $F_0$ = −50 pN, the positive twitch can be induced after the termination time of the pulse force, as shown in Fig. 10b1, which resembles that of Case-2-3 (Fig. 7b1). In phase plane, the distinction between phase trajectory of Case-1-2 and Case-2-3 happens from phase E to F. The phase trajectory of twitch for Case-1-2 does not run along the middle branch of X-nullcline while for Case-2-3 does, as shown in Figs. 10b2 and  7b2. Other parts of both trajectories exhibit similar characteristics.

The dynamic of fatigue of twitch

The fatigue effect of twitch is an interesting phenomenon induced by two force pulses, which has been observed in previous experiments, as shown in Fig. 8 in Ref Benser et al. (1996). Fatigue of twitch means that the amplitude of twitch caused by the latter pulse force is obviously smaller than that caused by the former pulse force. In the present subsection, the fatigue phenomenon of twitch is simulated, and the fatigue effect is explained using the nullclines and phase trajectory. The steady state for $K_{\mathit{SP}}$ = 1.8 mN/m, D = 60.9 nm, and S = 2 is chosen as representative. The duration of either of two pulse forces is 10 ms and the strength of the force is 50 pN. The interval between two force pulses is chosen as control parameter. The dynamics of the fatigue of twitch at different values of the interval between two force pulses are shown in Fig. 11.

Fig. 11.

Two force pulses (strength F = 50 pN, duration 10 ms) with different time interval induce fatigue of twitch from steady state for $K_{\mathit{SP}}$ = 1.8 mN/m, D = 60.9 nm, and S = 2. a1 and a2 Time interval 1 ms. Phase a represents the initial phase of the latter force pulse; b1 and b2 Time interval 2 ms. Phase b represents the initial phase of the latter force pulse; c1 and c2 Time interval 3 ms. Phase c represents the initial phase of the latter force pulse; d1 and d2 Time interval 4ms. Phase d represents the initial phase of the latter force pulse. Left: Time process of force (blue curve) and twitch (black curve). The initial position of the former twitch, peak of the former twitch, and the equilibrium point for F = 50 pN correspond to phases A, B, and C, respectively. Right: Phase trajectory of former and latter twitches (black curves) and nullclines (similar to Fig. 5). Phases A, B, and C are similar to the left panel. e1 Figs. a1–d1 are plotted in one figure; e2 Figs. a2–(d2) are plotted in one figure. (Color figure online)

Simulation of fatigue of twitch

The former force pulse begins from t = 9000 ms and terminates at t = 9010 ms, then, the amplitude of the former twitch (evoked by the former pulse force) is 13.12 nm, as shown in left column of Fig. 11. The latter twitch exhibits different amplitudes at different values of time interval between two forces pulses. For example, if the interval is chosen as 1 ms, 2 ms, 3 ms, and 4 ms, respectively, the amplitude of the latter twitch is 5.15 nm, 11.06 nm, 12.09 nm, and 12.35 nm (12.35/13.12 $\ast$ 100$\%$ = 94.85$\%$), as shown in Fig. 11a1, b1, c1 and d1, which manifests two characteristics. One is that all of 4 amplitudes of latter twitch are less than that of the former twitch, and the other is that the amplitude of the latter twitch increases with increasing the time interval between two force pulses. For example, when time interval between two force pulses is 4 ms, the amplitude of the latter twitch returns to 94.85$\%$. If the interval is chosen as a value long enough (not shown here), the amplitude of the latter twitch retuned to 13.12 nm (100$\%$). Such two characteristics can be found in Fig. 11e1, which is the superposition of Fig. 11a1, b1, c1, and d1. The third characteristic of fatigue of twitch can be found from Fig. 11e1. With increasing time interval between two force pulse, the value of X at beginning time of latter force pulse decreases, as shown by the phase a, b, c, and d, respectively. In fact, the latter twitches for different time intervals between two force pulses correspond to the transient behaviors beginning from different initial values (phases a, b, c, and d) for F = 50 pN.

Dynamical mechanism for the lower amplitude of the latter twitch

The cause for the fatigue of twitch can be well explained with phase trajectory and nullclines in phase plane ($X_a$, X), as shown in the right column of Fig. 11. Each panel of right column represents the dynamics of the left panel in phase plane.

For the trajectory in ($X_a$, X) phase space, the dynamics of the lower amplitude of the latter twitch can be obtained. The decrease of time interval causes the initial position (phases a, b, c, or d) of the latter twitch to move to the right in phase space, compared to the former twitch (phase A, red star). The shorter the time interval is, the more to the right the initial position is, as shown in the right column of Fig. 11. The movement to the right of the initial position causes trajectory to run across the X-nullclines at a lower position, resulting in the reduction of the amplitude of the latter twitch, which is the first characteristic of the fatigue of twitch. The second characteristic is that the amplitude of the latter twitch increases with increasing the time interval which can be explained by Fig. 12. The initial position of the trajectory moves to the left with the increase of time interval, which is equivalent to the transient behavior with different initial values. Determined by the vector fields (light red arrows) of the X-nullcline (blue curve) of F = 50 pN, the more left the initial value of the trajectory is (from phase a, to b, to c, and to d), the higher the position of the trajectory running across the X-nullcline is, i.e. the higher the latter twitch amplitude is, as shown in Fig. 12.

Fig. 12.

Trajectories of fatigue of twitch and vector fields (light red arrows) for $K_{\mathit{SP}}$ = 1.8 mN/m, D = 60.9 nm, and S = 2. Other labels are the same as those in Fig. 11 (e2). (Color figure online)

Force induces transition between steady state and oscillations for Case-1 instead of Case-2

Transition between Case-1 steady state and oscillations

For Case-1 steady state near Hopf bifurcation, pulse force can induce the transition between oscillations and steady state. For example, for the Case-1-2 steady state (S = 2.5 and D = 90 nm chosen as representative), the transition between steady state and oscillations with higher X value can be induced by positive pulse force with an amplitude of 30 pN and duration 200 ms, as shown in Fig. 13a. Similarly, for the Case-1-1 steady state (S = 0.7 and D = 80 nm), transition between steady state and oscillations with lower X values is induced by negative pulse force F = −30 pN, as shown in Fig. 13b. These simulation results closely match those observed in the experiment (Benser et al. 1996).

Fig. 13.

Transition between Case-1 steady state and oscillations induced by force. a Case-1-2: S = 2.5 and D = 90 nm, transition from steady state to sustained oscillations induced by positive external pulse force (F = 30 pN, duration 200 ms); b Case-1-1: S = 0.7 and D = 80 nm, transition from steady state to sustained oscillations induced by negative external pulse force (F = −30 pN, duration 200 ms)

Bifurcation mechanism for the transition induced by force

The mechanism of the transition for Case-1 steady state can be explained by bifurcation. For Case-1-2 steady state at S = 2.5 and D = 90 nm, as the positive external force F increases, displacement X changes from equilibrium point to limit cycle via a supercritical Hopf bifurcation at $F\approx$ 17.7 pN, as shown in Fig. 14a and insert figure of Fig. 14a. Then, positive force can induce the transition from steady state to sustained oscillations via the Hopf bifurcation. Such a transition appears for 17.7 pN $<F<$ 539.6 pN, due to that F = 539.6 pN corresponds to another supercritical Hopf bifurcation via which the limit cycle is changed to steady state. Therefore, force changed from 0 pN to 30 pN can induce transition from steady state to sustained oscillations.

Fig. 14.

Bifurcation of displacement X with respect to external force F. a D = 90 nm and S = 2.5, corresponding to Fig. 13a; a D = 80 nm and S = 0.7, corresponding to Fig. 13b

For the Case-1-1 steady state at S = 0.7 and D = 80 nm, as the negative force F decreases, X exhibits a subcritical Hopf bifurcation at F = −17.9 pN and a fold or saddle-node bifurcation of limit cycles at F = −17.8 pN, as shown in Fig. 14b and insert figure in Fig. 14b. Therefore, negative force can induce the transition from steady state to sustained oscillations via the Hopf bifurcation. Such a transition appears for −47.1 pN $<F<$ −17.9 pN, due to that $F\approx$ −47.1 pN corresponds to another subcritical Hopf bifurcation via which the limit cycle is changed to steady state. Therefore, force changed from 0 pN to −30 pN can induce transition from steady state to sustained oscillations.

Force does not induce transition between Case-2 steady state and sustained oscillations

For the Case-2 steady state, neither positive nor negative impulse force can induce the transition between steady state and sustained oscillations. For example, when S = 0.45 and D = 50 nm, the steady state belongs to Case-2-2. Positive pulse force with 50 pN cannot induce transition from steady state to sustained oscillations, as shown in Fig. 15a. Similarly, negative pulse with -50 pN cannot induce transition from steady state to sustained oscillations, as shown in Fig. 15b. The results of Case-2-1 and Case-2-3 are similar to that of Case-2-2 (not shown here to avoid repetations).

Fig. 15.

No transition from Case-2 steady state to sustained oscillations induced by force (Case-2-2 steady state with S = 0.45 and D = 50 nm to pulse force). a Positive pulse force 50 pN; b Negative pulse force −50 pN

No bifurcation with respect to force for Case-2 steady state

The mechanism of no transition from steady state to sustained oscillations for Case-2 steady state can be explained by no bifurcation of displacement X with respect to external force F. For the Case-2-2 steady state in Section "Force does not induce transition between Case-2 steady state and sustained oscillations", with increasing or decreasing external force F, displacement X appears to be globally stable without bifurcations, as shown in Fig. 16, which shows that no stable limit cycle appears. Therefore, it is impossible to induce the transition from steady state to sustained oscillations corresponding to stable limit cycle. The results of Case-2-1 and Case-2-3 are similar to that of Case-2-2. In a word, there is no transition between Case-2 steady state and sustained oscillations due to no bifurcations with respect to the external force.

Fig. 16.

Change of displacement X with respect to external force F for Case-2-2 steady state with D = 50 nm and S = 0.45

Threshold for Case-1 steady state and no threshold for Case-2 steady state

Threshold for Case-1 steady state

Similar to the nervous system, the dynamic behavior of the Case-1 steady state with Z-shaped X-nullcline shows obvious threshold phenomenon. For example, steady state for D = 60.9 nm and S = 0.54 belongs to Case-1-1, which exhibits a higher position X = -24.087 nm (close to the left ${\text{H}}_{\mathit{sub}}$ bifurcation point (hollow star) at $S\approx$ 0.5527 in Fig. 21a). At t = 20 ms, the X value is disturbed to −26nm, then the hair bundle returns to the steady state X = −24.087 nm, as shown in Fig. 17a1. However, when the X value is disturbed to −27 nm, X exhibits a downward spike with large amplitude (approximate −50 nm) and then returns to X = −24.087 nm, as shown in Fig. 17b1. Comparing Figs. 17a1, b1, it can be concluded that Fig. 17b1 corresponds to superthreshold behavior and Fig. 17a1 corresponds to subthreshold behavior. Therefore, there is threshold phenomenon for Case-1-1 steady state.

Fig. 21.

Hopf bifurcation curves in (S, D) plane at different values of external force F. a Hopf bifurcation curves at F = −30 (blue), −19 (green), 0 (black), 18 (magenta), and 30 pN (red); The blue dashed line ($D\approx$ 50.3 pN) represents the border between the Case-1 and Case-2 steady state. Hollow star and solid star correspond to the steady state with threshold phenomenon shown in Fig. 17 and Fig. 18, respectively; b Hopf bifurcation curves at F = 0 pN (black) and 30 pN (red). Green square and green circle correspond to Case-1-1 steady state of Fig. 9 and Case-1-2 steady state of Fig. 13a, respectively; c Hopf bifurcation curves at F = 0 pN (black) and −30 pN (blue). Green square and green circle correspond to Case-1-2 steady state of Fig. 10 and Case-1-1 steady state of Fig. 13b, respectively. (Color figure online)

Fig. 17.

Threshold dynamics for the Case-1-1 steady with D = 60.9 nm and S = 0.54. a1 Subthreshold behavior after X disturbed to be −26 nm; b1 Superthreshold behavior (a spike) after X disturbed to be −27 nm; a2 X-nullcline (black curve), $X_a$-nullcline (magenta curve) and vector fields (black arrows) plotted with subthreshold behavior (blue curve); b2 X-nullcline (black curve), $X_a$-nullcline (magenta curve) and vector fields (black arrows) plotted with superthreshold behavior (red curve). (Color figure online)

Case-1-2 steady for D = 60.9 nm and S = 0.99 (close to the left ${\text{H}}_{\mathit{sup}}$ bifurcation point (solid star) at $S\approx$ 0.9790 in Fig. 21a), which exhibits a lower position (X = −41.696 nm), as shown in Fig. 18a1, b1. If X is disturbed to be −40 nm at t = 10 ms, subthreshold oscillations (focus) appear and at last X returns to the steady state (X = −41.696 nm), as shown in Fig. 18a1. However, if X is disturbed to be −39 nm at 10 ms, a spike with large amplitude (approximate −27 nm) is evoked, as shown in Fig. 18b1. After the spike, damping subthreshold oscillations appear and at last return to the steady state. Then, threshold phenomenon exists for the Case-1-2 steady state.

Fig. 18.

Threshold dynamics for the Case-1-2 steady (D = 60.9 nm and S = 0.99) near the right Hopf bifurcation (The solid star in Fig. 21a). a1 Subthreshold behavior after X disturbed to be -40 nm; b1 Superthreshold behavior (a spike) after X disturbed to be −39 nm; a2 X-nullcline (black curve), $X_a$-nullcline (magenta curve) and vector fields (black arrows) plotted with subthreshold behavior (blue curve); b2 X-nullcline (black curve), $X_a$-nullcline (magenta curve) and vector fields (black arrows) plotted with superthreshold behavior (red curve). (Color figure online)

Threshold mechanism: coexistence of fast variable X on Z-shaped X-nullcline for Case-1 steady state

The threshold phenomenon of Case-1 steady state is due to the coexistence of X values, i.e. the Z-shaped X-nullcline. For Case-1-1 steady state, except the X value of equilibrium point on upper branch, as shown in Fig. 17a2, b2, there are two other X values on middle branch and lower branch at $X_a$ value of the equilibrium point. Therefore, if X is disturbed to a value upper to the middle branch (or lower but very close to the middle branch), the phase trajectory of the response to the weak disturbance runs along the vector fields and returns to the steady state, resulting in sub-threshold behavior, as shown in Fig. 17a2. If X is disturbed to a value lower to the middle branch to a certain extent, the phase trajectory after the strong disturbance runs along the vector fields to approach the lower branch to from a spike and then returns to the steady state, resulting in a spike with large amplitude, i.e. super-threshold oscillations. Such a dynamical mechanism of threshold resembles that of the action potential

For Case-1-2 steady state, there are also three X values on upper branch, middle branch, and lower branch, respectively, at $X_a$ value of the the equilibrium point, while the intersection point of the nullclines on the lower branch and the corresponding equilibrium point is focus. Therefore, if X is disturbed to a value lower to the middle branch (or upper but very close to the middle branch), the phase trajectory of the response to the weak disturbance runs along the vector fields and returns to the steady state, resulting in subthreshold behavior, as shown in Fig. 18a2. If X is disturbed to a value upper to the middle branch to a certain extent, the phase trajectory after the strong disturbance runs along the vector fields to approach the upper branch to from a spike and then returns to the steady state, resulting in a spike with large amplitude, i.e. superthreshold oscillations.

No threshold phenomenon for Case-2 steady state

There is no threshold phenomenon for Case-2 steady state. For example, when D = 50 nm and S = 0.32, Case-2-1 steady state exhibits a stable equilibrium point (green point, X = −34.699nm and $X_a$ = −123.410 nm), as shown in Fig. 19. If X is respectively disturbed to be −38 nm, −40 nm, and −45 nm at t = 5 ms, the value of X after disturbance decreases and returns to the equilibrium point, as shown by cyan, blue, and red curves in Fig. 19a1. The results imply that as the disturbance to X gradually increases, the maximal value of response to disturbance gradually increases, as shown in Fig. 19a1, i.e. no threshold phenomenon. On other hand, if the disturbance is applied to $X_a$ instead of X, the results are similar, as shown in Fig. 19b1. The cyan, blue, and red curves represent the changes of displacement X with respect to time as $X_a$ is disturbed to be −119 nm, −118 nm, and −115 nm at 5ms, and the maximal value of X after disturbance is −40.5 nm, −44.8 nm, and −46.7 nm, respectively, which changes gradually. Therefore, there is no threshold in $X_a$ direction.

Fig. 19.

No threshold phenomenon for Case-2-1 steady state (green point, X = −34.699 nm and $X_a$ = −123.410 nm) with D = 50 nm and S = 0.32. a1 The changes of displacement X with respect to time as X is changed to −38 nm (cyan), −40 nm (blue), and −45 nm (red) by transient stimulation applied at t = 5 ms; a2 Phase trajectories of behaviors of a1 plotted with vector fields and nullclines. Insert: Partial enlargement; b1 The changes of displacement X with respect to time as $X_a$ is changed to −119 nm (cyan), −118 nm (blue), and −115 nm (red) by transient stimulation applied at t = 5 ms; b2 Phase trajectories of behaviors of b1 plotted with vector fields and nullclines. (Color figure online)

For the steady state when D = 50 nm and S = 1, which belong to Case-2-3 steady state, the displacement of hair bundle is in steady state (green point, X = −46.773 nm and $X_a$ = −116.914 nm), as shown in Fig. 20. Similar to Case-2-1, continuous displacement stimuli in the X or $X_a$ direction can cause gradual change of the maximal value of displacement X, as shown in Fig. 20. Therefore, there is no threshold.

Fig. 20.

No threshold phenomenon for Case-2-3 steady state (green point, X = −46.773 nm and $X_a$ = −116.914 nm) with D = 50 nm and S = 1. a1 The changes of displacement X with respect to time as X is changed to −43 nm (cyan), −40 nm (blue), and −37 nm (red) by transient stimulation applied at t = 5ms; a2 Phase trajectories of behaviors of a1 plotted with vector fields and nullclines. Insert: Partial enlargement; b1 The changes of displacement X with respect to time as $X_a$ is changed to −120 nm (cyan), −123 nm (blue), and −125 nm (red) by transient stimulation applied at t = 5 ms; b2 Phase trajectories of behaviors of Fig. b1 plotted 1with vector fields and nullclines. (Color figure online)

No threshold mechanism: non-coexistence of fast variable X on N-shaped X-nullcline for Case-2 steady state

The X-nullcline of Case-2 steady state is N-shaped. There is no coexistence of fast variable X on X-nullcline. A stimulus in either the X direction or the $X_a$ direction cannot induce displacement X to change to another very different value. With the gradual increase of stimulus, the maximal value of displacement X changes gradually, as shown in the right column of Figs. 19 and  20, i.e. no threshold phenomenon.

The underlying bifurcation mechanism in (S, D) plane for the forced oscillations

The bifurcations in (S, D) plane at different values of external force are acquired, as shown in Fig. 21, which can be used to explain the forced oscillations such as the transition between resting state and sustained oscillations, threshold phenomenon, and twitch.

The Hopf bifurcations in (S, D) plane at different values of F

The Hopf bifurcation curves in the (S, D) plane at different value of external force F are shown in Fig. 21a. The blue, green, black, magenta, and red curves respectively represent the Hopf bifurcation curves at F = −30, −19, 0, 18, and 30 pN. At each value of F, spontaneous oscillations appear within the region surrounded by the Hopf bifurcation curve, and steady state locates outside of the Hopf bifurcation curve. With the change of external force, the Hopf bifurcation curve and the region of spontaneous oscillations surrounded by the Hopf bifurcation curve show the characteristics listed as follows:

1. If the external force is positive, with the increase of the external force, the region of spontaneous oscillations expands, due to that the right Hopf bifurcation curve moves to the right to a large extent, and the left Hopf bifurcation curve moves to right to a small extent.

2. If the external force is negative, with the increase of the absolute value of the external force, the region of spontaneous oscillations region shrinks, because that the right Hopf bifurcation curve moves to the left to a large extent, and the left Hopf bifurcation curve moves to the left to a small extent.

3. The blue dashed line ($D\approx$ 50.3 pN) represents the border between the Case-1 and Case-2 steady state, which remains unchanged with respect to the external force F (the detailed proof please refer to following Section "The unchanged boundary between case-1 and case-2 steady state in in (S, D) plane with respect to external force F".

Threshold for Case-1 steady state and no threshold for Case-2 steady state

The hollow star and the solid star in Fig. 21a correspond to the Case-1-1 and Case-1-2 steady states of Section "Threshold for Case-1 steady state", respectively, which locates near the Hopf bifurcation curve. A suitable disturbance induces a spike corresponding to the limit cycle near the Hopf bifurcation points, which is the threshold phenomenon. If the steady state is far from the Hopf bifurcation, the threshold phenomenon disappears. The Hopf bifurcation is the underlying dynamical mechanism of the threshold.

For the Case-2 steady state lower to the blue dashed line ($D\approx$ 50.3 nm), there is no bifurcations containing Hopf bifurcation. Therefore, no threshold phenomenon exists for the Case-2 steady state, which corresponds to the result of Section "No threshold phenomenon for Case-2 steady state".

Transition between steady state and sustained oscillations for Case-1 steady state and no transition for Case-2 steady state

When external force F = 0 pN, the hair bundle exhibits Case-1-2 steady state, as shown by the green circle in Fig. 21b. The green circle is outside of the oscillation region formed by the Hopf bifurcation curve of F = 0 pN (black curve), corresponding to Fig. 13a of Section "Transition between Case-1 steady state and oscillations". When external force is changed to F = 30 pN, the border of oscillation region changes to the Hopf bifurcation curve depicted by the red curve, resulting in that the green circle locates within the oscillation region for F = 30 pN. Therefore, the transition from steady state to sustained oscillations appears if F = 0 pN is changed to F = 30 pN. The change of the Hopf bifurcation curve with respect to external force F is the underlying mechanism for the transition. The result shows that the positive external force induces the transition behavior for the Case-1-2 steady state.

Similarly, for external force F = 0 pN, the Case-1-1 steady state shown by the green circle in Fig. 21c is outside (in left side) of the oscillation region formed by the Hopf bifurcation curve illustrated by black curve, corresponding to Fig. 13b of Section  "Transition between Cse-1 steady state and oscillations". When external force is changed to F = −30 pN, the border of oscillation region changes to the Hopf bifurcation curve depicted by blue curve. The green circle locates within the oscillation region for F = −30 pN. Therefore, the transition from steady state to sustained oscillations happens if F = 0 pN is changed to F = −30 pN, which shows that the negative external force induces the transition behavior for the Case-1-1 steady state.

For the Case-2 steady state locating in the region below the blue dashed line, as shown in Fig. 21a, no bifurcations containing Hopf bifurcation curve appear at different values of external force F. Therefore, change of external force cannot induce transition from steady state to sustained oscillations corresponding to limit cycles. No transitions from steady state to sustained oscillations for Case-2 steady state, which is addressed in Sects. "Force does not induce transition between Case-2 steady state and sustained oscillations" and "No bifurcation with respect to force for case-2 steady state", are well explained.

The parameter region of twitch For the Case-1 steady state far away from Hopf bifurcation, such as the green square in Fig. 21b, c, corresponding to the Case-1-1 and Case-1-2 steady states of Section "Twitch induced by pulse force for Case-1 steady state", the negative twitch and positive twitch can be induced by appropriate pulse force, respectively.

For the Case-2 steady state (below the blue dashed line) which is far away from Hopf bifurcation, three subcases of steady state are separated by the magenta curves in Fig. 1. Negative twitch, positive and negative twitches, and positive twitch can be induced by pulse force from Case-2-1, Case-2-2, and Case-2-3 steady states, respectively.

The unchanged boundary between Case-1 and Case-2 steady state in in (S, D) plane with respect to external force F

As shown in Fig. 21a, the blue dashed line ($D\approx$ 50.3 nm) represents the border between Case-1 steady state and Case-2 steady state. Interestingly, such a border does not change with respect to F. The reason for the unchanged border is explained in the following paragraphs.

Case-1 steady state exhibits a Z-shaped X-nullcline, which corresponds to that $X_a$ has a maximal and a minimal value at two different X values. Case-2 steady state represents a N-shaped X-nullcline, which corresponds to that $X_a$ has no local extreme values with respect to X. Therefore, the existence of local extreme values of $X_a$ is used to distinguish two shapes of X-nullclines. If $X_a$ has extreme values, the X-nullcline belongs to Z-shaped type, which corresponds to the Case-1 steady state or spontaneous oscillations. If $X_a$ has no extreme values, the X-nullcline belongs to N-shaped type, which corresponds to Case-2 steady state.

In theory, $X_a$ has extreme points or not corresponds to that the equation $d(X_a)/d(X)$ = 0 has solution or not, respectively. For the Z-shaped X-nullcline, the equation $d(X_a)/d(X)$ = 0 has two solutions. For the N-shaped X-nullcline, $d(X_a)/d(X)$ = 0 has no solution.

In the phase plane of ($X_a$, X), X-nullcline is the curve obeys the equation shown as follows:

$$\begin{array}{c}\hfill -K_{\mathit{GS}}(X-X_a-D{P}_0)-K_{\mathit{SP}}X+F=0\end{array}$$ 6

Then, $d(X_a)/d(X)$ = 0 is acquired as follows:

$$\begin{array}{c}\hfill \frac{d{X}_a}{\mathit{dX}}=\frac{(K_{\mathit{GS}}+K_{\mathit{SP}})(A^2+exp(\frac{2Z(X-X_a)}{k_{b}T}))+A(2K_{\mathit{SP}}+K_{\mathit{GS}}(2-\frac{\mathit{ZD}}{k_{b}T}))exp(\frac{Z(X-X_a)}{k_{b}T})}{K_{\mathit{GS}}(A^2+A(2-\frac{\mathit{ZD}}{k_{b}T}))exp(\frac{Z(X-X_a)}{k_{b}T})+exp(\frac{2Z(X-X_a)}{k_{b}T})}=0\end{array}$$ 7

Solve Eq. (7) by using the software Mathematica 8.0 to acquired that:

When $\frac{D^2{K}_{\mathit{GS}}^2}{(K_{\mathit{GS}}+K_{\mathit{SP}})k_{b}TN}>$ 4, $X_a$ has two different real solutions, which correspond to the Z-shaped X-nullcline;

When $\frac{D^2{K}_{\mathit{GS}}^2}{(K_{\mathit{GS}}+K_{\mathit{SP}})k_{b}TN}<$ 4, $X_a$ has no real solution, which correspond to the N-shaped X-nullcline.

Therefore, $\frac{D^2{K}_{\mathit{GS}}^2}{(K_{\mathit{GS}}+K_{\mathit{SP}})k_{b}TN}$ = 4 is the critical condition between Z-shaped X-nullcline and N-shaped X-nullcline.

In fact, the external force F disappears in equation of $\frac{d{X}_a}{\mathit{dX}}$ (Eq. (7)) due to F is a constant, which means that F does not affect the number of solutions of Eq. (7). Therefore, as the external force F changes, the boundary (blue dashed in Fig. 21a) between two types of X-nullcline (steady state) does not change.

Substitute the parameters in Section "Methods" into the $\frac{D^2{K}_{\mathit{GS}}^2}{(K_{\mathit{GS}}+K_{\mathit{SP}})k_{b}TN}$ = 4 to obtain $D\approx$ 50.3 nm, corresponding to the dashed blue line in Figs. 1 and 21a.

Discussion and conclusion

The nonlinear forced oscillations of hair bundle represented by twitch are important manifestations of fast adaptation, which is a potential mechanism of signal amplification for non-mammalian (Bozovic 2019), plays an important role in frequency selection (Fettiplace et al. 2001), and therefore is a prerequisite for auditory function. Researches on the forced oscillations of hair bundles make it possible to understand the important function of the rapid adaptation, such as filtering the response signal of the MET channel (Stepanyan and Frolenkov 2009), controlling the amount of energy transferred by hair bundle movement (Azimzadeh et al. 2018). In the present paper, the nonlinear dynamics of fast adaptation mediated by forced oscillation of hair bundle are acquired, which are helpful understanding and modulating the forced oscillations. The main significance exhibit in the following four aspects.

Firstly, the nonlinear dynamical mechanism for the diverse twitch observed in experiments (Benser et al. 1996) are obtained. The vector fields and nullclines widely used in the nervous system (Zhao et al. 2020; Wang et al. 2021) are adopted to explain the nonlinear dynamics of twitch. Under the action of pulse force, the phase trajectory of forced oscillations is identified to be an evolution process between two equilibrium points for zero force and pulse force, and the twitch is obtained as the behavior running along the nonlinear part of X-nullcline. Furthermore, the twitch observed in the experiments is classified into 6 types, which are build relationships to the 3 subcases of Case-2 steady states under action of positive or negative force. The results present comprehensive and deep understanding for the twitch, which are helpful to explain the diversity of twitch observed in the experiments.

Secondly, the nonlinear mechanism of fatigue of twitch observed in the experiment (Benser et al. 1996) are obtained as decreased amplitude of twitch beginning from different initial values. The former twitch begins from the initial value which is the equilibrium point of zero force, runs across the maximal value of X-nullcline for the pulse force, resulting in a large amplitude. However, the latter twitch begins from an initial value different from that of former one due to the short time interval between two pulses of external force, which leads to that the behavior cannot recover to the equilibrium point. The initial value of the latter twitch is a phase point on lower branch of X-nullcline for zero force. The twitch beginning from such an initial value runs along the vector fields and does not run across middle branch of X-nullcline for pulse force, resulting in an amplitude smaller than the former one, due to that X value of middle branch exhibit lower amplitude than the largest value of the upper branch. Therefore, the fatigue of twitch can be well explained with the nonlinear characteristic of X-nullcline and vector fields.

Thirdly, the experimental (Benser et al. 1996 observation of transition between sustained oscillations and steady state induced by pulse force was explained with the bifurcations related to Case-1 steady state. With respect to external force, Case-1 steady state exhibits Hopf bifurcation from steady state to limit cycle corresponding to sustained oscillations. Therefore, if a suitable pulse force is applied, transition from steady state to sustained oscillations can be induced. However, Case-2 steady state does not manifest bifurcations with respect to force, then pulse force cannot induce transition between steady state and subthreshold oscillations. The results present a comprehensive and deep understanding for the transition between sustained oscillations and steady state induced by pulse force.

Lastly, the threshold phenomenon induced by simple pulse stimulation is identified for Case-1 steady state instead of Case-2, which can be explained by shape of X-nullcline and bifurcations. For Case-1 steady state near Hopf bifurcation, the X-nullcline exhibits Z-shaped. The upper and lower branches of Z-shaped X-nullcline manifest coexisting behaviors of variable X near the middle branch. The stable equilibrium point of steady state locates on either upper branch or lower branch of X-nullcline. A suitable disturbance can induce the trajectory behavior which begins from the equilibrium point on one branch, runs rapidly across the middle branch along X direction due to X is fast variable, runs along the other branch to form a large oscillation, and recovers to the equilibrium point, resulting in a threshold phenomenon. However, for Case-2 steady state with N-shaped X-nullcline, no coexisting behavior of X values. Then, no threshold phenomenon appears for Case-2 steady state.

Although the nonlinear mechanical vibration of the hair bundle belonging to the auditory system has many similarities with the nervous system, its complex nonlinear dynamics needs to be further studied. For example, the specific dynamics of twitch, transition, and threshold affect hearing’s most important amplification function, nonlinear dynamics of different complex oscillation modes of hair bundles, the dynamical mechanism underlying the famous nonlinear phenomena such as two-tone suppression observed in biological experiments (Barral and Martin 2012), are urgent problems to be solved. Various external factors, including magnetic (memristor), light, and heat, which have been studied in the nervous system (Yang et al. 2021; Yao et al. 2021; Zhang et al. 2020), especially in the auditory nervous system (Zhou et al. 2021), should also be considered in the auditory system. In addition, the nonlinear behaviors such as resonance and synchronization that exist in the nervous system (Xie et al. 2014; Yao and Ma 2018) also exist in the oscillations of the auditory system (Han and Neiman 2010; Li et al. 2014; Zhang et al. 2015), which should be further studied. In addition to mechanical vibration of hair bundles, there are electrical oscillations (Meenderink et al. 2015; Rong and Wang 2019) in the hair bundle and auditory nerve, which is an important issue of future research. In addition to the mechanics of hair bundles of hair cells, cochlea mechanics is also an important research direction (Chen et al. 2017; Su et al. 2020), and the combination of hair bundle mechanics and cochlea mechanics is an important research content. Neurodynamic analysis methods such as nullclines analysis, the vector fields, bifurcation analysis, and fast/slow variable separation method (Duan et al. 2020; Xie et al. 2008), have been used effectively to analyze biomechanical problems such as touch, hearing, molecular mechanics of protein, etc (Wang et al. 2003, 2020; Yao and Wang 2019). Subsequent researches of the auditory nervous system should further utilize these neurodynamic analysis methods and expand the scope of neurodynamics and nonlinear dynamics.

Author Contributions

HG conceived the experiments, BC, HG, and RW conducted the experiments, BC, HG, and RW analyzed the results, BC and HG written the paper. All authors reviewed the manuscript.

Declarations

Conflict of interest

The authors declare no competing financial interests.