1 Introduction

The nervous system, an intricate and highly nonlinear component of biological systems, poses significant challenges for traditional neuroscience research due to limitations in experimental techniques and conditions [1]. Exploring the intricate patterns of electrical activity induction and cooperative mechanisms between neurons from a purely experimental standpoint is inherently tricky [2]. Thus, to gain insights into the physiological and cognitive behaviors of the nervous system, researchers often turn to mathematical, physical, or computer technologies for assistance [3, 4]. In recent decades, leveraging these advanced methodologies, researchers have developed numerous nonlinear neuron models based on clinical electrophysiological experiments, facilitating a deeper understanding of signal transmission within the nervous system. For instance, Hodgkin and Huxley formulated the Hodgkin-Huxley (HH) neuron model using neural stimulation potential data from squids in 1952 [5]. Subsequently, Fitzhugh [6] proposed the FitzHugh-Nagumo (FHN) model by simplifying the HH model, delineating fast subsystems to represent changes in neuronal membrane potential and slow subsystems for channel inactivation. Aibara et al. [7, 8] investigated the various responses of giant squid axons under external sinusoidal currents through numerical simulations and experiments. Similarly, Hindmarsh et al. [9] drawing on extensive data from voltage-clamp experiments conducted in 1984, introduced the Hindmarsh-Rose (HR) model. Notably, these neuron models can be translated into corresponding nonlinear circuits using appropriate nonlinear devices, facilitating the simulation of neural systems [10]. Integrating experimental data with mathematical modeling and technological advancements, this interdisciplinary approach has dramatically enriched our comprehension of the nervous system’s intricate dynamics [11, 12].

In the human auditory system, the sound is captured by the ear, where eardrum vibrations lead to the conversion of these vibrations into nerve signals by hair cells in the cochlea. These signals are then transmitted via the cochlear nerves and auditory pathways to the auditory cortex, resulting in sound perception [13]. Damage to hair cells caused by loud noise or drug abuse impedes the transmission of sound signals from the cochlea to the brain. Cochlear implants can bypass damaged hair cells by converting sound into electrical signals, directly stimulating the auditory nerve to restore or improve hearing [14]. However, damage to auditory nerves and neural circuits can block this conversion and signal propagation, leaving no current treatment for auditory nerve recovery. Future technologies may include artificial sensors or processors to mimic firing patterns, aiding the central auditory system [15].

Investigating how auditory neurons react to various stimuli is crucial for understanding their nonlinear dynamics and designing artificial auditory sensors. Recent biophysical research has aimed to enhance neuronal circuit functionality by incorporating physical devices into nonlinear circuits, allowing for sensing specific external signals. For instance, Liu et al. [16] explored the effects of light on neural activity patterns by integrating a photocell into the circuit. In contrast, Guo et al. [17] analyzed the synchronization stability of thermosensitive neurons. Yang et al. [18] enhanced a neuronal model by combining a photocell and thermistor to examine the influence of energy diversity on neuronal coupling. Embedding piezoelectric devices into nonlinear circuits has led to the design of auditory circuits sensitive to sound waves [19]. The PNS model using piezoelectric ceramics for acoustic-electric conversion can simulate cochlear nerve excitation and has potential applications in restoring hearing [15]. This paper focuses on the bifurcation phenomenon of the PNS for external stimulus intensity and resistance parameters.

Over recent decades, extensive research on neuronal model dynamics has emerged, highlighting the significance of numerical methods for their computational efficiency, versatility, and robustness in analyzing neuronal systems’ dynamic behaviors [18, 20]. Standard numerical integration methods like the Newmark-\(\beta \) method [21], Runge–Kutta (RK) method and finite difference method apply time integration to the differential equations, providing discrete numerical solutions that may not fully capture the system’s dynamics. Selecting inappropriate time steps can result in inaccurate dynamic analyses [22]. Moreover, nonlinear dynamics can exhibit complex behaviors such as multiple solutions, jumps, bifurcations, and instability, complicating quantitative analyses using numerical methods [23]. Analytical methods offer a distinct advantage by providing exact solution expressions, accurately depicting the system’s long-term behavior, and identifying stable intervals and bifurcations, thus deeply reflecting the nonlinear system’s inherent characteristics. However, obtaining simple analytical solutions for nonlinear problems is rare, with most solutions being approximate or represented as finite series [24]. As a result, analytical and semi-analytical methods, including the harmonic balance method [25] and incremental harmonic balance (IHB) method [26], have gained attention.

Recently, Liu et al. [27] presented a novel semi-analytical solution approach: the time-domain minimum residual method (TMRM), which theoretically enables the derivation of approximate solutions of arbitrary precision by adjusting the truncation series. This method’s application enhances the accuracy of computational approaches for nonlinear systems [28], crucial for developing reliable neuron models and exploring discharge rhythms. This study employs the TMRM to solve the piezoelectric neuron circuit system, analyzing its complex dynamics through time history, phase, and bifurcation diagrams and the stability of piezoelectric neuron system (PNS) was analyzed using Floquet theory [29]. Findings suggest that the TMRM-based solutions closely match those from the RK method, efficiently tracking unstable solutions. Moreover, by changing the bifurcation parameters or applying external stimuli, the electrical activity in the piezoelectric neuron circuit can exhibit diverse discharge patterns, revealing the potential biophysical mechanisms of auditory neurons’ multistable response under external stimuli.

This article is structured as follows: Sect. 2 outlines the principles and basic equations governing the piezoelectric neuron system (PNS). Section 3 delves into the foundational principles of the TMRM and discusses the general methodology for solving nonlinear systems. In Sect. 4, we first describe the procedure for applying the TMRM to solve the PNS, including presenting the semi-analytical solution obtained. Subsequently, bifurcation analysis of the PNS is carried out by receiving the semi-analytical solution. Finally, the conclusions drawn from our study are summarized in Sect. 5.

2 The equation of the piezoelectric neuron system

As delineated previously, the human auditory system captures sound through the outer ear, vibrating the tympanic membrane in the middle ear. Subsequently, the acoustic data is transformed into nerve impulses within the inner ear’s cochlea, relayed via the cochlear nerves and neural pathways to the auditory cortex of the cerebral cortex, culminating in the perception of hearing. In artificial auditory neurons, piezoelectric devices can be integrated into neural circuits to emulate the firing patterns observed in biological neurons. Upon exposure to external mechanical forces or sound wave vibrations, piezoelectric devices undergo continuous deformation. This deformation induces a potential difference across the terminals of the piezoelectric device, as described by the following equation [15]:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{} V_{PC}=\frac{F}{S} \frac{d_{33}}{\epsilon }h=Pgh \\ &{} P=\frac{F}{S}, g=\frac{d_{33}}{\epsilon }, d_{33}=\frac{Q}{F} \end{aligned} \end{array}\right. } \end{aligned}$$
(1)

where F symbolizes the external mechanical force, S refers to the cross-sectional area, \(\epsilon \) represents the dielectric constant, h indicates the thickness of the piezoelectric device, Q is the charge released, and \(d_{33}\) is the piezoelectric strain constant, which is associated with the properties of the piezoelectric material.

Fig. 1
figure 1

Piezoelectric neuron circuit model [15]

Full size image

A constant external mechanical force generates a stable voltage difference, whereas nonlinear vibration results in a time-varying voltage difference. The renowned FHN neuron model, proposed by Fitzhugh and Nagumo [6, 30] effectively describes the primary characteristics of neural firing behavior in response to suitable external stimuli. Consequently, as depicted in Fig. 1, a piezoelectric neuron circuit can be constructed by integrating a piezoelectric ceramic with the FHN neuron model. Utilizing Kirchhoff’s laws for current and voltage, the subsequent circuit equations can be derived:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{} C\frac{\textrm{d}V_C}{\textrm{d}t} =\frac{V_{PC}-V_C}{R_s}-i_L-i_{R_{N}} \\ &{} L\frac{\textrm{d}i_L}{\textrm{d}t} =V_C-Ri_L+E \end{aligned} \end{array}\right. } \end{aligned}$$
(2)

where C represents the capacitance in Fig. 1, and L represents the self inductance coefficient of the coil. \(V_C\) signifies the voltage across the capacitor, and \(V_{PC}\) denotes the voltage output across the piezoelectric ceramic. \(R_s\) and R are the linear resistances, while E is the constant voltage source that mimics the reversal potential in the ion channel. \(i_L\) represents the current through the inductor, and the current–voltage relationship across the nonlinear resistor \(R_N\) is given by:

$$\begin{aligned} \begin{aligned} i_{R_N}=- \frac{1}{\rho }\left( V-\frac{V^3}{3V_0^2}\right) \end{aligned} \end{aligned}$$
(3)

where \(\rho \) is the normalization parameter, and V and \(V_0\) denote the voltage across the nonlinear resistor and the cut-off voltage, respectively. To further simplify the above equation, the following form of dimensionless parameters can be introduced: \(x_1=\frac{V_C}{V_0}, x_2=\frac{\rho i_L}{V_0}, \tau =\frac{t}{\rho C}, u_{PC}=\frac{V_{PC}}{V_0}, \varphi =\frac{\rho }{R_s}, \alpha =\frac{E}{V_0}, \delta =\frac{R}{\rho }, \mu =\frac{\rho ^2 C}{L}\). From then on, the dimensionless dynamic equation of equivalent PNS (2) can be simplified as:

$$\begin{aligned} \begin{aligned}&\dot{x}_1=x_1(1- \varphi )-\frac{1}{3}x_1^3-x_2+\varphi u_{PC} \\&\dot{x}_2=\mu (x_1+\alpha -\delta x_2) \end{aligned} \end{aligned}$$
(4)

where the derivative of the state variable \(\varvec{x}=[x_1, x_2]\) in relation to the dimensionless time \(\tau \) is denoted by the dot above \(\varvec{x}\). It is widely known that the excitability of neurons will change with changes in the external stimulation current \(u_{PC}\), thereby inducing neurons to produce different discharge patterns. Under certain conditions, acoustic waves converted into electrical signals by piezoelectric ceramics can select different forms of signals, which can be a single periodic signal, a composite periodic signal, or a chaotic signal [19, 31]. In this paper, to illustrate the efficacy of the TMRM in neuronal computation and dynamics analysis succinctly, here we choose periodic signals as the electrical signal of the external stimulus, i.e. \(\varphi u_{PC}=A\cos (\Omega \tau )\). The following dynamic equations can be further obtained:

$$\begin{aligned} \begin{aligned}&\dot{x}_1-x_1(1- \varphi )+\frac{1}{3}x_1^3+x_2=A\cos (\Omega \tau ) \\&\dot{x}_2-\mu (x_1+\alpha -\delta x_2)=0 \end{aligned} \end{aligned}$$
(5)

For the convenience of subsequent derivation, the above equation (5) can be further simplified into matrix form:

$$\begin{aligned} \begin{aligned} \varvec{C}\dot{\varvec{X}}+\varvec{KX}+\varvec{N}(\dot{\varvec{X}},\varvec{X})=\varvec{F}(\varvec{\tau }) \end{aligned} \end{aligned}$$
(6)

where \(\dot{\varvec{X}}=\begin{pmatrix}{\dot{x}_1} \\ {\dot{x}_2}\\ \end{pmatrix}, \varvec{X}=\begin{pmatrix}{{x}_1}\\ {{x}_2}\\ \end{pmatrix}, \varvec{C}=\begin{pmatrix}1&{}0\\ 0&{}1\\ \end{pmatrix}, \varvec{K}=\begin{pmatrix}\varphi -1&{}1\\ - \mu &{}\delta \mu \\ \end{pmatrix}, \varvec{N}(\dot{\varvec{X}},\varvec{X})=\begin{pmatrix}\frac{1}{3}x_1^3\\ - \alpha \mu \\ \end{pmatrix}, \varvec{F}(\varvec{\tau })=\begin{pmatrix}A\cos (\Omega \tau )\\ 0\\ \end{pmatrix}\).

3 The process of solving piezoelectric neuron system using the TMRM

The TMRM, which blends semi-numerical and semi-analytical approaches, is applicable to both weak and strong nonlinear systems. The procedure for employing the TMRM to address PNS will be thoroughly explained.

3.1 Problem description

For the PNS as presented in Eq. (6), its periodic solution can be expressed as an infinite series in the following form:

$$\begin{aligned} \begin{aligned}&x_{i}(\tau )=a_{i0}+ \sum _{k=1}^{+\infty } [b_{ik} \cos (\Omega _k \tau )+c_{ik} \sin (\Omega _k \tau )],\\&\quad k=1,2,\cdots \end{aligned} \end{aligned}$$
(7)

where \(x_{i}(\tau )\) denotes the i-th degree-of-freedom (DOF) for \(i=1\) to 2, \(a_{i0}\) is a constant term, the coefficients for the cosine and sine terms are denoted by \(b_{ik}\) and \(c_{ik}\), respectively, while \(\Omega _k\) indicates the k-th order harmonic’s frequency. Typically, when the system is subject to an external excitation and the excitation frequency \(\Omega \) is near the system’s natural frequency (\(\Omega \approx \omega _0\)), the system will exhibit a primary harmonic response. Should the external force’s frequency approach n multiples of the fundamental frequency (\(\Omega \approx n\omega _0\)), given that n is a positive integer distinct from 1), the system may also exhibit a subharmonic response at a frequency of \(\Omega /n\). This relationship is captured as:

$$\begin{aligned} \Omega _{ k}={\left\{ \begin{array}{ll} \begin{aligned} &{} k\Omega , \quad \text {primary harmonic response}, \\ &{} k\frac{\Omega }{n}, \quad \text {subharmonic response}. \end{aligned} \end{array}\right. } \end{aligned}$$
(8)

Differentiating Eq. (7) with respect to dimensionless time \(\tau \) yields the first-order differential form of the response:

$$\begin{aligned} \dot{x}_{i}(\tau )= & {} \sum \limits _{k=1}^{+\infty } \left[ -b_{ik} {\Omega _k}\sin (\Omega _k\tau ) + c_{ik}{\Omega _k}\cos (\Omega _{ k}\tau )\right] ,\nonumber \\{} & {} \quad k=1,2,\cdots \end{aligned}$$
(9)

Considering computational practicality, the solution can be approximated by truncating the first N terms in Eq. (7), resulting in:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{} x_{i}(\tau ) \approx x_{i}^N(\tau )=a_{i0}+ \sum \limits _{k=1}^{N} [b_{ik} \cos (\Omega _{ k} \tau )\\ {} &{}+c_{ik} \sin (\Omega _{ k} \tau )]\\ &{} \dot{x}_{i}(\tau ) \approx \dot{x}_{i}^N(\tau )=\sum \limits _{k=1}^{N} [-b_{ik} {\Omega _k}\sin ({\Omega _k}\tau ) \\ {} &{}+ c_{ik}{\Omega _k}\cos (\Omega _{k}\tau )] \end{aligned} \end{array}\right. } k=1,2,\cdots ,N\nonumber \\ \end{aligned}$$
(10)

This series solution, as approximated in Eq. (10), introduces a residual when substituted back into Eq. (7):

$$\begin{aligned} \begin{aligned} \varvec{R}=\varvec{C}\dot{\varvec{X}}^{\varvec{N}}+\varvec{KX}^{\varvec{N}} +\varvec{N}(\dot{\varvec{X}}^{\varvec{N}},\varvec{X}^{\varvec{N}})-\varvec{F}(\varvec{\tau }) \end{aligned} \end{aligned}$$
(11)

while this approximate solution does not ensure the residual \(\varvec{R}\) vanishes across the entire time domain, selecting an appropriate N and harmonic coefficients \(\varvec{a}=[a_{i0}, b_{ik}, c_{ik}], i=1,2; k=1,2,\cdots ,N\) can minimize \(\varvec{R}\) as much as possible. Therefore, solving for a semi-analytical solution to the nonlinear system as outlined in Eq. (7) becomes a quest to determine harmonic coefficients \(\varvec{a}\) that reduce the residual \(\varvec{R}=\varvec{R}(\varvec{a}, \tau )\) to its minimum across the time span \(\tau _j\in [0,\tau _1,\tau _2,\dots , T]\), defined as:

$$\begin{aligned} \varvec{a}^*= & {} \arg \min _{a\in M}\hbar (\varvec{a},\tau )\nonumber \\:= & {} \min _{a\in M}\int _{0}^{T}\varvec{R}(\varvec{a},\tau )^T \varvec{R}(\varvec{a},\tau )\textrm{d}\tau \end{aligned}$$
(12)

where \(\hbar (\varvec{a},\tau )\) represents the nonlinear objective function and M indicates the allowable range for the harmonic coefficients \(\varvec{a}\), and \(\varvec{R}(\varvec{a}, \tau )\) is the system’s residual vector.

3.2 Unknown harmonic coefficients are solved by iteration

For the nonlinear least-squares optimization problem described as Eq. (12), an iterative method such as the enhanced response sensitivity approach is typically employed to solve it. This process involves initially selecting a suitable set of initial iteration values, denoted as \(\varvec{a}^{(0)}\). Subsequently, the iteration update \(\Delta \varvec{a}^{(j)}\) is calculated in a manner that minimizes \(\hbar (\bar{\varvec{a}}+\Delta \varvec{a})\) based on the current parameter values \(\bar{\varvec{a}}\in M\). The iteration proceeds according to the following methodology:

$$\begin{aligned} \begin{aligned} \varvec{a}^{(j)}=\varvec{a}^{(j-1)}+\Delta \varvec{a}^{(j)} \end{aligned} \end{aligned}$$
(13)

In accordance with the coefficient traversal principle, all unknown coefficients must be involved in the iterative process to incrementally correct and ultimately approximate the exact solution. Directly solving the original nonlinear objective function, as indicated by (12), presents challenges. Thus, the nonlinear objective function is linearized at discrete time nodes \(\tau _j\in [0,\tau _1,\tau _2,\dots , T]\). This process transforms the original nonlinear objective function \(\hbar (\bar{\varvec{a}},\tau )\) into an approximate linear objective function \(\hat{\hbar }(\bar{\varvec{a}}+\Delta \varvec{a}, \tau )\), delineated as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{} \min _{a\in M}\hat{\hbar }(\bar{\varvec{a}}+\Delta \varvec{a}, \tau )=||\Delta \varvec{R}(\bar{\varvec{a}},\tau )-\varvec{S}(\bar{\varvec{a}},\tau )\Delta \varvec{a}||^2\\ &{}\Delta \varvec{R}(\bar{\varvec{a}},\tau ):=\hbar (\bar{\varvec{a}},\tau )-\varvec{R}(\bar{\varvec{a}},\tau )\\ \end{aligned} \end{array}\right. } \end{aligned}$$
(14)

where \(||\cdot ||^2\) signifies the \(l^2\)-norm. The infinite series in Eq. (7) across the entire time domain aims to render the objective function \(\hbar (\varvec{a},\tau )\) equal to zero, hence \(\Delta \varvec{R}(\bar{\varvec{a}},\tau ):=0-\varvec{R}(\bar{\varvec{a}},\tau )\). \(\varvec{S}(\bar{\varvec{a}},\tau )\) is the first-order response sensitivity matrix, expressed as:

$$\begin{aligned} \varvec{S} (\bar{\varvec{a}},\tau )= & {} \nabla _{\bar{\varvec{a}}} \varvec{R}(\bar{\varvec{a}},\tau )\nonumber \\= & {} \begin{pmatrix} \frac{\partial \varvec{R}(\bar{\varvec{a}},\tau _1)}{\partial a_1},\frac{\partial \varvec{R}(\bar{\varvec{a}},\tau _1)}{\partial a_2}\dots \frac{\partial \varvec{R}(\bar{\varvec{a}},\tau _1)}{\partial a_{(4N+2)}} \\ \frac{\partial \varvec{R}(\bar{\varvec{a}},\tau _2)}{\partial a_1},\frac{\partial \varvec{R}(\bar{\varvec{a}},\tau _2)}{\partial a_2}\dots \frac{\partial \varvec{R}(\bar{\varvec{a}},\tau _2)}{\partial a_{(4N+2)}}\\ \vdots \\ \frac{\partial \varvec{R}(\bar{\varvec{a}},\tau _l)}{\partial a_1},\frac{\partial \varvec{R}(\bar{\varvec{a}},\tau _l)}{\partial a_2}\dots \frac{\partial \varvec{R}(\bar{\varvec{a}},\tau _l)}{\partial a_{(4N+2)}}\end{pmatrix} \end{aligned}$$
(15)

Due to the nonlinearity and discretization of the system equation, the iteration process may encounter ill-posed problems, such as having more unknowns than data points, potentially leading to an ill-posed approximate objective function \(\hat{\hbar }(\bar{\varvec{a}}+\Delta \varvec{a})\). To address possible ill-posed issues, a Tikhonov regularization process is introduced, modifying Eq. (14) as follows:

$$\begin{aligned} \begin{aligned}&\Delta \varvec{a}_{\lambda } =\arg \min _{\Delta \varvec{a}\in M-\bar{\varvec{a}}}||\Delta \varvec{R}(\bar{\varvec{a}},\tau )-\varvec{S}(\bar{\varvec{a}},\tau )\Delta \varvec{a}||^2 + \lambda ||\Delta \varvec{a}||^2\\&\quad =\big [S^T(\bar{\varvec{a}},\tau )S(\bar{\varvec{a}},\tau )+\lambda \varvec{I}\big ]^{-1}S^T(\bar{\varvec{a}},\tau )\Delta \varvec{R}(\bar{\varvec{a}},\tau ) \end{aligned}\nonumber \\ \end{aligned}$$
(16)

where \(\varvec{I}\) denotes the identity matrix. The regularization parameter \(\lambda (\lambda \ge 0)\), which influences the iterative update quantity \(\Delta \varvec{a}\), can be determined using the L-curve method [28]. The process to ascertain \(\lambda \) begins with the singular value decomposition of \(S(\bar{\varvec{a}},\tau )\):

$$\begin{aligned} \begin{aligned} S(\bar{\varvec{a}},\tau )=U\Sigma V^T =\sum \limits _{i=1}^{N} \vartheta _i u_i v_i^T \end{aligned} \end{aligned}$$
(17)

where U and V represent orthogonal matrices, with \(\Sigma =diag(\xi _1,\xi _2,...,\xi _N)\) denoting the diagonal matrix whose diagonal elements are arranged in descending order: \(\xi _1\ge \xi _2\ge \dots \ge \xi _N\). The L-curve technique is employed for identifying the best regularization parameters by plotting the norm of residuals \(\zeta (\lambda )=||\Delta \varvec{R}(\bar{a},\tau )-\varvec{S}(\bar{\varvec{a}},\tau )\Delta \varvec{a}||^2\) against the norm of the iterative update quantity \(\Psi (\lambda )=||\Delta \varvec{a}||^2\). A suitable regularization parameter \(\lambda \) balances these two norms effectively. The expressions for \(\zeta (\lambda )\) and \(\Psi (\lambda )\) are given by:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} \zeta (\lambda )&{}=\Vert \Delta \varvec{R}(\bar{\varvec{a}},\tau )-\varvec{S}(\bar{\varvec{a}},\tau )\Delta \varvec{a}\Vert ^2\\ &{}=\Vert (\Sigma \Sigma _{\lambda }^{+}-\varvec{I})u_i^T\Delta \varvec{R}(\bar{\varvec{a}},\tau )\Vert ^2\\ &{}=\sum \limits _{i=1}^{N}\left( \frac{\lambda }{\vartheta _i^2 +\lambda }u_i^T\Delta \varvec{R}\left( \bar{\varvec{a}},\tau \right) \right) ^2\\ \Psi (\lambda )&{}=\Vert \Delta \varvec{a}\Vert ^2=\Vert \Sigma _{\lambda }^{+}u_i^T\Delta \varvec{R}(\bar{\varvec{a}},\tau )\Vert ^2\\ &{}=\sum \limits _{i=1}^{N}\left( \frac{\vartheta _i}{\vartheta _i^2 +\lambda }u_i^T\Delta \varvec{R}\left( \bar{\varvec{a}},\tau \right) \right) ^2\\ \end{aligned} \end{array}\right. } \end{aligned}$$
(18)

The optimal regularization parameter \(\lambda \) is determined at the point of maximum curvature on the L-curve, denoted as \(\varphi _{max}=\varphi (\lambda )\). The maximum curvature is calculated using:

$$\begin{aligned} \begin{aligned} \varphi (\lambda )=\frac{2\zeta \Psi (\lambda \Psi ^{\prime } \zeta +2\sqrt{\lambda }\zeta \Psi +\lambda ^2\Psi \Psi ^{\prime })}{\Psi ^{\prime }(\lambda ^2\Psi ^2+\zeta ^2)^{3/2}} \end{aligned} \end{aligned}$$
(19)

where \(\Psi ^{\prime }\) denotes the derivative of \(\Psi \). Following these steps, the suitable iterative update quantity \(\Delta \varvec{a}_{\lambda }\) is obtained as:

$$\begin{aligned} \begin{aligned} \Delta \varvec{a}_{\lambda }&=V\Sigma _{\lambda }^{+}U^T\Delta \varvec{R}(\bar{\varvec{a}},\tau )\\&=\sum _{i=1}^{N}\frac{\vartheta _i}{\vartheta _i^2 +\lambda }u_i^T\Delta \varvec{R}(\bar{\varvec{a}},\tau )v_i \end{aligned} \end{aligned}$$
(20)

The L-curve method primarily assesses the linearized objective function \(\hat{\hbar }(\bar{\varvec{a}}+\Delta \varvec{a}, \tau )\) without direct consideration of the original nonlinear objective function \(\hbar (\bar{\varvec{a}},\tau )\). To evaluate the appropriateness of the iterative update quantity \(\Delta \varvec{a}_{\lambda }\) the agreement indicator is introduced, defined as:

$$\begin{aligned} \begin{aligned} \varpi (\Delta \varvec{a},\bar{\varvec{a}})&=\frac{\hbar (\bar{\varvec{a}})-\hbar (\bar{\varvec{a}}+\Delta \varvec{a})}{\hat{\hbar }(0,\bar{\varvec{a}})-\hat{\hbar }(\Delta \varvec{a},\bar{\varvec{a}})}\\&=\frac{\Vert \Delta \varvec{R}(\bar{\varvec{a}},\tau )\Vert ^{2}-\Vert \Delta \varvec{R}(\bar{\varvec{a}}+\Delta \varvec{a},\tau )\Vert ^{2}}{\Vert \Delta \varvec{R}(\bar{\varvec{a}},\tau )\Vert ^{2}-\Vert \Delta \varvec{R}(\bar{\varvec{a}},\tau )-\varvec{S}(\bar{\varvec{a}},\tau )\Delta \varvec{a}\Vert ^{2}} \end{aligned}\nonumber \\ \end{aligned}$$
(21)

\(\varpi (\Delta \varvec{a},\bar{\varvec{a}})\ge \varpi _{cr}\in [0.25, 0.75]\) indicates a strong correspondence between the linearized and original nonlinear objective functions.

4 Results and discussion

Subsequently, the TMRM described in Sect. 3 will be applied to derive the semi-analytical solution for the PNS as indicated by (6), and the stability and bifurcation analysis of the PNS will be conducted based on the obtained semi-analytical solution combined with the Floquet theory.

4.1 The semi-analytical solution of the PNS obtained by the TMRM

Inserting the semi-analytical solution approximation from Eq. (10) into PNS (5) and rearranging yields the system’s residual equation as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{} R_1=\dot{x}_1^N-x_1^N(1-\varphi )+\frac{1}{3}\left( x_1^N\right) ^3\\ &{}\quad \quad +x_2^N-A\cos (\Omega \tau ) \\ &{} R_2=\dot{x}_2^N-\mu \left( x_1^N+\alpha -\delta x_2^N\right) \end{aligned} \end{array}\right. } \end{aligned}$$
(22)
Fig. 2
figure 2

Time histories and phase diagrams. a, b time histories, c, d Phase diagram

Full size image

Using the above residual equation with the semi-analytical approximation, the frequency \(\Omega _k=k\Omega \) is given by Eq. (8), leading to the determination of \(4N+2\) unknowns, namely \(\varvec{a}=[a_{10},a_{20},b_{11},b_{12},\dots ,b_{1N},c_{11},c_{12}, \dots ,c_{1N},b_{21},b_{22},\dots ,b_{2N},c_{21},c_{22},\dots ,c_{2N}]\). This sets the stage for converting the task of computing the system’s approximate analytical solution \(\varvec{X}^N=[x_1^N, x_2^N]^T\) as per (6) into a least-squares optimization challenge. The goal is to adjust the unknown \(\varvec{a}\) to minimize the residual \(\varvec{R}=[R_1, R_2]^T\) across the time intervals \(\tau _j\in [0,\tau _1,\tau _2,\dots , T]\), as shown in Eq. (12). The sensitivity of the state variable \(\varvec{X}^N, \dot{\varvec{X}}^N\) concerning the unknown variable \(a_k\in \varvec{a}\) can be expressed as follows:

$$\begin{aligned} \begin{aligned}&a_k\in \varvec{a}\sim \frac{\partial \varvec{x}_{i}^N}{\partial a_{k}},\frac{\partial \dot{\varvec{x}}_{i}^N}{\partial a_{k}},\\&\quad i=1,2,\quad k=1,2,\dots ,4N+2 \end{aligned} \end{aligned}$$
(23)

For example, the sensitivity to the constant \(a_{10}\) and the harmonic coefficients \(b_{1k}\) and \(c_{1k}\) is:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{} \frac{\partial x_{1}^N}{\partial a_{10}}=1, \frac{\partial x_{1}^N}{\partial b_{1k}}=\cos (\Omega _k\tau ),\frac{\partial {x_{1}^N}}{\partial c_{1k}}=\sin (\Omega _k\tau )\\ &{} \frac{\partial \dot{x}_{1}^N}{\partial a_{10}}=0, \frac{\partial \dot{x}_{1}^N}{\partial b_{1k}}=-\Omega _k\sin (\Omega _k\tau ),\frac{\partial {\dot{x}_{1}^N}}{\partial c_{1k}}\\ &{}=\Omega _k\cos (\Omega _k\tau ) \end{aligned} k=1,2,\cdots ,N\\ \end{array}\right. }\nonumber \\ \end{aligned}$$
(24)

For other parameters, their sensitivity can be obtained in the same way. Since then, the sensitivity of residual \(\varvec{R}=[R_1, R_2]^T\) for unknown parameters \(a_k\in \varvec{a}\), there are:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{} \frac{\partial R_1}{\partial a_{k}}=\frac{\partial {\dot{x}}_{1}^N}{\partial a_{k}}-\frac{\partial x_{1}^N}{\partial a_{k}}(1-\varphi )\\ &{}+\frac{\partial x_{1}^N}{\partial a_{k}}\left( x_1^N\right) ^2+\frac{\partial x_{2}^N}{\partial a_{k}}\\ &{} \frac{\partial R_2}{\partial a_{k}}=\frac{\partial {\dot{x}}_{2}^N}{\partial a_{k}}-\mu \left( \frac{\partial x_{1}^N}{\partial a_{k}}-\delta \frac{\partial x_{2}^N}{\partial a_{k}}\right) \end{aligned} k=1,2,\cdots ,N \end{array}\right. } \end{aligned}$$
(25)

From Eq. (25), the first-order response sensitivity matrix \(\varvec{S} (\varvec{a},\tau )\) is derived. The parameters for the PNS as detailed in (6) are established at \(\alpha =0.7, \delta =0.8, \mu =0.1, \varphi =0.15, A=1\) and \(\Omega =2\) [15]. In the TMRM framework, initial constants are selected with \(a_{10}=-1, a_{20} = -0.5\), while the first set of harmonic coefficient initial values are determined as \(b_{11}=0, c_{12}=0.5, b_{11}=0, c_{12}=0\), and zeroes are assigned as the initial values for the remaining parameters.

Firstly, Fig. 2 has shown the system time histories and phase diagrams obtained by the TMRM with truncated series \(N=10\). Figure 2a, b represent the time histories of \(x_1\) and \(x_2\), respectively, while Fig. 2c, d represent the corresponding phase diagrams. Additionally, the system’s numerical solution was computed using the fourth-order Runge–Kutta method (4-th RK method, ’ode45’ function in MATLAB), and the comparison shown in the figure indicates a close agreement between the semi-analytical solution derived from the TMRM and the numerical solution represented by black dots. Table 1 details the values of the harmonic coefficients for the semi-analytical solution calculated through the TMRM.

Table 1 The harmonic coefficients of the PNS with \(N=10\) obtained by the TMRM
Full size table

Furthermore, to assess the precision of the semi-analytical solution generated by the TMRM more precisely, we plot the system’s residual curve. This involves reinserting the semi-analytical solutions from Table 1 into the system Eq. (6) and computing the curve of residuals. The proximity of the residual \(\varvec{R}=[R_1, R_2]^T\) to zero directly correlates with the solution’s precision. For this purpose, truncation parameters \(N=5, N=10\) and \(N=15\) were chosen for the residual curve illustration in Fig. 3a–c. Figure 3a demonstrates that with only the 5-th harmonic included, the residual for the TMRM is on the order of \(10^{-4}\). By including additional harmonics, the solution’s precision is enhanced further. When \(N=10\), it reaches the order of magnitude of \(10^{-9}\) (displayed in Fig. 3b), while when \(N=15\), it has already reached the order of magnitude of \(10^{-13}\) (displayed in Fig. 3c), which fully meets the accuracy requirements in engineering. An interesting point can also be observed in Fig. 3, where the residual of \(x_1\) (red lines) is significantly greater than that of \(x_2\) (blue lines), as the nonlinearity of the PNS (5) mainly comes from \(x_1\).

Fig. 3
figure 3

Residuals (red lines represent \(x_1\), blue lines corresponding to \(x_2\)) with different truncation series \(N=5\), \(N=10\) and \(N=15\)

Full size image

It is important to note that besides the TMRM for deriving the semi-analytical solution of the PNS as outlined in (6), alternative approaches like the incremental harmonic balance (IHB) method [32] is also capable of yielding semi-analytical solutions in the format presented in Eq. (10). The reason for selecting TMRM in this paper is that this method has good convergence (without the need for deliberately selecting initial values of unknown parameters), computational efficiency, and the accuracy of its semi-analytical solution is not lower than other methods mentioned above. Nevertheless, we still calculated the semi-analytical solution of the PNS using the IHB method under the same set of conditions (including system parameters and truncation order). Figure 3 displays the residual curves for both the TMRM and the IHB method at \(N=5, N=10\) and \(N=15\). Observations from the figure reveal that the residual from the TMRM (Fig. 3a–c) is an order of magnitude smaller compared to the IHB method’s residual (Fig. 3d–f), suggesting a superior accuracy of the semi-analytical solution derived via the TMRM under the specified conditions.

4.2 Stability and bifurcation analysis

Fig. 4
figure 4

Floquet multiplier path diagram. a Three ways for Floquet multipliers to penetrate the circle, b Floquet multipliers varying with amplitude A

Full size image

In this section, a bifurcation analysis of the PNS is performed, leveraging the semi-analytical solutions derived via the TMRM in conjunction with Floquet theory. Floquet theory plays a crucial role in identifying bifurcation types and assessing the stability of periodic solutions. Illustrated in Fig. 4a, a periodic solution is considered asymptotically stable if all Floquet multipliers are within the unit circle in the complex plane. Should a single Floquet multiplier be positioned on the unit circle, while the others are within it, the solution is deemed to be critically stable. On the other hand, if all Floquet multipliers fall outside the unit circle, the solution is regarded as unstable. Furthermore, Floquet theory aids in identifying the bifurcation type of periodic solutions. A saddle-node bifurcation leading to instability arises when a Floquet multiplier traverses the point (1, 0) and moves beyond the boundary of the unit circle, as indicated by the pink arrow in Fig. 4a, while the remaining multipliers stay inside the circle. When a conjugate pair of Floquet multipliers crosses the unit circle, with the remaining multipliers staying inside, the system experiences either a Hopf or a quadratic Hopf bifurcation, illustrated by the blue arrow. Lastly, a periodic-doubling bifurcation is observed when at least one Floquet multiplier crosses through the point \((-1, 0)\) with all others inside the unit circle, marked by the red arrow in Fig. 4a.

In next analysis, the system parameters are set as \(\alpha =1.1, \delta =0.8, \mu =0.3, \varphi =0.15, \Omega =0.942\), with the amplitude A of the external excitation serving as the bifurcation parameter for examining the PNS’s bifurcation behavior. Figure 4b illustrates the variation of the Floquet multiplier in relation to A within the range \(A\in [0.5, 2]\). As evident from Fig. 4b, for \(A\in [0.5, 0.94)\), every Floquet multiplier is positioned within the unit circle, signifying stability in the system’s periodic solution. Nonetheless, once A surpasses 0.94, a Floquet multiplier exits the unit circle, passing through the point \((-1, 0)\) on the complex plane, signifying the system’s entrance into the first periodic-doubling bifurcation, resulting in the loss of stability for the periodic solution. As A further increases beyond 1.46, the Floquet multiplier re-enters the unit circle through the point \((-1,0)\), suggesting the system undergoes a reverse periodic-doubling bifurcation, thereby restoring stability to the previously unstable periodic solution.

Fig. 5
figure 5

Bifurcation diagram of the PNS varying with amplitude A

Full size image
Fig. 6
figure 6

The time histories and phase diagram of the PNS with the truncation series \(N=30\). a Time histories, b phase diagram

Full size image

The findings from the preceding analysis are corroborated by constructing the bifurcation diagram of the PNS, as depicted in Fig. 5. This diagram, generated from the high-precision semi-analytical solutions derived through the TMRM, plots the extreme values of \(x_1\) (Fig. 5a) and \(x_2\) (Fig. 5b) on the vertical axis. In Fig. 5a, the stable periodic-1 solutions are indicated by a solid black line, whereas a dashed line denotes the unstable periodic-1 solutions, and the stable periodic-2 solutions are marked with blue dots. The diagram reveals that the stable periodic-1 solution undergoes bifurcation into a stable periodic-2 solution when A surpasses 0.94. Notably, while the system retains a periodic-1 solution, this solution becomes unstable. It is essential to highlight that while numerical methods can obtain stable periodic-2 solutions, unstable periodic-1 solutions require semi-analytical approaches like the TMRM. The bifurcation diagram for \(x_2\) presented in Fig. 5b leads to analogous conclusions, thereby aligning perfectly with the observations in Fig. 4b.

Fig. 7
figure 7

Logarithmic spectrogram of the stable periodic-2 response (\(x_2\))

Full size image

Figure 5 illustrates that the system exhibits a subharmonic response when the amplitude A falls within the range (0.94, 1.46). Specifically, at \(A=1.3\), the stable periodic-2 solution can be determined using the TMRM. The time histories and the corresponding phase diagram for the PNS are showcased in Fig. 6. Here, the solid red (\(x_1\)) and blue (\(x_2\)) lines depict the outcomes computed by the TMRM with a truncation series of \(N=30\). Meanwhile, the black dots denote the responses acquired via the 4-th RK method. Observations from the figure reveal that both \(x_1\) and \(x_2\) exhibit dual extreme values within their time histories, correlating to the two blue dots during the interval \(A\in (0.94, 1.46)\) in Fig. 5. The phase diagram in Fig. 6b demonstrates two rotations, indicative of the periodic-2 behavior. Further analysis involves performing a spectral transformation on the system’s periodic-2 response to scrutinize the harmonic content. Figure 7 displays a logarithmic spectrogram density analysis of \(x_2\), derived from executing a fast Fourier transform (FFT) on the stable periodic-2 response and subsequently applying a logarithmic scale to the amplitude. The analysis reveals that besides the primary harmonic response at a frequency of \(\Omega \), a subharmonic response emerges at a frequency of \(\frac{\Omega }{2}\). This subharmonic frequency, corresponding to the stable periodic-2 solution as expressed by \(\Omega _k=\frac{k\Omega }{2}\), underscores the intricate dynamical behavior of the system within the specified amplitude range.

Fig. 8
figure 8

Bifurcation diagram of the system varying with \(\varphi \)

Full size image

With \(\varphi \) is selected as the bifurcation parameter from Eq. (5) for system stability analysis, the other parameters are set as \(\alpha =0, \delta =0.8, \mu =0.1, A=1\), and \(\Omega =0.942\), with \(\varphi \) decreasing progressively from 0.6 to 0. This approach mirrors the bifurcation analysis conducted for the external excitation amplitude A, where the semi-analytical solution is initially derived through the TMRM. Subsequently, a bifurcation diagram depicting the response of \(x_1\) as a function of \(\varphi \) is generated, as illustrated in Fig. 8. This diagram reveals that as \(\varphi \) diminishes, the PNS displays increasingly complex dynamical behavior. Specifically, a unique stable periodic-1 solution (represented by black dots) is observed as \(\varphi \) decreases from 0.6 to 0.394 (highlighted by the red point \(P_1\) in Fig. 8b). Below \(\varphi =0.394\), a critically stable periodic-3 solution emerges (depicted by pink red dots), coexisting with the stable periodic-1 solution.

Fig. 9
figure 9

Floquet multipliers of the PNS varying with \(\varphi \)

Full size image

As \(\varphi \) crosses 0.374 (marked by the black point \(P_2\)), the PNS generates a saddle-node bifurcation. Specifically, the critically stable periodic-3 solution bifurcates, yielding two periodic-3 solutions with differing amplitudes: one stable (blue dots) and the other unstable (green dots). Thus, in the range \(\varphi \in (0.330, 0.374)\), the system simultaneously exhibits three periodic solutions: a stable periodic-1 solution, along with both stable and unstable periodic-3 solutions. The point \(\varphi =0.330\) (the pink point \(P_3\)) is identified as a critical juncture where the stable periodic-1 solution becomes unstable. When \(\varphi \) decreases from 0.330 to 0, the system’s stability status remains consistent, maintaining an unstable periodic-1 solution alongside both a stable and an unstable periodic-3 solution.

Fig. 10
figure 10

Phase diagrams of the \(x_1\) with different \(\varphi \)

Full size image

The stability of these solutions is further elucidated by analyzing the corresponding Floquet multipliers. Figure 9a showcases the trend of Floquet multipliers for the periodic-1 solution, indicating stability with all multipliers inside the unit circle for \(\varphi >0.330\). Upon crossing point \(P_3\), the Floquet multipliers move outside the unit circle, indicating that all subsequent periodic-1 solutions become unstable. Figure 9b focuses on the Floquet multiplier within \(\varphi \in (0.374, 0.393)\), where the system maintains one multiplier on the unit circle, especially at (1,0) on the complex plane, while the other remains inside, suggesting the periodic-3 solution’s critical stability at this juncture. Figure 9c is based on the stable periodic-3 solution depicted in Fig. 8, with Floquet multipliers for the blue dots entirely within the unit circle, affirming these solutions’ asymptotic stability. Calculation of Floquet multipliers for the periodic-3 solution (green dots in Fig. 8) reveals that the modulus of the maximum Floquet multiplier consistently exceeds 1 within this range (Fig. 9d), indicating the instability of these periodic-3 solutions.

To further elucidate the system’s dynamics, phase diagrams of \(x_1\) for \(\varphi =0.38, \varphi =0.35\) and \(\varphi =0.20\) are depicted in Fig. 10. For \(\varphi =0.38\), Fig. 10a, b illustrate the stable periodic-1 solution and the critically stable periodic-3 solution, as identified in Fig. 8. When \(\varphi \) is adjusted to 0.35, Fig. 10c–e display a trio of solutions coexisting within the system: a stable periodic-1 solution, a stable periodic-3 solution, and an unstable periodic-3 solution. These are visually represented by a black solid line, a blue solid line, and a green dashed line, respectively, corresponding to the black, blue, and green dots in Fig. 8 for \(\varphi =0.35\). Similarly, Figs. 10f–h portray the conditions at \(\varphi =0.20\), showcasing the coexistence of an unstable periodic-1 solution, a stable periodic-3 solution, and an unstable periodic-3 solution. The phase diagrams are complemented by the numerical solutions calculated using the 4-th RK method, highlighted with black dots. The correspondence of the semi-analytical solutions obtained through the TMRM with numerical results highlights the compatibility of the methods. Notably, the amplitude of the periodic-1 solution exhibits minor variations across different \(\varphi \) values, in contrast to the periodic-3 solution, where significant amplitude discrepancies are observed, highlighting the system’s sensitive response to changes in \(\varphi \) (cell membrane resistance). Due to the fact that the PNS model of piezoelectric ceramics for acoustic-electric conversion can simulate cochlear nerve excitation, the bifurcation phenomenon of PNS under the influence of resistance parameters will have great potential applications in restoring hearing.

5 Conclusion

This paper presents the high-accuracy semi-analytical solution of the piezoelectric neuron system (PNS) via the time-domain minimum residual method (TMRM) and performs an in-depth bifurcation analysis of the PNS utilizing the derived semi-analytical solution. Specifically, we analyzed the periodic-doubling bifurcation phenomenon that occurs when the amplitude of external excitation (the amplitude of external stimuli) changes, as well as the periodic instability phenomenon caused by changes in parameter \(\varphi \) (cell membrane resistance). Based on the analysis results, the following conclusions are drawn:

  • The TMRM efficiently yields the periodic solution of the PNS, with the precision of the semi-analytical solution enhancing as the truncation series is extended. With identical parameter configurations, the solution’s accuracy via the TMRM surpasses that achieved through the IHB method.

  • As the amplitude of the external excitation increases, the system exhibits a periodic-doubling bifurcation phenomenon, which means that in addition to an unstable periodic-1 solution, there is also a stable periodic-2 solution with the subharmonic frequencies \(\Omega _k=\frac{k\Omega }{2}\), and the relevant Floquet multiplier similarly exits the unit circle, passing through the point \((-1, 0)\) on the complex plane.

  • As the amplitude of the external excitation further increases, the Floquet multiplier moves back into the unit circle via the point \((-1,0)\), indicating the system experiences a periodic-doubling bifurcation in reverse.

  • The reduction of \(\varphi \) brings about complex dynamic phenomena. As \(\varphi \) decreases, the system first exhibits a critical stable periodic-3 solution. As \(\varphi \) further decreases, the PNS exhibits a saddle-node bifurcation; that is, the system has generated a stable periodic-3 solution and an unstable periodic-3 solution.

  • The change of \(\varphi \) has further led to the transformation of the stable periodic-1 solution into unstable, and the corresponding periodic-1 solution will cross the unit circle. Currently, the system has stable/unstable periodic-1 solutions and periodic-3 solutions.