Dynamical mechanism of neuronal firing in the Hodgkin-Huxley model

Dynamical mechanism of neuronal firing in the Hodgkin-Huxley model | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Dynamical mechanism of neuronal firing in the Hodgkin-Huxley model Xue Zhong, Yaze Liu, Tumurpurev Namnan, Hexi Baoyin This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7322686/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The Hodgkin-Huxley (H-H) model remains a cornerstone in the study of neuronal dynamics, yet a global understanding of its behavior, particularly regarding transitions between silent, bistable, and repetitive firing states, has been limited. In this study, we employ the cell mapping method to systematically investigate the global dynamics of the H-H model under varying external currents —— an approach that overcomes the limitations of traditional local analysis methods. Our analysis reveals two critical novel findings, a previously unrecognized bistable regime at high currents and the “reversion to silence” phenomenon where the neuron returns to a silent state under extremely high currents. The results indicate that the model’s states are governed by the emergence and disappearance of distinct attractors as external current varies, with state transitions exhibiting asymmetric dynamics. Additionally, the system exhibits extreme sensitivity to initial conditions, particularly in bistable regimes. These findings advance our understanding of neuronal excitability, with implications for neuroengineering and clinical neuromodulation, and offer a robust framework for exploring global dynamics in nonlinear neuronal systems. Nonlinear dynamics Hodgkin-Huxley model Cell mapping method Attractor evolution Bifurcation Figures Figure 1 Figure 2 Figure 3 1. Introduction The Hodgkin-Huxley (H-H) model, introduced in 1952, is widely regarded as a groundbreaking contribution to neuroscience and nonlinear dynamics. This model quantitatively describes the ionic mechanisms underlying the generation and propagation of action potential in neurons [1–7]. Although developed over seven decades ago, it remains a cornerstone in biology and computational neuroscience, shaping contemporary research on neural activity [8–10]. The model’s influence continues to shape contemporary research, with numerous studies focusing on its properties, applications, and computational implementations [11–20]. By establishing a direct relationship between neuronal activity and ion channel behavior, the H-H model provides a mechanistic foundation for understanding action potential generation. This mechanistic insight is critical for conducting physiological experiments and exploring neuronal firing patterns [21–23]. Beyond its experimental relevance, the model also exemplifies a highly nonlinear dynamical system, which offers a platform for studying complex neuronal behaviors. Researchers extensively explored the nonlinear characteristics of the H-H model, including bifurcation, chaos, and bistability, which significantly impact neural dynamics [24–33]. These nonlinear properties are not only crucial for understanding biological electrical activity but also for uncovering the broader applications of the H-H model in nonlinear dynamics. The nonlinear nature of the H-H model ensures its sensitivity to external current, which induce distinct dynamical states. Extensive investigations into the model’s bifurcation and chaotic behaviors have revealed its potential to generate diverse electrical activity patterns, enhancing our understanding of neuronal functions and the pathological mechanisms underlying neurological disorders [34–49]. Notably, while prior research has elucidated the impact of parameters and external inputs on neuronal dynamics, the mechanisms governing the transition from a silence state to a repetitive firing state remain poorly understood. Wang et al. [38] demonstrated that the resting state corresponds to a stable equilibrium, whereas repetitive firing aligns with a stable limit cycle. They also identified an unstable limit cycle near the bifurcation point, which marks the transition between these states. However, the precise mechanism leading to limit cycle formation remains unresolved. Moreover, existing studies predominantly focus on neuronal dynamics from a resting state, leaving gaps in our understanding of global behaviors, including how attractors evolve across the entire state space and how initial conditions influence long-term dynamics. In reality, as a highly nonlinear system, the dynamical properties of the H-H model exhibit considerable sensitivity to variations in initial conditions. This underscores the necessity for a comprehensive, global perspective on the system’s dynamics, which can reveal essential insights into its complex behavior across different initial states. Such an approach is crucial for advancing our understanding of neuronal activity beyond the constraints of resting state models, offering a more nuanced view of the system’s full dynamical range. The cell mapping method has proven to be a powerful tool for analyzing the global characteristics of nonlinear dynamical systems, such as the H-H model [50]. Originally developed by Hsu [51, 52] in the 1980s, this method has since been refined to analyze systems with multiple steady states, allowing precise determination of attractive domain boundaries [53–57]. When applied to the H-H model, the cell mapping method offers a unique opportunity to uncover the global dynamical features underlying neuronal firing patterns, thus providing a robust framework for understanding the intricate dynamics of neuronal behavior across a range of initial conditions. In this study, we apply the cell mapping method to: (1) systematically characterize global dynamics of the H-H model under varying external currents; (2) identify novel dynamical regimes; (3) reveal the mechanisms of state transitions, emphasizing attractor emergence and disappearance over stability changes. 2. Hodgkin-Huxley model The H-H model is a seminal mathematical framework that describes the ionic mechanisms underlying action potential initiation and propagation in neurons. Developed through groundbreaking experiments on the squid giant axon, the model explains how membrane potential dynamics are influenced by three primary ionic currents: sodium ( Na + ), potassium ( K + ), and a chloride-induced leakage current ( L ). The movement of Na + and K + ions across the membrane is regulated by voltage-dependent ion channels, with the system mathematically represented by four coupled nonlinear differential equations: $$\left\{ \begin{gathered} C\frac{{dV}}{{dt}}=I - {g_K}{n^4}\left( {V - {E_K}} \right) - {g_{Na}}{m^3}h\left( {V - {E_{Na}}} \right) - {g_L}\left( {V - {E_L}} \right) \hfill \\ \frac{{dn}}{{dt}}={\alpha _n}\left( V \right)\left( {1 - n} \right) - {\beta _n}\left( V \right)n \hfill \\ \frac{{dm}}{{dt}}={\alpha _m}\left( V \right)\left( {1 - m} \right) - {\beta _m}\left( V \right)m \hfill \\ \frac{{dh}}{{dt}}={\alpha _h}\left( V \right)\left( {1 - h} \right) - {\beta _h}\left( V \right)h \hfill \\ \end{gathered} \right.$$ 1 where C is the membrane capacitance, V represents the membrane potential, and I denotes the external current density. The variables n and m correspond to the activation dynamics of K + and Na + ion channels, respectively, while h represents the inactivation dynamics of Na + ion channel. The parameters E K , E Na , and E L denote the reversal potentials for potassium, sodium, and leakage currents, respectively, and g Na , g K , and g L are their respective maximal conductances. In neurodynamics studies [29, 46], these parameters typically adopt standard values: C = 1 µF /cm 2 , E Na = 50 mV, E K = -77 mV, E L = -54.4 mV, g Na = 120 mS/cm 2 , g K = 36 mS/cm 2 , and g L = 0.3 mS/cm 2 . The voltage-dependent parameters α n ( V ), β n ( V ), α m ( V ), β m ( V ), α h ( V ), and β h ( V ) are described the by following nonlinear functions: $$\begin{array}{*{20}{l}} {{\alpha _n}\left( V \right)=\frac{{0.01V+0.55}}{{1 - \exp \left( { - 0.1V - 5.5} \right)}},}&{{\beta _n}\left( V \right)=0.125\exp \left( {\frac{{ - V - 65}}{{80}}} \right),} \\ {{\alpha _m}\left( V \right)=\frac{{0.1V+4}}{{1 - \exp \left( { - 0.1V - 4} \right)}},}&{{\beta _m}\left( V \right)=4\exp \left( {\frac{{ - V - 65}}{{18}}} \right),} \\ {{\alpha _h}\left( V \right)=0.07\exp \left( {\frac{{ - V - 65}}{{20}}} \right),}&{{\beta _h}\left( V \right)=\frac{1}{{1+\exp \left( { - 0.1V - 3.5} \right)}}.} \end{array}$$ 2 The interplay of these parameters results in three fundamental dynamical regimes in neurons: silence, bistability, and repetitive firing [29]. These regimes have been extensively investigated, with numerous studies examining the model’s nonlinear characteristics from various perspectives [34–45]. Despite significant progress in characterizing these behaviors, the mechanisms underlying the emergence of these nonlinear features remain poorly understood, warranting further exploration into the global dynamics of the H-H model. 3. Dynamical behaviors of H-H neurons Understanding neuronal dynamics is key to unraveling the mechanisms underlying neuronal firing, a critical process for proper nervous system function. A global dynamical analysis of the H-H model allows investigation of how neuronal firing patterns evolve under varying conditions, particularly in response to external factors like neurotoxin exposure. 3.1 Influence of external current on steady state To explore the impact of a constant external current on neuronal dynamics, we used the H-H model as defined by Eq. ( 1 ). This analysis examines the influence of external current on the steady-state behavior of neurons, starting with the identification of equilibrium points. In dynamical systems theory, an equilibrium point is a state where the system remains constant over time, provided it starts there. Mathematically, it is a point where the system’s derivatives equal to zero. Setting the left-hand side of Eq. ( 1 ) to zero yields the necessary and sufficient conditions for equilibrium points in the H-H model. At equilibrium, the membrane potential V and input current I satisfy the following equation: $$I={g_K}{\left( {\frac{{{\alpha _n}}}{{{\alpha _n}+{\beta _n}}}} \right)^4}\left( {V - {E_K}} \right)+{g_{Na}}{\left( {\frac{{{\alpha _m}}}{{{\alpha _m}+{\beta _m}}}} \right)^3}\left( {\frac{{{\alpha _h}}}{{{\alpha _h}+{\beta _h}}}} \right)\left( {V - {E_{Na}}} \right)+{g_L}\left( {V - {E_L}} \right)$$ 3 Here, α n , β n , α m , β m , α h , and β h are functions of the membrane potential V , as defined in Eq. ( 2 ). Notably, when V = -55 mV, V = -40 mV, or V = -35 mV, the denominators of α n , α m , and β h become zero, indicating that these membrane potential values cannot correspond to equilibrium states. To further analyze the steady-state behavior, we solved the H-H equations using the adaptive step-size Runge-Kutta method for external input currents ranging from 0 to 160 mA. We assumed the stability of equilibrium points by evaluating the eigenvalues of the corresponding Jacobian matrix. The results were visualized in the phase plane defined by the membrane potential V and the potassium channel activation variable n , as shown in Fig. 1 . The chosen range of external currents covers diverse dynamical regimes of neuronal activity, offering a comprehensive view of how external current influences neuronal behavior through attractor emergence and disappearance. The analysis revealed critical dynamical transitions in the H-H model as a function of external current, such as the emergence of bistable states within specific current ranges. Table 1 summarizes the steady-state responses of the H-H model under varying external currents. Table 1 Types of steady-state responses in the H-H neuron model Types Range of external current (mA) Attractor dynamics Silence state (0, 6.264) Stable equilibrium point Bistable state (6.264, 9.778) Stable equilibrium point and periodic solution Firing state (9.778, 154.485) Stable periodic solution Bistable state (154.485, 154.568) Stable equilibrium point and periodic solution Silence state (154.568,160.00) Stable equilibrium point When the neuron is in a silence state, the H-H model maintains a stable fixed-point attractor, no periodic attractor is present. Consistent with previous findings [29], the neuron remains at rest for input currents ranging from 0 mA to 6.264 mA. However, a novel observation in this study reveals that the neuron returns to the silence state when the external current exceeds 154.568 mA, a phenomenon not reported in previous research. In the bistable state, the H-H model is characterized by the coexistence of both attractors: a stable fixed-point and a periodic solution. Our findings identify two distinct current ranges which correspond to bistable behavior: 6.246 mA < I < 9.778 mA, and 154.485 mA < I < 154.568 mA. While the former range aligns with prior studies [29], the latter represents a newly identified regime. Furthermore, within the firing state (9.778 mA < I < 154.485 mA), the neuron exhibits stable periodic oscillations driven by the constant external current, where the fixed-point attractor has vanished and only the periodic attractor remains. Interestingly, our analysis also uncovered a unique external current range (27.237 mA < I < 27.238 mA), within which the H-H model does not exhibit any equilibrium state. This occurs because, in this narrow range, the membrane potential approaches − 55 mV, causing the denominator of α n in Eq. ( 2 ) to approach zero. As illustrated in Fig. 1 , increasing external current induces transitions in the H-H neuron from a silence state to a firing state and then back to a silence state. Both transitions involve bistable states as intermediate phases, but the underlying mechanisms differ significantly. During the transition from silence to firing, the system exhibits an abrupt “jump” from a fixed point to a periodic solution. In contrast, the reverse transition, from firing to silence, involves a gradual “contraction” of the periodic solution into a fixed point. While the latter transition aligns with intuitive dynamical processes, the abrupt jump observed in the former transition warrants further investigation. To gain deeper insights into the mechanisms driving these transitions, the following analysis explores the evolution of equilibrium points and the global dynamical characteristics of the system, providing a nuanced understanding of the interplay between local and global dynamics in shaping neuronal behavior under varying external currents. 3.2 Evolution of equilibrium points The properties of equilibrium points in dynamical systems are typically classified based on their stability and local behavior, which can be analyzed using the linearization of the system and the eigenvalues of the Jacobian matrix. For two-dimensional systems, equilibrium points are typically categorized as saddle points, nodes, spiral points, centers, and other types based on the eigenvalue structure. However, the H-H model is a four-dimensional system, and all its fixed points exhibit mixed characteristics because of the higher-dimensional state space. In this study, we identified two distinct types of equilibrium points in the H-H model, referred to as the stable spiral-node and the high-dimensional saddle. The stable spiral-node is characterized by a Jacobian matrix with a pair of complex conjugate eigenvalues having negative real parts, plus two additional negative real eigenvalues. This configuration causes some phase trajectories to converge linearly, while others spiral inward before eventually stabilizing. In contrast, the high-dimensional saddle corresponds to a Jacobian matrix with a pair of complex conjugate eigenvalues having positive real parts, alongside two negative real eigenvalues. In this case, some phase trajectories spiral outward and diverge from the fixed point with oscillatory behavior, while others converge linearly. The dynamics of these equilibrium points under varying external current intensities (0mA − 160mA) were systematically analyzed. Figure 2 illustrates the evolution of the eigenvalues as a function of external current, with the color bar representing current intensity. The system’s four eigenvalues consist of two real eigenvalues, λ 1 and λ 2 , and a pair of complex conjugate eigenvalues, λ 3 and λ 4 . As the external current increases, λ 1 and λ 2 change from − 0.1207 and − 4.6753, respectively, to -0.3153 and − 9.5116. Simultaneously, the complex eigenvalues λ 3 and λ 4 evolve from − 0.2027 ± 0.3831 i to -0.0246 ± 1.0746 i . Importantly, as the external current reaches 9.778 mA, the real parts of λ 3 and λ 4 transition from negative to positive, signifying the disappearance of the stable fixed-point attractor. At 154.485mA, the real parts of λ 3 and λ 4 revert to negative, marking the re-emergence of the stable fixed-point attractor with periodic attractors still present in the bistale regime. The corresponding bifurcation values and types of equilibrium points are summarized in Table 2 . Between 0 mA and 9.778 mA, the equilibrium point is a stable spiral-node. At 9.778 mA, a bifurcation occurs, and the equilibrium point transitions into a high-dimensional saddle, which persists until 27.237 mA. Notably, no equilibrium points exist in the narrow range from 27.237 mA to 27.238 mA. Beyond 27.238 mA, the high-dimensional saddle reappears and remains unstable until 154.485 mA. At this point, the equilibrium transitions back to a stable spiral-node and persists until 160 mA. We observe a clear relationship between equilibrium types and neuronal behavior. The silence state of the H-H neuron corresponds to a stable spiral-node, whereas its repetitive firing state aligns with a high-dimensional saddle. During transitions between equilibrium types, the neuron exhibits bistable behavior. Specifically, as the external current increases, the equilibrium shifts from a spiral-node to a high-dimensional saddle, causing the silence state to disappear. When the external current exceeds 154.485 mA, the equilibrium transitions back to a stable spiral-node, restoring the neuron to the silence state. Table 2 Types of equilibrium points External current (mA) Types of the points Stability of the fixed point (0, 9.778) stable spiral-node stable (9.778, 27.237) high dimensional saddle unstable (27.237, 27.238) no equilibrium points —— (27.238, 154.485) high dimensional saddle unstable (154.485, 160.00) stable spiral-node stable It is important to emphasize that the above analysis primarily focuses on the local properties of equilibrium points, which provide valuable insights into the mechanisms driving the disappearance of the silence state. However, a complete understanding of the neuron’s behavior requires investigating its global dynamics, which can capture the broader characteristics of the system under varying external conditions. 3.3 Global dynamics of the H-H neuron Previous studies, in conjunction with the results of this research, have demonstrated that under different external current conditions, the H-H neuron exhibits different dynamic states, including silence, firing, and bistability. These states correspond to the equilibrium, periodic, and coexistence solutions of the H-H equations. We explored the evolution of the equilibrium points by analyzing the eigenvalues of the system. To uncover the mechanisms governing the evolution of periodic solutions, we employed the cell mapping technique [51,52] to compute the global dynamics of the H-H neuron, with a focus on attractor emergence and disappearance. The detailed computational procedure is as follows: (a) Partition the state space Ω into n subspaces Ω i for i = 1, 2, ..., n ; (b) For an arbitrary initial state x i (0) in subspace Ω i , compute the dynamic response x i ( t ) according to Eq. ( 1 ) until a steady state is reached; (c) Count the number of times the dynamic response x i ( t ), excluding the initial values, passes through each subspace Ω j for j = 1, 2, ..., n . A count of 1 is given if x i ( t ) passes through Ω j (regardless of how many times), and 0 if it does not; (d) Repeat steps (b) and (c) until the dynamical responses x i ( t ) from all subspaces Ω i for i = 1, 2, ..., n have been detected. Using this approach, we examined the global dynamics of the H-H neuron in a four-dimensional state space, where V \(\in\) [-80, 80], n \(\in\) [0, 1], m \(\in\) [0, 1], and h \(\in\) [0, 1], discretized into 200×200×200×200 subspaces. To simplify and improve readability, we fixed the values of m and h , presenting the global dynamics analysis on the state plane of V and n . The global dynamic behaviors of the neuron in the silence, firing, and bistable states are depicted in Fig. 3 . The colorbar in these figures represents the trends of solutions starting from different initial values within the state space. A value of 1 on the colorbar indicates that solutions originating from all initial states within Ω converge to that subspace, while a value of 0 indicates that no solutions pass through it. Intermediate values between 0 and 1 represent regions where solutions pass through, with higher values indicating a greater frequency of solutions traversing the subspace. The analysis of the global dynamics of the H-H neuron in the silence state (Fig. 3 a) reveals that, regardless of the initial conditions within the state space Ω , all trajectories asymptotically converge to a unique equilibrium point at V e = -62.668, n e = 0.354, m e = 0.069, h e = 0.513. This suggests that the system exhibits stable long-term behavior, with the equilibrium point acting as a global attractor for all initial conditions. Interestingly, not all trajectories converge directly to the equilibrium point. Solutions originating from subspaces near the equilibrium point converge more rapidly, while those starting further away initially explore a broader region of the phase space, characterized by colorbar values between 0.1 and 0.2. In this intermediate region, the system’s dynamics may exhibit transient oscillations or slower convergence toward a narrower region, marked by higher colorbar values. Eventually, all trajectories are drawn toward a slender region with a colorbar value of 1, corresponding to the equilibrium point. These findings confirm that the H-H neuron in the silence state is governed by a single fixed-point attractor, ensuring that, regardless of initial conditions within Ω , all solutions ultimately converge to this equilibrium configuration. Importantly, the absence of unstable periodic solutions is evident as the presence of such solutions would preclude the intermediate states observed in Fig. 3 a. In the case of the firing state (Fig. 3 b), the system’s trajectory ultimately converges to a periodic solution, regardless of the initial conditions within the state space Ω . This indicates the neuron’s inherent repetitive spiking behavior in the firing state. Upon further examination, it is evident that solutions from any initial condition within Ω are initially drawn to subspaces with colorbar values fluctuating between 0.1 and 0.2, representing the transient phase before the system settles into a stable periodic rhythm. Once these subspaces are reached, trajectories converge to the periodic solution, as illustrated in left diagram of Fig. 3 b. The time series plots for the membrane potential V and the potassium ion channel activation variable n , shown in right diagram of Fig. 3 b, further elucidate the system’s dynamics over time. The green background in the time series indicates the stable periodic solutions, while the phase trajectories in the V-n plane provide a geometric representation of the system’s oscillatory behavior. These results suggest that the repetitive firing behavior of the H-H neuron model, induced by external currents ranging from 9.775 mA to 154.485 mA, is governed by a single periodic attractor which dictates the system’s long-term periodic spiking behavior. In the Fig. 3 c we highlighting the coexistence of both a fixed-point attractor and a periodic attractor. Depending on the initial conditions, trajectories within Ω may converge to either attractors, reflecting the system’s complex long-term behavior under different initial states. The evolution of the system is influenced by the location of the initial state within the attraction domains of these attractors. The boundaries of these attraction domains are also delineated in left diagram of Fig. 3 c. To offer a more intuitive representation of the evolution of solutions from different attraction domains, phase trajectories initiated from various initial values in the V - n plane are shown in the middle and right diagrams of Fig. 3 c. These observations underscore the critical role of both attractors in governing the system’s behavior in the bistable state. The fixed-point attractor corresponds to a stable equilibrium, while the periodic attractor represents the system’s oscillatory behavior. The coexistence of these two attractors leads to rich dynamic behavior, marking the system highly sensitive to initial conditions and showcasing typical bistable characteristics. Preceding analysis suggest a dynamic explanation for the abrupt emergence of periodic solutions at an external current of 6.264 mA. This value marks a critical threshold, above which the input energy becomes sufficient to sustain periodic solutions. Below this threshold, the input energy is inadequate, with the system transitioning toward the equilibrium point, as shown in Fig. 3 a. As the external current exceeds this critical value, further increases in input energy lead to a gradual contraction of the periodic solution toward the equilibrium point. At an external current of 154.485 mA, the periodic solution fully converges into the equilibrium point. This dynamic behavior highlights a fundamental distinction in the mechanisms governing state transitions in the H-H neuron model. Specifically, the transition from silence to firing is characterized by a critical "jump" to periodic solutions, whereas the reverse transition involves a gradual collapse of the periodic solution into the equilibrium state. This asymmetry underscores the intricate interplay between energy input and system dynamics in shaping the transitions between different states. 4. Conclusion This study provides a comprehensive exploration of the global dynamics of the H-H model, offering novel insights into the mechanisms that govern neuronal firing patterns. By employing the cell mapping method, we have uncovered a range of dynamical behaviors, including bistability, silent, and repetitive firing states, along with critical bifurcation points that were previously underexplored. Importantly, our findings emphasize that transitions between neuronal states are primarily driven by the appearance and disappearance of attractors, rather than changes in stability. This perspective shifts the understanding of neuronal behavior, underscoring the pivotal role of attractor dynamics in state switching. Additionally, we highlight the potential applications of high-current bistability (154.485–154.568 mA) in clinical neuromodulation, neuroengineering, and biological protection. Small current adjustments, within this bistable region, can switch neuronal states between therapeutic and non-therapeutic conditions, enable robust transitions between silent and firing state under extreme inputs, and prevent excessive firing or ion channel fatigue under pathological conditions. These findings not only deepen our understanding of neuronal excitability but also provide a foundational framework for future investigations into the complex dynamics of neuronal systems and their potential applications in neuromodulation and neuroengineering. Declarations Funding Declaration This work was supported by the National Natural Science Foundation of China (12402051), and the Natural Science Foundation of Inner Mongolia Autonomous Region of China (2024QN01002, 2024MS01006). Acknowledgments Thanks to the support of "Inner Mongolia Academician Workstation for New Energy Intelligent Equipment and Operation Maintenance in IMUT" for this research. References Hodgkin, A. L., Huxley, A. 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C., et al.: Parameter identification of dynamical systems based on short-term prediction by the generalized cell mapping method with deep learning. Nonlinear Dyn. 113 , 4031-4044 (2025). https://doi.org/10.1007/s11071-024-09943-8 Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7322686","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":506027034,"identity":"9fda9470-b9fb-4ea4-9a4d-6788e50b9bbc","order_by":0,"name":"Xue Zhong","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA10lEQVRIiWNgGAWjYJACZhDBzwzhMDYQrUWyGUgcIEmLwQFitRjcSD78uaDijt3m49xp0h8YbGQ3HGB+9gCfFskZaQnGM848S952mHebxAGGNOMNB9jMDfBp4ZfIMUjmbTucbAbRcjhxwwEeNgl8Wtgk8j8cBmkxbgZr+U9YC9AWxmagFjsDZrCWA4S1SPY8M2bmOXM4QeIw72aLMwbJxjMPs5nh1WJwPPnxZ56Kw/b8/Wc33qiosJPtO978DK8WGEhsgJjAAI0mIoA9kepGwSgYBaNgJAIA1k1Iz5+ExPYAAAAASUVORK5CYII=","orcid":"","institution":"Inner Mongolia University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Xue","middleName":"","lastName":"Zhong","suffix":""},{"id":506027035,"identity":"4f4034c3-d04f-404e-adfe-07b2d2173664","order_by":1,"name":"Yaze Liu","email":"","orcid":"","institution":"Inner Mongolia University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Yaze","middleName":"","lastName":"Liu","suffix":""},{"id":506027036,"identity":"d8c89c71-9fbe-4d03-a08a-ff4427f4af26","order_by":2,"name":"Tumurpurev Namnan","email":"","orcid":"","institution":"Mongolian university of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Tumurpurev","middleName":"","lastName":"Namnan","suffix":""},{"id":506027037,"identity":"2b79eb1c-9182-4b96-8f09-c97c534b6247","order_by":3,"name":"Hexi Baoyin","email":"","orcid":"","institution":"Inner Mongolia Key Laboratory of New Energy and Energy Storage Technology","correspondingAuthor":false,"prefix":"","firstName":"Hexi","middleName":"","lastName":"Baoyin","suffix":""}],"badges":[],"createdAt":"2025-08-08 02:23:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7322686/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7322686/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":90040208,"identity":"5a040d22-44fc-42cf-9e7e-a8d93ddb1b25","added_by":"auto","created_at":"2025-08-27 16:49:16","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":173318,"visible":true,"origin":"","legend":"\u003cp\u003eSteady-state response of the H-H neuron model in the membrane potential \u003cem\u003eV\u003c/em\u003e and potassium activation variable \u003cem\u003en\u003c/em\u003e space\u003c/p\u003e","description":"","filename":"Fig1.png","url":"https://assets-eu.researchsquare.com/files/rs-7322686/v1/527c25dcd88825aaa899ffe2.png"},{"id":90040207,"identity":"ba520a40-c592-4fcc-9917-4c16fa443fd0","added_by":"auto","created_at":"2025-08-27 16:49:16","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":28363,"visible":true,"origin":"","legend":"\u003cp\u003eEvolution of the eigenvalues with the external current with the color bar representing the intensity of the external current (mA).\u003c/p\u003e","description":"","filename":"Fig2.png","url":"https://assets-eu.researchsquare.com/files/rs-7322686/v1/d833612e8f16a6a7825268fe.png"},{"id":90042146,"identity":"de092d09-564b-4d21-acb0-dd2af7f32e71","added_by":"auto","created_at":"2025-08-27 17:13:16","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":152101,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eGlobal dynamics of H-H neuron. a\u003c/strong\u003e, Fixed-point attractor in the silent state for \u003cem\u003eI \u003c/em\u003e= 3.298 mA, \u003cem\u003em \u003c/em\u003e= 0.069, \u003cem\u003eh \u003c/em\u003e= 0.513. \u003cstrong\u003eb\u003c/strong\u003e, Periodic attractor in the firing state (left) and corresponding periodic solution (right) for \u003cem\u003eI\u003c/em\u003e = 38.770 mA, \u003cem\u003em\u003c/em\u003e = 0.192, \u003cem\u003eh\u003c/em\u003e = 0.213. \u003cstrong\u003ec\u003c/strong\u003e, Coexistence of fixed-point and periodic attractors in the bistable state (left), phase trajectories near the equilibrium point (middle), and global phase plane trajectories (right) for \u003cem\u003eI\u003c/em\u003e = 8.000 mA, \u003cem\u003em\u003c/em\u003e = 0.090, \u003cem\u003eh\u003c/em\u003e = 0.431. Initial point coordinates: E\u003csub\u003e1\u003c/sub\u003e (-59.990, 0.380), E\u003csub\u003e2\u003c/sub\u003e (-59.990, 0.379), E\u003csub\u003e3\u003c/sub\u003e (-60.030, 0.417), E\u003csub\u003e4\u003c/sub\u003e (-60.010, 0.418), P\u003csub\u003e1\u003c/sub\u003e (-59.990, 0.378), P\u003csub\u003e2\u003c/sub\u003e (-59.990, 0.377), P\u003csub\u003e3\u003c/sub\u003e (-60.020, 0.420), P\u003csub\u003e4\u003c/sub\u003e (-60.020, 0.419), P\u003csub\u003e5\u003c/sub\u003e (-60.510, 0.599), P\u003csub\u003e6\u003c/sub\u003e (-0.155, 0.401), P\u003csub\u003e7\u003c/sub\u003e (-1.451, 0.601), P\u003csub\u003e8\u003c/sub\u003e (17.150, 0.800).\u003c/p\u003e","description":"","filename":"Fig3.png","url":"https://assets-eu.researchsquare.com/files/rs-7322686/v1/99d9a1fa48f64fccfb929d7e.png"},{"id":91739313,"identity":"bca0f40d-d914-4c29-8e45-dd2272e3a930","added_by":"auto","created_at":"2025-09-19 18:16:34","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1006318,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7322686/v1/4404bfa5-8b74-4925-94c5-f0b62a225b93.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Dynamical mechanism of neuronal firing in the Hodgkin-Huxley model","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eThe Hodgkin-Huxley (H-H) model, introduced in 1952, is widely regarded as a groundbreaking contribution to neuroscience and nonlinear dynamics. This model quantitatively describes the ionic mechanisms underlying the generation and propagation of action potential in neurons [1\u0026ndash;7]. Although developed over seven decades ago, it remains a cornerstone in biology and computational neuroscience, shaping contemporary research on neural activity [8\u0026ndash;10]. The model\u0026rsquo;s influence continues to shape contemporary research, with numerous studies focusing on its properties, applications, and computational implementations [11\u0026ndash;20].\u003c/p\u003e\n\u003cp\u003eBy establishing a direct relationship between neuronal activity and ion channel behavior, the H-H model provides a mechanistic foundation for understanding action potential generation. This mechanistic insight is critical for conducting physiological experiments and exploring neuronal firing patterns [21\u0026ndash;23]. Beyond its experimental relevance, the model also exemplifies a highly nonlinear dynamical system, which offers a platform for studying complex neuronal behaviors. Researchers extensively explored the nonlinear characteristics of the H-H model, including bifurcation, chaos, and bistability, which significantly impact neural dynamics [24\u0026ndash;33]. These nonlinear properties are not only crucial for understanding biological electrical activity but also for uncovering the broader applications of the H-H model in nonlinear dynamics.\u003c/p\u003e\n\u003cp\u003eThe nonlinear nature of the H-H model ensures its sensitivity to external current, which induce distinct dynamical states. Extensive investigations into the model\u0026rsquo;s bifurcation and chaotic behaviors have revealed its potential to generate diverse electrical activity patterns, enhancing our understanding of neuronal functions and the pathological mechanisms underlying neurological disorders [34\u0026ndash;49]. Notably, while prior research has elucidated the impact of parameters and external inputs on neuronal dynamics, the mechanisms governing the transition from a silence state to a repetitive firing state remain poorly understood. Wang et al. [38] demonstrated that the resting state corresponds to a stable equilibrium, whereas repetitive firing aligns with a stable limit cycle. They also identified an unstable limit cycle near the bifurcation point, which marks the transition between these states. However, the precise mechanism leading to limit cycle formation remains unresolved.\u003c/p\u003e\n\u003cp\u003eMoreover, existing studies predominantly focus on neuronal dynamics from a resting state, leaving gaps in our understanding of global behaviors, including how attractors evolve across the entire state space and how initial conditions influence long-term dynamics. In reality, as a highly nonlinear system, the dynamical properties of the H-H model exhibit considerable sensitivity to variations in initial conditions. This underscores the necessity for a comprehensive, global perspective on the system\u0026rsquo;s dynamics, which can reveal essential insights into its complex behavior across different initial states. Such an approach is crucial for advancing our understanding of neuronal activity beyond the constraints of resting state models, offering a more nuanced view of the system\u0026rsquo;s full dynamical range.\u003c/p\u003e\n\u003cp\u003eThe cell mapping method has proven to be a powerful tool for analyzing the global characteristics of nonlinear dynamical systems, such as the H-H model [50]. Originally developed by Hsu [51, 52] in the 1980s, this method has since been refined to analyze systems with multiple steady states, allowing precise determination of attractive domain boundaries [53\u0026ndash;57]. When applied to the H-H model, the cell mapping method offers a unique opportunity to uncover the global dynamical features underlying neuronal firing patterns, thus providing a robust framework for understanding the intricate dynamics of neuronal behavior across a range of initial conditions.\u003c/p\u003e\n\u003cp\u003eIn this study, we apply the cell mapping method to: (1) systematically characterize global dynamics of the H-H model under varying external currents; (2) identify novel dynamical regimes; (3) reveal the mechanisms of state transitions, emphasizing attractor emergence and disappearance over stability changes.\u003c/p\u003e"},{"header":"2. Hodgkin-Huxley model","content":"\u003cp\u003eThe H-H model is a seminal mathematical framework that describes the ionic mechanisms underlying action potential initiation and propagation in neurons. Developed through groundbreaking experiments on the squid giant axon, the model explains how membrane potential dynamics are influenced by three primary ionic currents: sodium (\u003cem\u003eNa\u003c/em\u003e\u003csup\u003e+\u003c/sup\u003e), potassium (\u003cem\u003eK\u003c/em\u003e\u003csup\u003e+\u003c/sup\u003e), and a chloride-induced leakage current (\u003cem\u003eL\u003c/em\u003e). The movement of \u003cem\u003eNa\u003c/em\u003e\u003csup\u003e+\u003c/sup\u003e and \u003cem\u003eK\u003c/em\u003e\u003csup\u003e+\u003c/sup\u003e ions across the membrane is regulated by voltage-dependent ion channels, with the system mathematically represented by four coupled nonlinear differential equations:\u003c/p\u003e\n\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$$\\left\\{ \\begin{gathered} C\\frac{{dV}}{{dt}}=I - {g_K}{n^4}\\left( {V - {E_K}} \\right) - {g_{Na}}{m^3}h\\left( {V - {E_{Na}}} \\right) - {g_L}\\left( {V - {E_L}} \\right) \\hfill \\\\ \\frac{{dn}}{{dt}}={\\alpha _n}\\left( V \\right)\\left( {1 - n} \\right) - {\\beta _n}\\left( V \\right)n \\hfill \\\\ \\frac{{dm}}{{dt}}={\\alpha _m}\\left( V \\right)\\left( {1 - m} \\right) - {\\beta _m}\\left( V \\right)m \\hfill \\\\ \\frac{{dh}}{{dt}}={\\alpha _h}\\left( V \\right)\\left( {1 - h} \\right) - {\\beta _h}\\left( V \\right)h \\hfill \\\\ \\end{gathered} \\right.$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003eC\u003c/em\u003e is the membrane capacitance, \u003cem\u003eV\u003c/em\u003e represents the membrane potential, and \u003cem\u003eI\u003c/em\u003e denotes the external current density. The variables \u003cem\u003en\u003c/em\u003e and \u003cem\u003em\u003c/em\u003e correspond to the activation dynamics of \u003cem\u003eK\u003c/em\u003e\u003csup\u003e+\u003c/sup\u003e and \u003cem\u003eNa\u003c/em\u003e\u003csup\u003e+\u003c/sup\u003e ion channels, respectively, while \u003cem\u003eh\u003c/em\u003e represents the inactivation dynamics of \u003cem\u003eNa\u003c/em\u003e\u003csup\u003e+\u003c/sup\u003e ion channel. The parameters \u003cem\u003eE\u003c/em\u003e\u003csub\u003e\u003cem\u003eK\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eE\u003c/em\u003e\u003csub\u003e\u003cem\u003eNa\u003c/em\u003e\u003c/sub\u003e, and \u003cem\u003eE\u003c/em\u003e\u003csub\u003e\u003cem\u003eL\u003c/em\u003e\u003c/sub\u003e denote the reversal potentials for potassium, sodium, and leakage currents, respectively, and \u003cem\u003eg\u003c/em\u003e\u003csub\u003e\u003cem\u003eNa\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eg\u003c/em\u003e\u003csub\u003e\u003cem\u003eK\u003c/em\u003e\u003c/sub\u003e, and \u003cem\u003eg\u003c/em\u003e\u003csub\u003e\u003cem\u003eL\u003c/em\u003e\u003c/sub\u003e are their respective maximal conductances. In neurodynamics studies [29, 46], these parameters typically adopt standard values: \u003cem\u003eC\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1 \u003cem\u003e\u0026micro;F\u003c/em\u003e/cm\u003csup\u003e2\u003c/sup\u003e, \u003cem\u003eE\u003c/em\u003e\u003csub\u003e\u003cem\u003eNa\u003c/em\u003e\u003c/sub\u003e = 50 mV, \u003cem\u003eE\u003c/em\u003e\u003csub\u003eK\u003c/sub\u003e = -77 mV, \u003cem\u003eE\u003c/em\u003e\u003csub\u003eL\u003c/sub\u003e = -54.4 mV, \u003cem\u003eg\u003c/em\u003e\u003csub\u003e\u003cem\u003eNa\u003c/em\u003e\u003c/sub\u003e = 120 mS/cm\u003csup\u003e2\u003c/sup\u003e, \u003cem\u003eg\u003c/em\u003e\u003csub\u003e\u003cem\u003eK\u003c/em\u003e\u003c/sub\u003e = 36 mS/cm\u003csup\u003e2\u003c/sup\u003e, and \u003cem\u003eg\u003c/em\u003e\u003csub\u003e\u003cem\u003eL\u003c/em\u003e\u003c/sub\u003e = 0.3 mS/cm\u003csup\u003e2\u003c/sup\u003e. The voltage-dependent parameters \u003cem\u003e\u0026alpha;\u003c/em\u003e\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003eV\u003c/em\u003e), \u003cem\u003e\u0026beta;\u003c/em\u003e\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003eV\u003c/em\u003e), \u003cem\u003e\u0026alpha;\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003eV\u003c/em\u003e), \u003cem\u003e\u0026beta;\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003eV\u003c/em\u003e), \u003cem\u003e\u0026alpha;\u003c/em\u003e\u003csub\u003e\u003cem\u003eh\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003eV\u003c/em\u003e), and \u003cem\u003e\u0026beta;\u003c/em\u003e\u003csub\u003e\u003cem\u003eh\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003eV\u003c/em\u003e) are described the by following nonlinear functions:\u003c/p\u003e\n\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e$$\\begin{array}{*{20}{l}} {{\\alpha _n}\\left( V \\right)=\\frac{{0.01V+0.55}}{{1 - \\exp \\left( { - 0.1V - 5.5} \\right)}},}\u0026amp;{{\\beta _n}\\left( V \\right)=0.125\\exp \\left( {\\frac{{ - V - 65}}{{80}}} \\right),} \\\\ {{\\alpha _m}\\left( V \\right)=\\frac{{0.1V+4}}{{1 - \\exp \\left( { - 0.1V - 4} \\right)}},}\u0026amp;{{\\beta _m}\\left( V \\right)=4\\exp \\left( {\\frac{{ - V - 65}}{{18}}} \\right),} \\\\ {{\\alpha _h}\\left( V \\right)=0.07\\exp \\left( {\\frac{{ - V - 65}}{{20}}} \\right),}\u0026amp;{{\\beta _h}\\left( V \\right)=\\frac{1}{{1+\\exp \\left( { - 0.1V - 3.5} \\right)}}.} \\end{array}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe interplay of these parameters results in three fundamental dynamical regimes in neurons: silence, bistability, and repetitive firing [29]. These regimes have been extensively investigated, with numerous studies examining the model\u0026rsquo;s nonlinear characteristics from various perspectives [34\u0026ndash;45]. Despite significant progress in characterizing these behaviors, the mechanisms underlying the emergence of these nonlinear features remain poorly understood, warranting further exploration into the global dynamics of the H-H model.\u003c/p\u003e"},{"header":"3. Dynamical behaviors of H-H neurons","content":"\u003cp\u003eUnderstanding neuronal dynamics is key to unraveling the mechanisms underlying neuronal firing, a critical process for proper nervous system function. A global dynamical analysis of the H-H model allows investigation of how neuronal firing patterns evolve under varying conditions, particularly in response to external factors like neurotoxin exposure.\u003c/p\u003e\n\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n \u003ch2\u003e3.1 Influence of external current on steady state\u003c/h2\u003e\n \u003cp\u003eTo explore the impact of a constant external current on neuronal dynamics, we used the H-H model as defined by Eq. (\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e). This analysis examines the influence of external current on the steady-state behavior of neurons, starting with the identification of equilibrium points. In dynamical systems theory, an equilibrium point is a state where the system remains constant over time, provided it starts there. Mathematically, it is a point where the system\u0026rsquo;s derivatives equal to zero. Setting the left-hand side of Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e) to zero yields the necessary and sufficient conditions for equilibrium points in the H-H model. At equilibrium, the membrane potential \u003cem\u003eV\u003c/em\u003e and input current \u003cem\u003eI\u003c/em\u003e satisfy the following equation:\u003c/p\u003e\n \u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e$$I={g_K}{\\left( {\\frac{{{\\alpha _n}}}{{{\\alpha _n}+{\\beta _n}}}} \\right)^4}\\left( {V - {E_K}} \\right)+{g_{Na}}{\\left( {\\frac{{{\\alpha _m}}}{{{\\alpha _m}+{\\beta _m}}}} \\right)^3}\\left( {\\frac{{{\\alpha _h}}}{{{\\alpha _h}+{\\beta _h}}}} \\right)\\left( {V - {E_{Na}}} \\right)+{g_L}\\left( {V - {E_L}} \\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003eHere, \u003cem\u003e\u0026alpha;\u003c/em\u003e\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003e\u0026beta;\u003c/em\u003e\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003e\u0026alpha;\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003e\u0026beta;\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003e\u0026alpha;\u003c/em\u003e\u003csub\u003e\u003cem\u003eh\u003c/em\u003e\u003c/sub\u003e, and \u003cem\u003e\u0026beta;\u003c/em\u003e\u003csub\u003e\u003cem\u003eh\u003c/em\u003e\u003c/sub\u003e are functions of the membrane potential \u003cem\u003eV\u003c/em\u003e, as defined in Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e). Notably, when \u003cem\u003eV\u003c/em\u003e = -55 mV, \u003cem\u003eV\u003c/em\u003e = -40 mV, or \u003cem\u003eV\u003c/em\u003e = -35 mV, the denominators of \u003cem\u003e\u0026alpha;\u003c/em\u003e\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003e\u0026alpha;\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e, and \u003cem\u003e\u0026beta;\u003c/em\u003e\u003csub\u003e\u003cem\u003eh\u003c/em\u003e\u003c/sub\u003e become zero, indicating that these membrane potential values cannot correspond to equilibrium states.\u003c/p\u003e\n \u003cp\u003eTo further analyze the steady-state behavior, we solved the H-H equations using the adaptive step-size Runge-Kutta method for external input currents ranging from 0 to 160 mA. We assumed the stability of equilibrium points by evaluating the eigenvalues of the corresponding Jacobian matrix. The results were visualized in the phase plane defined by the membrane potential \u003cem\u003eV\u003c/em\u003e and the potassium channel activation variable \u003cem\u003en\u003c/em\u003e, as shown in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e. The chosen range of external currents covers diverse dynamical regimes of neuronal activity, offering a comprehensive view of how external current influences neuronal behavior through attractor emergence and disappearance. The analysis revealed critical dynamical transitions in the H-H model as a function of external current, such as the emergence of bistable states within specific current ranges. Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e summarizes the steady-state responses of the H-H model under varying external currents.\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\n \u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eTypes of steady-state responses in the H-H neuron model\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eTypes\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eRange of external current (mA)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eAttractor dynamics\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSilence state\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e(0, 6.264)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eStable equilibrium point\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBistable state\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e(6.264, 9.778)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eStable equilibrium point and periodic solution\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFiring state\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e(9.778, 154.485)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eStable periodic solution\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBistable state\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e(154.485, 154.568)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eStable equilibrium point and periodic solution\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSilence state\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e(154.568,160.00)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eStable equilibrium point\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eWhen the neuron is in a silence state, the H-H model maintains a stable fixed-point attractor, no periodic attractor is present. Consistent with previous findings [29], the neuron remains at rest for input currents ranging from 0 mA to 6.264 mA. However, a novel observation in this study reveals that the neuron returns to the silence state when the external current exceeds 154.568 mA, a phenomenon not reported in previous research. In the bistable state, the H-H model is characterized by the coexistence of both attractors: a stable fixed-point and a periodic solution. Our findings identify two distinct current ranges which correspond to bistable behavior: 6.246 mA\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eI\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;9.778 mA, and 154.485 mA\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eI\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;154.568 mA. While the former range aligns with prior studies [29], the latter represents a newly identified regime. Furthermore, within the firing state (9.778 mA\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eI\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;154.485 mA), the neuron exhibits stable periodic oscillations driven by the constant external current, where the fixed-point attractor has vanished and only the periodic attractor remains. Interestingly, our analysis also uncovered a unique external current range (27.237 mA\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eI\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;27.238 mA), within which the H-H model does not exhibit any equilibrium state. This occurs because, in this narrow range, the membrane potential approaches \u0026minus;\u0026thinsp;55 mV, causing the denominator of \u003cem\u003e\u0026alpha;\u003c/em\u003e\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e in Eq. (\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e) to approach zero.\u003c/p\u003e\n \u003cp\u003eAs illustrated in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e, increasing external current induces transitions in the H-H neuron from a silence state to a firing state and then back to a silence state. Both transitions involve bistable states as intermediate phases, but the underlying mechanisms differ significantly. During the transition from silence to firing, the system exhibits an abrupt \u0026ldquo;jump\u0026rdquo; from a fixed point to a periodic solution. In contrast, the reverse transition, from firing to silence, involves a gradual \u0026ldquo;contraction\u0026rdquo; of the periodic solution into a fixed point. While the latter transition aligns with intuitive dynamical processes, the abrupt jump observed in the former transition warrants further investigation. To gain deeper insights into the mechanisms driving these transitions, the following analysis explores the evolution of equilibrium points and the global dynamical characteristics of the system, providing a nuanced understanding of the interplay between local and global dynamics in shaping neuronal behavior under varying external currents.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\n \u003ch2\u003e3.2 Evolution of equilibrium points\u003c/h2\u003e\n \u003cp\u003eThe properties of equilibrium points in dynamical systems are typically classified based on their stability and local behavior, which can be analyzed using the linearization of the system and the eigenvalues of the Jacobian matrix. For two-dimensional systems, equilibrium points are typically categorized as saddle points, nodes, spiral points, centers, and other types based on the eigenvalue structure. However, the H-H model is a four-dimensional system, and all its fixed points exhibit mixed characteristics because of the higher-dimensional state space.\u003c/p\u003e\n \u003cp\u003eIn this study, we identified two distinct types of equilibrium points in the H-H model, referred to as the stable spiral-node and the high-dimensional saddle. The stable spiral-node is characterized by a Jacobian matrix with a pair of complex conjugate eigenvalues having negative real parts, plus two additional negative real eigenvalues. This configuration causes some phase trajectories to converge linearly, while others spiral inward before eventually stabilizing. In contrast, the high-dimensional saddle corresponds to a Jacobian matrix with a pair of complex conjugate eigenvalues having positive real parts, alongside two negative real eigenvalues. In this case, some phase trajectories spiral outward and diverge from the fixed point with oscillatory behavior, while others converge linearly.\u003c/p\u003e\n \u003cp\u003eThe dynamics of these equilibrium points under varying external current intensities (0mA \u0026minus;\u0026thinsp;160mA) were systematically analyzed. Figure \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e illustrates the evolution of the eigenvalues as a function of external current, with the color bar representing current intensity. The system\u0026rsquo;s four eigenvalues consist of two real eigenvalues, \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, and a pair of complex conjugate eigenvalues, \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e and \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e4\u003c/sub\u003e. As the external current increases, \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e change from \u0026minus;\u0026thinsp;0.1207 and \u0026minus;\u0026thinsp;4.6753, respectively, to -0.3153 and \u0026minus;\u0026thinsp;9.5116. Simultaneously, the complex eigenvalues \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e and \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e4\u003c/sub\u003e evolve from \u0026minus;\u0026thinsp;0.2027\u0026thinsp;\u0026plusmn;\u0026thinsp;0.3831\u003cem\u003ei\u003c/em\u003e to -0.0246\u0026thinsp;\u0026plusmn;\u0026thinsp;1.0746\u003cem\u003ei\u003c/em\u003e. Importantly, as the external current reaches 9.778 mA, the real parts of \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e and \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e4\u003c/sub\u003e transition from negative to positive, signifying the disappearance of the stable fixed-point attractor. At 154.485mA, the real parts of \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e and \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e4\u003c/sub\u003e revert to negative, marking the re-emergence of the stable fixed-point attractor with periodic attractors still present in the bistale regime.\u003c/p\u003e\n \u003cp\u003eThe corresponding bifurcation values and types of equilibrium points are summarized in Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. Between 0 mA and 9.778 mA, the equilibrium point is a stable spiral-node. At 9.778 mA, a bifurcation occurs, and the equilibrium point transitions into a high-dimensional saddle, which persists until 27.237 mA. Notably, no equilibrium points exist in the narrow range from 27.237 mA to 27.238 mA. Beyond 27.238 mA, the high-dimensional saddle reappears and remains unstable until 154.485 mA. At this point, the equilibrium transitions back to a stable spiral-node and persists until 160 mA.\u003c/p\u003e\n \u003cp\u003eWe observe a clear relationship between equilibrium types and neuronal behavior. The silence state of the H-H neuron corresponds to a stable spiral-node, whereas its repetitive firing state aligns with a high-dimensional saddle. During transitions between equilibrium types, the neuron exhibits bistable behavior. Specifically, as the external current increases, the equilibrium shifts from a spiral-node to a high-dimensional saddle, causing the silence state to disappear. When the external current exceeds 154.485 mA, the equilibrium transitions back to a stable spiral-node, restoring the neuron to the silence state.\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\n \u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eTypes of equilibrium points\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eExternal current (mA)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eTypes of the points\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eStability of the fixed point\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0, 9.778)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003estable spiral-node\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003estable\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(9.778, 27.237)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ehigh dimensional saddle\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eunstable\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(27.237, 27.238)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eno equilibrium points\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026mdash;\u0026mdash;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(27.238, 154.485)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ehigh dimensional saddle\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eunstable\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(154.485, 160.00)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003estable spiral-node\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003estable\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eIt is important to emphasize that the above analysis primarily focuses on the local properties of equilibrium points, which provide valuable insights into the mechanisms driving the disappearance of the silence state. However, a complete understanding of the neuron\u0026rsquo;s behavior requires investigating its global dynamics, which can capture the broader characteristics of the system under varying external conditions.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\n \u003ch2\u003e3.3 Global dynamics of the H-H neuron\u003c/h2\u003e\n \u003cp\u003ePrevious studies, in conjunction with the results of this research, have demonstrated that under different external current conditions, the H-H neuron exhibits different dynamic states, including silence, firing, and bistability. These states correspond to the equilibrium, periodic, and coexistence solutions of the H-H equations. We explored the evolution of the equilibrium points by analyzing the eigenvalues of the system. To uncover the mechanisms governing the evolution of periodic solutions, we employed the cell mapping technique [51,52] to compute the global dynamics of the H-H neuron, with a focus on attractor emergence and disappearance. The detailed computational procedure is as follows:\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\u003cspan\u003e\n \u003cp\u003e(a) Partition the state space \u003cem\u003eΩ\u003c/em\u003e into \u003cem\u003en\u003c/em\u003e subspaces \u003cem\u003eΩ\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e for \u003cem\u003ei\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, 2, ..., \u003cem\u003en\u003c/em\u003e;\u003c/p\u003e\n \u003c/span\u003e\u003cspan\u003e\n \u003cp\u003e(b) For an arbitrary initial state \u003cstrong\u003ex\u003c/strong\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e(0) in subspace \u003cem\u003eΩ\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e, compute the dynamic response \u003cstrong\u003ex\u003c/strong\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e) according to Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e) until a steady state is reached;\u003c/p\u003e\n \u003c/span\u003e\u003cspan\u003e\n \u003cp\u003e(c) Count the number of times the dynamic response \u003cstrong\u003ex\u003c/strong\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e), excluding the initial values, passes through each subspace \u003cem\u003eΩ\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e\u003c/sub\u003e for \u003cem\u003ej\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, 2, ..., \u003cem\u003en\u003c/em\u003e. A count of 1 is given if \u003cstrong\u003ex\u003c/strong\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e) passes through \u003cem\u003eΩ\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e\u003c/sub\u003e (regardless of how many times), and 0 if it does not;\u003c/p\u003e\n \u003c/span\u003e\u003cspan\u003e\n \u003cp\u003e(d) Repeat steps (b) and (c) until the dynamical responses \u003cstrong\u003ex\u003c/strong\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e) from all subspaces \u003cem\u003eΩ\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e for \u003cem\u003ei\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, 2, ..., \u003cem\u003en\u003c/em\u003e have been detected.\u003c/p\u003e\n \u003c/span\u003e\n \u003cp\u003eUsing this approach, we examined the global dynamics of the H-H neuron in a four-dimensional state space, where \u003cem\u003eV\u003c/em\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\in\\)\u003c/span\u003e\u003c/span\u003e[-80, 80], \u003cem\u003en\u003c/em\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\in\\)\u003c/span\u003e\u003c/span\u003e[0, 1], \u003cem\u003em\u003c/em\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\in\\)\u003c/span\u003e\u003c/span\u003e[0, 1], and \u003cem\u003eh\u003c/em\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\in\\)\u003c/span\u003e\u003c/span\u003e[0, 1], discretized into 200\u0026times;200\u0026times;200\u0026times;200 subspaces. To simplify and improve readability, we fixed the values of \u003cem\u003em\u003c/em\u003e and \u003cem\u003eh\u003c/em\u003e, presenting the global dynamics analysis on the state plane of \u003cem\u003eV\u003c/em\u003e and \u003cem\u003en\u003c/em\u003e. The global dynamic behaviors of the neuron in the silence, firing, and bistable states are depicted in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. The colorbar in these figures represents the trends of solutions starting from different initial values within the state space. A value of 1 on the colorbar indicates that solutions originating from all initial states within \u003cem\u003eΩ\u003c/em\u003e converge to that subspace, while a value of 0 indicates that no solutions pass through it. Intermediate values between 0 and 1 represent regions where solutions pass through, with higher values indicating a greater frequency of solutions traversing the subspace.\u003c/p\u003e\n \u003cp\u003eThe analysis of the global dynamics of the H-H neuron in the silence state (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ea) reveals that, regardless of the initial conditions within the state space \u003cem\u003eΩ\u003c/em\u003e, all trajectories asymptotically converge to a unique equilibrium point at \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e = -62.668, \u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e = 0.354, \u003cem\u003em\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e = 0.069, \u003cem\u003eh\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e = 0.513. This suggests that the system exhibits stable long-term behavior, with the equilibrium point acting as a global attractor for all initial conditions. Interestingly, not all trajectories converge directly to the equilibrium point. Solutions originating from subspaces near the equilibrium point converge more rapidly, while those starting further away initially explore a broader region of the phase space, characterized by colorbar values between 0.1 and 0.2. In this intermediate region, the system\u0026rsquo;s dynamics may exhibit transient oscillations or slower convergence toward a narrower region, marked by higher colorbar values. Eventually, all trajectories are drawn toward a slender region with a colorbar value of 1, corresponding to the equilibrium point. These findings confirm that the H-H neuron in the silence state is governed by a single fixed-point attractor, ensuring that, regardless of initial conditions within \u003cem\u003eΩ\u003c/em\u003e, all solutions ultimately converge to this equilibrium configuration. Importantly, the absence of unstable periodic solutions is evident as the presence of such solutions would preclude the intermediate states observed in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ea.\u003c/p\u003e\n \u003cp\u003eIn the case of the firing state (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eb), the system\u0026rsquo;s trajectory ultimately converges to a periodic solution, regardless of the initial conditions within the state space \u003cem\u003eΩ\u003c/em\u003e. This indicates the neuron\u0026rsquo;s inherent repetitive spiking behavior in the firing state. Upon further examination, it is evident that solutions from any initial condition within \u003cem\u003eΩ\u003c/em\u003e are initially drawn to subspaces with colorbar values fluctuating between 0.1 and 0.2, representing the transient phase before the system settles into a stable periodic rhythm. Once these subspaces are reached, trajectories converge to the periodic solution, as illustrated in left diagram of Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eb. The time series plots for the membrane potential \u003cem\u003eV\u003c/em\u003e and the potassium ion channel activation variable \u003cem\u003en\u003c/em\u003e, shown in right diagram of Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eb, further elucidate the system\u0026rsquo;s dynamics over time. The green background in the time series indicates the stable periodic solutions, while the phase trajectories in the \u003cem\u003eV-n\u003c/em\u003e plane provide a geometric representation of the system\u0026rsquo;s oscillatory behavior. These results suggest that the repetitive firing behavior of the H-H neuron model, induced by external currents ranging from 9.775 mA to 154.485 mA, is governed by a single periodic attractor which dictates the system\u0026rsquo;s long-term periodic spiking behavior.\u003c/p\u003e\n \u003cp\u003eIn the Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ec we highlighting the coexistence of both a fixed-point attractor and a periodic attractor. Depending on the initial conditions, trajectories within \u003cem\u003eΩ\u003c/em\u003e may converge to either attractors, reflecting the system\u0026rsquo;s complex long-term behavior under different initial states. The evolution of the system is influenced by the location of the initial state within the attraction domains of these attractors. The boundaries of these attraction domains are also delineated in left diagram of Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ec. To offer a more intuitive representation of the evolution of solutions from different attraction domains, phase trajectories initiated from various initial values in the \u003cem\u003eV\u003c/em\u003e-\u003cem\u003en\u003c/em\u003e plane are shown in the middle and right diagrams of Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ec. These observations underscore the critical role of both attractors in governing the system\u0026rsquo;s behavior in the bistable state. The fixed-point attractor corresponds to a stable equilibrium, while the periodic attractor represents the system\u0026rsquo;s oscillatory behavior. The coexistence of these two attractors leads to rich dynamic behavior, marking the system highly sensitive to initial conditions and showcasing typical bistable characteristics.\u003c/p\u003e\n \u003cp\u003ePreceding analysis suggest a dynamic explanation for the abrupt emergence of periodic solutions at an external current of 6.264 mA. This value marks a critical threshold, above which the input energy becomes sufficient to sustain periodic solutions. Below this threshold, the input energy is inadequate, with the system transitioning toward the equilibrium point, as shown in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ea. As the external current exceeds this critical value, further increases in input energy lead to a gradual contraction of the periodic solution toward the equilibrium point. At an external current of 154.485 mA, the periodic solution fully converges into the equilibrium point. This dynamic behavior highlights a fundamental distinction in the mechanisms governing state transitions in the H-H neuron model. Specifically, the transition from silence to firing is characterized by a critical \u0026quot;jump\u0026quot; to periodic solutions, whereas the reverse transition involves a gradual collapse of the periodic solution into the equilibrium state. This asymmetry underscores the intricate interplay between energy input and system dynamics in shaping the transitions between different states.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"4. Conclusion","content":"\u003cp\u003eThis study provides a comprehensive exploration of the global dynamics of the H-H model, offering novel insights into the mechanisms that govern neuronal firing patterns. By employing the cell mapping method, we have uncovered a range of dynamical behaviors, including bistability, silent, and repetitive firing states, along with critical bifurcation points that were previously underexplored. Importantly, our findings emphasize that transitions between neuronal states are primarily driven by the appearance and disappearance of attractors, rather than changes in stability. This perspective shifts the understanding of neuronal behavior, underscoring the pivotal role of attractor dynamics in state switching. Additionally, we highlight the potential applications of high-current bistability (154.485\u0026ndash;154.568 mA) in clinical neuromodulation, neuroengineering, and biological protection. Small current adjustments, within this bistable region, can switch neuronal states between therapeutic and non-therapeutic conditions, enable robust transitions between silent and firing state under extreme inputs, and prevent excessive firing or ion channel fatigue under pathological conditions. These findings not only deepen our understanding of neuronal excitability but also provide a foundational framework for future investigations into the complex dynamics of neuronal systems and their potential applications in neuromodulation and neuroengineering.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eFunding Declaration\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis work was supported by the National Natural Science Foundation of China (12402051), and the Natural Science Foundation of Inner Mongolia Autonomous Region of China (2024QN01002, 2024MS01006).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThanks to the support of \"Inner Mongolia Academician Workstation for New Energy Intelligent Equipment and Operation Maintenance in IMUT\" for this research.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eHodgkin, A. L., Huxley, A. F.: Action potentials recorded from inside a nerve fibre. \u003cem\u003eNat.\u003c/em\u003e\u003cstrong\u003e114\u003c/strong\u003e, 710-711 (1939). https://doi.org/10.1038/144710a0\u003c/li\u003e\n\u003cli\u003eHodgkin, A. L., Huxley, A. F.: Resting and action potentials in single nerve fibres. \u003cem\u003eJ. Physiol.\u003c/em\u003e\u003cstrong\u003e104\u003c/strong\u003e, 176-195 (1945). https://doi.org/10.1113/jphysiol.1945.sp004114\u003c/li\u003e\n\u003cli\u003eHodgkin, A. L., Huxley, A. F., Katz, B.: Measurement of current-voltage relations in the membrane of the giant axon of Loligo. \u003cem\u003eJ. Physiol.\u003c/em\u003e\u003cstrong\u003e116\u003c/strong\u003e, 428-448 (1952). https://doi.org/10.1113/jphysiol.1952.sp004716\u003c/li\u003e\n\u003cli\u003eHodgkin, A. L., Huxley, A. F.: Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. \u003cem\u003eJ. Physiol.\u003c/em\u003e\u003cstrong\u003e116\u003c/strong\u003e, 449-472 (1952). https://doi.org/10.1113/jphysiol.1952.sp004717\u003c/li\u003e\n\u003cli\u003eHodgkin, A. L., Huxley, A. F.: The components of membrane conductance in the giant axon of Loligo. \u003cem\u003eJ. Physiol.\u003c/em\u003e\u003cstrong\u003e116\u003c/strong\u003e, 473-496 (1952). https://doi.org/10.1113/jphysiol.1952.sp004718\u003c/li\u003e\n\u003cli\u003eHodgkin, A. L., Huxley, A. F.: The dual effect of membrane potential on sodium conductance in the giant axon of Loligo. \u003cem\u003eJ. Physiol.\u003c/em\u003e\u003cstrong\u003e116\u003c/strong\u003e, 497-506 (1952). https://doi.org/10.1113/jphysiol.1952.sp004719\u003c/li\u003e\n\u003cli\u003eHodgkin, A. L., Huxley, A. F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. \u003cem\u003eJ. 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C., et al.: Parameter identification of dynamical systems based on short-term prediction by the generalized cell mapping method with deep learning. \u003cem\u003eNonlinear Dyn.\u003c/em\u003e\u003cstrong\u003e113\u003c/strong\u003e, 4031-4044 (2025). https://doi.org/10.1007/s11071-024-09943-8\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Nonlinear dynamics, Hodgkin-Huxley model, Cell mapping method, Attractor evolution, Bifurcation","lastPublishedDoi":"10.21203/rs.3.rs-7322686/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7322686/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe Hodgkin-Huxley (H-H) model remains a cornerstone in the study of neuronal dynamics, yet a global understanding of its behavior, particularly regarding transitions between silent, bistable, and repetitive firing states, has been limited. In this study, we employ the cell mapping method to systematically investigate the global dynamics of the H-H model under varying external currents \u0026mdash;\u0026mdash; an approach that overcomes the limitations of traditional local analysis methods. Our analysis reveals two critical novel findings, a previously unrecognized bistable regime at high currents and the \u0026ldquo;reversion to silence\u0026rdquo; phenomenon where the neuron returns to a silent state under extremely high currents. The results indicate that the model\u0026rsquo;s states are governed by the emergence and disappearance of distinct attractors as external current varies, with state transitions exhibiting asymmetric dynamics. Additionally, the system exhibits extreme sensitivity to initial conditions, particularly in bistable regimes. These findings advance our understanding of neuronal excitability, with implications for neuroengineering and clinical neuromodulation, and offer a robust framework for exploring global dynamics in nonlinear neuronal systems.\u003c/p\u003e","manuscriptTitle":"Dynamical mechanism of neuronal firing in the Hodgkin-Huxley model","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-27 16:49:11","doi":"10.21203/rs.3.rs-7322686/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"9d1f7684-d92c-41d3-aebf-75c4e0cabc4d","owner":[],"postedDate":"August 27th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-09-19T18:08:28+00:00","versionOfRecord":[],"versionCreatedAt":"2025-08-27 16:49:11","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7322686","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7322686","identity":"rs-7322686","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}