Adding the latency period to a muscle contraction model coupled to a membrane action potential model

Introduction: Skeletal muscle is responsible for multiple functions for maintaining energy homeostasis and daily activities. Muscle contraction is activated by nerve signals, causing calcium release and interaction with myofibrils. It is important to understand muscle behavior and its impact on medical conditions, like in the presence of some diseases and their treatment, such as cancer, which can affect muscle architecture, leading to deficits in its function. For instance, it is known that radiotherapy and chemotherapy also have effects on healthy tissues, leading to a reduction in the rate of force development and the atrophy of muscle fibers. The main aim is to reproduce the behavior of muscle contraction using a coupled model of force generation and the action potential of the cell membrane, inserting the latency period observed between action potential and force generation in the motor unit.

Methods: Mathematical models for calcium dynamics and muscle contraction are described, incorporating the role of calcium ions and rates of reaction. An action potential initiates muscle contraction, as described by the Hodgkin–Huxley model. The numerical method used to solve the equations is the forward Euler method.

Results and Discussion: The results show dynamic calcium release and force generation, aligning with previous research results, and the time interval between membrane excitation and force generation was accomplished. Future work should suggest simulating more motor units at the actual scale for the possibility of a comparison with real data collected from both healthy individuals and those who have undergone cancer treatment.

1 Introduction

Skeletal muscle is an efficient and adaptable tissue responsible for multiple functions for maintaining energy homeostasis and daily activities [1]. It is composed of blood vessels, connective tissues, and muscle cells known as muscle fibers. A muscle fiber is a long and slender cell whose primary component is the myofibril. Myofibrils play a central role in initiating muscle cell contractions due to the presence of two vital filament types: actin and myosin [2].

Muscle fibers produce force through electrical activation by the nervous system, allowing for muscle contraction and, consequently, motion (Peterson and Bronzino [3]). The interaction between a nerve and a muscle fiber is known as a synapse, and the entire process is initiated by the arrival of an action potential. A motor unit (MU) comprises all the muscle cells controlled by a single nerve fiber. The nerves responsible for controlling muscles are known as motor neurons [2].

Motor units are controlled through synchronous recruitment by the central nervous system. This type of recruitment, as shown in Figure 1, recruits one unit at a time to maintain constant tension and, if necessary, recruits different motor units simultaneously to generate greater tension in the muscle [4]. Farina et al. [5] stated that recruitment begins with the smallest muscle fibers, which typically exhibit the lowest conduction velocity. The controlled activation of motor unit populations accomplishes movement, as described in [6], which was based on the understanding of motor unit physiology through computational and experimental studies. However, muscles do not develop tension immediately; instead, there is a brief period known as the latency period before tension is generated [2].

FIGURE 1

FIGURE 1. Tension developed in each MU, resulting in a constant tension in the muscle.

The contractile system of skeletal muscles is regulated by ions Ca²⁺, which are stored in the sarcoplasmic reticulum (SR). It was stated in [2] that the release of ions Ca²⁺ into the cytoplasm of muscle cells is triggered by the arrival of an action potential at the motor end plate, followed by neurotransmitter release and uptake by the muscle cells. When Ca²⁺ is removed from the cytoplasm, the contraction process ceases, and the muscle returns to its initial length. The net effect of a single action potential results in a transient contraction of the motor unit, commonly referred to as a twitch.

Studying skeletal muscle and its physiological functioning can be important for understanding its behavior in medical conditions, such as muscle fatigue [8,9], bone diseases [10,11], muscular damage [12], dystrophies [13], and diseases of treatment sequelae [14–19]. Commonly used cancer treatments, including chemotherapy [20–22] and radiation therapy [23], affect skeletal muscle, inducing a decrease in the rate of force development and loss in muscle fibers, which leads to a change in muscle function and promotion of its inflammation [17]. It was stated in [18] and [19]that cancer survivors experience several treatment-related symptoms, muscular weakness, and reduced mobility, thereby compromising their quality of life. Furthermore, after pelvic radiotherapy, the exposure of the anal canal and nerves of the sacral plexus to radiation may be associated with the deterioration of sphincter function [24], and changes are observed in the composition of the pelvic floor muscle structure, which was maintained even after 4 years of treatment for prostate or colorectal cancer [25]. In addition, functional modifications in the pelvic floor have been reported, such as reduced pressures at rest and during maximum contraction after radiotherapy, for up to 1 year after treatment [26]. It was suggested in [23] that ionizing irradiation leads to a reduction in the perimeter and contractility of muscle fibers as well as a lower amount of skin fiber renewal. Late effects of radiotherapy include gastrointestinal, urological, female reproductive tract, skeletal, and vascular toxicity, secondary malignancies, and quality-of-life issues [27–29].

Recent studies focused on the association between skeletal muscle quality and the prognosis of patients with gynecological cancer [14–16,30]. In [15], it was shown that maintaining muscle mass can prolong survival in cancer patients, and the work in [16] related that visceral obesity before radiotherapy and chemotherapy has a protective effect on the prognosis of patients with stage IVB cervical cancer, while a low muscle index and low visceral-to-subcutaneous adipose tissue area ratio are associated with worse prognosis. Research studies also identified a low pretreatment skeletal muscle index as a prognostic factor for overall survival in patients diagnosed with cervical or ovarian cancer [14,31]. On the other hand, in [30], data from the skeletal muscle area were reviewed to identify skeletal muscle mass loss (sarcopenia) in patients with cervical, endometrial, and ovarian cancers, appointing that the limited literature data seem to suggest that baseline muscle indexes have an uncertain prognostic pertinence, whereas their changes during treatment often correspond with chances of patient survival. Although a novel sarcopenia measure combining quantity and quality of muscle is important to spread the basis to explain the relationship between sarcopenia and solid tumor aftereffects considering high-risk patients [31], there is a lack of studies that analyze muscle changes during cancer treatments, which might be justified by the discrepancy in measurement methods of muscle depletion across research studies [17].

The overall electrical activity of skeletal muscles can be measured by electromyography (EMG). However, EMG signals are difficult to interpret since they are controlled by the nervous system and are dependent on the anatomical and physiological properties of muscles [32]. Additionally, for breast cancer patients, surgery and radiation therapies impact shoulder muscle health throughout changes in muscle morphology and neuromuscular function. Notwithstanding, besides the conflicting results, EMG amplitudes obtained during motion activities demonstrate that the neuromuscular strategy and control may be dependent on the treatment received [33]. Mathematical and computational modeling can help investigate the characteristics of EMG signals and test their accuracy and validity. Mathematical modeling allows the estimation of parameters that are not directly accessible for measurements, for example, related to the description of the spatial and temporal recruitment of motor units [34]. Moreover, it can help in developing tools to measure the force developed by a muscle [35].

The objective here is to reproduce the behavior of muscle contraction using a coupled model of force generation and the action potential of the cell membrane, associated with Ca²⁺ regulation, inserting the latency period observed between the action potential in the membrane and force generation in the motor unit.

2 Materials and methods

2.1 Calcium dynamics and muscle contraction models

The mathematical model of calcium dynamics and muscle contraction was based on the work in [36] that appears in the work in [37] and [38], which uses simple mass action kinetics to describe calcium dynamics in the muscle. The model is shown in Figure 2. It describes the relationship between concentrations of free calcium ions [c], unbound SR calcium binding [s], unbound contractile filaments (CFs) calcium binding [f], calcium-bound SR sites [cs], and calcium-bound CF sites [cf].

FIGURE 2

FIGURE 2. Model of calcium dynamics. Adapted from McMillen, Williams, and Holmes [36] licensed Creative Commons Attribution 4.0 International.

The rates at which reactants act are represented by parameters k_i. The rates k₁ and k₂ operate similar to a switch, dependent on the value of the action potential of muscle V_m, as shown by the following equations:

$k_1\left(V_m\left(t-T\right)\right)=\left\{\begin{array}{cc}{k}_1,\hfill & \text{if\hspace{0.17em}}{V}_m\left(t-T\right)>V_{\mathit{min}}\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}\right.,(1)$

$k_2\left(V_m\left(t-T\right)\right)=\left\{\begin{array}{cc}{k}_2,\hfill & \text{if\hspace{0.17em}}{V}_m\left(t-T\right)<V_{\mathit{min}}\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}\right.,(2)$

where V_m is the action potential in the membrane, V_min is its minimum value to activate the contraction process, and T is the latency period (in ms) between the action potential in the membrane and the onset of contraction. The inclusion of delay was implemented by a delay-differential equation (DDE), as evaluated in [39], which achieved dynamics similar to the original FitzHugh–Nagumo and Hodgkin–Huxley models using a single DDE formulation in each case.

When a muscle is activated, k₁ represents the rate constant for the release of calcium from the SR and k₂ = 0. When the muscle is not activated, k₁ = 0 and k₂ represent the rate constant for the binding of calcium to the SR. Likewise, the rate of binding of calcium-bound CF sites is proportional to the concentration of both free calcium ions and unbound calcium-binding sites with rate constant k₃. The reversible process occurs with rate constant k₄, and it is proportional to the concentration of both bound and unbound CF sites. In [38], it was explained that it was necessary to introduce some cooperativity in the release of calcium so that the relaxation process does not begin abruptly. Although different from the relationship of k₁ and k₂, k₃ and k₄ can both be non-zero at the same time. The differential equations of calcium dynamics are as follows:

$\frac{d\left[c\right]}{dt}=k_1\left[cs\right]-k_2\left[c\right]\left[s\right]-k_3\left[c\right]\left[f\right]+k_4\left[cf\right]\left[f\right],(3)$

$\frac{d\left[cs\right]}{dt}=-k_1\left[cs\right]+k_2\left[c\right]\left[s\right],(4)$

$\frac{d\left[s\right]}{dt}=k_1\left[cs\right]-k_2\left[c\right]\left[s\right],(5)$

$\frac{d\left[cf\right]}{dt}=k_3\left[c\right]\left[f\right]-k_4\left[cf\right]\left[f\right],(6)$

$\frac{d\left[f\right]}{dt}=-k_3\left[c\right]\left[f\right]+k_4\left[cf\right]\left[f\right].(7)$

It is assumed the total number of calcium ions (C), SR-binding sites (S), and filament-binding sites (F) remains constant, following mass conservation laws:

$\left[c\right]+\left[cf\right]+\left[cs\right]=C,(8)$

$\left[s\right]+\left[cs\right]=S,(9)$

$\left[f\right]+\left[cf\right]=F.(10)$

Combining the differential equations from mass action kinetics and mass conservation laws yields

$\begin{array}{cc}\hfill \frac{d\left[c\right]}{dt}& =k_1\left(C-\left[c\right]-\left[cf\right]\right)-k_2\left[c\right]\left(S-C+\left[c\right]+\left[cf\right]\right)-k_3\left[c\right]\left(F-\left[cf\right]\right)\hfill \\ \hfill & +k_4\left[cf\right]\left(F-\left[cf\right]\right)\hfill \end{array},(11)$

$\frac{d\left[cf\right]}{dt}=k_3\left[c\right]\left(F-\left[cf\right]\right)-k_4\left[cf\right]\left(F-\left[cf\right]\right).(12)$

Accordingly to the work in [3], there are three general classes of models for predicting muscle force: biochemical models, constitutive models, or Hill’s models. In [36], Hill’s model based on the work in [40] was used, which describes the muscle as a contractile element in series with a linearly spring element, as shown in Figure 3. The model says that the total length of muscle corresponds to the length of the contractile element l_c plus the length of the linearly spring element l_s:

$L=l_c+l_s.(13)$

FIGURE 3

FIGURE 3. Simplified Hill's model for predicting muscle force.

According to Hooke’s law, the total force in a linearly elastic body is proportional to the final length of that body minus the initial length:

$P_s={\mu }_s\left(l_s-l_{s_0}\right),(14)$

where P_s is the applied force, l_s is the final length, $l_{s_0}$ is the initial length, and the proportionality constant μ_s is Young’s modulus, or stiffness constant. In Hill’s model, that stiffness varies when muscle exerts force from total relaxation.

${\mu }_s={\mu }_0+{\mu }_1\left[cf\right].(15)$

Combining Eq. 13 with Eq. 14 and isolating l_c yield

$l_c=L-\frac{P_s}{{\mu }_s}+l_{s_0}.(16)$

Taking the time derivative of Eq. 16 yields

$v_c=V\left(t\right)-\frac{d{P}_s}{dt}\frac{1}{{\mu }_s}+\frac{{\mu }_1{P}_s}{{{\mu }_s}^2}\frac{d\left[cf\right]}{dt},(17)$

where v_c and V(t) are the time derivatives of l_c and L, respectively.

Assuming that the muscle performs an isometric contraction, its total length L(t) is constant, and time derivative V(t) is null.

As assumed in [36], the applied force on the contractile element (P_c) is proportional to independent multiplicative factors of its length (l_c) and velocity (v_c):

$P_c=P_0\lambda \left(l_c\right)\alpha \left(v_c\right)\left[cf\right].(18)$

Furthermore, dividing Eq. 18 by P₀ provides a non-dimensional value for P_c, as was carried out in [38].

$P_c=\lambda \left(l_c\right)\alpha \left(v_c\right)\left[cf\right].(19)$

The functions λ(l_c) and α(v_c) were measured in [36], which provided a linear function for α and a quadratic function for λ as follows:

$\alpha \left(v_c\right)=1+\left\{\begin{array}{cc}{\alpha }_m{v}_c\hfill & \text{if\hspace{0.17em}}{v}_c<0\hfill \\ {\alpha }_p{v}_c\hfill & \text{if\hspace{0.17em}}{v}_c\ge 0\hfill \end{array}\right.,(20)$

$\lambda \left(l_c\right)=1+{\lambda }_2{\left(l_c-l_{c0}\right)}^2.(21)$

These functions are restricted such that 0 ≤ α(v_c) ≤ α_max and 0 ≤ λ(l_c) ≤ 1. According to [3], when muscle performs concentric contractions, i.e., the shortening of fiber muscle occurs, the relationship between force and velocity is nearly hyperbolic and relatively lower than when it performs eccentric contractions (lengthening of fiber muscle). This fact reflects α_p > α_m > 0.

In a steady state, P_s and P_c must be equal. So, the transfer of force from the contractile element to the spring element was modeled by simple linear kinetics:

$\frac{d{P}_s}{dt}=k_5\left(P_c-P_s\right),(22)$

where k₅ is a selected parameter to approximate P_c to P_s.

To prevent instability, Eq. 17 was combined with Eq. 19 and Eq. 20:

$P_c=\lambda \left(1-\frac{d{P}_s}{dt}\frac{\alpha }{{\mu }_s}+\frac{\alpha {\mu }_1{P}_s}{{{\mu }_s}^2}\frac{d\left[cf\right]}{dt}\right)\left[cf\right],(23)$

$\alpha \left(v_c\right)=\left\{\begin{array}{cc}{\alpha }_m\hfill & \text{if\hspace{0.17em}}{v}_c<0\hfill \\ {\alpha }_p\hfill & \text{otherwise}.\hfill \end{array}\right..(24)$

Finally, Eq. 23 was combined with Eq. 22, and isolating the time derivative $\frac{dP}{dt}$, the model of muscle force was obtained:

$\frac{d{P}_s}{dt}=\frac{\lambda \left[cf\right]\left(1+\frac{\alpha {\mu }_1}{{\mu }_s^2}\frac{d\left[cf\right]}{dt}\right)-P_s}{\frac{1}{k_5}+\frac{\lambda \alpha \left[cf\right]}{{\mu }_s}}.(25)$

The constants used for calcium dynamics and muscle contraction models are given in Table 1. The value of V_min was arbitrarily chosen by the authors to represent a minimum threshold that must be reached for the contraction to actually occur. In the future, it is possible to compare it with real data to adjust this parameter. Although the model used here shares its foundation with the model in [36], it introduces a previously unaccounted latency period (T). As explained in [2], this latency period represents the delay between the arrival of the action potential and the release of Ca⁺² in the muscle cell, typically averaging between 3 and 10 ms. Thus, the value of T was derived from the established literature.

TABLE 1

TABLE 1. Input parameters for the action potential model.

2.2 Action potential model

The contraction in a motor unit begins with an action potential reaching the motor end plate and neurotransmitters being released in the synaptic cleft. This causes Ca²⁺ to be released into the cytoplasm of the muscle cell. In [36] and [38], a high-frequency sequence of individual stimuli, called tetanic stimulus [38], was used, while in [37], stimuli were created based on square and exponential functions to represent tetanic stimulus and individual electric impulse.

Here, the stimulus in the cell membrane that activates the calcium dynamic is the action potential from the model described by [41], explained in the Appendices. The Hodgkin–Huxley model was chosen due to its accurate representation of the action potential, as well as being a simpler model (with only four differential equations) compared to more recent ones. Considering the potential for simulating multiple motor units, larger models become computationally expensive. The FitzHugh–Nagumo model was also assessed for having only two equations; however, its representation is less faithful than that of the Hodgkin–Huxley model [42]. Moreover, many recent studies were based on the formalism of the Hodgkin–Huxley model, while many others employ an even simpler formulation based on a transfer function, as indicated in [43]. As shown in [37], the stimulus is responsible for changing the rates k₁ and k₂.

2.3 Numerical method

The numerical method applied to solve Eqs 11, 12, 25, A1–A–A4 was Euler’s method, or forward Euler, which replaces the derivative term by the approximation presented in the following equation [44]:

$\frac{dU}{dt}=\frac{U^{i+1}-U^i}{k},(26)$

where U is the variable of interest, k is the discretization time step, and i indicates which time step the variable U is in. This way, it was possible to approximate variable U in time i + 1, starting from the time t = 0, in which the state of the variables is known. So, starting with a relaxed muscle, the membrane is at rest (without an action potential), all calcium is in the SR, and, consequently, there is no force developed. Thus, Table 2 shows the initial conditions used in this paper. The conditions were chosen based on the idea that there is no free calcium or calcium bound to filaments initially, and the cell membrane is at rest. The value of V_m was chosen according to the work in [45] and the variables m, n, and h according to the work in [42].

TABLE 2

TABLE 2. Initial conditions for each variable.

Compared to the backward Euler and Crank–Nicolson model, it is faster, which is great for optimizing time simulations. The system of equations was solved using an algorithm implemented in Python with k = 0.001, and their exploitation about how it was implemented is presented in the following equations.

The ODE refers to free calcium ions:

$\left[c^{i+1}\right]=\left[c^i\right]+k\left[k_1\left(C-\left[c^i\right]-\left[c{f}^i\right]\right)-k_2\left[c^i\right]\left(S-C+\left[c^i\right]+\left[c{f}^i\right]\right)-k_3\left[c^i\right]\left(F-\left[c{f}^i\right]\right)\right].(27)$

The ODE refers to calcium-bound CF sites:

$\left[c{f}^{i+1}\right]=\left[c{f}^i\right]+k\left[k_3\left[c{f}^i\right]\left(F-\left[c{f}^i\right]\right)-k_4\left[c{f}^i\right]\left(F-\left[c{f}^i\right]\right)\right].(28)$

The ODE refers to the force developed by the muscle:

$P_s^{i+1}=P_s^i+k\left[\frac{\lambda \left[c{f}^i\right]\left(1+\frac{\alpha {\mu }_1{P}_s^i}{{\mu }_s^2}\frac{d\left[c{f}^i\right]}{dt}\right)-P_s^i}{\frac{1}{k_5}+\frac{\lambda \alpha \left[c{f}^i\right]}{{\mu }_s}}\right].(29)$

The ODE refers to an action potential in the membrane:

$V_m^{i+1}=V_m^i+k\left(\frac{I_{\mathit{app}}^i-I_{Na}^i-I_K^i-I_L^i}{C_m}\right),(30)$

where

$I_{Na}^{i+1}={\bar{g}}_{Na}{m}^3{h}^i\left(V_m^i-V_{Na}\right),(31)$

$I_K^{i+1}={\bar{g}}_K{n}^4{h}^i\left(V_m^i-V_K\right),(32)$

$I_L^{i+1}={\bar{g}}_L\left(V_m^i-V_L\right).(33)$

The ODE refers to auxiliary variables:

$n^{i+1}=n^i+k\left[{\alpha }_n\left(1-n^i\right)-{\beta }_n{n}^i\right],(34)$

$m^{i+1}=m^i+k\left[{\alpha }_m\left(1-m^i\right)-{\beta }_m{m}^i\right],(35)$

$h^{i+1}=h^i+k\left[{\alpha }_h\left(1-h^i\right)-{\beta }_h{h}^i\right].(36)$

3 Results

The contraction process begins with a command from the nervous system, represented here by applied current I_app, in the first part of Figure 4. When the stimulus is large enough to generate the action potential, calcium is released, leading to the contraction of the motor unit.

FIGURE 4

FIGURE 4. In the first part, the current applied to the Hodgkin–Huxley model is indicated in red, representing the nervous system command to the cell. In the middle, the action potential curve in the cell membrane is denoted in blue, derived from the Hodgkin–Huxley model. In the lower part, indicated in green, the force generation curve in the motor unit temporally shifted due to the latency period.

As shown in blue in Figure 4, where current I_app has two consecutive stimuli before 25 ms, the Hodgkin–Huxley model has a refractory period that prevents another action potential from occurring while one was already being developed. Consequently, there was no force generation due to this second current stimulus. However, stimuli applied after the refractory period generate an action potential and, after the latency period, force in the motor unit once again. Since it was set that calcium would be released starting at a value of −20 mV, the 10-ms latency period was observed from this point. The selection of different values for V_min would cause the contraction to start earlier or later, depending on the value. Here, the value was arbitrary to represent the threshold of the muscle cell.

Figure 5 illustrates the dynamic initiated by an action potential; first, all the calcium ions are stored in the SR, and once the potential is generated, they are released and bind to the filaments. After some time, the process is reversed, and all the calcium returns to SR.

FIGURE 5

FIGURE 5. Curves representing the dynamics of calcium release and reuptake by the SR, the presence of free calcium, and calcium bound to filaments during the contraction process initiated by the action potential of the membrane.

Finally, Figure 6 shows that a muscle stimulus was simulated using two motor units to verify the total tension generated in the muscle. It can be considered that when a high load is demanded in major time, lots of motor units are stimulated, resulting in a higher or longer-lasting tension developed in the muscle.

FIGURE 6

FIGURE 6. The first part shows the action potential curves of two different motor units being stimulated. The second part shows the forces developed in both. The third part shows the combined force of both, representing how it would be in a muscle.

4 Discussion

A simulation coupling the Hodgkin–Huxley action potential model to the models demonstrated in [36], [38], and [37] was presented, adding the latency period between the stimulus and muscle contraction to approximate the model to the real behavior of the muscle. Although the Hodgkin–Huxley model is relatively old, it remains a significant global reference and is integrated into numerous research studies across the vast field of the electrophysiology community. Despite its use, certain biological processes such as the activation of ion channels in the SR by the entry of calcium for subsequent release were simplified by delay T because that model does not include those processes.

The control over the rates was also different from that shown in [37] because here, the Hodgkin–Huxley model was used to determine the nervous stimulus associated with the release and resorption of calcium by the SR. Therefore, the Hodgkin–Huxley model introduced the dynamics of the sodium and potassium ions as electrical current components, along with the calcium dynamics in the model, which is an important component of the skeletal muscle contraction behavior [46].

The focus of this study was on muscle behavior, simulating reduced-size motor units. Nevertheless, it paves the way to associate and distinguish the influence of electrical stimuli and calcium ion dynamics on healthy muscle contraction. The question remains as to whether alterations in muscle contraction dynamics during certain illnesses are attributed to changes in electrical conduction or disruptions in calcium dynamics. Hence, future efforts should involve comparing results obtained from full-sized scenarios with a higher number of motor units, as well as incorporating data collected from both healthy individuals and those who have undergone cancer treatment.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author contributions

NK, AG, GL, RW, SV, and FB: writing–original draft. AG and GL: Software.

Funding

The authors declare financial support was received for the research, authorship, and/or publication of this article. This work was supported by the Brazilian research funding agencies FAPEMIG, CAPES, CNPq, and UFJF. CAPES - Processo 88881.708850/2022-0.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Appendix A

Hodgkin–Huxley model

The conductance-based model describes the potential using the currents passing through the membrane. Figure A1 shows the electrical circuit that represents this phenomenon, which presents a capacitor representing the membrane (C_m), resistors and sources to represent the ion channels (potassium V_K, sodium V_Na, and others V_L), and an applied current to indicate the stimulus from the nervous system (I_app).

FIGURE A1

FIGURE A1. Electric circuit representing the cellular membrane, as described in [41].

Kirchhoff’s first law [47] was applied to the circuit shown in Figure A1 to determinate Eq. A1, which describes the action potential in the membrane, where ${\bar{g}}_{Na}$, ${\bar{g}}_K$, and ${\bar{g}}_L$ are the conductance of the channels of sodium, potassium, and other ions, V_Na, V_K, and V_L are the potential differences of these channels, and C_m is the membrane conductance. Eqs A2–A4 represent the auxiliary variables n, m, and h, as described in [41]. These equations use alpha (α) and beta (β) functions, as also described in [41], which are given in Eqs A5–A10. All constants used for the membrane action potential model are given in Table A1.

$\frac{d\left[V_m\right]}{dt}=\frac{1}{C_m}\left[I_{\mathit{app}}-{\bar{g}}_k{n}^4\left(V_m-V_k\right)-{\bar{g}}_{Na}{m}^{3}h\left(V_m-V_{Na}\right)-{\bar{g}}_l\left(V_m-V_l\right)\right],(\mathrm{A1})$

$\frac{dn}{dt}={\alpha }_n\left(1-n\right)-{\beta }_{n}n,(\mathrm{A2})$

$\frac{dm}{dt}={\alpha }_m\left(1-m\right)-{\beta }_{m}m,(\mathrm{A3})$

$\frac{dh}{dt}={\alpha }_h\left(1-h\right)-{\beta }_{h}h,(\mathrm{A4})$

${\alpha }_n=\frac{0.01\left(10-V_m\right)}{e^{\frac{10-V_m}{10}}-1},(\mathrm{A5})$

${\beta }_n=0.125e^{\frac{-V_m}{80}},(\mathrm{A6})$

${\alpha }_m=\frac{0.1\left(25-V_m\right)}{e^{\frac{25-V_m}{10}}-1},(\mathrm{A7})$

${\beta }_m=4e^{\frac{-V_m}{18}},(\mathrm{A8})$

${\alpha }_h=0.07e^{\frac{-V_m}{20}},(\mathrm{A9})$

${\beta }_h=\frac{1}{e^{\frac{-V_m}{10}}+1}.(\mathrm{A10})$

TABLE A1

TABLE A1. Input parameters for the action potential model.