Introduction

A living cell has an electrical potential difference between its inside and outside. This voltage difference is referred to as the membrane potential (hereafter, MP), and it is generally negative1. The outside of the membrane is charged with positive ions (mainly Na cations) and its inside charged with negative ions (organic anions). This negative charge of inside is partly compensated by positive charge, mainly K cations in the cell. When a cell is at rest, the MP is called the resting potential. The resting potential changes from cell to cell and can range from − 80 mV to -40 mV2, even it varies quite a bit more than this range. For example, the red blood cells have a resting membrane potential of -10 mV, although they are non-excitable cells but sudden changes in the MP occur3. On the other hand, a typical nerve cell has a resting potential of about − 70 mV though this varies. The resting potential is generally constant in case of absence of any stimulation. In other words, the resting potential is the energy reservoir for the cells which supplies for the occurrence of electrical signals. It is also described as a standard reference point for other potentials of the cell membranes. The cell membrane is composed of lipid bilayer which behaves in much as a leaky capacitor. The lipid bilayer is not permeable to ions and it keeps ions from easily crossing. However, the membrane contains channel proteins that allow for the ions to move through it1,2,4,5.

Two types of ion channel are observed in a membrane; namely gated and nongated channels. The gated channels can open and close depending on the MP, whereas nongated channels are always open. The gated channels are typically selective for a single ion and referred to voltage-gated channels. The membrane permeability to a specific ion depends on the number of selective open channels. Under normal conditions, nearly all gated channels are closed at rest. In this case, the nongated channels are firstly responsible for creating the resting potential. In order to generate an action potential, the gated channels should be open for allowing the ion flux across the membrane. When the relevant channels are open, Na cations and Cl anions tend to diffuse into the cell, oppositely K cations show a tendency to diffuse outward due to concentration differences; of course, the energy is not required to maintain this tendency. However, a cell has the capacity to experience depolarization process by which the MP gets less negative. After that, the repolarization process starts. In this case, an outward stream corresponds to a cation (i.e., K+) leaving the cell and/or an opposite stream refers to an anion (i.e., Cl−) entering the cell. In such a case, the cell gets hyperpolarized (until − 90 mV or more negative: -105 mV). This means the hyperpolarization process results in an overshoot in the cell potential1,6. Unhappily, some mutations of ion channels may cause channelopathies such as migraine, epilepsy, diabetes, hypertension, cardiac arrhythmia, hyperekplexia, blindness, asthma, deafness, cancer, and irritable bowel syndrome7,8.

As reported by Ermentrout & Terman1numerous excellent studies have been carried out to give detailed information on the mathematical aspects of cellular biophysics. For example, John Carew Eccles, Alan Lloyd Hodgkin, and Andrew Fielding Huxley shared the 1963 Nobel Prize in Physiology/Medicine for their fundamental discoveries on the ionic mechanisms of the nerve cells. As is known very well, a stimulus is anything that can trigger a behavioral and/or physical change to begin. An action potential at the cell membrane can be generated in response to a stimulus. We assume the midsection of selectivity filter of ion channels within the membrane to be a hypothetical plane indicating the existence of an entry potential (\(\:\zeta\:\)). This is the first study to prove that the \(\:\zeta\:\) potential at the selectivity filter of ion channels can be used as a benchmark to estimate a threshold voltage of stimulus which is enough strength of a stimulant to depolarize the cell membrane to the threshold level. Although this paper is a purely simulation/hypothetical study, the literature data support our predictions that the stimulus voltage is modulated by resting potential. In other words, this new approach may have a bias, but it provides a rough guideline at best. The focus on the potential drop in voltage-dependent ion channels of lipid bilayer of a cell membrane is an important and novel approach. While most studies focus on the action potential in the excitable cells, understanding the potential drop in the voltage-dependent ion channels is critical for mitigating channelopathies in the cases of emergency.

The equations of electro-kinetic processes have been extensively used in the past to study transport phenomena through ion channels. There are many well-executed publications that present the transport aspects covered in this paper in much more detail3,9,10,11,12. The electro-kinetic phenomena include various processes such as electrophoresis, electro-osmosis, streaming potential, sedimentation potential, etc. Many studies on electro-kinetic phenomena are typically carried out using electro-kinetic equations having applications as diverse as agriculture, medical science, industrial chemistry, and biotechnology. In this context, electrophoresis and electro-osmosis techniques are frequently used by researchers to understand migration/behavior of charged particles. In other words, migration of such particles can experimentally be determined by electro-kinetic techniques. As is known, the particle behavior depends on charge density, size/density/shape of the suspended particles, ion species, electrolyte concentration, pH, temperature, viscosity, and dielectric permittivity of the medium. Hereafter, it is better to focus on the function/activity of ion channels within a cell membrane. In this regard, the transport of ions across cellular and sub-cellular membranes is a biological issue of the first magnitude13. The use of Hückel–Onsager equation to estimate stimulus voltage for a cell is sound. The presented approach in the study is easy to follow and effectively communicates the dynamics of voltage-dependent ion channels, and provides a scientific basis for electrical stimulation contributing to the pre-treatment of some channelopathies in the emergency cases. For example, the outputs of this study would offer practical measures for intervention to the epileptic patients. In other words, the determination of stimulus voltage would have potential implications for reducing the severity of epileptic seizures, particularly in the context of refractory status epilepticus in emergency departments/intensive care units.

Methods

As mentioned before, when a cell is depolarized, the Na+ channels are open and allow more Na cations to enter the cell (Fig. 1a). This state is the upstroke of the action potential. Eventually, the Na+ channels switch from open to close state before the depolarization is turned off. It is important to know that a Na+ channel has two gates (Fig. 1b): an activation gate (the fast one) and an inactivation gate (the slow one). Therefore, the flow of Na cations is transient. In brief, a spike of 105 mV (from − 70 to + 35) is usually generated by a temporary increase in Na cations current14. Meanwhile, the depolarization opens K+ channels; then, repolarization occurs and K cations exit the cell. Even further, this process hyperpolarizes the cell. The cell membrane is refractory until the voltage-gated K+ channels close up once again. In summary, inward stream of Na cations depolarizes cells, whereas outward currents of K cations hyperpolarize1,15. Here, definitions of the membrane potentials with respect to the resting potential would help non-expert readers to understand the process (Fig. 1c).

For typical negative MP, a positive work is necessary to push a cation from the interior to the exterior. However, the fluxes of permeating ions are determined by both their concentration gradient and the electrical potential across the membrane16. At the threshold potential, the outward and inward currents of K and Na cations, respectively, are opposite and equal. This means a change in local charge separation (not in concentration) is required for an action potential. In other words, any changes in current must be due to either the leak or the opening and closing of voltage-gated membrane channels. In order to determine the effects of ion concentration on cell potential, the Nernst equation (which calculates the equilibrium potential of one type of ions) is used1.

On the other hand, the ion channels can be described as narrow and water-filled tubes through the lipid bilayer or the membrane. Therefore, we assume that the electrophoretic mechanism may work through the voltage-gated channels and electrical field strength may play a role. As mentioned before, the cell membrane is composed of lipid bilayer which behaves in much as a poor capacitor. This means the two sides of the membrane to be the two layers of charge of a simple parallel plate capacitor. In related studies, one can utilize the capacitor formulae of elementary physics to obtain a number of different biophysical variables such as the membrane potential (or voltage difference between inside and outside of a cell), the electric field intensity across voltage-gated ion channels, etc17,18.

Within the scope of colloid and interface science, theoretical applications of electrostatic layers have usually been based on the Gouy-Chapman hypothesis describing a diffuse layer which is formed to compensate the excess charges. In their theory, the negative charges are assumed to be constant as well as to be dissipated uniformly over the particle surfaces19. This picture is similar to cell membrane, in spite of the fact that they originate within the cell.

The inside surface of the membrane together with multiple organic anions may act as a negatively charged plane since the membrane is impermeable to these organic ions. When the voltage-gated channels are open, the neutralizing counterions are attracted to the inside surface (see Figs. 1a and b). In other words, negative surface charge tends to pull the cations inward, and push anions outward. The electrostatic attraction and repulsion forces may create a diffuse layer across the ion channels of the lipid bilayer if the voltage-gated channels are open, although the thickness of the diffuse layer is rather undefined in this paradigm. Here, the electrical potential at the midsection of selectivity filter of ion channels, which is the narrowest part of the channel and next to Na+ entryway from extracellular side, is called as entry potential (\(\:\zeta\:\)). More clearly, the potential drop at the core of selectivity filter has been defined as the \(\:\zeta\:\) potential. Let us comment further on the \(\:{\upzeta\:}\) potential for its theoretical expressions and exercises in a physiological medium. The shape of symbolic potential curve across ion channel reminds Helmholtz’s theory (Fig. 1d). We intended to calculate the \(\:\zeta\:\)potential at a hypothetical plane in the selectivity filter of ion channels of bilayer using the necessary parameter and/or variable derived from the literature, even if the conditions are not the same with those of the typical electro-kinetic processes.

The most appropriate equations for calculating the potential drop at shear plane are Helmholtz-Smoluchowski (HS) model and Hückel–Onsager (HO) equation. The HS equation is frequently used for large particles whereas the HO model is usually suggested for small particles20,21. Thus, in this study, the Hückel–Onsager equation is better to employ and given as:

$$\:\zeta\:=\frac{3{u}_{e}\eta\:}{2{ \varepsilon }_{r}{ \varepsilon }_{o}}$$
(1)

where \(\:\zeta\:\) is originally called as zeta potential at the shear plane (V). Herein, this term is called as entry potential at a hypothetical plane in the selectivity filter of ion channels. This means that we introduce an electrical potential at the entrance to the port to estimate its effects on channel behavior. We believe that this potential would well play a role in channel activity, but it is just not the zeta potential calculated by the above equation. Going back to the equation; \(\:{u}_{e}\) is the electrophoretic mobility (m2/(V.s)), \(\:\eta\:\) is the viscosity of the medium (Pascal.s), \(\:{ \varepsilon }_{r}\) is the relative permittivity (or dielectric constant) of the medium (dimensionless), and \(\:{ \varepsilon }_{o}\) is the electric permittivity of a vacuum (or free space) (Farad/m). Note that \(\:{\varepsilon}_{o}\) value is about 8.85 × 10− 12 F/m as expressed by Ermentrout & Terma1and \(\:{ \varepsilon }_{r}{\varepsilon}_{o}\) is called as the actual permittivity of the medium, \(\: \varepsilon\) (in F/m). The pore size is in the HO regime, although the HO equation results in a potential that is 1.5 times that of the HS equation for the same electrophoretic mobility value. Using Henry equation -subsequently simplified by Ohshima- in which the Debye length is considered, one can relate the HO and HS approximations21, if necessary. The SI Base-units of parameters and/or variables are given in Table 1.

The mobility \(\:{u}_{e}\) is defined as:

$$\:{u}_{e}=\frac{v}{E}$$
(2)

where \(\:v\) is the velocity of particle and/or ion (m/s), and \(\:E\) is the electrical field strength (V/m).

Electrical field intensity can be calculated for a given membrane potential from simple parameters in the following way:

$$\:E=\frac{MP}{l}$$
(3)

in which, \(\:MP\) is the cell membrane potential (V) and it refers to the voltage difference between its inside and outside or electrical potential difference across an ion channel, \(\:l\) is the membrane thickness (m). In the physiological literature, the MP is defined as the potential measured between cytoplasm and medium.

The following expression can be proposed to estimate stimulus voltage:

$$\:\varPsi\:=-\frac{{\zeta\:}_{r}E}{{E}_{r}}$$
(4)

where \(\:\varPsi\:\) and \(\:E\) are the threshold voltage of stimulus (V) and intensity of electrical field (as described before, V/m), respectively, for membrane (or resting) potential of any excitable cell; \(\:{\zeta\:}_{r}\) and \(\:{E}_{r}\) are the reference entry potential (V) and corresponding electrical field intensity (V/m), respectively, determined for a membrane potential of -70 mV. Here, using the potential of -70 mV was a convenient way to compute \(\:{\zeta\:}_{r}\) potential from which stimulus voltage is calculated, because the resting membrane and corresponding threshold potentials of a nerve cell are very well known and usually around − 70 and − 55 mV, respectively14. This proportional assessment may not always give the most correct outcomes for all excitable cells, but can provide a reasonable estimate underlying the necessary magnitude of the stimulus strength to generate an action potential.

The outer membrane of the neurons is composed of a lipid bilayer about 8–10 nm thick, which acts as a leaky capacitor. The dielectric constant of the lipid bilayer is likely to be in the range 2-622,23. Beside this, the lipid bilayer embeds proteins24. The channel proteins have a dielectric constant of 1525, whereas the dielectric constant of pore water is about 7826. In this study, it was assumed that the dielectric constant of channel water is similar to that of pore water. In addition, Adrien et al.27 reported that the mean viscosity value of the bilayer with embedded membrane protein was 255 (mPa.s). However, in our computations, we preferred to use the viscosity of liquid water (0.888 mPa.s) given by Hafner et al.28.

The electrical field intensity through an open ion-channel can easily be calculated with the use of corresponding membrane potential. Afterwards, the electrical (or electrophoretic) mobility is derived by using the ion velocity and electrical field strength. This all to say, in order to find a solution for quantifying a stimulus voltage, we needed to use the calculated \(\:{\zeta\:}_{r}\) potential at the rest, and to consider the minimum stimulus strength for this point to be also a reference quantity. Thus, the ratio of \(\:{\zeta\:}_{r}E\) to \(\:{E}_{r}\) would correctly reflect the sufficient level of stimulus strength (see Eqs. 1–4). As in the present study, if the terminal velocity of ions moving through water-filled channel (as part of the electrical field across the membrane) is unknown, first we should determine the corresponding number of ions passing through one open channel by the way of trial to calculate the reference entry potential (\(\:{\zeta\:}_{r}\)) which was assumed to be about − 15 mV. Thus, we indirectly calculated the velocity of a single ion from the total number of ions matching the \(\:{\zeta\:}_{r}\) potential (-15 mV) by the use of Eq. 1, via excel program. Then, we computed the stimulus voltage for any resting (or membrane) potential using Eq. 4, and assumed that the absolute value of entry potential is equal to stimulus voltage for the same electrical field intensity. Thereafter, using Eq. 1 again, the velocity of a single ion which was deduced from the total number of ions matching the obtained entry potential for a given MP was calculated by the same way. In summary, meaningful simplifications can be achieved by considering key physical properties of the cell membrane/ion channel such as membrane thickness, dielectric constant, and viscosity to obtain entry potential/stimulus voltage for different membrane (or resting) potentials. Similarly, the number of ions passing through one open channel and the velocity of a single ion moving through channel length can be calculated indirectly for an estimated entry potential. This means choosing this way allows us to find a solution with satisfying results.

Fig. 1
figure 1

Schematic view of extracellular and intracellular sides of an excitable cell membrane showing a voltage-gated sodium channel (a), locations of activation and inactivation gates, and selectivity filter in ion channel (b), definitions of the membrane potentials with respect to the resting potential (c), and negative potential drop through ion channel from inside to outside of the membrane (d).

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Table 1 Elementary constants and units of parameters and/or variables.
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Results and discussion

The resting potential and corresponding stimulus voltage (Ψ) are presented in Fig. 2. The Hückel–Onsager model generates negative \(\:\zeta\:\) values for negative membrane potentials. The calculated \(\:\zeta\:\) potential was − 15.01 mV for a resting potential of -70 mV as soon as a few voltage-gated Na+ channels open. However, the Ψ exhibits positive values and its magnitude depends strongly on resting membrane potential. It suggests that a positive work (or stimulus voltage) is necessary to open the voltage-gated ion channels initially for entering the cations (such as Na+) into the pore, from extracellular to intracellular at the beginning of depolarization. The relationship between stimulus voltage and resting potential shows an explicit expression, and the threshold stimulus increases as the negative values of resting potential increases. As mentioned earlier, the resting potential changes from cell to cell and can vary between − 40 mV and − 80 mV in the excitable cells2. The results of this study indicate that the corresponding Ψ voltages ranged from 8.58 to 17.16 mV, respectively. Surprisingly, the literature data support our predictions that the stimulus voltage is modulated by resting potential. For example, Filatov et al.29 reported that the stimulating voltage was about 20 mV for a resting potential of -85 mV, and threshold stimulus for action potential initiation was lesser at a resting potential of -67 mV.

A high negative value of resting potential suggests that the necessary stimulus strength is also high. Thus, we have a physical explanation of why the higher stimulus voltage will be needed for a higher negative resting potential to generate an action. The strength should be enough to open some ion channels by attracting or repelling forces, initially. However, ion diffusion may be reduced somewhat by a counter ion concentration of the same ions moving back toward opposite direction. In this context, there are also some special cases; for example, it is quite difficult to active a nerve cell during hyperpolarization phase probably due to closing of the activation gate in Na+ channels as well as the continued outward efflux of K cations. In such circumstances, subsequently, the activated N/K pump prevents the excess ionic gradients by the closing of all voltage-gated ion channels, and pumps the ions from one side of the membrane to the other side. The pump exports 3Na cations out of the cell, then imports 2 K cations into the cell in each pump cycle at a roughly rate of 200 cycles/s to re-establish the resting potential ultimately1,30.

Fig. 2
figure 2

Stimulus voltage as a function of resting membrane potential of the excitable cells.

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The threshold and resting potentials are compared in Fig. 3. The different excitable cells appear to have different resting potentials. The equivalent voltage of a trigger action is rather undefined in the literature and depends on the value of resting potential. A stimulant should have enough strength to depolarize the cell membrane to the threshold level. This means that a stronger stimulus is needed to excite a cell for a more negative resting potential. More clearly, it is apparent from the illustration that the higher negative resting potential the stronger is the stimulus. In other words, a rapid change can start in action potential evoked by a higher quantity of stimulus strength, when the cell is at rest. It is very well known that the action potential acts upon the all-or-none law1. For this reason, determination of Ψ has also should be considered a good index of the magnitude of difference between resting and threshold potentials. A low negative value of resting potential suggests that the necessary stimulus voltage is also low indicating that a single cell with a low negative value of resting potential may respond to a slight stimulus. The lower value of Ψ suggests that the attractive interaction between organic anions and Na and/or K cations is lesser or vice versa. As cations are adsorbed positively by the organic anions, the other anions such as Cl− are repelled and relegated from intracellular to extracellular. This new paradigm may have a bias, but it provides a rough guideline at best. These quantities could even correctly reflect the sufficient level of stimulus strength.

Fig. 3
figure 3

Histogram of threshold potential versus resting potential of the excitable cells (please remember that sum of resting potential and stimulus voltage gives threshold potential).

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The number of ions passing through one open channel versus the electrical field strength is presented in Fig. 4. To evaluate the relationship between the field intensity and the number of ions, the simple regression analysis was applied. The number of ions exhibited a power growth with the rising electrical field strength. The estimations in this study are in agreement with the literature data indicating that the passing number of ions through an open channel is in the range of 1 to 100-million ions per second. For example, Milo et al.31 reported that this number was typically 10-million ions each second, based on the measurements. In our study, the passing number of ions changed between 2.2 and 8.8-million/s under the absolute values of electrical field intensities of 4.4 × 106 and 8.9 × 106 mV/mm, respectively.

Fig. 4
figure 4

Relationship between electrical field intensity and number of ions passing through one open channel.

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Although ion species were ignored, the depolarization phase of action potential is continued basically by Na cations. Apart from the electrical field strength, the movement of ions across the channels depends on ionic strength (concentration), ion species, channel selectivity, phases of action potential, and viscosity and dielectric permittivity of the medium. On the other hand, the nerve cells at rest are permeable to K and Na cations, and Cl anions. The Na cations and Cl anions move into the cell, whereas K cations move outward due to their concentration differences. For example, the concentrations of Na+ and Cl− ions are much lower inside the cell than outside whilst the concentration of K+ inside a cell is about 10 times higher than that in the extracellular fluid1. However, a balance between these fluxes is noted at the resting potential, depending on the membrane permeability to each of the ions and the concentrations of the ions both sides of the cell, although many more K+ channels are open, compared with Na+ channels. The membrane of a cell is not equally permeable to different ions; thus, the ion concentrations are not the same on both inside and outside of the cell. Otherwise, no charge or potential difference would be existed. Actually, the resting potential is maintained basically by the Nernst potentials of K cations and Cl anions1,15.

The velocity of a single ion moving through membrane thickness exhibited a power rise with increasing electrical field intensity (Fig. 5). Higher electrical field strength would accelerate ion passage through the channel. In this comparison, the behaviors of ions in the channels were not taken into consideration, because our approach does not appear to be applicable for underlying the mobility mechanisms of different ions. But it is known very well from the literature, K cations could be pulled by organic anions more strongly compared with Na cations, probably due to its smaller hydrated radius32. However, we considered the electrical field intensity to be high to push or pull ions through an open voltage-dependent channel. Unfortunately, there are some special cases or reasons limiting the applications of our approach. For example, during repolarization (actually at hyperpolarization phase), the continued outward efflux of K cations builds up an excess negative charge on the inside of the cell as well as a higher electrical field strength. In spite of this fact, the K cations continue to move out of the cell so much that the new MP becomes more negative than the resting potential33. However, the proteins impede the further efflux of K cations and transport them to opposite direction by using the free energy of ATP. Even so, our new model is reasonable for many circumstances. For example, the membrane potential of erythrocyte and gland cells, which are not excitable, is around − 25 mV. This may be related to the membrane potentials of the cells that are not stimulated, probably because of too low electrical field strength. It is also important to know that the value of \(\:\zeta\:\) potentials should be greater than a specific level to assure a complete electrostatic stabilization34.

Consequently, the Hückel–Onsager equation may contribute to the understanding of the electrochemical and electro-kinetic behavior of the cell membranes. We understand that the conceptualizing voltage generated by a stimulus is just work to move the Na cations from outside to into cell, and then to migrate K+ in opposite direction as soon as the related voltage-gated channels open. Our approach may not be valid for the process of ion exchange equilibrium but is valid for estimating attraction and/or repulsion forces based on electrical field strength except in special cases.

Fig. 5
figure 5

Relationship between electrical field intensity and ion velocity across an open channel.

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In summary, the manuscript presents a theoretical exploration and the application of the entry potential within the selectivity filter of ion channels, and proposes a novel approach to approximate the threshold voltage of stimulus in excitable cells. This study provides a valuable and original contribution to understanding of the dynamics of voltage-dependent ion channels, particularly in the context of electrical field intensity through an open ion-channel. This model of the neural membrane stimulation can create practical solutions for devising new approaches to prevent/reduce epileptic seizures or status epilepticus. The paper has potential implications far beyond nerve cells and into all excitable cells. However, future research should aim at experimental validation, since there is no enough corroborating experimental evidence from literature.

Conclusion

The exact mathematical application of entry potential (\(\:\zeta\:\)) at a hypothetical plane in the selectivity filter of ion channels and of stimulus voltage (Ψ) may be complex but it can, by approximation, be greatly simplified when the potential and thickness of the cell membrane as well as dielectric constant and viscosity of the medium, are taken into account in analyzing excitation or stimulus voltage. Thus, the Hückel–Onsager equation can be used for this purpose since the \(\:\zeta\:\) potential at the midsection of selectivity filter of ion channels could be calculated theoretically. Then, the reference entry potential (\(\:{\zeta\:}_{r}\)) can be utilized to obtain stimulating voltage for a cell. Consequently, the determined quantities of stimulus voltage would correctly reflect the sufficient level of stimulus strength to generate an action potential. Therefore, this work would apparently create new possibilities in the research on the bio-membrane phenomena and contribute to biophysical interpretation for some present or newly discovered neurological pathologies. For example, the presented approach can contribute valuable insight into early management of some channelopathies and potentially improve strategies mainly for reducing the severity of status epilepticus. Similarly, our approach may also be helpful to understand clearly the mechanism of how an implantable device of the pacemaker contributes to the treatment (therapy) of some neurological diseases. In this regard, a low electrical current by applying electrical stimulation to the brain via the electrodes may be recommended as early treatment to manage epileptic seizures in the emergency departments, after implementation of the necessary legislative regulations.