The possibility of anesthesia stimulated by a train of current pulses

This study considers a simple theoretical model of blocking the passage of signals (action potentials) from sensory neurons, thereby affecting anesthesia without the use of anesthetics as a result of a sequence of unipolar current pulses generated by an external source. The proposed model allows the selection of parameters and the required frequency of the repetition of current pulses for the possible implementation of anesthesia, depending on the electrical characteristics of the skin and the conductivity of the saline solution in which the myelinated nerve fibers are located.


1. Introduction
Acupuncture as a means to relieve pain has been known since the third millennium BC and has remained essentially the same as the present day [1][2][3][4][5]. However, one of the modern innovations in the traditional acupuncture technique is the use of weak alternating current sources connected to needles (electroacupuncture). This has been shown to significantly improve the effects of anesthesia [6][7][8]. It should be noted that [9] considered the possibility of anesthesia associated with the redistribution of transmembrane ion channels in the membranes of nerve cells in the acoustic field of an ultrasound source or in the field of stimulated longitudinal ultrasonic oscillations resulting from the interaction of a charged membrane with microwaves.
The research in [10][11][12][13][14][15][16][17] shows the effectiveness of restoring control over the lower extremities during spinal stimulation of the lumbosacral areas of the spinal cord below the site of spinal cord injury using non-invasive electrodes placed directly on the skin above the lower spinal cord. In a recent work [18], a numerical model was considered for blocking the compound nerve fiber, consisting of many myelinated axons, through kilohertz voltage pulses applied to electrodes adjacent to the nerve. However, in this proposed approach, along with blocking, there are always fascicles in which the action potential is initiated. In other words, complete or controlled blocking of the compound nerve cannot be achieved within the framework of the model [18].
A simple theoretical model of the excitation of action potentials of multiple motor pools by stimulating current pulses over the lumbosacral regions of the spinal cord was considered in [19]. This paper shows that a similar approach makes it possible, in principle, to block the passage of signals from sensory neurons with the help of external electrodes, and thus to perform anesthesia without the use of anesthetics. An example of such an approach is finger anesthesia (figure 1(A)), in which signals from receptors on the fingertips, such as heat receptors, are blocked. This approach, which allows experimental testing of the theory, entails a dielectric cylinder with ring electrodes in direct contact with the skin, worn, for example, on a finger. Let us consider a model problem in which a nerve fiber (myelinated axon) of radius a is located on the axis of a hollow dielectric cylinder of radius R 0 ( )  a R 0 and filled with a conductive liquid (saline) with a specific conductivity s ( figure 1(B)). The currents inside the cylinder generated by the ring electrodes are fed with short unipolar current pulses, as shown in figure 2. Section 2 of this paper shows that the problem of generating currents in saline inside a dielectric cylinder can be considered within the framework of a stationary continuity equation, similar to that used by us, the authors, earlier in stimulation theory [19]. In section 3, the corresponding distributions of the potential and current inside a dielectric cylinder with built-in ring electrodes are presented. Section 4 shows that the potential distribution found in section 3 can lead to the blocking of action potential propagation in the myelinated nerve fiber, as shown in figure 1(B).

2. Theoretical model
Saline, the fluid in which neurons are located in living organisms, is a highly conductive electrolyte, with s -W -m 1 3 1 1 [20]. Therefore, the relaxation time of the volume charge perturbation in saline s [21], where e 0 is the dielectric constant of the vacuum, and e » 80 is the relative dielectric permittivity of water, which is shorter than the characteristic time of the excitation and relaxation of the action potential in the axons by 5-6 orders of magnitude. The size of the non-quasi-neutral region near the membrane of the axons was determined using the Debye length Here, k B is the Boltzmann constant, T is the temperature, and n j is the density of ions with charge q . j In the case of saline, we can assume that all the ions in the liquid are singly charged and that their total density is of the order » ⋅ n 2 10 j 26 m −3 [22]. At » T K 300 , l » 0.5 D nm, and this value is much smaller than the typical radius of the axon m »a 1 8 m [23]. Therefore, the violation of quasi-neutrality cannot be considered, and the potential distribution in the vicinity of a neuron can be determined from the equation of current continuity.
Because the conductivity s of the electrolyte is uniform and constant over time, and j = -  E , the problem of determining the potential is reduced to the Laplace equation: Let us estimate the charging time of the myelinated axon using the currents induced inside the dielectric cylinder ( figure 1(B)). Since the action potential is  ap 0 and J 0 are the current pulse duration, duty cycle, and amplitude, respectively. Since the duration of the current pulse that blocks the propagation of the action potential must be no less than the time of initiation of the action potential t 1 1~m s [20], it is natural to assume that t t.
ap 1 Since the relaxation time of the action potential t -10 15 relax~m s [20], the duty cycle of the current pulses t t .
relax 02 generated in unmyelinated sections, nodes of Ranvier, we will consider the charging of the membrane in the nodes of Ranvier, in which the capacitance per unit [20] and the typical radius Following [19], the estimation of the charging time of a cylindrical capacitor of radius a, capacitance per unit area c , m and conductivity s is:  [24], and its thickness [25]. Assume that for the estimates, = b 2 mm. Substituting these values into For definiteness, we take the radius of the outer cylinder Substituting these values into (3), we find that the charging time of the external cylindrical capacitor is = t 0.05 ec μs. That is, the charging time of the Ranvier interception membranes in the internal nerve fiber and external cylindrical capacitor is at least three orders of magnitude less than the duration of the current pulse t ap ms shown in figure 2. Therefore, we do not consider the charging process itself when the radial current is different from 0, but instead consider the steady state when the radial currents charging the cylindrical capacitor become equal to 0 and the radial potential distribution near the fiber surface is equipotential.
Note that the surface of the myelin fiber can be considered an impenetrable dielectric surface only until the potential on the membrane at the nodes of Ranvier reaches the threshold of the excitation of the action potential. Below, we assume that this condition is satisfied.

3. Potential and current distribution inside the outer dielectric cylinder
Equation (2) in cylindrical coordinates has the form: As noted in the previous part of the work, during most of the stimulated current pulses, the charge on the myelin fiber membrane and on the dielectric surface of the outer cylinder can be considered constant. In this case, we can assume that the radial currents are equal to zero.
Since  a R , 0 the influence of the nerve fiber on the distribution of the currents induced by external electrodes inside the dielectric cylinder is insignificant. Therefore, the presence of myelin fibers was neglected.
From the symmetry of the problem of current distribution inside a dielectric cylinder with ring electrodes ( figure 1(B)) (without myelin fibers on the axis), it follows that the radial current on the axis is.

4. Discussion
According to the results of the calculations for the considered example, the radial current ( figure 3(A)) near the axis of the external dielectric cylinder tends to zero and the potential ( figure 3(C)) is practically independent of the radius. Thus, the change in potential across the membranes of the nodes of Ranvier ( figure 1(B)) can be considered equal to the change in potential along the axis of the considered dielectric cylinder. Therefore, in accordance with the results of the calculations, the potential on the membrane of the nodes of Ranvier in the vicinity of the longitudinal coordinate corresponding to the left electrode becomes approximately 20 mV higher than the resting potential, and on the right, 20 mV lower. Consider the case of action potential propagation along the fiber from right to left. It was shown in [26,27] that when an action potential is excited in the nth node of Ranvier, the induced potential in the adjacent (non-excited) nodes of Ranvier depends on the length of the myelinated segments. If the myelinated areas are sufficiently long, then the action potential excitation threshold of 40-45 mV will be only at the (n + 1)st node of Ranvier, as shown schematically in figure 4.
The maximum amplitudes of the action potential and the time dependences of the action potential at the (n+1)th node of Ranvier depending on the length of the myelin-coated sections are shown in figures 5 and 6, respectively. The calculation was performed based on the model [26] with the same parameters as the myelinated axon. The capacitance and resistance per  The time dependence of the action potential in the nodes of Ranvier is approximated by the Gaussian curve, as shown in figure 7.
In the absence of external stimulation, the value of the resting potential ∼−80 mv. In the case of the considered initiation of currents by an external source, the nerve fiber is charged, for example, by 20 mV in absolute value. Then, the value of the resting potential in the vicinity of the left electrode (see figure 3(C)) becomes ∼−60 mV, and in the vicinity of the right electrode is ∼−100 mV. Correspondingly, the action potential excitation threshold in the vicinity of the right electrode decreases from 40-45 mV to 20-25 mV and, respectively, increases from 40 to 45 mV to 60-65 mV in the vicinity of the right electrode. Therefore, the propagation of the action potential near the right electrode was blocked. An action potential propagating from left to right was also blocked at the right electrode.
Let us now consider the case where the myelinated fiber is not on the cylinder axis. From the graphs shown in figure 3, it can be seen that when moving to the right from the right electrode at a distance equal to the electrode size > + z l l 2 1 2 , c 0 ⎛ ⎝ ⎞ ⎠ the current components J r and J z are close to 0, regardless of r, while the potential j remains practically constant and independent of. Therefore, blocking of the action potential propagating from right to left can be expected to occur regardless of the radial position of the myelinated fiber. The same reasoning is also valid in the case of the  propagation of the action potential from left to right; it will also be blocked behind the right electrode regardless of the radial position of the myelinated fiber.
We note a significant difference between the initiation of the action potential considered in [19] and the opposite task, blocking the action potential, presented in this work. In [19], a nerve fiber with axons located perpendicular to the currents induced in a conducting fluid was considered, and this, as shown in [28], greatly changes the structure of the currents near the axon. In this study, we consider the case in which mainly longitudinal currents are induced and the radial current is close to zero. Therefore, it does not affect the potential and its distribution near the nerve fiber, because the fiber is charged only by currents perpendicular to it. For a fiber that is not on the axis of the considered dielectric cylinder with ring electrodes, all the considerations given above remain applicable.

5. Conclusions
We considered a possible approach to anesthesia without the use of anesthetics. It has been shown that with a sequence of current pulse excitations in the vicinity of the nerve fiber, it is possible to block the propagation of the action potential, that is, a reversible loss of sensitivity, without the use of anesthetics. However, it will be possible to draw final conclusions and discuss the application of the considered approach only after experimental verification of the theory.