Neuron with well-designed ionic system

Abstract

Neurons have an ionic system with several types of ion pumps and ion channels on their membranes. Each ion pump creates a specific difference in ion concentration inside and outside the neuron, and the energy resulting from this difference in concentration is maintained inside the neuron as a resting potential. Each ion channel senses the necessary situation, opens the channel, and allows the corresponding ion to pass through to perform its corresponding role. This ionic system realizes important functions such as (i) fast conduction of action potentials, (ii) achieving synaptic integration in response to several inputs with a time lag, and (iii) the information processing functions by neural circuits. However, the mechanisms by which these functions are realized have remained unclear. Therefore, based on the reports on various highly polymeric ion pumps, ion channels, cell membranes, and other components that have been elucidated so far, author analyzed how this ionic system can realize the above important functions from an electrical circuit designer point of view. As a result of a series of analyses, it was found that neurons realize each function by making full use of high-density packaging technology based on basic electrical principles and making maximum use of the extremely high dielectric properties of the ionic fluid of neurons. In other words, neuron looks to equip well designed ionic system which is the collaboration by designers of proteins and membranes that perform advanced functions and designers of electrical circuits that utilize them to achieve important functions electrically.

Significance

Neurons have an ionic system that consists of several types of ion pumps and ion channels with high-order functions on their membranes. Through a series of research, the author discovered that this ionic system is well designed to realize important functions with energy saving such as fast conduction of action potentials, establishment of synaptic integration by several inputs with time lags, and information processing by making full use of ultra-high density packaging technology based on basic electrical principles and the extremely high dielectric properties of ionic fluids in neurons.

 Introduction

From the standpoint of an electrical circuit designer, author has attempted to elucidate the mechanism behind the well-designed ionic system of neurons that realizes important functions of neurons. First, author has begun to elucidate the mechanism behind the high speed saltatory conduction of action potentials, which is highly questionable from the perspective of electrical circuits [1,2]. Then, he elucidated the mechanism that enables the achievement of synaptic integration by several inputs with time lag, which is the next important issue. As a result, it was found that the same factors that realize fast conduction of action potentials also contribute greatly to the achievement of synaptic integration [3]. Furthermore, by using the equation derived from the conditions for achieving this synaptic integration, he was able to elucidate the information processing function of neural circuits that had not been elucidated until now [4].

As shown below, in a series of research [1,3,4], it is very fortunate that the results of the just before research have become clues to the solution of the next research. As the first research topic, author chose to elucidate the mechanism of high-speed conduction of action potentials in the long axons of motor and sensory neurons. In 1939, J. B. Hursh reported that the conduction velocity v of an action potential was about 120 m/s (about 430 km/H) for an axon with a diameter of 20 μm [5]. However, the axon is composed of ionic fluids similar to seawater, fibers made of proteins, and a lipid bilayer membrane that encases them [6]. Since the diffusion rate of ions in ionic fluids is slow [7–9], it was very difficult and interesting to understand the mechanism of high-speed conduction of action potentials in axons that are not metallic conductors from an electrical circuit perspective [10]. Therefore, author proceeded with the analysis by calculating backward from the conditions for high-speed conduction based on the reports on axons that have been studied so far. As a result, it was discovered that neurons have axons with a structure that, based on basic electrical principles, enables action potentials to be transmitted at high speed with little attenuation, and that the high conduction speeds are achieved by effectively utilizing the extremely high dielectric properties of the ionic fluids within the axon, which far exceed imagination [1]. As the next research topic, author started to elucidate the mechanism by which synaptic integration is established for several inputs with a time lag. Conventionally, the condition for synaptic integration is that the sum of pulse-like excitatory postsynaptic potentials (EPSPs) generated by excitatory AMPA receptors present in spines connected to dendrites [9,11] receiving excitatory neurotransmitters exceeds a certain value. Therefore, in order to enable the summation of several pulse-like EPSPs, it was required that the EPSPs occur almost simultaneously [12,13].

However, synaptic integration also needs to be established by EPSPs that occur discretely based on information with a time lag, such as conversation, reading, and recalling past memories. For this need, there is a phenomenon in which Ca²⁺ ions taken up by excitatory NMDA receptors, which are activated by an increase of the potential within the spine after the generation of an EPSP, cannot pass through the spine neck and are retained in the spine for a long time. This phenomenon is called the compartment state of Ca²⁺ ions, and has been expected to be an effective phenomenon for the establishment of synaptic integration for several inputs with a time lag [14,15]. However, the mechanism by which synaptic integration is achieved through this phenomenon remained unclear so for. However, fortunately this research was able to elucidate that the electrical structure of the spine head, spine neck, and dendrites, as well as the extremely high dielectric property of the ionic fluid within neurons which was discovered by the just before research, achieves the synaptic integration [3]. Furthermore, as the next research topic, author investigated the extent to which neural circuits can execute with repeatability functions necessary for information processing. Although the types of inhibitory neurons and their input-output relationships have been well studied to date [16], the specific contribution of inhibitory neurons to information processing in neural circuits remains unclear. However, from the condition equation for synaptic integration obtained by the just before research, it became possible to concretely demonstrate the inhibitory effect of the negative charge of Cl^– ions taken up by inhibitory receptors. As a result, in addition to the functions of AND, OR, and addition, it became possible for neural circuits to have exclusive OR (EOR), NOR, and subtraction functions due to the inhibitory effect of inhibitory neurons, demonstrating that the basic functions necessary for information processing could be realized with repeatability [4].

 Basic Understanding and Electrical Principles behind the Series of Research

 Concentration Difference and Main Role of each Ion in Ionic System

Neurons create a concentration difference between the inside and outside of the neuron by several types of ion pumps on the membrane. Energy proportional to the concentration difference of the ions is stored inside the neuron, which appears as a potential difference between the inside and outside of the neuron [12,13]. The potential determined by Goldman equation [17], which takes into account the concentration of each ion and the relative permeability of each ion to the cell membrane shown in Table 1 [12], is called the resting potential and is approximately –65 mV [12,13]. The ion channel corresponding to each ion knows the timing to operate by sensing the neurotransmitter or voltage, etc., and opens the channel to allow the corresponding ion to pass. When the corresponding ion passes through the channel and moves from the high concentration to the low concentration, the ion plays the corresponding role as shown in the rightmost column of Table 1.

Table 1 Concentration of Ion and main roles achieved by ion movements [12]

Ion	Outside (mM)	Inside (mM)	Concentration Outside : Inside	The main roles achieved by ion movements
K+	5	100	1 : 20	Suppression of action potential amplitude
Na+	150	15	10 : 1	Generation of EPSP Generation of action potentials at axon and node of Ranvier
Ca2+	2	0.0002	10,000 : 1	Increase in electrical potential within the spine Increase in electrical potential within axonal terminl
Cl–	150	13	11.5 : 1	Inhibition of establishment of synaptic integration by negative charges

In other words, neurons can be said to have four types of independent pumped-storage power generation systems using ion pumps and ion channels that correspond to each ion in single neuron. It is major advantage that this ion system is based on ions that are abundant in seawater. The diffusion rate of each ion is slow [7–9], but the electric field caused by the ion’s charge propagates quickly. Therefore, the role of the dielectric constant, which has a large influence on the propagation of the electric field, is large.

 Basic Electrical Principles used by Ionic Systems

One of the important electrical principles used by the ionic system of neurons is the relationship between electric charges and electric fields. The electric field E is inversely proportional to the square of the distance from the charge Q and the dielectric constant of the medium surrounding the charge Q [18].

  

$$\mathrm{E}=\mathrm{Q}/4\mathrm{\pi }{\mathrm{r}}^2\mathrm{\epsilon }=\mathrm{Q}/4\mathrm{\pi }{\mathrm{r}}^2{\mathrm{\epsilon }}_0{\mathrm{\epsilon }}_{\mathrm{r}}$$(1)

The ε is the dielectric constant that indicates the degree of convergence of the electric field lines generated by the charge Q. The ε₀ is the dielectric constant of a vacuum and is 8.854×10^–12 F/m. However, the dielectric constant ε of a general medium is large because the electric field lines polarize the medium [18,19]. In other words, dielectric constant ε of a general medium is proportional to the polarizability of the medium and differs from medium to medium. Therefore, the diffusion of the electric field lines does not necessarily spread radially, but depends on the dielectric constant ε of the medium [18,19]. To make it easier to understand the dielectric constant ε of each medium, the ratio of the dielectric constant ε of each medium to the dielectric constant of a vacuum ε₀ is called the relative dielectric constant εᵣ and is expressed by the following equation [18].

  

$${\mathrm{\epsilon }}_{\mathrm{r}}=\mathrm{\epsilon }/{\mathrm{\epsilon }}_0$$(2)

At normal temperature, the relative dielectric constant of poor conductors of common solids, liquids, gases, compounds, and mixtures in is less than 10, except for water and barium titanate etc. [20]. The amount of electricity Q stored in a capacitance C due to polarization of medium and the potential difference V of both end of the capacitance C are represented by the following equation.

  

$$\mathrm{V}=\mathrm{Q}/\mathrm{C}$$(3)

The capacitance C is determined by the shape of the electrode that induces polarization within medium and the shape envelopes the medium with dielectric constant ε. For example, the capacitance C of a medium with dielectric constant ε sandwiched between two parallel electrode plates is given by the following equation.

  

$$\mathrm{C}=\mathrm{\epsilon }\mathrm{S}/\mathrm{d}={\mathrm{\epsilon }}_0{\mathrm{\epsilon }}_{\mathrm{r}}\mathrm{S}/\mathrm{d}$$(4)

S is the area of the electrodes and d is the distance between the electrodes. The capacitance C of a cylindrical membrane with dielectric constant ε, such as the membrane of an axon, is given by the following equation [18].

  

$$\mathrm{C}=2\mathrm{\pi }\mathrm{\epsilon }\mathrm{L}/\mathrm{l}\mathrm{n}\left(\mathrm{D}\mathrm{o}/\mathrm{D}\mathrm{i}\right)=2\mathrm{\pi }{\mathrm{\epsilon }}_0{\mathrm{\epsilon }}_{\mathrm{r}}\mathrm{L}/\mathrm{l}\mathrm{n}\left(\mathrm{D}\mathrm{o}/\mathrm{D}\mathrm{i}\right)$$(5)

The capacitance C of a cylindrical membrane is proportional to the length L of the cylinder and inversely proportional to the logarithm of the ratio (Do/Di) of the outer diameter to the inner diameter of the cylinder. As mentioned above, the actual substance of the capacitance is the dielectric constant ε. Next important electrical principle used by the ionic system of neurons is displacement current which is induced the product of capacitance and voltage changing speed. It is given by the following equation [18].

  

$$\mathrm{I}=\mathrm{C}\mathrm{d}\mathrm{V}/\mathrm{d}\mathrm{t}$$(6)

 MKS Units

The basic units used in this study are M (meter), Kg (K gram), and S (second). In other words, they are described in MKS units. The unit of charge Q is q (coulomb), the unit of current I is A (ampere), the unit of voltage V is V (Volt), the unit of resistance R is Ω (ohm), and the unit of capacitance C is F (farad). Current I is the flow of charge Q, and the amount of charge Q that flows in 1 second of 1A of current I is defined as 1 q (coulomb). The dielectric constant ε is expressed as the capacity per unit length (m). On the other hand, the definition of ion concentration, which is often mentioned in Table 1 and elsewhere, is the number of moles M of molecules or ions per liter (10 cm)³ rather than per cubic meter (1 m)³. Therefore, it should be expressed as M/ℓ, but in the case of concentration, it is often written simply as M. The number of molecules in 1M is 6.02×10²³, known as Avogadro’s number, and is the same regardless of the type of molecule or ion [12].

 Results

The results of a series of studies on ionic systems are described below: (A) fast conduction of action potentials, (B) achieving synaptic integration in response to several inputs with a time lag, and (C) the information processing functions by neural circuits.

 A. Elucidation of the Mechanism of Fast Conduction of Action Potentials

In 1938, Ichiji Tazaki discovered the phenomenon of saltatory conduction of action potentials in myelinated nerves [2]. In 1939, J. B. Hursh reported that the conduction velocity v of action potentials in myelinated nerves is proportional to the diameter of the axon, and that when the axon has a diameter of 20 μm, the conduction velocity v is approximately 120 m/s [5]. Furthermore, he reported that the spacing of the nodes of Ranvier that relay the action potential [6] is proportional to the diameter of the axon, and that the relay spacing for a diameter of 20 μm is approximately 2 mm [5]. Therefore, achieving a conduction velocity of 120 m/s means that the conduction is relayed by approximately 60,000 nodes of Ranvier per second [6]. Therefore, to determine the conduction velocity v, it is necessary to consider not only the raw velocity Vr of the action potential, but also the fact that the action potential is relayed at the nodes of Ranvier using a relay time τ [6]. Therefore, the conduction velocity v is calculated using the following equation, which takes into account the propagation time t due to the raw velocity Vr of the action potential and the delay time τ due to the relay.

  

$$\mathrm{V}=\mathrm{x}/(\mathrm{t}+\mathrm{\tau })=\mathrm{x}/(\mathrm{x}/\mathrm{V}\mathrm{r}+\mathrm{\tau })$$(7)

x is the relay interval, t is the time it takes for the action potential to reach the relay point at raw velocity Vr, and τ is the relay time. The relay time τ is proportional to the rise time of the action potential. From this equation, to increase the conduction velocity v, it is necessary to lengthen the relay interval x, speed up the raw velocity Vr of the action potential, and shorten the relay time τ. If the attenuation of the action potential due to propagation is small, the action potential can pass through several nodes of Ranvier [6].

 Analysis of the Mechanism of Fast Conduction using Equivalent Circuits

To know the attenuation due to the propagation of the action potential and the raw velocity Vr, it is necessary to set up an equivalent circuit for the axon and create a wave equation for the action potential [18]. Then, from the wave equation, the attenuation constant α, which determines the attenuation due to the propagation of the action potential, and the phase constant β, which determines the raw velocity Vr, are calculated [18]. Here, the rise time to the peak value W of the action potential is considered to be the first 1/4 period of the sine wave. In addition, since the equivalent circuit includes a capacitance that causes a phase shift due to a voltage change, the function Ve^jωt, which is used for general electric circuits, is used for the action potential as a function that can handle the phase shift [18]. ω is the angular velocity and is defined as 2πf for frequency f [18]. It has been reported that the rise time of the action potential during fast conduction is short, about 125 microseconds [21]. Therefore, 2 kHz is used to get the maximum conduction velocity v [1]. Figure 1(a) shows the equivalent circuit of an axon between the nodes of Ranvier, which was used by Frankenhaeuser-Huxley et al. [6]. The displacement current I (leakage current) due to the membrane capacitance C2 of the axon is taken into account, but the axial capacitance C1 of the axon has been ignored because it is considered too small. However, since the conduction velocity v of the action potential cannot be ensured in the conventional circuit shown in Figure 1(a), author attached the capacitance C1 in parallel with axon, which has been ignored until now, as shown in Figure 1(b), so that displacement current I in the axial direction could be calculated [1].

Figure 1  Equivalent circuit of an axon. (a) Conventional axon equivalent circuit. (b) Equivalent circuit with axial capacitance C1 added. Both circuits are distributed constant circuits, and the circuit parameters axial resistance R1 (3,500 MΩ/m), axon membrane resistance R2 (0.32 MΩm), and axon membrane capacitance C2 (1,300 pF/m) were set by converting the values compiled by A. L. Hodgkin (1964) to a diameter of 20 μm [1,22].

Then, author took an approach of calculating backwards what kind of axial capacitance C1 is required in the equivalent circuit of Figure 1(b) so that the conduction velocity v of the action potential with a rise time of 125μs (at 2 kHz) reaches 120 m/s [1,2]. And then, if the value of the axial capacitance C1 required for achieving a conduction velocity v of 120 m/s coincides with the value of the axial capacitance of the actual axon, it will become clear that the axial capacitance C1 of the axon is a major factor in achieving a high conduction velocity v. Here, in both axon equivalent circuits, the circuit parameters other than the axial capacitance C1 use the same values based on the circuit parameters summarized by A. L. Hodgkin in 1964 [22]. In addition, for an axon with a diameter of 20 μm, the length of the uniform myelin sheath is 2 mm, while the length of the node of Ranvier is approximately 2 μm, which is a ratio of approximately 1,000:1 [5], and since the interior of the axon at the node of Ranvier is continuous [6], then axon is able to be considered to be a uniform transmission path, so the calculations were performed as a distributed constant circuit [18].

In both axon equivalent circuits, the action potential is generated at point A in the form of a function f(v)=Ve^jωt that can be calculated including the phase shift, and a voltage drops between infinitesimal segments AB, and a current corresponding to that voltage flows between BC [18].

In the conventional circuit, this is expressed by the following equations:

  

$$-\mathrm{d}\mathrm{V}/\mathrm{d}\mathrm{x}=\mathrm{R}1\mathrm{I}=\mathrm{Z}\mathrm{I}$$(8)

  

$$-\mathrm{d}\mathrm{I}/\mathrm{d}\mathrm{x}=\mathrm{V}/\mathrm{R}2+\mathrm{C}2\left(\mathrm{d}\mathrm{V}/\mathrm{d}\mathrm{t}\right)=(1/\mathrm{R}2+\mathrm{j}\mathrm{\omega }\mathrm{C}2)\mathrm{V}=\mathrm{Y}\mathrm{V}$$(9)

In the circuit with C1 added, this is expressed by the following equations:

  

$$-\mathrm{d}\mathrm{V}/\mathrm{d}\mathrm{x}=\mathrm{I}/(1/\mathrm{R}1+1/{\mathrm{Z}}_{\mathrm{C}1})=\mathrm{I}/(1/\mathrm{R}1+\mathrm{j}\mathrm{\omega }\mathrm{C}1)=\mathrm{Z}\mathrm{I}$$(10)

  

$$-\mathrm{d}\mathrm{I}/\mathrm{d}\mathrm{x}=\mathrm{V}/\mathrm{R}2+\mathrm{C}2\left(\mathrm{d}\mathrm{V}/\mathrm{d}\mathrm{t}\right)=(1/\mathrm{R}2+\mathrm{j}\mathrm{\omega }\mathrm{C}2)\mathrm{V}=\mathrm{Y}\mathrm{V}$$(11)

Here, Z_C1 means the impedance of C1 for the function f (v). For these two circuits, by differentiating equations (8) and (10) with respect to x and substituting dI/dx on the right side of the obtained equation into the left side of equations (9) and (11), the following simple form of partial differential equation propagation equation is obtained [18].

  

$${\mathrm{d}}^2\mathrm{V}/{\mathrm{d}\mathrm{x}}^2=\mathrm{Z}\mathrm{Y}\mathrm{V}$$(12)

In general, it is difficult to find a solution to a partial differential equation, but it is easier to derive the solution by considering the above ZY as a constant. As in the cited references [1], the tendency of the solution can be easily shown by plotting while changing the angular velocity ω, i.e., frequency f, included in ZY. As a result, as shown in Table 2, the raw velocity Vr was calculated to be 81.7 m/s and the conduction velocity v was calculated to be 42.5 m/s at the action potential frequency of 2 kHz from the wave equation for the conventional equivalent circuit (Figure 1(a)). This value of conduction velocity v is significantly different from J. B. Hursh’s measured value of 120 m/s.

Table 2 Approach to calculating the required axonal capacitance C1 using an equivalent circuit [1]

Circuit type	Attenuation constant	Maximu reachable distance (L=ln(W/w)/α)	Phase constant	Raw velocity (Vr=2πf/β)	Conduction velocity	Axonal capacitance
	α	L (cm)	β	Vr (m/s)	v (m/s)	C1 (Fm)
Conventional circuit (Figure1(a))	185.9	0.746	153.7	81.7	42.5	Non
Circuit with C1 (Figure1(b))	130.5	1.06	7.1	1769	120	7.4×10–14

W is the amplitude of the action potential and w is the threshold voltage of Na⁺ ion channel from the rest potential.

In response to this, author calculated backward the axial capacitance C1 which achieves a conduction velocity v of 120 m/s at 2 kHz in the axon equivalent circuit (Figure 1(b)) with the addition of the axial capacitance C1.

As a result, the raw velocity Vr was significantly improved from 81.7 m/s to 1769 m/s, and the axial capacitance C1 required to achieve a conduction velocity v of 120 m/s was calculated to be 7.40×10^–14Fm. Therefore, if this value matches the axial capacitance C1 of the actual axon, it can be concluded that the axial capacitance C1 is the factor that realizes the faster conduction velocity v of the action potential. However, unfortunately there has been no measurement data on the axial capacitance C1 of the axon until now. Therefore, author decided to investigate the dielectric constant ε of the medium within the axon, which makes the capacitance C1. The axial capacitance C1 can be calculated using the following equation, so the dielectric constant ε and relative dielectric constant εr can be calculated backward from the calculated axial capacitance C1 value above.

  

$$\mathrm{C}1=2\mathrm{\epsilon }\mathrm{S}/\mathrm{d}=2{\mathrm{\epsilon }}_0{\mathrm{\epsilon }}_{\mathrm{r}}\mathrm{\pi }{\mathrm{r}}^2/\mathrm{d}$$(13)

The electrode for the axial capacitance C1 of the axon is a lump of Na⁺ ions that flow in all at once from the Na⁺ ion channels that exist at high density around the nodes of Ranvier. The electric field created by the lump of Na⁺ ions spreads to the left and right, so a coefficient of 2 is required [13]. S is the cross-sectional area of the axon, r is the radius of the axon, and d is the unit length (m). Therefore, the dielectric constant ε of the axial capacitance C1 of an axon with a radius of 10 μm that was required is calculated to be 1.18×10^–4 F/m from equation (13), and the relative dielectric constant ε_r with respect to the dielectric constant ε₀ of a vacuum is 1.33×10⁷. However, the relative dielectric constant ε_r of a typical medium is less than 10 [20]. Therefore, it can be said that such a relative dielectric constant ε_r is a value far larger than generally expected. Therefore, we searched for previous reports of a large relative dielectric constant ε_r in a living body. However, it is difficult to analyze the behavior of ions in the KCl and NaCl solutions contained in the axonal fluid, and the mechanism behind the dielectric constant has not been fully elucidated [23]. Fortunately, the references [24,25] reported that the relative dielectric constants ε_r of the gray matter (neuron group) of the cerebral cortex and a 0.1 M NaCl solution at 2 kHz were very large, about 10⁵ and 10⁶, respectively. Furthermore, it was reported that the relative permittivity ε_r is proportional to the concentration of the NaCl solution and inversely proportional to the frequency. From this data [24,25], it was believed that the extremely high relative dielectric constant ε_r in the axon was due to the relative dielectric constant ε_r of the ionic fluid in the axon.

From the above, it was found that the factor that increases the speed of the conduction velocity v of the action potential is the axial capacitance C1 of the axon, which is composed of an ionic liquid with an extremely large relative dielectric constant ε_r. Furthermore, J. B. Hursch’s measurements [5] have reported that the conduction velocity v of the action potential is proportional to the axon diameter. As described in detail in the references [1], it was confirmed that the conduction velocity v proportional to the diameter could be calculated by corresponding the circuit parameters to the diameter for axon diameters of 6 μm, 13 μm, and 20 μm. Therefore, the analysis algorithm was also proven to be correct.

 Neural Structure for Fast Conduction of Action Potentials

In a typical axon diagram, the ratio of the length of the myelin sheath to the length of the nodes of Ranvier to does not appear to be very large, but as mentioned above, in an axon with a diameter of 20 μm, the ratio of the two lengths is about 1,000:1 [5]. The long myelin sheath about 4 μm thick is made of a very thin myelin sheet wound uniformly more than several hundred times [22]. This is similar to a transmission cable with uniform properties. In addition, by wrapping a 4 μm myelin sheath around a 5 nm thick axon membrane [26], the capacitance C of the cylindrical axon membrane is greatly reduced. In other words, the outer diameter/inner diameter of the axon with a 4 μm thick myelin sheath is 28/20, but the outer diameter/inner diameter without the myelin sheath is 20.01/20. Therefore, if we calculate the cylindrical membrane capacitance C using equation (5) [18], the myelin sheath reduces the axon’s membrane capacitance C to about 1/670. This reduces the amount of displacement current I (leakage current) that leaks out of the axon, which is proportional to the product of the action potential’s voltage change dV/dt and the membrane capacitance C, to about 1/670, as shown in equation (6).

On the other hand, to make the action potential travel fast and far, a large displacement current I is required in the axial direction. This requires a large action potential with a large voltage change rate dV/dt and a large axial capacitance C1. For this reason, voltage-gated Na⁺ ion channels are present at an ultra-high density of 12,000/μm² around the 2 μm-wide node of Ranvier that generates the action potential, so that a large amount of Na⁺ ions can enter the axon at the same time [6,27]. Details are given in the reference materials [1], but this generates an action potential with a large amplitude of approximately 100 mV and a short rise time of approximately 125 μs, which, when multiplied with the large axial capacitance C1 mentioned above, generates a large displacement current I. Moreover, the above-mentioned structure, which uses the myelin sheath made of several hundred thin and wide sheets and ultra-high density mounting technology of Na⁺ ion channels, is superior even from the perspective of current highly mounting technology, and its structure reduces attenuation due to propagation and conducts action potentials at high speed with little energy.

 Relationship between Relay Interval x and Relay Time τ

In the case of Figure 1(b), the raw speed Vr is calculated as 1769 m/s as shown in Table 2, and at that speed, it is calculated that the distance of 120 m is reached in 0.0678 seconds [1]. That is, in equation (7), if x is 120 m, the arrival time t is 0.0678 seconds, while the total relay time τ is 0.9322 seconds. Therefore, the relay time τ becomes very important factor. Intuitively, it is thought that the maximum conduction velocity v of the action potential would be relayed by the node of Ranvier near the maximum reach of the action potential from equation (7) [1]. Therefore, in the case of a maximum reach distance of 10.6 mm, it was thought that the fifth node of Ranvier, which is 2 mm interval, would be optimal for relaying. However, in the case of the maximum reach distance, it is the distance at which the peak value W of the action potential at the time of generation attenuates to the threshold w of the Na⁺ ion channel. On the other hand, when an action potential is relayed by a node of Ranvier located at a distance before the maximum reach where the attenuation of the action potential is small, the relay time τ is small because the waveform is in the middle of rising, but conversely, the relay interval x is short. For this reason, equation (7) should be modified as follows so that the attenuation due to propagation (1/e^αx), the trade-off between the relay interval x and the relay time τ should be calculated [1].

  

$$\mathrm{V}=\mathrm{x}/(\mathrm{t}+\mathrm{\tau })=\mathrm{x}/\{\left(\mathrm{x}/\mathrm{V}\mathrm{r}\right)+\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathrm{w}/\mathrm{W}/\left(1/{\mathrm{e}}^{\mathrm{\alpha }\mathrm{x}}\right)\right)/\mathrm{\omega }\}$$(14)

W in the second term of the denominator is the peak value of the action potential from the resting potential, w is the threshold of the Na⁺ ion channel, and α is the attenuation constant. Asin is an arcsine function that calculates the angle of the argument value. Since ω is the angular velocity, τ is the time to the angle calculated by Asin, that is, the relay time τ. Details are given in the references [1], but the relationship between the node of Ranvier number # and the conduction velocity v is #1: 72.87 m/s, #2: 108.44 m/s, #3: 121.28 m/s, #4: 118.71 m/s, #5: 100.98 m/s. In reality, all nodes of Ranvier from #1 to #5 generate action potentials, but the action potential of the third node of Ranvier precedes it. Note that 120 m/s is the average value when the third and fourth nodes of Ranvier are relayed [1]. The existence of multiple nodes of Ranvier within the maximum reach is considered to be meaningful in terms of maintaining the safety of the conduction of the activity signal and the stability of the conduction velocity [28].

 B. Elucidation of the Mechanism of Synaptic Integration by Several Inputs with Time Lag

Fortunately, the results of the immediately preceding study, presented in previous section, contribute significantly to elucidate this mechanism, as shown below.

 Background of Focusing on the State of Ca²⁺ Ion Compartments

According to conventional theory, there is a strict time constraint that excitatory AMPA receptors on multiple spines must receive excitatory neurotransmitters from other neurons at almost the same time, so that the waveforms of multiple EPSPs are added together as shown in Figure 2(a) [12,13]. However, if thinking and judgment are a chain of the establishment of synaptic integration, synaptic integration must also be established by EPSPs caused by the release of excitatory neurotransmitters with time lags due to information from long periods of reading, conversation, or past memories, as shown in Figure 2(b) [3]. To solve this situation, it is necessary to memorize those neurotransmitters with time lags were received, which were released discretely. In response to this need, it has been suggested that the state in which Ca²⁺ ions taken up into the spine by NMDA receptors in the spine after EPSP generation cannot pass through the spine neck and remain in the spine, i.e., the compartment state of Ca²⁺ ions, may contribute to the establishment of synaptic integration [29–31].

Figure 2  (a) Sum of EPSPs under t <˂ T. (b) Sum of EPSP under t >> T. (c) Membrane potential is increased by sum of Ca²⁺ions restrained in the spines and EPSPs [3].

However, the mechanism by which this state establishes the synaptic integration has remained unclear. If, as shown in Figure 2(c), multiple spines that take up Ca²⁺ ions by NMDA receptors after EPSP generation and become compartment states of Ca²⁺ ions can raise the membrane potential to the threshold of the voltage-gated Na⁺ ion channel on the axon hillock, synaptic integration can be established by multiple inputs with a time lag [3]. Therefore, author examined this possibility.

 Necessity of Analysis of the Effect of Spine Neck Capacitance Cn

The membrane of the dendrite is very thin, about 5 nm [26], because there is no thick myelin sheath. Therefore, the ratio of the outer diameter to the inner diameter of the dendrite is almost 1, and the membrane capacitance of the dendrite is very large according to equation (5). The spine that holds Ca²⁺ ions has a Ca²⁺ ion concentration close to the outside of the neuron, and is separated from the inside of the dendrite by the spine neck. Therefore, the inside of the dendrite is affected by the charge of the high concentration of Ca²⁺ ions in the spine through the spine neck capacitance Cn. However, since the length of the spine neck is 0.5 μm on average, and the average radius of the spine neck is also 0.18 μm, the spine neck capacitance Cn is considered to be negligibly small compared to the membrane capacitance due to the total membrane area of the dendritic membrane [11]. However, as found in the research in the previous chapter, the relative dielectric constant ε_r of the ionic liquid inside the neuron is remarkably high, about 10⁷ [1]. Furthermore, since the dielectric constant ε_r of ionic liquid is inversely proportional to the frequency of the change in the electric field, the dielectric constant ε_r becomes even higher under the electric field due to the charge of Ca²⁺ ions that does not change in the spine [1,24,25]. On the other hand, the dielectric constant ε_r of the lipid bilayer membrane, which determines the membrane capacitance of the dendrite, is thought to be less than 10 as normal materials [20]. Therefore, the spine neck capacitance Cn, which has been intuitively thought to be negligible until now, includes the spine neck fluid, which is an intracellular fluid with a significantly higher dielectric constant ε_r, so it has become necessary to specifically analyze the effect of the spine neck capacitance Cn [3].

 Analysis Considering the Effect of Spine Neck Capacitance Cn

The concentration of Ca²⁺ ions in a spine that holds Ca²⁺ ions can be up to 2 mM, which is almost the same as the outside. Therefore, the effect of the difference in Ca²⁺ ion concentration between the inside of a spine that holds Ca²⁺ ions and the outside is small. However, there is an effect of a concentration difference of about 10,000 times between the inside of the spine and the inside of the dendrite, where the concentration of Ca²⁺ ions is 0.2 μM, across the spine neck [3,12]. To specifically analyze this effect, it is necessary to clarify the path by which the electric field caused by the excess charge Q in the spine diffuses outside the cell. Therefore, as shown in Figure 3, the connection of all media in the diffusion path of the electric field E caused by the charge Q in the spine is converted to a connection of capacitance C and expressed as an equivalent circuit [3]. In addition, since the voltage-gated Na⁺ ion channel that senses the membrane potential corresponding to the observation point is located in the axon hillock of the soma, the capacitance beyond the axon hillock may be ignored.

Figure 3  Diagram of the capacitance connection for the charge in the spine. Csm is the spine membrane capacitance, Cn is the spine neck capacitance, Cd is the axial capacitance of the dendrite, Csp is the spine capacitance, Cdm is the dendrite membrane capacitance, and Csom is the soma membrane capacitance. n out of m spines hold excess Ca²⁺ ions. The series membrane capacitance Cns is composed of the series connection of Cn, Csp, and Csm, and the series capacitance Cs is composed of the series connection of Cn, Cd, and the total membrane capacitance ΣCm [3].

Figure 3 shows that m spines with identical characteristics are connected to a dendrite, of which n spines hold excess Ca²⁺ ion charge Q due to the Ca²⁺ ion compartment state, and m-n spines do not hold excess Ca²⁺ ions. Each spine has a pathway that connects to the outside of the cell via the spine membrane capacitance Csm and a pathway that connects to the axial capacitance Cd of the dendrite via the spine neck capacitance Cn. Furthermore, the axial capacitance Cd of the dendrite has a pathway that connects to the outside of the cell via the dendrite membrane capacitance Cdm and the membrane capacitance Csom of the soma to which the dendrite is connected. In addition, if the capacitance of a spine that does not hold Ca²⁺ ions is Csp, the spine membrane capacitance Csm, spine capacitance Csp, and spine neck capacitance Cn form a series membrane capacitance Cns and are connected to the axial capacitance Cd of the dendrite as a membrane capacitance. Therefore, the axial capacitance Cd of the dendrite has in parallel as total membrane capacitance ΣCm; the membrane capacitance Cdm of the dendrite, the soma membrane capacitance Csom, and (m-n) series membrane capacitances Cns. Meanwhile, the electric field E due to the charge Q in the spine diffuses outside the cell via the series capacitance Cs composed with spine neck capacitance Cn, the axial capacitance Cd of the dendrite, and the total membrane capacitance ΣCm [3]. The series membrane capacitance Cns and the series capacitance Cs are respectively expressed by the following equations.

  

$$1/\mathrm{C}\mathrm{n}\mathrm{s}=1/\mathrm{C}\mathrm{n}+1/\mathrm{C}\mathrm{s}\mathrm{p}+1/\mathrm{C}\mathrm{s}\mathrm{m}$$(15)

  

$$1/\mathrm{C}\mathrm{s}=1/\mathrm{C}\mathrm{n}+1/\mathrm{C}\mathrm{d}+1/((\mathrm{m}-\mathrm{n})\mathrm{C}\mathrm{n}\mathrm{s}+\mathrm{C}\mathrm{d}\mathrm{m}+\mathrm{C}\mathrm{s}\mathrm{o}\mathrm{m})=1/\mathrm{C}\mathrm{n}+1/\mathrm{C}\mathrm{d}+1/\mathrm{\Sigma }\mathrm{C}\mathrm{m}$$(16)

The concentration of Ca²⁺ ions inside the spine that holds Ca²⁺ ions is almost the same as that outside the cell, so the potential is the same inside and outside. Also, the spine membrane capacitance Csm is very small, about 1/100, compared to the series membrane capacitance Cs via the spine neck capacitance Cn, so the effect of the membrane capacitance Csm on spines that hold excess Ca²⁺ ions can be ignored. Therefore, the electric field E due to the charge Q inside the spine is analyzed for the path that diffuses outside the cell only via the series capacitance Cs. Therefore, the potential difference V between the inside of the spine and the outside of the cell due to the charge Q inside the spine is obtained from equations (3) and (16).

  

$$\mathrm{V}=\mathrm{Q}/\mathrm{C}\mathrm{s}=\mathrm{Q}(1/\mathrm{C}\mathrm{n}+1/\mathrm{C}\mathrm{d}+1/\mathrm{\Sigma }\mathrm{C}\mathrm{m})=\mathrm{Q}/\mathrm{C}\mathrm{n}+\mathrm{Q}/\mathrm{C}\mathrm{d}+\mathrm{Q}/\mathrm{\Sigma }\mathrm{C}\mathrm{m}=\mathrm{V}1+\mathrm{V}2+\mathrm{V}3$$(17)

When n spines, which have the same charge Q, are connected in parallel to the axial capacitance Cd of the dendrite, the potential difference V between the inside of the spine and the outside of the cell is as shown in Figure 3 [3].

  

$$\begin{array}{cc}\mathrm{V}\hfill & =\mathrm{n}\mathrm{Q}/\mathrm{\Sigma }\mathrm{C}\mathrm{s}=\mathrm{n}\mathrm{Q}(1/\mathrm{n}\mathrm{C}\mathrm{n}+1/\mathrm{C}\mathrm{d}+1/((\mathrm{m}-\mathrm{n})\mathrm{C}\mathrm{n}\mathrm{s}+\mathrm{C}\mathrm{d}\mathrm{m}+\mathrm{C}\mathrm{s}\mathrm{o}\mathrm{m}))\hfill \\ & =\mathrm{Q}/\mathrm{C}\mathrm{n}+\mathrm{n}\mathrm{Q}/\mathrm{C}\mathrm{d}+\mathrm{n}\mathrm{Q}/\mathrm{\Sigma }\mathrm{C}\mathrm{m}=\mathrm{V}1+\mathrm{V}2+\mathrm{V}3\hfill \end{array}$$(18)

If A dendrite with similar spines are connected to the cell body, the total membrane capacitance ΣCm increases and the total charge ΣQ also increases to AnQ, so the potential difference V is as follows.

  

$$\begin{array}{cc}\mathrm{V}\hfill & =\mathrm{A}\mathrm{n}\mathrm{Q}/\mathrm{\Sigma }\mathrm{C}\mathrm{s}=\mathrm{A}\mathrm{n}\mathrm{Q}(1/\mathrm{A}\mathrm{n}\mathrm{C}\mathrm{n}+1/\mathrm{A}\mathrm{C}\mathrm{d}+1/\left((\mathrm{A}(\mathrm{m}-\mathrm{n})\mathrm{C}\mathrm{n}\mathrm{s}+\mathrm{A}\mathrm{C}\mathrm{d}\mathrm{m}+\mathrm{C}\mathrm{s}\mathrm{o}\mathrm{m})\right)\hfill \\ & =\mathrm{Q}/\mathrm{C}\mathrm{n}+\mathrm{n}\mathrm{Q}/\mathrm{C}\mathrm{d}+\mathrm{A}\mathrm{n}\mathrm{Q}/\mathrm{\Sigma }\mathrm{C}\mathrm{m}=\mathrm{Q}/\mathrm{C}\mathrm{n}+\mathrm{n}\mathrm{Q}/\mathrm{C}\mathrm{d}+\mathrm{\Sigma }\mathrm{Q}/\mathrm{\Sigma }\mathrm{C}\mathrm{m}=\mathrm{V}1+\mathrm{V}2+\mathrm{V}3\hfill \end{array}$$(19)

For the charge of Ca²⁺ ions in the spine, V1 is the potential difference across the spine neck capacitance Cn, V2 is the potential difference between the spine neck capacitance Cn and the total membrane capacitance ΣCm, and V3 is the potential difference across the total membrane capacitance ΣCm. In other words, V3 is the increase in the membrane potential inside the cell, which is determined by the total charge amount ΣQ and the total membrane capacitance ΣCm in the spine relative to the outside of the cell. Therefore, the resting potential inside a normal neuron is –65 mV, and the threshold of the voltage-gated Na⁺ ion channel that generates an action potential is –50 mV to –40 mV. If the voltage Vup of the increase in membrane potential caused by V3, then if the voltage Vup is 15 mV to 25 mV or more, the voltage-gated Na⁺ ion channel will operate and an action potential will be generated. If the spine neck capacitance Cn is the same and the charge Q held in the spine is the same, the charge ΣQ will increase in proportion to the number of spines holding charges. Therefore, if the resting potential is –65 mV and the threshold of the voltage-gated Na⁺ ion channel is –45 mV, the condition for an action potential to be generated by the sum of ΣQ Coulombs of multiple spines holding charge Q is as follows [3].

  

$$\mathrm{\Sigma }\mathrm{Q}/\mathrm{\Sigma }\mathrm{C}\mathrm{m}\ge \mathrm{V}\mathrm{u}\mathrm{p}=-45\mathrm{–}\left(-65\right)=20\mathrm{m}\mathrm{V}$$(20)

That is, the total charge ΣQ (coulombs) required for synaptic integration to occur is calculated by the following equation.

  

$$\mathrm{\Sigma }\mathrm{Q}\ge \mathrm{V}\mathrm{u}\mathrm{p}\times \mathrm{\Sigma }\mathrm{C}\mathrm{m}=20\mathrm{m}\mathrm{V}\times \mathrm{\Sigma }\mathrm{C}\mathrm{m}$$(21)

If each spine holds the same amount of charge, the total charge ΣQ coulombs that satisfy formula (21) divided by the charge per spine gives the number of spines in the Ca²⁺ ion compartment state required for synaptic integration to occur. Immediately after synaptic integration occurs, even if the charge of the Ca²⁺ ions in the spine moves to the dendrite side, the charge in the total membrane capacity ΣCm is the same, so the potential of V3 is the same. Therefore, the output action potential pulse continues for the period during which the total charge ΣQ satisfies equation (21). The excess Ca²⁺ ions that move to the dendrite side are expelled by the Ca²⁺ ion channels in the dendritic membrane, and equation (21) is no longer satisfied, and the output stops.

 Verification of Synaptic Integration Conditions by Concrete Values

The rise in membrane potential required for the synaptic integration can be expressed as the ratio of the total membrane capacitance ΣCm to the total charge ΣQ of the spine that holds Ca²⁺ ions, as shown in equation (20). To verify whether this relationship holds in actual neurons, it is necessary to apply specific values. For this purpose, we referred to a representative hippocampal CA1 pyramidal cell that has a well-studied dendrite with a diameter of 1.0 μm and a length of 130 μm [9,11]. As shown in the reference for details [3], the spine membrane capacitance Csm was calculated to be 0.009 pF, the spine capacitance Csp was 32 pF, the axial capacitance Cd of the dendrite was 10.7 pF, the membrane capacitance Cdm of the dendrite was 5.45 pF, and the membrane capacitance Csom of the soma with a diameter of 20 μm was 11.3 pF. The important spine neck capacitance Cn is set to 0.45 pF to 0.9 pF, because the ratio of ionic fluid in the spine neck volume is low and the resistance of the spine neck is 500 MΩ to 1 GΩ, which is much higher than the resistance of the axon (140 MΩ/cm). First, author will verify it at 0.45 pF, and later consider the case of 0.9 pF. With these capacitances, the series capacitance Cs is 0.42 pF and the series membrane capacitance Cns is 0.0088 pF [3].

Therefore, the total membrane capacitance ΣCm, including the 130 μm dendrite with 130 spines connected at 1 μm intervals, and a soma is 17.8 pF. Assuming that the potential inside the spine that has taken up Ca²⁺ ions rises from the resting potential of –65 mV to 0 mV, the same as outside the cell, within the spine will be 65 mV higher than the potential inside the cell. Therefore, by taking in Ca²⁺ ions, the spine will hold an excess of 2.73×10^–14q (coulombs), which is the product of 65 mV and the series capacitance Cs of 0.42 pF. Since the total membrane capacitance ΣCm is 17.8 pF, the total charge ΣQ required to establish synaptic integration is 3.56×10^–13q from equation (20). Therefore, the number of spines holding Ca²⁺ ions required for synaptic integration to be established can be calculated using the following equation.

  

$$\mathrm{\Sigma }\mathrm{Q}/\mathrm{q}/\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{e}=3.56\times {10}^{-13}\mathrm{q}/2.73\times {10}^{-14}\mathrm{q}/\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{e}=13.0\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}$$(22)

In other words, for synaptic integration to be established, 13 or more spines out of 130 spines must be in a Ca²⁺ ion compartment state holding Ca²⁺ ions. Therefore, if the spine neck capacitance Cn is 0.9pF, 7 or more will be required.

Some pyramidal cells in the hippocampal CA1 have about 30 dendrites. If each dendrite is the same length and has the same number 130 of spines connected to it, there will be a total of 3900 spines. Therefore, the total membrane capacitance ΣCm is calculated from equation (19).

  

$$\begin{array}{cc}\mathrm{\Sigma }\mathrm{C}\mathrm{m}\hfill & =\mathrm{A}(\mathrm{m}-\mathrm{n})\mathrm{C}\mathrm{n}\mathrm{s}+\mathrm{A}\mathrm{C}\mathrm{d}\mathrm{m}+\mathrm{C}\mathrm{s}\mathrm{o}\mathrm{m}=30\times (130-13)\times 0.0088\mathrm{p}\mathrm{F}+30\times 5.45\mathrm{p}\mathrm{F}+11.3\mathrm{p}\mathrm{F}\hfill \\ & =205.69\mathrm{p}\mathrm{F}\hfill \end{array}$$(23)

The total charge ΣQ required for synaptic integration is calculated from equation (21) to be 4.11×10^–12 coulombs. Since the total membrane capacitance ΣCm has increased from 17.8 pF to 205.7 pF, the series capacitance Cs has increased from 0.42 pF to 0.43 pF, so the charge that one spine can hold is 2.795×10^–14q. Therefore, the number of spines that hold Ca²⁺ ions required for synaptic integration to occur is

  

$$\mathrm{\Sigma }\mathrm{Q}/\mathrm{q}/\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{e}=4.11\times {10}^{-12}\mathrm{q}/2.795\times {10}^{-14}\mathrm{q}/\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{e}=147.0\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}$$(24)

In other words, for synaptic integration to occur, 147 or more spines out of 3900 spines must be in a Ca²⁺ ion compartment state that holds Ca²⁺ ions. Therefore, if the spine neck capacitance Cn is 0.9 pF, 74 or more spines are required. If the number of spines in the Ca²⁺ ion compartment state exceeds the number of conditions for synaptic integration, the output period of the pulsed action potential becomes longer according to the amount of excess. That is, in a neuron with one dendrite with 130 spines, synaptic integration occurs when from 7 to13 spines hold Ca²⁺ ions, and in a neuron with 30 dendrites and 3,900 spines, synaptic integration occurs when from 74 to147 spines hold Ca²⁺ ions. This means that a neuron with 130 spines can handle an almost infinite number of combinations of inputs, with from ₁₃₀C₇ to ₁₃₀C₁₃ (from 1.06×10¹¹ to 2.62×10¹⁷), and a neuron with 3900 spines can do as same with from ₃₉₀₀C₇₄ to ₃₉₀₀C₁₄₇ (from 8.3×10¹⁵⁷ to 2.75×10²⁷⁰). In addition, because the total membrane capacitance ΣCm is small in small neurons, synaptic integration can be achieved even if the number of spines holding Ca²⁺ion or the amount of charge that the spines can hold is small [3].

 Elucidation of Existing Questions by This Approach

The above analysis revealed that several spines in the Ca²⁺ ion compartment state enable to the establishment of synaptic integration by inputs with a time lag. In addition, this analytical approach has enabled us to clarify the following important questions that had not been clarified until now.

First, equation (20) can be applied to the total charge ΣQ not only for the charge of Ca²⁺ ions, but also for the charge of activated proteins such as long-term potentiation (LTP) [31] and the sum of EPSPs due to Na⁺ ions. However, since EPSPs are puls-like, there are strict time constraints. In other words, the total charge ΣQ is the sum of excess positive charges within the area of the total membrane capacitance ΣCm.

Second, the relationship between the shape of the spine head and spine neck and their effects had not been specifically clarified until now [32–34]. This analysis revealed that the contribution of spines to the establishment of synaptic integration is proportional to the amount of Ca²⁺ ion charge that can be accumulated. It was also verified that the determining factor in the ability of spines to store Ca²⁺ ion charges is the spine neck capacitance Cn. The spine neck capacitance Cn is determined by the cross-sectional area and length of the spine neck, as shown in equation (4). Therefore, it became clear that spines with thick and short spine necks can store a large amount of charge. On the other hand, the voltage V in a spine with a thin and long spine neck increases significantly even with a small amount of Ca²⁺ ion charge Q according to the equation (3) because the capacitance C is small. Next, it was found that when a spine takes in and stores a large amount of Ca²⁺ ions, the Ca²⁺ ion concentration in the spine increases, so a spine head with a large volume is required to keep it below the concentration outside the neuron. Incidentally, as shown in the reference [3], when one spine stores a charge of 2.73×10^–14 coulombs, the Ca²⁺ ion concentration in the spine was calculated to be 1.42 mM compared to the concentration outside 2 mM. However, this is the case when the entire volume of the spine head is filled with ionic fluid, and if about 70% of the volume of the spine head is filled with ionic fluid, it is thought that the concentration will be approximately the same as that on the outside.

Third, it was unclear whether there was any difference in the effect of the position where the spine that holds Ca²⁺ ions connects to the long, thin dendrite. As mentioned above, contribution for the establishment of synaptic integration depends on the amount of charge Q in each spine. Therefore, we investigated the relationship between the series capacitance Cs of each spine, which is the element that stores charge, and the position where the spine connects to the dendrite. The details of the calculation are described in the reference material [3], but the change was very small, at 0.43 pF, 0.42 pF, and 0.41pF, at distances from the cell body of 1 μm, 100 μm, and 200 μm. In other words, it was found that the position where the spine connects to the dendrite is not greatly affected by the distance from the cell body [3]. Therefore, it was understood that many spines can be connected to the tip of a long, thin dendrite.

Fourth, because a 1 μm-diameter dendrite has more resistance and leakage current than a 20 μm-diameter axon with a thick myelin sheath, we investigated the extent to which puls-like wave such as EPSP and the backpropagation of action potential are attenuated by propagating through the dendrite. The details of the calculations are described in the reference [3], but the attenuation of the amplitude of these pulse waveforms was 0.92, 0.74, 0.56, and 0.41 at distances of 50 μm, 100 μm, 150 μm, and 200 μm from the source. Therefore, it seems reasonable that the length of each dendrite of a CA1 pyramidal cell is about 130 μm [9,31].

Fifth, it remained unclear why the backpropagation waveform of action potentials is transmitted from the parent dendrite through the spine head with almost no attenuation [30]. Considering the impedance Zc (=1/ωC) of each capacitance for a pulse waveform with changing voltage, the ratio of the series impedance (Zcn+Zcsp+Zcsm) from the parent dendrite to the outside of the cell via the spine to the impedance Zcsm due to the membrane capacitance Ccsm of the spine membrane is the ratio of the voltage V of the parent dendrite to the voltage Vb inside the spine. Therefore, it was calculated that there is only 2% attenuation as shown below [31].

  

$$\begin{array}{cc}\mathrm{V}\mathrm{b}/\mathrm{V}\hfill & =\mathrm{Z}\mathrm{c}\mathrm{s}\mathrm{m}/(\mathrm{Z}\mathrm{c}\mathrm{n}+\mathrm{Z}\mathrm{c}\mathrm{s}\mathrm{p}+\mathrm{Z}\mathrm{c}\mathrm{s}\mathrm{m})=1/\mathrm{C}\mathrm{s}\mathrm{m}/(1/\mathrm{C}\mathrm{n}+1/\mathrm{C}\mathrm{s}\mathrm{p}+1/\mathrm{C}\mathrm{s}\mathrm{m})\hfill \\ & =1/0.009\mathrm{p}\mathrm{F}/(1/0.45\mathrm{p}\mathrm{F}+1/32\mathrm{p}\mathrm{F}+1/0.009\mathrm{p}\mathrm{F})=111.11/113.36=0.98\hfill \end{array}$$(25)

The fact that the backpropagation waveform of the action potential reaches the spine through the spine neck with little attenuation suggests that it is useful for generating the plasticity of the spine, since it is possible to recognize that each spine that holds a charge contributed to the establishment of synaptic integration.

 Effectiveness and Release of the Compartment State of Ca²⁺ Ions

Neurons have four types of pumped-storage systems that use ion pumps and ion channels corresponding to each ion. In particular, for Ca²⁺ ions, there are two stage pumped-storage ponds. The first stage pond is outside the neuron and the second stage pond is inside the spine. The accumulation of Ca²⁺ ions in the spine, which is the second stage pond, serves a role of remembering that the spine has received an input, and of continuing to raise the membrane potential by the charge of the accumulated Ca²⁺ ions, by an effective energy-saving means.

However, if the compartment state of Ca²⁺ ions in spines is not released correctly, the number of spines in this state will increase one after another every time an output is generated due to the establishment of synaptic integration, and the number of neurons that do not function properly will rapidly increase. Therefore, it is important that the compartment state of Ca²⁺ ions is released, but the mechanism has not been clarified. On the other hand, in order for synaptic integration to be established by multiple inputs over a long period of time, the compartment state of Ca²⁺ ions must continue for a long period of time. It is speculated that the most convenient timing to release the compartment state to meet the above two needs is when the backpropagation waveform of the action potential generated by the establishment of synaptic integration reaches the spine. This is because after synaptic integration is established and the purpose is achieved, the nerve cell can function correctly again in response to subsequent inputs by initializing all spines of the nerve cell. In other words, it seems that the Ca²⁺ ion channel in the spine neck senses the potential of the backpropagation waveform of the action potential generated by the establishment of synaptic integration, opens the channel, and moves the excess Ca²⁺ ions in the spine to the dendrite side, which is a natural sequence in terms of energy conservation. Therefore, when the backpropagation waveform of the action potential occurs, it is thought that the voltage transitions of the parent dendrite and the voltage in the spine will match even after they match [31]. Therefore, elucidating the mechanism of the release of the compartment state of this Ca²⁺ ion is an important issue. However, since the charged particles passing through the spine neck are Ca²⁺ ions, not electrons, it is necessary to focus on the change in the permeability of the spine neck to Ca²⁺ ions, rather than the change in the direct current resistance of the spine neck. In addition, from equation (20), LTP by proteins activated with positive charges in the spine also contributes to the establishment of synaptic integration [35], but the relationship between the initialization of the spine in the LTP state and the backpropagation waveform of the action potential is unclear.

 C. Elucidation of Information Processing Functions by Neural Circuits

In digital circuits such as computers, the basic functions required for information processing are logical operation functions such as NAND, AND, OR, and exclusive OR (EOR) [36] and data storage functions such as flip-flops. Digital circuits perform repeatable and accurate operations according to the circuit, and continuously output a level of 0 or 1. On the other hand, there are two types of neurons: excitatory neurons that output excitatory neurotransmitters and inhibitory neurons that output inhibitory neurotransmitters [12]. However, the output of either neuron is pulsed and does not last long. In addition, either neuron will not generate an output unless the conditions for synaptic integration are met. Therefore, it seems difficult to build a repeatable circuit by combining neurons that do not output continuously or that may not output even when there is input. However, the cerebrum shows relatively the same response to the same situation. Therefore, author investigated how the cerebrum realizes repeatable the functions of information processing [4]. Fortunately, the results of the immediately preceding study, presented in previous section, induced to elucidate this mechanism too, as shown below.

 Quantification of the Inhibitory Effect of Inhibitory Neurons

In recent years, it has become known that there are a large number of inhibitory neurons and that there are many types of them depending on their purposes. It is also known that the output destinations of inhibitory neurons have synaptic connections to dendrites, somas, spines, and axon terminals, etc. [37–39]. However, the inhibitory effect of inhibitory neurons on the establishment of synaptic integration has not been quantitatively demonstrated until now. In contrast, the condition for the establishment of synaptic integration shown in equation (20) in the previous chapter also includes the inhibitory effect of inhibitory neurons. In other words, the condition for the establishment of synaptic integration is that the rate of the total positive charge ΣQ to the total membrane capacitance ΣCm exceeds a specified value [3]. This means that the establishment of synaptic integration can be suppressed by inserting a negative charge -Q into the total positive charge ΣQ. Therefore, it can be said that the inhibitory receptors on the membrane of the dendrites or soma receive inhibitory neurotransmitters from other inhibitory neurons, and by taking in negatively charged Cl^– ions into the neuron, the total positive charge ΣQ can be cancelled, thereby inhibiting the establishment of synaptic integration [4]. Therefore, if the charge of the Cl^– ion is Q_Cl^– and the sum is ΣQ_Cl^–, equation (20) becomes the following equation. However, the resting potential is –65 mV, and the threshold of the voltage-gated Na⁺ ion channel is –45 mV.

  

$$(\mathrm{\Sigma }\mathrm{Q}-\mathrm{\Sigma }{\mathrm{Q}}_{{\mathrm{C}\mathrm{l}}^{-}})/\mathrm{\Sigma }\mathrm{C}\mathrm{m}\ge 20\mathrm{m}\mathrm{V}$$(26)

This inhibitory effect continues until the excess Cl^– ions taken in are expelled by the ion pump. The Cl^– ion concentration inside the neuron is about 1/10 or less than that outside (Table 1) and the Cl^– ion retention capacity inside the dendrite is proportion to the total membrane capacity ΣCm and the volume of the dendrite and cell body, so it is possible to exert an inhibitory effect on synaptic integration for a long period of time. Furthermore, the amount of Cl^– ions taken up by the inhibitory receptors of individual spines or axon terminals is considered to be the amount that inhibits only that spine or axon terminal. As above, the equation (26) induced from the equation (20) makes it possible to quantitatively demonstrate the inhibitory effect of inhibitory neurons.

 Circuit Symbols and Operation Rules for Basic Elements

As mentioned above, digital circuits can perform all the functions required for information processing using logical operations such as NAND, AND, OR, and exclusive OR (EOR) and basic functions such as flip-flop type data storage. These basic functions can be realized by combining only NAND elements. Therefore, circuits in general Large Scale Integration (LSI) chips are made up of a huge number of NAND elements. Figure 4 shows the circuit symbols and functions of digital elements, and Figure 5 shows the circuit symbols and functions of neurons. In Figure 5, the short lines connected vertically to the dendrites are spines, all of which have the same characteristics. When a spine receives input, the effective number of the spine increases by one. The number inside the soma is the threshold, and when the sum of the effective numbers by several spines exceeds the threshold, a pulse-like output is output from the axon terminal. The duration of the pulse is proportional to the amount by which the sum of the effective numbers exceeds the threshold. The circles on the dendrites, spines, soma, and axon terminals represent inhibitory receptors that receive output from inhibitory neurons [4]. When an inhibitory receptor receives input, the negative effective number –1 increases by one. If the sum of the positive effective numbers and the negative effective numbers is equal to or greater than the threshold value, an output is produced [4]. If the effective number for the same input is two or more, multiple spines (lines) or inhibitory receptors (circles) are connected in parallel to represent it. The total membrane capacitance ΣCm of a neuron is a key factor in the establishment of synaptic integration. The total membrane capacitance ΣCm tends to be proportional to the size of the neuron. The threshold number in the soma in Figure 5 reflects that total membrane capacitance ΣCm [4].

Figure 4  Circuit symbols and operations of digital circuit. (a); AND circuit outputs 1, only when all inputs are 1. (b); NAND circuit outputs 0, only when all inputs are 1.

Figure 5  Circuit symbols and operations of neural circuits. (a); Excitatory neuron outputs an excitatory neurotransmitters when the conditions for synaptic integration are met. (b); Inhibitory neuron outputs inhibitory neurotransmitters when the conditions for synaptic integration are met. In both circuits, short line segments connected to dendrite represent spines. Number in the soma indicates thresholds. Circles on spines, dendrites, soma, and axon terminals represent inhibitory receptors those receive the output of inhibitory neurons [4].

 Comparison of Basic Operations by Digital Circuits and Neural Circuits

Figure 6 shows the truth table of basic logical operations of digital circuits and each basic logical operation represented by a combination of NAND elements. The logical operations of NAND, AND, and OR are well known. On the other hand, exclusive OR (EOR) is a logical operation that is 1 only when two inputs are different, and is an essential logical operation for detecting data inconsistencies and arithmetic operations.

Figure 6  Digital circuits for basic operations. (a); Truth table for each basic logical operation. (b); Digital circuit for each basic logical operation using NAND elements [4].

Figure 7 shows the truth table for each operation type of the neural circuit and each operation represented using the circuit symbols of excitatory neurons and inhibitory neurons [4].

Figure 7  Neural circuits for basic operations. (a); Truth table for each operation. (b); Neural circuit for each operation.

As shown in Figure 7, neural circuits have two types of neurons, excitatory neurons and inhibitory neurons, and thus realize the basic operations required for information processing, although they are specific. In addition, all operations are determined by the threshold value written in the soma and the effective number of positive and negative inputs. Therefore, it is speculated that there exist unique neurons with threshold values corresponding to various purposes and corresponding spines and inhibitory receptors.

 Examples of Encoder and Decoder using Digital and Neural Circuits

It is said that the cerebrum has about 16 billion neurons [40]. Although it is said that an amount that cannot be used in a lifetime, but if one neuron is considered to be 1 bit, it is only 2 gigabytes. Recently, there are inexpensive USB memory sticks on the market that have a capacity of 4terabytes, which is about 2000 times of the number of neuron of the cerebrum. Therefore, it is thought the brain utilizes neurons effectively. In computer data processing, a method is often used in which data is encoded in binary, utilized effectively, and finally decoded back to the original. For example, if the audible range of sound is up to 20,000 Hz, encoding is a method that uses only 15 neurons (2¹⁵) corresponding to the exponent of the binary exponential function that represents more than 20,000, rather than using 20,000 neurons. Therefore, the author tried to see if we could build at least such a binary coding circuit using neurons. The encoder circuits for digital and neural circuits are shown in Figure 8 (a) and (b). Next, the decoder circuits for the digital circuit and neural circuit are shown in Figure 9 (a) and (b).

Figure 8  Encoder circuit. (a); Digital encoder circuit. (b); Neural encoder circuit [4].

Figure 9  Decoder circuit. (a); Digital decoder circuit. (b); Neural decoder circuit [4].

In the cerebrum, the encoding function is suitable for generating data that efficiently sends olfactory, gustatory, auditory, visual, tactile information [12,11] and existing memory information to the required areas, while the decoding function is suitable for selecting specific motor neurons from the encoded data in the motor cortex [12,13]. However, as can be seen from the circuit, in order for the output of the encoder and decoder to be stable, the data input to the circuit must be stable. The input data of a digital circuit is stable between clocks. Therefore, like the clock of a digital circuit, the data input to the encoder and decoder in neurons must be synchronized. For this reason, it is speculated that the existence of neurons for the clock signal distributed to many cells is necessary. This suggests a relationship between the synchronized output of neurons due to such a synchronization signal and brain waves [4].

 Circuit for Drawing Image Contours (Application Example)

To recognize an image, the first step is to capture the contour of the image. To draw a contour, it is necessary to connect the points where the brightness of the target image changes. To do this, it is necessary to compare the brightness of the target pixel with the adjacent pixels. To compare the inputs, it is necessary to use the EOR operation to compare the difference in brightness between the two inputs. In reality, it is necessary to compare (EOR) the target pixel with the adjacent pixels above, below, left and right. First, the plane on which the image exists is defined as the V plane, with the

In the cerebrum, the encoding function is suitable for generating data that efficiently sends olfactory, gustatory, auditory, visual, tactile information [12,11] and existing memory information to the required areas, while the decoding function is suitable for selecting specific motor neurons from the encoded data in the motor cortex [12,13]. However, as can be seen from the circuit, in order for the output of the encoder and decoder to be stable, the data input to the circuit must be stable. The input data of a digital circuit is stable between clocks. Therefore, like the clock of a digital circuit, the data input to the encoder and decoder in neurons must be synchronized. For this reason, it is speculated that the existence of neurons for the clock signal distributed to many cells is necessary. This suggests a relationship between the synchronized output of neurons due to such a synchronization signal and brain waves [4].

 Circuit for Drawing Image Contours (Application Example)

To recognize an image, the first step is to capture the contour of the image. To draw a contour, it is necessary to connect the points where the brightness of the target image changes. To do this, it is necessary to compare the brightness of the target pixel with the adjacent pixels. To compare the inputs, it is necessary to use the EOR operation to compare the difference in brightness between the two inputs. In reality, it is necessary to compare (EOR) the target pixel with the adjacent pixels above, below, left and right. First, the plane on which the image exists is defined as the V plane, with the x axis defined as horizontal and the y axis defined as vertical. If the xy coordinates of the target pixel are V(n,n), the coordinates of the pixels above, below, left and right of the target pixel are V(n,n+1), V(n,n-1), V(n-1,n), and V(n+1,n), respectively. Then, using the EOR operation, the comparison result between the target pixel of the image on the V plane and the pixels above, below, left and right, and right and left are projected to the coordinate W(n,n) on the W plane. Figure 10 shows a circuit that compares the target pixel with the pixels above, below, left and right when the target pixel in the V plane shown in Figure 11 is 1. Figure 10(a) is a digital circuit, while Figures 10(b) and 10(c) are neural circuits. Figure 10(b) shows that there are inhibitory receptors in each spine as an EOR operation, and Figure 10(c) shows that there are four inhibitory receptors in the dendrite as a subtraction operation [4].

Figure 10  Contour detection circuits. (a); Digital contour detection circuit using an EOR function. (b); Neural contour detection circuit using an EOR function. (c); Neural contour detection circuit using a subtraction function [4].

Figure 11  Images of a square, a cross, a triangle, and a diagonal line on the V-plane [4].

The result of a binary logic operation by a digital circuit is ultimately either 1 or 0. However, the result of an operation by a neural circuit may exceed 1 because arithmetic operations are performed with the output of an excitatory neuron as 1 and the output of an inhibitory neuron as –1. The following equation faithfully reflects the digital circuit in Figure 10 (a) in binary logic. The superscript bars indicate inverters. Therefore, by applying De-Morgan’s law [41], we can express it in an easy-to-calculate form as equation (27).

If any of the pixels above, below, left, or right are 0, the coordinate W (n, n) is 1.

  

$$\begin{array}{cc}\mathrm{W}(\mathrm{n},\mathrm{n})\hfill & =\overline{\overline{(\mathrm{V}(\mathrm{n},\mathrm{n})·\overline{V(n,n+1)})}·\overline{(\mathrm{V}(\mathrm{n},\mathrm{n})·\overline{V(n,n-1)})}·\overline{(\mathrm{V}(\mathrm{n},\mathrm{n})·\overline{V(n-1,n)})}·\overline{(\mathrm{V}(\mathrm{n},\mathrm{n})·\overline{V(n+1,n)})}}\hfill \\ & =(\mathrm{V}(\mathrm{n},\mathrm{n})·\overline{\mathrm{V}(\mathrm{n},\mathrm{n}+1)}+(\mathrm{V}(\mathrm{n},\mathrm{n})·\overline{\mathrm{V}(\mathrm{n},\mathrm{n}-1)}+(\mathrm{V}(\mathrm{n},\mathrm{n})·\overline{\mathrm{V}(\mathrm{n}-1,\mathrm{n})}+(\mathrm{V}(\mathrm{n},\mathrm{n})·\overline{\mathrm{V}(\mathrm{n}+1,\mathrm{n})}\hfill \end{array}$$(27)

The following equation is for neural circuits. Since the superscripted bars indicate inhibitory neurons of –1, arithmetic operation is executed by +1 and –1.

  

$$\mathrm{W}(\mathrm{n},\mathrm{n})=(\mathrm{V}(\mathrm{n},\mathrm{n})+\overline{\mathrm{V}(\mathrm{n},\mathrm{n}+1)})+(\mathrm{V}(\mathrm{n},\mathrm{n})+\overline{\mathrm{V}(\mathrm{n},\mathrm{n}-1)})+(\mathrm{V}(\mathrm{n},\mathrm{n})+\overline{\mathrm{V}(\mathrm{n}-1,\mathrm{n})})+(\mathrm{V}(\mathrm{n},\mathrm{n})+\overline{\mathrm{V}(\mathrm{n}+1,\mathrm{n})})$$(28)

  

$$\begin{array}{cc}& =4\times \mathrm{V}(\mathrm{n},\mathrm{n})-(\mathrm{V}(\mathrm{n},\mathrm{n}+1)+\mathrm{V}(\mathrm{n},\mathrm{n}-1)+\mathrm{V}(\mathrm{n}-1,\mathrm{n})+\mathrm{V}(\mathrm{n}+1,\mathrm{n}))\hfill \end{array}$$(29)

Equation (28) is for the neural circuit in Figure 10 (b), and equation (29) is for the neural circuit in Figure 10 (c). If the threshold number is 1 and the number of inhibitory neurons for the negative effective number-1 of pixels on the top, bottom, left, and right are 0, 1, 2, 3, and 4, the above equations are arithmetic calculations, so when V(n,n) is 1, the output of W(n,n) is 4, 3, 2, 1, and 0. The V plane in Figure 11 shows a square, triangle, cross, and diagonal image. Figure 12(a) shows the W plane, which is the result of applying the digital circuit equation (27) to all pixels in the V plane. Figure 12 (b) shows the W plane, which is the result of applying the neuron circuit equations (28) and (29) to all pixels in the V plane. Both calculations show the same contour. The output of the contour (W plane) calculated by the digital circuit is all 1. On the other hand, the output of the contour (W plane) calculated by the neuron circuit varies depending on the location of the image. In particular, the output of the diagonal line, corner, and tip of the V plane image is strong. In this way, neurons perform mixed operation with EOR and arithmetic operations. In addition, when the target is a lot of image data that changes rapidly, such as visual images, it is thought that the output due to the sum of the effective number of EPSPs generated almost simultaneously by the Na⁺ ion charges from multiple spines contributes. In particular, the contour of an object that expands, shrinks, and moves is thought to be due to the sum of the EPSP charges [4].

Figure 12  Contours on the W plane. (a); Contours generated by a digital circuit. (b); Contours generated by a neural circuit [4].

As described above, information can be processed by the basic arithmetic functions of neural circuits, but they have their own unique characteristics. Interestingly, the contour drawn by the neural circuit in Figure 12(b) resembles the response characteristics of complex cells in the striate cortex of the monkey visual cortex [42].

 Discussion

 Well-Designed Ionic System Achieving Fast Conduction Velocity on Action Potential

Neurons have an excellent ionic system consisting of various highly functional proteins such as ion pumps, ion channels, and cellular components, and electrical circuits that utilize them to realize important functions. Author has investigated this ionic system from the standpoint of electrical circuit design, and was surprised at how well designed it was. In order to realize high-speed conduction of action potentials in axons, neurons utilize a displacement current I, which propagates rapidly in the axial direction instead for ions with slow diffusion speeds. As shown in equation (6), the displacement current I is generated in proportion to the product of the capacitance C and the voltage changing speed (dV/dt). For this reason, voltage-gated Na⁺ ion channels are present at an ultra-high density of 12,000/μm² [27] at the node of Ranvier, and a large amount of Na⁺ ions is injected into the axon at once, generating action potentials with a large voltage changing speed (dV/dt). And, the product of the action potential and the very large axon capacitance C1, which contains the axon fluid with an extraordinarily large relative dielectric constant ε_r, which is more than a million times to the general medium, generates a large displacement current I in the propagation direction of the action potential.

However, if the membrane capacitance C2 of the axon membrane is large, a large displacement current I (leakage current) toward outside is generated between it with the action potential with a large voltage change rate (dV/dt), resulting in the inconvenience of attenuating the action potential. In response to this, in order to reduce the capacitance C2 of the cylindrical axon membrane, a thick myelin sheath is wrapped around the axon based on the principle of equation (5), increasing the ratio of the outer diameter to the inner diameter of the axon, thereby reducing the axon membrane capacitance C to about 1/670, and greatly reducing the leakage current from the axon membrane [22]. Moreover, such axon structures were realized by ultra-high density packaging technology that exceeds imagination. However, from the standpoint of designing an electric circuit, it was able to be predicted a little that such axon morphology is suitable for realizing high-speed conduction of action potentials. However, the extraordinarily large relative dielectric constant ε_r of the axon fluid could not be predicted at all. Fortunately, however, author was able to derive an extraordinarily large value for the relative dielectric constant ε_r by a deductive analytical approach using the wave equation, and then it was able to be verified the validity of this value [1,24,25].

 The Relation between Two Factors that determine Membrane Potential with The Sleep

The resting potential of a neuron is due to the thermal motion of ions, and is calculated using Goldman’s equation, which takes into account the concentration difference between inside and outside the neuron, temperature, and relative membrane permeability, as shown in Table 1 [17]. On the other hand, as shown in equation (1) derived from Coulomb’s law, when an electric charge Q exists, an electric field E is generated around the charge Q, which is inversely proportional to the dielectric constant ε of the surrounding medium [18]. The potential due to this electric field E and the resting potential due to the thermal motion of ions are generated independently. Therefore, when calculating the potential inside a neuron, the effects of both must be calculated separately. Therefore, when calculating the membrane potential of a cell membrane in which a voltage-gated Na⁺ ion channel that generates an action potential exists, the resting potential inside the neuron and the increase in potential due to the electric field E created by the electric charge Q held by the spine must be considered. In other words, equation (20), which is the condition for synaptic integration, is a mathematical expression of the effects of both. However, if action potentials continue to occur frequently, the Na⁺ ion concentration will increase near the voltage-gated Na⁺ ion channels due to the Na⁺ ions taken in, and the K⁺ ion concentration will decrease near the K⁺ ion channels due to the K⁺ ions expelled. Therefore, according to the Goldman’s equation, the potential around these channels will gradually increase, so Vup in equation (20) will become smaller and it will become easier to establish synaptic integration. However, if this state continues, the balance of ion concentrations for the rest potential will be greatly disrupted and neurons will no longer function properly. To prevent this state, it is necessary to keep inputs such as perception under suppressed, and to consume ATP energy to activate the Na⁺/K⁺ ion pumps of each neuron whose ion concentration has changed significantly due to frequent occurrences of action potentials that day, and to restore the concentration at the resting potential state. The author speculates that sleep plays an important role as an effective recovery means for this. However, the relationship between the recovery of ion concentrations inside neuron to the state of the rest potential and sleep appears to have not been investigated so far [43].

 Changes in Conductance of Spine Neck to Ca²⁺ ions

A series of researches has revealed that the relative dielectric constant ε_r of ionic liquids in neuron is very high. It has been found that the dynamic properties of the relative dielectric constant ε_r are utilized for the fast conduction of action potentials, while the static properties of the relative dielectric constant ε_r are utilized for the establishment of synaptic integration. In the industrial field, such a medium with a very large relative dielectric constant ε_r is thought to be valuable as an internal liquid of a storage battery, but unfortunately, since the conductance of ionic liquids is large [24,25], the loss due to internal discharge is large and it is not practical. However, neurons reduce the conductance to Ca²⁺ ions by preventing charged Ca²⁺ ions in the spine from passing through the spine neck, and maintain the charge of the Ca²⁺ ions in the spine while maintaining a large spine neck capacitance Cn between the spine and the parent dendrite. This allows the spine neck to retain the charge of the Ca²⁺ ions in the spine for a long period of time, allowing it to contribute to the increase in membrane potential for a long period of time with little energy consumption [3]. As mentioned in previous section, the author speculates that the conductance of the spine neck to Ca²⁺ ions, which contributes to the establishment of such synaptic integration, changes rapidly and greatly due to the operation of Ca²⁺ ion channels that sense and open the potential of the backpropagation waveform of the action potential. If we focus only on resistance, which is the conductance to electrons, it appears that the resistance of the spine neck increases rapidly when the spine takes in Ca²⁺ ions and the voltage in the spine increases rapidly [29]. However, if the conductance (ease of passage) of the spine neck for Ca²⁺ ions is low, even if the resistance value for electrons is the same, the potential inside the spine will rise when Ca²⁺ ions are taken up into the spine. Therefore, it is the opposite reason for the above [31]. Note that conductance here refers to the degree of ease of passage for all charged particles, including electrons and ions. Note that conductance here means a measure of the ease of passage for all charged particles, including electrons and ions.

 Conclusion

Neurons have a morphology that allows for high speed, energy saving and miniaturization, and this morphology is passed down from generation to generation. When developing a computer system, the development of circuit elements, wiring between circuit elements, wiring capacitance, clock distribution, power supply and cooling are all set in detail with the goal of high speed, energy saving and miniaturization, and design documents are handed down. Therefore, for understanding the morphology of neurons and their associated behavior, knowledge used in computer system design is useful a little. However, the development of highly functional and highly accurate basic elements such as ion pumps and ion channels, the technology to implement voltage-gated Na⁺ ion channels at ultra-high density in the nodes of Ranvier, and the manufacturing technology of ultra-high-precision myelin sheaths made by wrapping uniform myelin sheets several nanometers thick and 2 mm wide hundreds of times are comparable or more to current development technologies. Moreover, one cannot help but be in awe of the evolutionary history that accomplished the above things before hundreds of thousands of years using algorithms that utilize materials abundant in seawater.

 Conflict of Interest

Author declares no conflicts of interest. Author declares no funding sources.

 Author Contributions

Author wrote the manuscript, prepared figures and tables, and checked all of them.

 Data Availability

The all data used in this Regular Article is available from corresponding references.

 Acknowledgements

The author would like to express their gratitude to Dr. Masanori Okuyama, Professor Emeritus at Osaka University, and Dr. Takumi Washio, Project Researcher at the Future Center Initiative, University of Tokyo.

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