A number of different types of neurons have been studied and modelled, for example, squid axons [

5], frog [

4], dog [

6], rabbit [

2], cat [

3], and the Purkinje cell [

7]. The common feature of these models is that they are based on the membrane potentials of the cells and the ion channel dynamics. In general, these models take the form:

where

C_{m} is the membrane capacitance,

I is the current,

V is the membrane potential in mV, and

t is time. The various forms of the mathematical models are based on the structure and form of

$f\left(\theta ,\overline{V}\right)$. This term is a function of (a) the probabilities of the opening and closing of an ion channel, (b) the conductivity of the ion channel, and (c) the potential difference between the membrane and the ion channel (given by

V). In general,

$f\left(\theta ,\overline{V}\right)$ is an algebraic sum of the currents associated with the various ion channels, and, thus, a specific ion,

i, would take the form:

where the variable

${g}_{i}$ is a function of the probabilities of the opening and closing of channels, and the conductance of that particular channel. The membrane potential dynamics activate and deactivate the channels (see

Section 2). If the neuron dynamics are restricted to sodium and potassium channels, these become:

where

n, m, and

h are representations of the fractions of the open and closed channels for the different ions. Thus, the membrane potential based model takes the form:

where

$\psi $ are given by

n,

m, and

h. This model describes the time behavior of the intracellular membrane potential and the currents through the channels. It is possible to explain observed phenomena accurately, and the change of voltages and currents on the nerve cell membrane can be analyzed quantitatively [

26]. For the channels under consideration, the parameters given in Equations (4)–(6) are defined as follows:

Based on the relationships between permeability, and conductance within the neuron [

5], Equation (2) can be obtained for the current generated within a particular ion channel. So, the three states, for sodium, potassium, and leakage are given by the following algebraic equations:

In these equations,

V is the trans-membrane potential.

I_{inj} is the sum of external and synaptic currents.

I_{Na} is the current in the sodium channel and

I_{k} in the potassium.

I_{L} is the leakage current. The gating variables indicating activation and inactivation of the sodium ion current are 0 ≤

m ≤ 1 and 0 ≤

h ≤ 1, respectively. The gating variable showing activation of potassium ion current is 0 ≤

n ≤ 1. The membrane capacitance is

${C}_{m}$ = 1.0 μF/cm

^{2}.

V_{Na} and

V_{K} are the equilibrium potentials for the sodium and potassium ions. For channels that conduct a single type of ion, the equilibrium potential can be easily determined. This equilibrium potential point has a direct relation with the each ion and can give via the Nernst equation. This equation can be used to predict the membrane voltage of a cell in which the plasma membrane is permeable to one ion only.

#### 3.1. Generalized Form of Neurons

In this section, a generalized form of a neuron is presented for the

N channels and

m gates. The reason is to rewrite a mathematically pure form. First, consider Equations (1) and (4)–(7). In these equations the part

${g}_{i}\left(V-{V}_{i}\right)$ is common for all channels and, when

$i=0,$ we get a leakage current. For ion currents the activation and deactivation gates can be rewritten as follows:

These equations calculate the probability of the opening/closing channel for sodium and potassium, respectively. Using this information, the generalized form of a neuron with

N channels and

m ionic gates is in the form:

where each part of

${g}_{i}{\psi}_{i}\left(y\right)\left(V-{V}_{i}\right)$ models a specific ion channel. The channel status is denoted by variable

$y$. In the original Hodgkin–Huxley model, with two channels and three gates, the variable y = (

${y}_{1}$,

${y}_{2},{y}_{3}$) and the functions of

$\psi $ are

${\psi}_{1}={y}_{1}^{3}{y}_{3}$ and

${\psi}_{2}={y}_{2}^{4}$. These functions are defined as probabilities and are in the range [0–1]. For this model,

${y}_{1}=n,$ ${y}_{2}=m,$ and

${y}_{3}=h$.

From Equation (4) we get:

By dividing and multiplying the right side of Equation (21) with

${\alpha}_{n}\left(v\right)+{\beta}_{n}\left(v\right)$, we get:

If

${y}_{i}=n,{\sigma}_{i}\left(v\right)=\frac{{\alpha}_{n}\left(v\right)}{{\alpha}_{n}\left(v\right)+{\beta}_{n}\left(v\right)}$ and

${\mathsf{\delta}}_{i}\left(v\right)={\alpha}_{n}\left(v\right)+{\beta}_{n}\left(v\right),$ Equation (21) is rewritten as follows:

Finally, the generalized form of a neuron model with

N channels and

m gates can be rewritten as follows [

27]:

where the parameters

p = (

${g}_{0},\dots ,{g}_{N},{V}_{0},\dots ,{V}_{N},I$) and the constant

T > 0 are dependent on temperature. The dynamic of each gate variable

${y}_{j}$ depends only on itself, the voltage,

$V,$ by smooth function

${\sigma}_{i}$ and the diagonal matrix

${\delta}_{i}\left(v\right)$ for all values of

V. Each of the terms of

${g}_{i}{\psi}_{i}\left({y}_{j}\right)\left(V-{V}_{i}\right)$ in Equation (24), with a constant

${g}_{i},$ refer to an ionic channel. This adjusts the voltage,

$V,$ across the nerve cell’s membrane and makes the dynamics of the

$i$th channel. So, the generalized form of the neuron model in Equation (24) represents

N channels and

m gates where

N and

m are not necessarily equal. The model in Equation (24) is suitable for shaping a single neuron’s behavior. However, when two or more neurons in a network work together they are coupled by synapse spaces, which are not referred to in Equation (24). The missing link here is the coupling phenomenon, which is discussed in the next section.

#### 3.2. Coupled Type Equations

Coupling in the neurons is done via synapses. A live neuron is an oscillator that can be coupled with the chain of neurons (see

Figure 3). A synapse can be explained as a site where a neuron makes a communicating connection with the next neuron. On one side of the synapse is a neuron that transmits the signal via the axon terminal, which is called the presynaptic cell and on the other side is another neuron, or a surface of an effector, that receives the signal and is called the postsynaptic cell. In nervous systems, there are three general types of synaptic connections among neurons.

All kinds of synapses are shown in

Figure 4. Electrical connections are also known as gap junctions. Other forms of synapses are two types of chemical connections, excitatory and inhibitory. This study constructed neuron pair models by electrical synapse. The electrical connections are usually axon-to-axon, or dendrite-to-dendrite and are shaped by channel proteins that span the membranes of both connected neurons. Electrical coupling is ubiquitous in the brain, in particular among the dendritic trees of inhibitory interneurons. This kind of direct non-synaptic interaction allows for electrical communication between neurons. All models with electrical coupling necessarily involve a single neuron model that can represent the shape of an action potential.

The experiments in this study show that when two equations of neuron models are coupled, their solutions seem to synchronize. This paper investigates two of the same action potential equations, coupled only with the electrical potential of each neuron.

Choosing a large enough coupling strength forces the neuron to have the same behavior regardless of the initial condition. Mathematical models for such systems are frequently very complicated. Consider a pair of equations for a neuron model with partial coupling and coupling constants,

p_{1} and

p_{2} ≥ 0, using Equation (24) [

28]:

Later, we will show that for sufficiently large

p_{1} and

p_{2} the solutions of the above equation always synchronize. Synchronicity is tackled in

Section 3.3.

#### 3.3. Synchronization in Coupled Neurons

In this section, the mathematical background for the synchronization of coupled neurons is presented. Synchronization is a phenomenon that can be seen in two or more coupled neurons. It is achieved by an adjustment of rhythms and their oscillations. Synchronization is one of the important features of nonlinear systems. Nonlinear systems can show behaviors that are impossible in linear systems [

29]. Synchronization analysis is a principle to discover interactions between nonlinear oscillators [

29]. In the synchronization between two neurons, their action potentials have relatively equal frequencies. This closeness relies on the strength of the coupling. Therefore, spike synchronization is crucially dependent on the inter spike frequency. The general case of two coupled neurons can be characterized as follows:

where

${\lambda}_{i}$ = (

g_{0}, …,

g_{N},

V_{0}, …,

V_{N},

I), which are dependent on parameter

$\lambda $. Perfect synchronization happens when

${v}_{1}\left(t\right)={v}_{2}\left(t\right)$ for all times

t (Labouriau & Rodrigues, 2003). This coupled system is symmetric if

${\lambda}_{1}={\lambda}_{2}$ and asymmetric if

${\lambda}_{1}\ne {\lambda}_{2}$. Perfect synchrony usually does not happen in asymmetric systems. Consider Equation (25), in this equation, when

p_{1} =

p_{2}, the coupled system is symmetric [

28]. The solution of Equation (26) is synchronized if

${v}_{1}\left(t\right)$ and

${v}_{2}\left(t\right)$ remain close to each other in the next periods of action potentials. This means that, if there is a constant

$\gamma >0$ and a continuous function like

$f\left(t\right)\ge 0$ defined for

$t\ge {t}_{0}$ in which

$\underset{t\to \infty}{\mathrm{lim}}f\left(t\right)=0$ in a way that for all

$t\ge {t}_{0}$, the following are true [

28]:

After a short interaction between the two neurons the synchronicity between them becomes higher. Generally, for the symmetric case, there is an exponential decay to perfect synchronization [

28].

#### 3.4. The Region of Synchronicity

Coupled neurons often possess symmetries; these behaviors are important for understanding dynamic effects in systems. The simplest symmetric system contains two coupled neurons.

Figure 5 presents two coupled neurons.

When the input current

I_{inj}_{1} is applied it integrates into the axon hillock of the first neuron. These synaptic inputs cause the membrane to depolarize; that is, they cause the membrane potential to rise. If this polarization causes the membrane potential to rise to the threshold, an action potential

V_{1} can be raised. When an action potential is triggered, it abruptly generates a dendritic current that flows through the axon

I_{c} in the output of the first neuron. By considering Equations (1)–(3) and Equations (14)–(16), the output of the first neuron for original Hodgkin–Huxley model is:

Similarly, for the second neuron, the output is:

In entirely normal conditions, if neuron number two is stimulated only by neuron number one and does not get any other stimuli from other neurons, the outputs of Equations (28) and (29) should be equal.

Equation (30) means to make synchronization between two neurons the minimum condition and should be respected.