Phase Synchronization and Dynamic Behavior of a Novel Small Heterogeneous Coupled Network

Abstract

Studying the firing dynamics and phase synchronization behavior of heterogeneous coupled networks helps us understand the mechanism of human brain activity. In this study, we propose a novel small heterogeneous coupled network in which the 2D Hopfield neural network (HNN) and the 2D Hindmarsh–Rose (HR) neuron are coupled through a locally active memristor. The simulation results show that the network exhibits complex dynamic behavior and is different from the usual phase synchronization. More specifically, the membrane potential of the 2D HR neuron exhibits five stable firing modes as the coupling parameter $k_1$ changes. In addition, it is found that in the local region of $k_1$, the number of spikes in bursting firing increases with the increase in $k_1$. More interestingly, the network gradually changes from synchronous to asynchronous during the increase in the coupling parameter $k_1$ but suddenly becomes synchronous around the coupling parameter $k_1$ = 1.96. As far as we know, this abnormal synchronization behavior is different from the existing findings. This research is inspired by the fact that the episodic synchronous abnormal firing of excitatory neurons in the hippocampus of the brain can lead to diseases such as epilepsy. This helps us further understand the mechanism of brain activity and build bionic systems. Finally, we design the simulation circuit of the network and implement it on an STM32 microcontroller.

1. Introduction

A memristor [1,2,3,4,5], as the fourth basic electronic component, describes the relationship between charge and magnetic flux. Due to its nanoscale, memorability, nonlinearity, and excellent bionic properties, a large number of researchers have introduced it into neural networks as a synapse [6,7,8,9] or an autapse [10], generating rich dynamic behaviors, such as coexistence [11,12] and multistability [13,14]. Memristors and chaotic systems have potential application value in the industry. He et al. applied the memristor and chaotic system to handwritten digit recognition [15] and sensor location optimization within a wireless sensor network [16], respectively. Remarkable results were reported [15,16].

As the main component of the biological nervous system, the neuron has been widely studied by many scholars. Some existing neuron models include Hodgkin–Huxley (HH) neuron [17], Hindmarsh–Rose (HR) neuron [18,19], Fitzhugh–Nagumo (FHN) neuron [20], Morris–Lecar (ML) neuron [21], Rulkov model [22], and Hopfield neural network (HNN) model [23,24,25]. Homogeneous neuron coupling and heterogeneous neuron coupling have received extensive attention from researchers. Li et al. [26] constructed a homogenous coupled network containing two HR neurons and studied the dynamic behavior, Hamiltonian energy, and phase synchronization behavior of the network. The results showed that Hamiltonian energy calculation could not only reveal the firing mechanism of the neural network but also be used to explore the synchronization control applications of the neural network. Li et al. coupled HR neurons and FHN neurons through memristors to construct a small heterogeneous coupled network, and studied the effects of time delay [27] and external radiation [28] on the heterogeneous coupled network, respectively. The former shows that the two time delays will make the stable equilibrium point unstable, resulting in periodic oscillations known as Hopf bifurcation. Such time delays affect the firing activity of the neural network, and the time delay on different neurons has different effects on synchronization. The latter shows that different initial conditions will generate different bifurcation paths in the network, resulting in various coexisting firing patterns. When the intensity of electromagnetic radiation changes, the network produces opposite bifurcation paths. In addition, it is observed that when the coupling intensity is reduced to a negative value, the two neurons can realize phase synchronization. Njitacke et al. [29] studied the effect of electromagnetic radiation on a heterogeneous coupled network, which has no equilibrium point and, thus, shows hidden dynamic behavior. Ref. [29] also proved that the network has Hamiltonian energy to maintain the electrical activity of neurons. In [30], HR and FHN neurons are coupled into a heterogeneous coupled network through a multistable memristor, which has no equilibrium point and also shows extreme multistability.

Most of the existing studies on heterogeneous coupling focus on HR neurons and FHN neurons. There is a lack of studies on the heterogeneous coupling between HNN and other neurons. Wang et al. [31] realized the heterogeneous coupling of 3D HNN and 2D HR neurons through a memristor and studied the firing behavior and multistability phenomenon of this novel heterogeneous coupled network. In this study, the 2D HNN and 2D HR neurons are coupled through a locally active memristor, and the firing behavior, coexistence phenomenon, and phase synchronization of the network are studied. More specifically, we use a 2D HNN to simulate a specific functional brain region in the brain, HR neurons to simulate their connected neurons, and a memristor to simulate the synapses between two neurons. Based on this, we build a small heterogeneous coupled network that exhibits multiple firing modes, multiple periodic bursters with different spikes, and coexistence. Interestingly, we also find that the phase synchronization behavior of the network is different from other networks, that is, when the coupling intensity gradually increases, the network changes from synchronization to asynchronous and suddenly changes to synchronization when the coupling intensity $k_1$ is near the value of 1.96. We believe that this abnormal phase synchronization behavior is related to epilepsy and other diseases. This helps us further understand the mechanism of biological brain activity. However, because the number of HNN neurons examined in this study is too small, while the number of neurons in the brain is large, the behavior of a small network is sometimes different from that of a large network, so some statements in this paper are conjecture and need to be further verified by experiments. This work also lays a foundation for the next stage of our research on fraction-order heterogeneous coupled networks, where we will compare the similarities and differences between integer-order heterogeneous coupled networks and fraction-order heterogeneous coupled networks, and explore the potential uses of both. In addition, we also design the simulation circuit of the network to verify the physical realizability of the system.

The rest of this paper is organized as follows: In Section 2, a novel small heterogeneous coupled network is proposed. Section 3 reveals the firing modes and phase synchronization of the network. In Section 4, the network is simulated in a Multisim environment and, subsequently, implemented on an STM32 microcomputer. Section 5 summarizes this article.

2. Mathematical Model and Equilibrium Point Studies

2.1. Mathematical Model

Neurons in the human brain are numerous and complex. Information needs to be transmitted between different brain regions or between brain regions and external neurons. This process often involves different types of neurons. Thus, the study of a heterogeneous coupled network is essential. We use a locally active memristor as a memristive synapse to couple 2D HNN and 2D HR neurons to construct a heterogeneous coupled network. Equation (1) is the memristor’s mathematical model.

$$\left\{\begin{array}{c}i=k_1(tanh\left(x\right)+c_0)v\hfill \\ \frac{dx}{dt}=v\hfill \end{array}\right.$$

(1)

To facilitate the study of the properties of the memristor, we assume $c_0$ = 0.01 and $k_1$ = 1 ($c_0$ is a constant and $k_1$ is a variable.) In Equation (1), v, i, and t are considered the input voltage, output current, and time constant, respectively. The pinched hysteresis loops of the memristor model are shown in Figure 1, where A, F, and $x\left(0\right)$ stand for the amplitude, frequency, and initial value, respectively. Keeping A = 1.9, F = 0.5, and the initial state of the memristor unchanged. The results show that the memristor is locally active.

The mathematical expression of a 2D HR neuron is given as follows:

$$\left\{\begin{array}{c}\frac{dx}{dt}=y-a{x}^3+b{x}^2+I\hfill \\ \frac{dy}{dt}=c-d{x}^2-y\hfill \end{array}\right.$$

(2)

In Equation (2), x, y, and I denote the membrane potential, relevant recovery variable, and external stimulation current, respectively, where a = 1, b = 3, c = 1, and d = 5. The generic HNN model is expressed as follows:

$$C_i\dot{x_i}=-\frac{x_i}{R_i}+\sum _{j=1}^n{w}_{ij}tanh\left(x_j\right)+I_i\hspace{1em}\hspace{1em}(i,j\in N^{*})$$

(3)

where the resistance, voltage, and capacitance on the membrane of the neuron are denoted using $R_i$, $x_i$, and $C_i$, respectively. In Equation (3), n = 2, $tanh(.)$, $w_{ij}$, and $I_i$ stand for the neuron activation function, synaptic weight (represents how closely two neurons are connected to each other), and current entering the network, respectively. In Equation (4), we obtain the novel small heterogeneous coupled network’s mathematical expression by setting $C_i$ = 1, $R_i$ = 1, and $I_i$ = 0 in Equation (3).

$$\left\{\begin{array}{c}\frac{dx}{dt}=y-x^3+3x^2+I+k_1(tanh\left(n\right)+0.01)(x-z)\hfill \\ \frac{dy}{dt}=1-5x^2-y\hfill \\ \frac{dz}{dt}=-z-3.5tanh\left(z\right)+0.1tanh\left(u\right)-k_1(tanh\left(n\right)+0.01)(x-z)\hfill \\ \frac{du}{dt}=-u-0.1tanh\left(z\right)+0.6tanh\left(u\right)\hfill \\ \frac{dn}{dt}=x-z\hfill \end{array}\right.$$

(4)

In order to facilitate readers to more intuitively understand the network listed in Equation (4), we draw a schematic diagram of its general structure in Figure 2. In Equation (4), the closeness between the HR neuron and HNN is denoted using $k_1$. The voltage and recovery variable on the membrane of the HR neuron is denoted using x and y, respectively. I is the external stimulation current, and the voltages on the cell membrane of the neuron 1 and 2 in Figure 2 are denoted using z and u, respectively. The magnetic flux within the network is denoted using u. We set I = 0.

2.2. The Equilibrium Points of the Small Heterogeneous Coupled Network

The calculation results of the equilibrium point will reveal the dynamic characteristics of the network. In the same way that self-excited dynamics correspond without a stable equilibrium point, the absence of the equilibrium point often means that a system exhibits hidden dynamics. In order to calculate the network’s equilibrium point, we let $\dot{x}$ = $\dot{y}$ = $\dot{z}$ = $\dot{u}$ = $\dot{n}$ = 0; then, Equation (5) is obtained:

$$\left\{\begin{array}{c}y-x^3+3x^2+k_1(tanh\left(n\right)+0.01)(x-z)=0\hfill \\ 1-5x^2-y=0\hfill \\ -z-3.5tanh\left(z\right)+0.1tanh\left(u\right)-k_1(tanh\left(n\right)+0.01)(x-z)=0\hfill \\ -u-0.1tanh\left(z\right)+0.6tanh\left(u\right)=0\hfill \\ x-z=0\hfill \end{array}\right.$$

(5)

The Jacobian matrix is shown in Equation (6).

$$J=\left(\begin{array}{ccccc}{W}_1& 1& -k_1(tanh\left(n\right)+0.01)& 0& W_3\\ -10x& -1& 0& 0& 0\\ -k_1(tanh\left(n\right)+0.01)& 0& W_2& 0.1(1-tan{h}^2\left(u\right))& -W_3\\ 0& 0& -0.1(1-tan{h}^2\left(z\right))& -1+0.6(1-tan{h}^2\left(u\right))& 0\\ 1& 0& -1& 0& 0\\ \end{array}\right)$$

(6)

$$\left\{\begin{array}{c}{W}_1=6x-3x^2+k_1(tanh\left(n\right)+0.01)\hfill \\ W_2=-1-3.5(1-tan{h}^2\left(z\right))+k_1(tanh\left(n\right)+0.01)\hfill \\ W_3=k_1(x-z)(1-tan{h}^2\left(n\right))\hfill \end{array}\right.$$

(7)

The result of solving Equation (5) illustrates that we cannot find the real solution of the network. Therefore, the network will present hidden dynamic behavior because it is without an equilibrium point.

3. Numerical Simulation

3.1. Firing Activity of the Small Heterogeneous Coupled Network

For revealing the fundamental properties of the network, some classic nonlinear analytical means are indispensable. The use of graphs helps to visually show the impact of certain parameters on the network. We introduce a bifurcation diagram to illustrate the dynamic behavior of an HR neuron inside the network, where x stands for the membrane potential on the neuron in Figure 3a. Apparently, the system is period, and then enters into chaos through forward period-doubling bifurcation, then exits from chaos through reverse period-doubling bifurcation and enters into period behavior. In Figure 3b, the maximum Lyapunov exponent is a positive number when $0.69<k_1<1.55$, which is powerful evidence that the network is in chaos. In Figure 4, the number of spikes in bursting firing can be controlled by changing the value of $k_1$, that is, the coupling strength of synapses affects the number of spikes in bursting firing. Obviously, the coupling strength of synapses between neurons affects the excitability of the nervous system, which provides a way to understand how the excitability of the nervous system affects the duration of emotions. In Figure 5, the increase in synaptic connection strength $k_1$ causes the network to generate five firing modes, i.e., periodic spiking, chaotic spiking, stochastic bursting, chaotic bursting, and periodic bursting. A disparate attractor shape implies a disparate firing mode, and five disparate attractors are displayed in Figure 6. Figure 7a shows the coexisting attractor phase diagram of the network with disparate initial values. The green curve in the figure illustrates that the network is in chaos, and it is in the period state when the blue curve appears. Figure 7b shows the basin of attraction for the coexisting behavior. The dark blue represents that the system is in a periodic state, and the light blue represents that the system is in a chaotic state.

3.2. Synchronization Behavior of the Small Heterogeneous Coupled Network

Abnormal synchronized behavior in the brain is thought to have certain adverse effects on the nervous system, possibly leading to illnesses such as epilepsy. In this study, we examine the phase synchronization of the heterogeneous neuron coupled system and find that the system shows abnormal synchronization behavior in a certain interval. We have conducted a preliminary study on this abnormal behavior and tried to eliminate this abnormal synchronous behavior by introducing external stimuli. Let us introduce the definition of the burst phase:

$$\theta \left(t\right)=2\pi n+2\pi \frac{t-t_n}{t_{n+1}-t_n}\hspace{1em}\hspace{1em}(t_n<t<t_{n+1})$$

(8)

where $t_n$ is the time when the n-th burst emerges, and $t_{n+1}-t_n$ denotes the burst interval. According to the above definition, every burst leads to a phase increase of $2\pi$. Hence, phase synchronization can be detected when the absolute value of the phase difference $|\Delta \theta (t\left)\right|$ = $|{\theta }_1\left(t\right)-{\theta }_2\left(t\right)|$ between two neurons is bounded with the value of $2\pi$.

We selected some $k_1$ values to simulate the phase synchronization of the system at these coupling intensities. As shown in Figure 8, when the coupling strength is very small, the system tends to be synchronous, and when the coupling strength increases, the system changes to be asynchronous. Different from the results of previous studies, the system suddenly changes into the synchronous state when $k_1$ = 1.96. Inspired by the fact that the episodic synchronous abnormal firing of excitatory neurons in the hippocampus of the brain can lead to diseases such as epilepsy, it is reasonable to think that this abnormal synchronous behavior can have some adverse effects on the nervous system. We think that adding radiation to the neurons may be able to weaken the possible adverse effects of this abnormal synchronization, but this needs to be confirmed by further experiments.

4. Circuit Simulation and Hardware Implementation

The physical realization of the network would facilitate its application to industrial production. Therefore, we choose to design the analog circuit of the network on Multisim and verify whether the results of the circuit are consistent with the MATLAB 2021a simulation results. The network will be further implemented on the STM32 microcontroller if they are consistent.

4.1. Circuit Simulation

Through the equivalent substitution of Equation (4), we obtain the circuit expression of the network:

$$\left\{\begin{array}{c}C\frac{d{v}_x}{dt}=\frac{v_y}{R_1}-\frac{g^2{v}_x^3}{R_2}+\frac{g{v}_x^2}{R_3}+\frac{g_{k_1}(tanh\left(v_n\right)+0.01e)(v_x-v_z)}{R_L}\hfill \\ C\frac{d{v}_y}{dt}=\frac{e}{R_s}-\frac{g{v}_x^2}{R_4}-\frac{v_y}{R_5}\hfill \\ C\frac{d{v}_z}{dt}=-\frac{v_z}{R}+\frac{tanh\left(v_z\right)}{R_6}+\frac{tanh\left(v_u\right)}{R_7}-\frac{g_{k_1}(tanh\left(v_n\right)+0.01e)(v_x-v_z)}{R_L}\hfill \\ C\frac{d{v}_u}{dt}=-\frac{v_u}{R}+\frac{tanh\left(v_z\right)}{R_9}+\frac{tanh\left(v_u\right)}{R_{10}}\hfill \\ C\frac{d{v}_n}{dt}=\frac{v_x-v_z}{R_k}\hfill \end{array}\right.$$

(9)

Equation (10) illustrates the circuit elements values of the network:

$$\left\{\begin{array}{c}VCC=5 \mathrm{V},VEE=-5 \mathrm{V},R_C=1 \mathrm{K}\Omega ,R_F=520 \Omega ,I_0=1.1 \mathrm{mA}\hfill \\ RC=100 \mathrm{us},R_0=10 \mathrm{K}\Omega ,C=10 \mathrm{nF},g=1,g_{k_1}=k_1\hfill \\ R_1=10 \mathrm{K}\Omega ,R_2=10 \mathrm{K}\Omega ,R_3=3.33 \mathrm{K}\Omega ,R_L=10 \mathrm{K}\Omega ,R_k=10 \mathrm{K}\Omega \hfill \\ R_s=10 \mathrm{K}\Omega ,R_4=2 \mathrm{K}\Omega ,R_5=10 \mathrm{K}\Omega ,E_1=1 \mathrm{V},E_2=0.01 \mathrm{V}\hfill \\ R_6=2.857 \mathrm{K}\Omega ,R_7=100 \mathrm{K}\Omega ,R_8=100 \mathrm{K}\Omega ,R_9=16.667 \mathrm{K}\Omega \hfill \end{array}\right.$$

(10)

The simulation circuit of the network is shown in Figure 9 and Figure 10. Figure 9 is the circuit diagram of the hyperbolic tangent function, and the values of the components in Figure 9 and Figure 10 are shown in Equation (10). The amplifiers used in Figure 9 and Figure 10 are TL082CP, and the supply voltages are +15 V and −15 V. The circuit’s frequency up to 3 MHz. The multiplier used is the analog multiplier, g represents the multiplier of the two inputs of the factor, and the transistor is 2N2222. The $-tanh$ module in Figure 10 is the circuit in Figure 9. The simulation results of the Multisim circuit are shown in Figure 11. Obviously, they are consistent with the simulation results of MATLAB 2021a. It is reasonable to think that the circuit in Figure 10 can physically realize the heterogeneous coupled network of Equation (4).

4.2. Microcontroller Implementation

A microcontroller is widely used in industry because of its advantages of high speed, compactness, and low cost. In this section, we will use the STM32F407ZGT6 microcontroller to implement this small heterogeneous coupled network and use C language to write programs. It is produced by STMicroelectronics which was formed by the merger of SGS Microelectronics of Italy and Thomson Semiconductor of France, and its headquarters are in Switzerland. It has a frequency up to 168 MHz. The signal generated via the STM32 microcontroller will be displayed on the oscilloscope, and the hardware layout diagram is shown in Figure 12. This small heterogeneous coupled network is mainly implemented using the fourth-order Runge–Kutta algorithm, the approximate flowchart of which is shown in Figure 13. The experimental setup of the small heterogeneous coupled network on the STM32 single-chip microcomputer is shown in Figure 14, and the result is consistent with that in Figure 5. It is reasonable to think that the network is implemented on the single-chip microcomputer.

5. Conclusions

This study proposes a novel small heterogeneous coupled network, using a locally active memristor as a memristive synapse to couple 2D HNN and 2D HR neurons. The calculation results show that the network has no equilibrium point and shows hidden dynamics. The novel small heterogeneous coupled network exhibits five stable firing modes as $k_1$ changes. In addition, in the local region of $k_1$, the number of spikes in bursting firing increases with the increase in $k_1$. The above phenomenon shows that the firing mode is controlled by the coupling strength and is consistent with the behavior of neurons. More interestingly, when the coupling intensity $k_1$ increases, the network changes from synchronous to asynchronous, but suddenly changes to synchronous around the coupling parameter $k_1$ = 1.96. This abnormal synchronization behavior is different from previous studies, which provides a new way for us to further understand the mechanism of brain activity. Finally, the network is simulated in Multisim environment and implemented experimentally on STM32.