Alternating chimera states and synchronization in multilayer neuronal networks with ephaptic intralayer coupling

Abstract

Over the past decade, most of researches on the communication between the neurons are based on synapses. However, the changes in action potentials in neurons may produce complex electromagnetic fields in the media, which may also have an impact on the electrical activity of neurons. To explore this factor, we construct a two-layer neuronal network composed of identical Hindmarsh–Rose neurons. Each neuron is connected with its neighbors in the layer via magnetic connections and a neuron in the corresponding position of the other layer via electrical synapse. By adjusting the electrical coupling strength and magnetic coupling strength, we find the appearance of alternating chimera states and transient chimera states whenever the intralayer coupling is nonlocal and local, respectively. According to our study, these phenomena have not been studied in multilayer networks of this structure. And it is found that the transient chimera states only could occur when the number of coupled neighbors is small. In addition, the states of two independent networks will affect the final states of networks applying the same sufficiently large interlayer coupling strength. Our study reveals a possible effect of electrical coupling and ephaptic coupling produced together on the dynamic behavior of the neuronal networks. Meanwhile, our results suggest that it makes sense to take electromagnetic induction into neuronal models.

Introduction

Complex networks have attracted wide attention in many fields, including physics, biology, computer science, and epidemiology et al. (Freitas et al. 2015; Xiong et al. 2018; Guan and Guo 2021; Rehman et al. 2021; Zhu and He 2022). Complex network is a network with non-trivial topological features and these features often occur in networks representing real systems (Ait Rai et al. 2023). By making some reasonable simplifications of the actual systems, we can construct corresponding network models that can reflect many important properties of the actual systems (Zhang and Zheng 2012). In 1998, Watts and Strogatz discovered the small-world networks, which can transmit information both locally and globally and have shorter path lengths and larger clustering coefficients (Watts and Strogatz 1998; Latora and Marchiori 2001; Hu et al. 2021). An et al. have analyzed the current research situation of complex networks (An and Yu 2020). There are many classic complex network models worth studying, such as local-world evolving network model (Wen et al. 2011; Yan 2023), weighted evolving network model (Barrat et al. 2004; Zhang et al. 2015), etc. The brain is the most complex structural and functional organ in the human body, and it is also one of the most complex systems in nature. The neural system that constitutes the human brain is a nonlinear dynamical system, and the neuronal network is an important complex network that is connected by neurons. There’s a lot of research on the neuronal networks from dynamics perspective (Liu et al. 2018; Lin et al. 2024; Deng et al. 2024; Xu et al. 2024). However, the dynamical mechanisms of many functions of the brain are still unknown.

The synchronization of coupled oscillators on complex networks has attracted great attention in recent years (Wang and Liu 2020; Lin et al. 2021). Synchronization is a widespread and very important self-organizing phenomenon in nature. Complex networks show many synchronization phenomena, such as full synchronization and phase synchronization (Lei and Jia 2024). Synchronization plays an important role in the brain during information transmission (Dayani et al. 2023). Abnormal synchrony may lead to the occurrence of many brain diseases. For example, during seizures, specific areas of the brain are extremely synchronized, while the rest are asynchronized. The electroencephalogram (EEG) signals of Alzheimer's patients exhibit significantly low synchrony in the brain (Zheng et al. 2023). Another significant self-organized phenomenon is the chimera state, which is a behavior that in a system composed of identical units, part of the oscillator appears as coherent in space and the rest is incoherent (Zhai and Zheng 2020). The chimera state was first discovered by Kuramoto and Battogtokh (2002). In 2004, Abrams and Strogatz named this counter-intuitive phenomenon as the chimera state (Abrams and Strogatz 2004). An important aspect of the chimera state is to describe the microscopic mechanisms of unihemispheric sleep in some birds and marine mammals. It is shown that during the sleep, one hemisphere of their brains is asleep and the other hemisphere is awake, that is to say, half of the neurons are synchronized and the other half are asynchronized. This phenomenon is called unihemispheric sleep (Glaze et al. 2016). Animals can remain more alert during the unihemispheric sleep to avoid being attacked (Mathews et al. 2006). In 2016, experiments confirmed that this similar mechanism exists in humans, called the first-night effect, that is, when we come to an unfamiliar environment, the first night of sleep is often unstable, and the synchronization of the left hemisphere is significantly lower than that of the right hemisphere (Tamaki et al. 2016). In addition, studies have shown that chimera states are related to some brain diseases, such as Parkinson's disease, Alzheimer's disease, epilepsy, and so on (Majhi et al. 2017). In fact, many scholars have studied the chimera states, and they also developed different types of chimera states according to the different characteristics of the obtained, such as globally clustered chimera states (Sheeba et al. 2009), multi-chimera states (Omelchenko et al. 2015), traveling chimera states (Bera et al. 2016a), breathing chimera states (Suda and Okuda 2020), amplitude-mediated chimera states (Bi and Fukai 2022), imperfect chimera states (Parastesh et al. 2018), etc. In a word, it is necessary to study the chimera states in neuronal networks.

Neurons are the basic constituent units of the brain. In order to analyze and interpret experimental data, some mathematical models of neurons have been established, such as Hodgkin–Huxley (HH) model (Hodgkin and Huxley 1990), FitzHugh–Nagumo model (Fitzhugh 1962; Nagumo et al. 1962), Hindmarsh–Rose (HR) model (Hindmarsh and Rose 1997), etc. The HR neuron model has a relatively simple structure and has the ability to show the rich firing phenomena of neurons, so it is studied by the majority of researchers (Usha and Subha 2019; Zhou et al. 2021; Aghababaei et al. 2021; Qiao and Gao 2022; Messee Goulefack et al. 2023).

Synapse is the basis for neurons to realize their specific functions, and neurons transfer information through synapses. According to the structure and electrophysiological characteristics of the synapse, it can be divided into electrical synapse and chemical synapse. Many scholars have used electrical synapse or chemical synapse to construct neuronal networks in past studies. Some scholars have studied hybrid synapses. Bera et al. (2016b) constructed a monolayer network of HR neurons with nonlocal coupling between neurons by chemical synapses. Li et al. (2019) constructed a two-layer network of HR neurons in which intralayer neurons and interlayer neurons are bidirectional and unidirectionally coupled, respectively, through electrical synapses. A two-layer network of HR neurons in which neurons both within and between layers are coupled by chemical synapses was constructed by Yuan et al. (2022a). In fact, however, the electrical activity of neurons is so complex that we ignore many factors. According to Faraday electromagnetic induction, when the action potential of the neuron changes, a magnetic field is created in the medium, which then adjusts the electrical activity of the neurons. Therefore, it is necessary to consider the effects of the magnetic flux and electromagnetic induction on the electrical activity of neurons into the neuronal network. In 2016, an improved HR neuron model was proposed by Lv et al. (2016). They introduced magnetic flux as an additional variable and regulated the membrane potential through a memristor, thus taking the effects of electromagnetic induction into the model. Later, the improved HR neuron model has been studied by many scholars (Korotkov et al. 2019; Liu et al. 2021). Although some scholars have studied the neuronal networks considering ephaptic interaction (Shafiei et al. 2020; Yuan et al. 2022b; Wan et al. 2022; Shang et al. 2023; Lin et al. 2023), there is still a lot of room to study more complex chimera states produced by the simultaneous interaction of synapses and ephaptic communication (Majhi et al. 2019).

In this paper, we constructed a two-layer network of HR neurons, where intralayer connections and interlayer connections are deemed to occur through the ephaptic couplings and electrical couplings, respectively. We observed the dynamical states of the two-layer network by regulating the electrical coupling strength and magnetic coupling strength. We have found the alternating chimera states and transient chimera states whenever intralayer connection was the nonlocal coupling and local coupling, respectively. In addition, it is found that for independent two layers, when the intralayer coupling strength is different, applying a sufficiently large interlayer coupling strength on the network may lead to different states of the two-layer network. In addition, the transient chimera states only appear with a small number of nonlocal coupling. Moreover, the increase of interlayer electrical coupling strength will increase the synchronization between the two layers.

The remainder of this paper is organized as follows. In Sect. 2 we will introduce the dynamics model of the network and some statistics that can quantitatively judge the states of the network. Section 3 presents the results of the numerical simulations. Our conclusion is given in Sect. 4.

Method

We construct a two-layer neuronal network composed of identical HR neurons. The reason for choosing the HR neuron model is that it is relatively convenient to calculate and it can show rich neuronal electrical activity, such as periodic spiking, periodic bursting, chaotic bursting, etc. The neuron model is described by the following equation (Hindmarsh and Rose 1997):

$$\left\{\begin{array}{c}\dot{x}=y-a{x}^3+b{x}^2-z+I,\hfill \\ \dot{y}=c-d{x}^2-y,\hfill \\ \dot{z}=r\left[s\left(x-{\chi }_0\right)-z\right],\hfill \end{array}\right)$$ 1

where $x$, $y$, and $z$ represent the membrane potential, the transport of ions across the membrane via the fast and slow channels, respectively. Since the network takes into account the effects of electromagnetic induction generated by the electrical activity of neurons, we need the modified HR neuron model mentioned above (Lv et al. 2016):

$$\left\{\begin{array}{c}\dot{x}=y-a{x}^3+b{x}^2-z+I-g_m\rho \left(\varphi \right)x,\hfill \\ \dot{y}=c-d{x}^2-y,\hfill \\ \dot{z}=r\left[s,\left(x,-,{\chi }_0\right),-,z\right],\hfill \\ \dot{\varphi }=k_{1}x-k_2\varphi ,\hfill \end{array}\right)$$ 2

where $\varphi$ represent the magnetic flux across the membrane, and $\rho \left(\varphi \right)=\alpha +3\beta {\varphi }^2$ is the memory conductance of the flux-controlled memristor, which shows the coupling between the membrane potential and the magnetic flux. $g_m$ and $k_1$ are two parameters that govern the mutual effects between the membrane potential and magnetic flux, respectively. $k_2$ represents the intensity of variation in magnetic flux induced by membrane potential. The different electrical activity can be shown when we change the external forcing current $I$. To select appropriate external forcing current, we plot the bifurcation diagram as shown in Fig. 1. Thus we fix the parameters $a=1$, $b=3$, $c=1$, $d=5$, $r=0.006$, $s=4$, $k_1=0.9$, $k_2=0.5$, $\alpha =0.4$, $\beta =0.02$, and ${\chi }_0=-1.6$(Majhi and Ghosh 2018). When $I=3.25$, the individual neuron displays multi-scale chaotic bursting as shown in Fig. 1.

Fig. 1.

a Bifurcation diagram associated with external forcing current. ISI denotes the inter-spike interval in membrane potential series. b The time series of membrane potential in isolated neuron for

We construct a two-layer network composed of the Hindmarsh–Rose neuron models. Each neuron in one layer is connected to its neighbors in the same layer via bidirectional magnetic connections and its corresponding neighboring neuron in the other layer via electrical synapses. In other words, the neurons in same layer interact through ephaptic coupling and each neuron is connected via electrical connections to its corresponding neighboring neuron in other layer. Thus, each neuron is subject to two kinds of coupling, one is the ephaptic coupling from neurons in the same layer, and the other is electrical coupling from the neuron in other layer. We use a modified Hindmarsh–Rose model because of the effects of the magnetic field. Figure 2 shows a schematic of this network. The dynamics of this two-layer network are described as the following equations:

$$\left\{\begin{array}{c}{\dot{x}}_{i.l}=y_{i,l}-a{x}_{i,l}^3+b{x}_{i,l}^2-z_{i,l}+I-{\text{g}}_{\mathit{ml}}w\left({\varphi }_{i,l}\right)x_{i,l}+{\text{g}}_e\left(x_{i,k},-,x_{i,l}\right),\hfill \\ {\dot{y}}_{i,l}=c-d{x}_{i,l}^2-y_{i,l},\hfill \\ {\dot{z}}_{i,l}=r\left[s,\left(x_{i,l},-,{\chi }_0\right),-,z_{i,l}\right],\hfill \\ {\dot{\varphi }}_{i,l}=k_1{x}_{i,l}-k_2{\varphi }_{i,l}+{\sum }_{j=i-P}^{j=i+P}\left({\varphi }_{j,l},-,{\varphi }_{i,l}\right),i,j=\text{1,2},\cdots ,N,\hfill \end{array}\right)$$ 3

where $N$ is the total number of neurons in each layer of the network. The subscript $i$, $j$, $l$ and $k$ represent the $i$th neuron, $j$th neuron, $l$th layer, and $k$th layer, respectively $\left(k,l=1,2,k\ne l\right)$. Thus, $x_{i,k}$ and $x_{i,l}$ represent the membrane potential of the $i$th neuron in the $k$th layer and $l$th layer, respectively. Then $x_{i,k}-x_{i,l}$ represent the electrical coupling between two neurons in the different layer. ${\varphi }_{j,l}-{\varphi }_{i,l}$ represent the magnetic coupling between two neurons in the same layer.$P$ specifies the number of neighbors in each direction on a ring. $g_{\mathit{ml}}$ $\left(l=1,2\right)$ and $g_e$ are the intralayer magnetic coupling strength and the interlayer electrical coupling strength, respectively.

Fig. 2.

The schematic diagram of the two-layer network. The black lines represent intralayer ephaptic coupling strength and the rad lines represent interlayer electrical coupling strength

The initial conditions for system (3) are given as follows: $x_{i0}=0.01\times \left(N\left./\phantom{N2}\right)2-i\right)$,$y_{i0}=0.02\times \left(N\left./\phantom{N2}\right)2-i\right)$, $z_{i0}=0.03\times \left(N\left./\phantom{N2}\right)2-i\right)$ for $i=1,2,\cdots ,N\left./\phantom{N2}\right)2$, and $x_{i0}=0.012\times \left(i-N\left./\phantom{N2}\right)2\right)$, $y_{i0}=0.024\times \left(i-N\left./\phantom{N2}\right)2\right)$, $z_{i0}=0.035\times \left(i-N\left./\phantom{N2}\right)2\right)$ for the remaining neurons. The initial magnetic flux ${\varphi }_{i0}=0$. The initial conditions are the same for both layers.

Statistics are very important in the study of nonlinear dynamics. According to the values of statistics, we can clearly judge the dynamics states. In order to make a quantitative judgment of the state obtained in this paper, we will introduce the following statistics. We can verify the appearance of the chimera state by computing local curvature $L_i\left(t\right)$ or Laplacian $l_i\left(t\right)$. The local curvature of $x_i$ at time $t$ is defined as:

$$L_i\left(t\right)=\left|l_i\left(t\right)\right|=\left|x_{i+1}\left(t\right)+x_{i-1}\left(t\right)-2x_i\left(t\right)\right|.$$ 4

Smooth local curvature plot represents spatial synchronization, whereas distinct non-zero values represent disorder. Therefore, the coexistence of both represents the chimera state.

Here, in order to further clarify the appearance and characteristics of the alternating chimera state and the transient chimera state, we will introduce two other statistics, the spatial correlation measure $C_{\mathit{sp}}\left(t\right)$ and the time-correlation $C_{\mathit{tm}}$. As the name suggests, $C_{\mathit{sp}}\left(t\right)$ reflects the degree of spatial correlation of the network at time $t$, while $C_{\mathit{tm}}$ reflects the degree of temporal correlation of the nodes in the network. The spatial correlation measure is defined as

$$C_{\mathit{sp}}\left(t\right)=\frac{1}{N}\sum _{i=1}^{N}f\left(L_i\left(t\right)\right),$$ 5

where the function $f$ of $L_i\left(t\right)$ is defined as

$$f\left(L_i\left(t\right)\right)=\left\{\begin{array}{cc}1\hfill & \text{if}{L}_i\left(t\right)\le {\delta }_1,\\ 0\hfill & \text{otherwise}\end{array}.\right)$$ 6

To calculate the time-measure $C_{\mathit{tm}}$, we introduce the time-correlation coefficient ${\sigma }_{\mathit{ij}}$. We use $m_i$, $m_j$ and $s_i$, $s_j$ to represent the temporal means and standard deviations of the membrane potentials $x_i$ and $x_j$, respectively. Then the time-correlation coefficient ${\sigma }_{\mathit{ij}}$ is defined as

$${\sigma }_{\mathit{ij}}=〈\left(x_i-m_i\right)\left(x_j-m_j\right)〉\left./\phantom{〈\left(x_i-m_i\right)\left(x_j-m_j\right)〉\left(s_i{s}_j\right)}\right)\left(s_i{s}_j\right).$$ 7

If ${\sigma }_{\mathit{ij}}\cong 1$ or ${\sigma }_{\mathit{ij}}\cong -1$, we can conclude that $x_i$ and $x_j$ are linearly time-correlated or anti-correlated. Then we introduce a function $g$ of $\left|{\sigma }_{\mathit{ij}}\right|$, which is defined as

$$g\left(\left|{\sigma }_{\mathit{ij}}\right|\right)=\left\{\begin{array}{cc}1\hfill & \text{if}\left|{\sigma }_{\mathit{ij}}\right|>{\delta }_2,\\ 0\hfill & \text{otherwise}\end{array}.\right)$$ 8

The time-correlation measure $C_{\mathit{tm}}$ can be calculated in this way:

$$C_{\mathit{tm}}=\sqrt{\frac{1}{N\left(N-1\right)}\sum _{i,j=1\left(i\ne j\right)}^{N}g\left(\left|{\sigma }_{\mathit{ij}}\right|\right)}.$$ 9

The threshold values are fixed at ${\delta }_1=0.04$ and ${\delta }_2=0.90$ here.

After introducing the definition of statistics, we will show the significance of these statistic values. When the network is in the steady alternating chimera state, $C_{\mathit{tm}}>0$ and $C_{\mathit{sp}}\left(t\right)\in \left(0,1\right),\forall t$, and the latter value is oscillating. When the network is in the transient chimera state, $\forall t$, $\exists \text{T}>0$, when $t<T$, $C_{\mathit{sp}}\left(t\right)\in \left(0,1\right)$. When $t>T$, $C_{\mathit{sp}}\left(t\right)=0$ or $C_{\mathit{sp}}\left(t\right)=1$. So far, we can verify the alternating chimera state and the transient chimera state by the values of these statistics. In addition, here the oscillation of the value of $C_{\mathit{sp}}\left(t\right)$ does indicate the appearance of the alternating chimera state.

Results and discussion

In this section, the time step is set as 0.01 and system (3) is numerical simulated by the ode45 solver in the MATLAB. First, we consider the case of intralayer nonlocal coupling. We fix the parameters $N=100$, $P=30$, $g_{m1}=3.0$ and $g_{m2}=0.1$. Figure 3 show the spatiotemporal diagrams and the membrane potential distribution diagrams of the two-layer network when the electrical synaptic coupling strength $g_e=0,0.15,1.0,5.0$, respectively. When $g_e=0$, the two-layer networks are independent from each other. From Fig. 3 and Fig. 3, we can see that the first layer network is in amplitude death, while from Fig. 3 and Fig. 3, it is found that the membrane potential of the second layer network is disordered and therefore it is in the incoherent state. For $g_e=0.15$, we can obtain from Fig. 3 that the first layer network is in a coherent state, while the membrane potential of the second layer network can be divided into two parts, one part is synchronous, the other part is disordered, that is, the second layer network is in the chimera state. Figure 4 show the distribution of membrane potential of the second layer of neurons at different times when $g_e=0.15$, and Fig. 4 show the corresponding local curvature $L_i\left(t\right)$. We can see that when $t=6777$ and $t=6871.1$, the coherent and incoherent domains of neuronal membrane potential exchange. When $t=6966.5$, the exchange occurs again. That is to say, the coherent and incoherent domains of the membrane potential at the time are the same as those at $t=6777$. The chimera states with this characteristic are called alternating chimera states. For further quantitative judgment, we calculate the time-correlation measure $C_{\mathit{tm}}$ and spatial correlation measure $C_{\mathit{sp}}\left(t\right)$ of the second layer when $g_e=0.15$, as shown in Fig. 5. We can see that $C_{\mathit{sp}}\left(t\right)$ changes repeatedly and $C_{\mathit{tm}}<C_{\mathit{sp}}\left(t\right)$, which further verifies our conclusion. For $g_e=1.0$, we can find that both layers of the network are in alternating chimera states from Fig. 3. Unlike $g_e=0.15$, we can divide the membrane potential into seven regions, but we can still find that the coherence of different regions is exchanged at different times. As shown in Fig. 6, the coherence of the second region at $t=6843.1$ is swapped with that at $t=6884.8$. The coherence of the third region at $t=6884.8$ is swapped with that at $t=6922.7$. For $g_e=5.0$, according to Fig. 3, we can conclude that the two-layer network is in a semi-alternating singular state, that is, part of the oscillator is in an alternating chimera state, while the other part is in a synchronized cluster state. This feature is clearly reflected in Fig. 7, from which it can be seen that about the 30th to the 70th neurons are in the alternating chimera state. When $t=6656.9$, $t=6718.7$, $t=6782.5$, the coherent and incoherent domains of the membrane potential of these neurons are exchanged, respectively. Finally, we plot the variation of the synchronization level $S=\frac{1}{N}\sum _{i=1}^N\left|x_{i,1}-x_{i,2}\right|$ with the electrical coupling strength, as shown in Fig. 8. From Fig. 8, it is more clearly shown that with the increase of electrical synaptic coupling strength, the membrane potential synchronicity of the first layer and the second layer becomes higher, that is, the electrical synaptic coupling between the layers promotes the synchronization of the two layers.

Fig. 3.

Different patterns of the non-locally two-layer neuronal network with $g_{m1}=3.0,g_{m2}=0.1$ and $P=30$ by increasing the interlayer electrical coupling strength $g_e$. a–c $g_e=0.0$: amplitude death for layer I, while incoherent state for layer II. d–f $g_e=0.15$: coherent state for layer I, while alternating chimera state for layer II. g–i $g_e=1.0$: both layers show the alternating chimera states. j–l $g_e=5.0$: both layers show the semi-alternating chimera states. a, d, g and j spatiotemporal plots for layer I; b, e, h and k spatiotemporal plots for layer II. c, f, i and l membrane potential of the two-layer network at $t=7000$

Fig. 4.

Snapshot of the membrane potentials $x_{i2}$ depicting the alternating chimera states at a $t=6777$, b $t=6871.1$, and c $t=6966.5$. The corresponding local curvature $L_i$ are, respectively, shown in d, e, and f. Here, $P=30$, $g_{m1}=3.0$, $g_{m2}=0.1$, and $g_e=0.15$

Fig. 5.

Spatial correlation measure $C_{\mathit{sp}}\left(t\right)$ as a function of time and the time-correlation measure $C_{\mathit{tm}}$ calculated over the time interval $t\in \left[6000,7000\right]$. Here, $P=30$, $g_{m1}=3.0$, $g_{m2}=0.1$, and $g_e=0.15$

Fig. 6.

Snapshot of the membrane potentials $x_{i2}$ depicting the alternating chimera states at a $t=6843.1$, b $t=6884.8$, and c $t=6922.7$. Here, $P=30$, $g_{m1}=3.0$, $g_{m2}=0.1$, and $g_e=1.0$

Fig. 7.

Snapshot of the membrane potentials $x_{i2}$ depicting the semi-alternating chimera states at a $t=6656.9$, b $t=6718.7$, and c $t=6782.5$. Here, $P=30$, $g_{m1}=3.0$, $g_{m2}=0.1$, and $g_e=5.0$

Fig. 8.

Variation of the degree of interlayer synchronization for different values of interlayer coupling strength in the two-layer network. Here $P=30$, $g_{m1}=3.0$, $g_{m2}=0.1$

Then, we change a set of parameters, that is, $g_{m1}=3.0$ and $g_{m2}=0.5$. We observe the changes of the two-layer network when $g_e=0.0$ and $g_e=5.0$, as shown in Fig. 9. When $g_e=0.0$, we can see that the two layers are in amplitude death state and alternating chimera state respectively from Fig. 9. When $g_e=5.0$, we can see that both layers of networks are in synchronized cluster state from Fig. 9. From the comparison between Figs. 3 and 9, it is found that the final states of the two-layer network with sufficient interlayer coupling are different. According to our analysis, this may be caused by the different states of the two-layer networks without interlayer coupling. That is, the different states of the two layers of independent networks may result in different states after coupling with each other. In order to make a clearer judgment on this conclusion, we fix $N=100$, $P=30$, and $g_e=5.0$, and change the intralayer magnetic coupling strength $g_{m1}$ and $g_{m2}$ of the two layers to plot a two-parameter phase diagram, as shown in Fig. 10. In Fig. 10, AD represents amplitude death, COH represents coherent states, CLT represents synchronized cluster states, SAC represents semi-alternating chimera states, AC represents alternating chimera states, and, ICH represents incoherent states, respectively.

Fig. 9.

Snapshots of spatiotemporal and the distribution of membrane potentials of neurons in the two-layer network. a–c $g_e=0$: amplitude death state for layer I, while alternating chimera state for layer II. d–f $g_e=5.0$: both layers show the synchronized cluster states. a and d spatiotemporal plots for layer I; b and e spatiotemporal plots for layer II. c and f membrane potential of the two-layer network at $t=7000$. Here $g_{m1}=3.0$, $g_{m2}=0.5$, $P=30$

Fig. 10.

Two-parameter phase diagram in the $g_{m1}-g_{m2}$ plane where dark blue, blue, cyan, yellow, orange, and red colors, respectively, correspond to the incoherent (ICH), alternating chimera (AC), semi-alternating chimera (SAC), cluster (CLT), coherent (COH), and amplitude death (AD) states

Next, we study the case of intralayer local coupling, that is, $P=1$. We fix $g_{m1}=3.0$ and $g_{m2}=2.0$.

Figure 11 shows the spatiotemporal diagrams when $g_e=0.0,0.5,1.5,3.0,5.0$. When $g_e=0.0$, we can judge from

Fig. 11.

Snapshot of spatiotemporal. a and f $g_e=0.0$: amplitude death for layer I, while incoherent state for layer II. b and g $g_e=0.5$: incoherent states for two layers. c–e and h–j $g_e=1.5$, $3.0$, $5.0$: both layers show the transient chimera states. a–e spatiotemporal plots for layer I; f–j spatiotemporal plots for layer II. Here, $g_{m1}=3.0$, $g_{m2}=2.0$ and $P=1$

Figure 11 that the two layers of independent networks are in amplitude dead state and incoherent state respectively. When $g_e=0.5$, we can judge from

Figure 11 that both layers of networks are in incoherent states. When $g_e=3.0$, Fig. 12 and Fig. 12 show the membrane potential distribution diagrams of the two-layer network at $t=3000$ and $t=5000$ respectively, and Fig. 12 and Fig. 12 show the corresponding local curvature diagrams. We can see that when $t=3000$, both layers of the network are in the chimera states. When $t=5000$, both layers of the network are in the disordered states. In other words, with the change of time, the network changes from a chimera state to a disordered state, which is called the transient chimera state. We calculated the spatial correlation measure $C_{\mathit{sp}}\left(t\right)$ when $g_e=3.0$, as shown in Fig. 13, and it can be seen that the value of $C_{\mathit{sp}}\left(t\right)$ gradually decreases to near 0 with time, which further verifies our judgment. From

Fig. 12.

Snapshot of the membrane potentials depicting the transient chimera state at a $t=3000$, b $t=5000$. The corresponding local curvature $L_i$ for layer II are shown in c and d. Here, $P=1$, $g_{m1}=3.0$, $g_{m2}=2.0$, and $g_e=3.0$

Fig. 13.

Spatial correlation measure $C_{\mathit{sp}}\left(t\right)$ as a function of time over the interval $t\in \left[3000,5000\right]$. Here $g_{m1}=3.0$, $g_{m2}=2.0$, $g_e=3.0$, $P=1$

Figure 11, it can be seen that with the increase of electrical coupling strength, the time for the chimera state to become disordered state in the network becomes longer.

Finally, we want to consider the influence of the number of nonlocal couplings $P$ on the network. We fix $g_{m1}=3.1$ and $g_e=5.0$, and change intralayer magnetic coupling strength $g_{m2}$ and $P$. We observe the corresponding states, and plot a two-parameter phase diagram, as shown in Fig. 14. In the figure, TC represents the transient chimera state, and the rest are the same as above. As can be seen from Fig. 14, the transient chimera state appears only when the number of nonlocal couplings $P$ is small. In addition, we find that when $P$ is greater than a certain value, incoherent states no longer appear in the figure, which may be because magnetic coupling strength of the first layer is so large that when the electrical coupling strength $g_e=5.0$, even the small coupling strength of the second layer will cause the two layers to be in the alternating chimera states.

Fig. 14.

Two-parameter phase diagram in the $P-g_{m2}$ plane where dark blue, light blue, blue, green, orange, and yellow colors, respectively, correspond to the incoherent (ICH), alternating chimera (AC), transient chimera (TC), cluster (CLT), coherent (COH), and amplitude death (AD) states

Conclusion

To sum up, in this paper, we have constructed a two-layer network composed of HR neuron models. Since we would like to take into account the effect of electromagnetic induction generated by the electrical activity of neurons and two kinds coupling produced together on the dynamic behavior of the neuronal network, the network was characterized by each intralayer neuron was connected to its neighbor via magnetic connections and coupling between two layers through electrical synapses.

In particular, we have found alternating chimera states and transient chimera states. According to our research, these have not yet been found in multilayer networks in this coupled manner. In addition, we have found that the state of the network without interlayer coupling may affect the final state of the two-layer network with sufficiently large interlayer coupling. In addition, it has been shown that the transient chimera state may only occurred when the number of intralayer couplings was small.

Our results may be useful for studying the dynamic behavior of real neuronal networks, and we hope that our results will be useful for studying information transmission in the brain and brain diseases. Finally, there are still many problems worthy of further study, such as the effect of time delay on multilayer memristor neural networks and the possible dynamic states in other network topologies.

Author contribution

All authors contributed to the study conception and design. Methodology, software, validation, formal analysis, investigation, data Curation, writing—original draft and visualization were performed by HL. Conceptualization, investigation, resources, writing—review and editing, visualization, supervision, project administration and funding acquisition were performed by YX. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.