Defects formation and spiral waves in a network of neurons in presence of electromagnetic induction

Abstract

Complex anatomical and physiological structure of an excitable tissue (e.g., cardiac tissue) in the body can represent different electrical activities through normal or abnormal behavior. Abnormalities of the excitable tissue coming from different biological reasons can lead to formation of some defects. Such defects can cause some successive waves that may end up to some additional reorganizing beating behaviors like spiral waves or target waves. In this study, formation of defects and the resulting emitted waves in an excitable tissue are investigated. We have considered a square array network of neurons with nearest-neighbor connections to describe the excitable tissue. Fundamentally, electrophysiological properties of ion currents in the body are responsible for exhibition of electrical spatiotemporal patterns. More precisely, fluctuation of accumulated ions inside and outside of cell causes variable electrical and magnetic field. Considering undeniable mutual effects of electrical field and magnetic field, we have proposed the new Hindmarsh–Rose (HR) neuronal model for the local dynamics of each individual neuron in the network. In this new neuronal model, the influence of magnetic flow on membrane potential is defined. This improved model holds more bifurcation parameters. Moreover, the dynamical behavior of the tissue is investigated in different states of quiescent, spiking, bursting and even chaotic state. The resulting spatiotemporal patterns are represented and the time series of some sampled neurons are displayed, as well.

Introduction

Formation of spatiotemporal patterns is an important phenomenon in various natural phenomena caused by interactions between the components. Emergence of different patterns in a system is in direct relation with its dynamics. Hence, studying pattern formation and pattern selection have been noticed increasingly, in various physical, chemical, and biological systems (Bertram et al. 2003; Chen et al. 2014b; Hu et al. 2013; Jakubith et al. 1990; Ma et al. 2013; Van Der Heide et al. 2010; Winfree 1972, 1987; Zhao et al. 2013), which can lead to better understanding of dynamic laws that govern the intended systems.

Spiral wave is one of the elegant patterns chosen by nature. It can be found in galaxies, hurricanes, weather patterns, design of many types of plants and animals, plant tendrils, flowers and leaves, pinecones, and so on. There are some evidences supporting this fact that spiral waves play an essential role in some biological systems (Cherry and Fenton 2008; Davidenko and Pertsov 1992; Gray et al. 1998; Lechleiter et al. 1991). Furthermore, spiral pattern can take place in many excitable media including the oxidation of CO on platinum (Bertram et al. 2003; Beta et al. 2003, 2004; Jakubith et al. 1990). Also there have been some studies focusing on spiral tip trajectory and its possible stability or instability (Fenton et al. 2002), meandering behaviors of spiral seed (Ma et al. 2008), pinned spiral waves (Chen et al. 2014a; Liu et al. 2014), defects in the tissue (Hildebrand et al. 1995; Ma et al. 2016b; Qin et al. 2015) and resulting spatiotemporal patterns including spiral patterns in some excitable tissues including cardiac tissue (Clayton et al. 2011; Fenton et al. 2002). Besides, the effects of presence of different couplings like field coupling and the resulting collective behaviors is an interesting approach to investigate spatiotemporal pattern and signal propagation (Guo et al. 2017).

Spiral wave is essential to understand due to the impression it has on complex cardiac electrical dynamics like tachycardia and ventricular fibrillation (Cherry and Fenton 2008). A spontaneous, asynchronous constriction of cardiac muscle fibers is called ventricular fibrillation (Davidenko and Pertsov 1992; Gray et al. 1998). According to the results of different studies on cardiac arrhythmias, it can be said that fibrillation is not just a random phenomenon (Gray et al. 1998). Although, it is still controversial to have a precise determination of what mechanisms are responsible for these electrical dynamics, there are some experimental results that can prove there is a relationship between initiation of spiral waves and ventricular fibrillation and tachycardia (Cherry and Fenton 2008).

Neurons are the basic blocks of neuronal system, coming together with connections and interactions among them. Information transference between different parts of a neuronal network is possible through a complete or partial synchronization of neurons (Moujahid et al. 2011). Indeed, what we see from the neuronal system, which controls different biological behaviors, is about the collective behaviors of neurons. Pattern recognition based on these collective behaviors is the most useful path to get into a deep insight into the secondary consequences of information transference, which is the primary reason for any normal or abnormal biological behavior. Spiral wave is a unique self-organized (Ma et al. 2012) spatiotemporal pattern, which can affect regulation of collective behaviors in neuronal network. Moreover, there have been evidences confirming the emergence of spiral wave in slices of neocortex (Huang et al. 2004; Schiff et al. 2007). It is found that spiral waves can have a kind of regulating role for ongoing cortical activities (Huang et al. 2010).

Generally, wave propagation in an excitable media can encounter some defects caused by initiation of reentry in a specific time span (Cherry and Fenton 2008). Considering cardiac tissue as an excitable media, reentrant waves can cause extremely fast, self-sustained cardiac activities (Pertsov et al. 1993). It may be due to different abnormalities often caused by lack of blood in a local area of the excitable tissue (for example cardiac tissue), tumor-induced pressure on blood vessels or nerves, or influences of penetration of drug into the body. Reentrant excitation in ventricular muscle may lead to some life-threatening arrhythmias like tachycardia and ventricular fibrillation (Pertsov et al. 1993). It seems that possibility of initiation of reentry depends on the level of excitability of the media associated with memory effects of the system, which reminds self-adaptation (Ma et al. 2016a; Xin-Lin et al. 2015) nature of biological systems. Therefore, spiral wave and its related mechanisms are not separable from memory effects of biological systems (Bueno-Orovio et al. 2008), which is one of the most significant properties of these systems.

Furthermore, it is also possible that the tissue encounter some deformations because of abnormalities caused by electromagnetic radiation exposure to the tissue, which is so harmful (Ma et al. 2016b). Considering physical phenomenon in presence of electromagnetic radiation, which can bring the components of the tissue some dielectric polarizations, some polarization currents can be formed in the tissue. Formation of some internal defects due to these polarization currents can be possible under some particular circumstances. In this study, we have focused on this kind of deformations called internal defects (Ma et al. 2016b). Hence, some additional currents can start to flow, because of dielectric polarization leading the tissue encounter some internal defects. These defects can lead to some beating activities including target waves or spiral waves, reorganizing collective behaviors of the neurons. An external force can define the generated current, which is capable of initiating some wave fronts. For simplicity, we consider periodical external force induced on the boundary of the two-dimensional neuronal network to show the effect of electromagnetic radiation. Besides, we have investigated influence of this external force under a number of excitability levels of the tissue. We have discovered that the level of excitability of the tissue and conditions provided by the power of the intended external force can cause limitations for reentrant waves. Subsequently, emergence of the spiral seed can be confined.

As soon as a spiral seed finds existence in an excitable tissue, it becomes a high frequency source rotating around an organizing center and propagating fortified wave fronts in all the directions (Cherry and Fenton 2008). It can affect regular waves and patterns and build up its own new spatial distribution of electrical activities and then there will occur related subsequent consequences. As a result, it can act as a pacemaker impressing the regular behavior of the excitable tissue (Ma et al. 2010), which was running before the existence of the spiral seed. It can also be said that emergence of the spiral seed is possible when the wave front and the wave back get together (Cherry and Fenton 2008).

The rest of the paper is organized as follows:

In the next section, neuronal model equations with respective variables and parameters are described. In addition, some useful explanations are given to illustrate the advantages of proposed model. In section three, we explain the method for our numerical study and simulations. The results of our simulations are depicted and the observations are described in details. Finally, the conclusions are given in section four.

Model and description

Various electrical activities of neurons have been studied through neuronal models. The four-variable Hodgkin-Huxley model (Hodgkin and Huxley 1952) with its nonlinear equations is the basic neuronal model. After that, some simplified neuronal models like Hindmarsh–Rose model (HR) have been proposed and used in many studies and the bifurcation parameters are investigated, as well (Gu et al. 2014). HR model (Hindmarsh and Rose 1982) is one of the successful ones in reproducing different states of quiescent, spiking, bursting, and also chaotic behaviors in neuronal dynamics. HR neuronal model mostly is known for its spiking-bursting behavior. Generally, spiking-bursting activities rely on recurrent transition between resting state and firing state of neuron (Wu et al. 2017). There are very good studies on spatiotemporal waves in an excitable tissue considering different circumstances (Fenton et al. 2002; Jun et al. 2009; Ma et al. 2010, 2011, 2013; Qin et al. 2014; Xu et al. 2015). However, the real electrical activity of a neuron shows far more complexities. In fact, electrical activities of the body are not such simple to be considered as electrical activities from an electrical circuit (which is all about electrons). Here in biological systems, there are ions instead of electrons that are responsible for what we see as electrical activity in the body, which makes it more complicated. Continuous displacement of ions across the membrane and effects of mechanisms related to accumulation of ions cause continuous time-varying electromagnetic field (Xu et al. 2017) that has mutual effects on electrical field, as well. Obviously, finding a way to get in touch with the reality that governs neuron’s activities is truly required. Accordingly, many researchers believe that we should consider more bifurcation parameters to get a more beneficial neuronal model (Gu and Pan 2015; Moujahid et al. 2010, 2011; Rech 2012; Torrealdea et al. 2009). Therefore, considering the complex electrophysiological properties of the neuron, it is necessary not to ignore the influence of magnetic field from the electric polarization and fluctuation of the membrane potential (Lv et al. 2016; Ma et al. 2016a, 2017). It is important to detect changes of electromagnetic field affecting membrane potential and consequently the electrical behavior of neuron (Ma and Tang 2015).

Considering the electromagnetic radiation (Ma et al. 2017), here we propose an improved HR neuronal model. In this model a magnet flux is introduced as the fourth variable in addition to the three variables of the original HR model, which is studied in a number of researches (Lv and Ma 2016; Lv et al. 2016; Wu et al. 2017). By using the application of memristor and considering magnet flux (Lv et al. 2016), the new four variable HR model for an isolated neuron is described as follow;

$$\begin{array}{cc}& \frac{\mathit{dx}}{\mathit{dt}}=y-a{x}^3+b{x}^2-z+I_{\mathit{ex}}-k_{1}w\left(\phi \right)x\hfill \\ \hfill & \frac{\mathit{dy}}{\mathit{dt}}=c-d{x}^2-y\hfill \\ \hfill & \frac{\mathit{dz}}{\mathit{dt}}=r\left(s\left(x-x_0\right)-z\right)\hfill \\ \hfill & \frac{d\phi }{\mathit{dt}}=x-k_2\phi \hfill \\ \hfill \end{array}$$ 1

The variable $x$ represents membrane potential, $y$ is the slow current for recovery variable, the variable $z$ is adaptation current, and $\phi$ denotes a variable for magnetic flux across the membrane (Lv et al. 2016). Here $k_1$ and $k_2$ are used to describe the interaction between magnetic flux and membrane potential (Lv et al. 2016). We set $k_1=1$ and $k_2=0.5$.

$w\left(\phi \right)$ is memory conductance of a memristor controlled by magnetic flux (Lv et al. 2016), which is defining the relationship between magnetic flux and membrane potential. $w\left(\phi \right)$ is often described as:

$$w\left(\phi \right)=\alpha +3\beta {\phi }^2$$ 2

$\alpha =0.1$ and $\beta =0.02$ are fixed parameters. In Eq. 1, we set $a=1$, $b=3$, $c=1$, $d=5$, $s=4$, $r=0.006$, $x_0=-1.6$ as fixed parameters, and $I_{\mathit{ex}}$ as control parameter. $I_{\mathit{ex}}$ shows external current, $b$ is an intrinsic parameter, $r$ denotes rate of activation for some current and $x_0$ is parameter affecting the activation of adaptation current (Zhang et al. 2015).

In order to have a better intuition of the term $k_{1}w\left(\phi \right)x$ it is beneficial to show the description for it based on Faraday law of electromagnetic induction (Lv et al. 2016),

$$i=\frac{dq\left(\phi \right)}{\mathit{dt}}=\frac{dq\left(\phi \right)}{d\left(\phi \right)}\frac{d\phi }{\mathit{dt}}=w\left(\phi \right)V=k_{1}w\left(\phi \right)x$$ 3

where the parameter $k_1$ is the feedback gain and the variable $V$ is the electromotive force, holding a same physical unit (Lv and Ma 2016).

In order to study wave propagation and spatiotemporal behavior of neurons, we develop Eq. 1 to a large number of neurons forming a square array network. Therefore, the equations can be described as follow;

$$\begin{array}{cc}& \frac{d{x}_{\mathit{ij}}}{\mathit{dt}}=y_{\mathit{ij}}-a{x}_{\mathit{ij}}^3+b{x}_{\mathit{ij}}^2-z_{\mathit{ij}}+I_{\mathit{ex}}-k_{1}w\left({\phi }_{\mathit{ij}}\right)x_{\mathit{ij}}+D\left(x_{i+1j}+x_{i-1j}+x_{ij+1}+x_{ij-1}-4x_{\mathit{ij}}\right)+f{\delta }_{i\alpha }{\delta }_{j\beta }\hfill \\ \hfill & \frac{d{y}_{\mathit{ij}}}{\mathit{dt}}=c-d{x}_{\mathit{ij}}^2-y_{\mathit{ij}}\hfill \\ \hfill & \frac{d{z}_{\mathit{ij}}}{\mathit{dt}}=r\left(s\left(x_{\mathit{ij}}-x_0\right)-z_{\mathit{ij}}\right)\hfill \\ \hfill & \frac{d{\phi }_{\mathit{ij}}}{\mathit{dt}}=x_{\mathit{ij}}-k_2{\phi }_{\mathit{ij}}\hfill \\ \hfill \end{array}$$ 4

in which,

$$w\left({\phi }_{\mathit{ij}}\right)=\alpha +3\beta {\phi }_{\mathit{ij}}^2$$ 5

where the subscript $\mathit{ij}$ shows position of each neuron in the two-dimensional network. Parameter $D$ denotes coupling intensity between the neurons. We set $D=1$ as a constant coupling intensity. $f$ is an external force. ${\delta }_{i\alpha }=1$ for $i=\alpha$, ${\delta }_{i\alpha }=0$ for $i\ne \alpha$; ${\delta }_{j\beta }=1$ for $j=\beta$, ${\delta }_{j\beta }=0$ for $j\ne \beta$. We can impose $f$ on the left boundary by setting $\beta =1.2$ and $\alpha =1:110$, or have a local imposing in the center of the network by $\alpha =\beta =53:56$. For simplicity, we choose a periodical behavior for $f$ as follow;

$$f=Acos\left(\omega t\right)$$ 6

In neuroscience perspective, $I_{\mathit{ex}}$ refers to supraspinal cord or background environment (Zhang et al. 2015). In HR model adjusting $I_{\mathit{ex}}$ parameter can bring different manners of dynamical behaviors, like quiescent, spiking, and bursting behaviors, as a determinative parameter for the local behavior of each isolated neuron. Moreover, it is possible to attain different responses from the large array of the neurons in excitable tissue in each case by adjusting $A$ and $\omega$ parameters.

Numerical results and discussion

Calculation of numerical study is under Neumann (no flux) boundary condition and by utilizing the Euler forward algorithm. We consider a network with nearest-neighbor connection composed of a square array of 110 × 110 neurons in which the local dynamic of each neuron complies with the intended four variable HR model. In our numerical calculation, we use ($x.y.z.\phi$) = (0.01, 0.02, 0.003, 1.01) as initial values of the variables. Considering appropriate $I_{\mathit{ex}}$, we investigate the formation of the existed wave patterns by applying external periodical force on the left boundary of the plane with different values of $A$ and $\omega$.

Actually, assigning a specific $I_{\mathit{ex}}$ to the excitable tissue, determines the level of excitability of the tissue. It plays a decisive role and turns out to be a conducting operator on its own, and is also related to provided possibility of reentrant waves. It means that the media would be able to transmit the stimulation initiated of any reason from one region to another. This transmission within its speed and its power and the frequency of reproducing the next excitation will all play a decisive role in what the tissue exhibits as a wave pattern. In other words, each time the moment in which an individual neuron faces a new stimulation, passed by the neighbor neurons, can make that neuron take a specific level of voltage and a specific pattern for its own action. This pattern selection relies on neuron’s situation itself, too. This is what we call interaction between each neuron and other neurons in its neighborhood. We can develop the case to a large number of neurons with a similar explanation. In this way, if we collect the behavior of all those neurons and display resulting distribution of electrical activities in color, there will be a colorful plane representing a spatiotemporal pattern by the colors. Now, among possible spatiotemporal patterns, if the spiral seed find existence under appropriate circumstances, and the spiral wave take emergence, there will appear a unique appearance of tissue from the rhythm of colors (level of voltages). Gradually, if there is no other active operator to break down this existed spiral seed before it comes to power, the propagated spiral wave can cover the entire tissue. Consequently, it can govern the dynamics of every single neurons of the tissue and reorganize the neurons based on its own dynamics. It is worth mentioning the lethal arrhythmias caused by pinned spiral waves, which are life threatening (Pan et al. 2016) and have been studied in a number of researches (Liu et al. 2014; Zemlin and Pertsov 2012). Rotating spiral seed and propagating strong spiral waves are due to level of excitability and other factors that have made proper conditions for this occurrence.

For examining different levels of excitability of the tissue, first we set $I_{\mathit{ex}}=1$ which is supposed to bring a quiescent state for an isolated magnetic Hindmarsh–Rose neuron (Lv et al. 2016). It is found in Fig. 1 that no propagation take place by having $I_{\mathit{ex}}=1$ even when we induce the external force with higher amounts of $A$ and $\omega$.

Fig. 1.

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network at $time=1000$ time units for $I_{\mathit{ex}}=1$, $\beta =1.2$ and $\alpha =1:110$. For a $A=1$, $\omega =0.0001$; b $A=1$, $\omega =0.001$; c $A=2$, $\omega =0.0001$; d $A=2$, $\omega =0.001$

In fact, by setting the value of $I_{\mathit{ex}}$ below a certain threshold, the media cannot be able to support any propagation. In other words, excitability of the media is not large enough to stimulate neurons and get them fired to represent a bursting behavior by the induced force on the left boundary. Furthermore, when we pick a higher amount for parameter $A$ or $\omega$, all it does is to excite only the left boundary neurons, but there is no propagation. In addition, some slight deformations on the left boundary are visible in Fig. 1c, d, caused by increase of external force amplitude. As it is clear in Fig. 1, there is no propagation by having a lower or even higher amplitude or frequency of the external force, either, since the level of excitability for the whole tissue is $I_{\mathit{ex}}=1$. In order to have a closer point of view, Fig. 2 shows the time series of node (55, 55) as an instance, which is placed in the center of the square array network. As it is clear in Fig. 2, the quiescent state persists even after inducing the external force with this level of excitability.

Fig. 2.

Sampled time series from node (55, 55) (example for central area) in the square array network for $I_{\mathit{ex}}=1$, $\beta =1.2$ and $\alpha =1:110$. For a $A=1$, $\omega =0.0001$; b $A=1$, $\omega =0.001$; c $A=2$, $\omega =0.0001$; d $A=2$, $\omega =0.001$

As we know, magnetic HR neuron model can show spiking behavior by taking appropriate $I_{\mathit{ex}}$. In order to investigate not only the behavior of a single neuron, but the spatiotemporal pattern from a large array of spiking neurons coupling together, we applied $I_{\mathit{ex}}=2$ (Figs. 3, 4) and $I_{\mathit{ex}}=2.3$ (Figs. 5, 6) to the whole tissue. We found that, propagation can take place when we let $I_{\mathit{ex}}$ take greater amounts (Figs. 3, 5). Besides, there will be observed periodicity in the time series (Figs. 4, 6). Although, both amounts of 2 and 2.3 for $I_{\mathit{ex}}$ regard to spiking state of magnet HR neuron, a small increase in level of excitability of the tissue (from $I_{\mathit{ex}}=2$ to $I_{\mathit{ex}}=2.3$) makes tissue more favorable for holding spiral seeds. This can be understood by comparing the snapshots in Figs. 3 and 5. The details of behavioral observations of the neurons are described in the next few paragraphs.

Fig. 3.

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network at $time=15000$ time units for $I_{\mathit{ex}}=2$, $\beta =1.2$ and $\alpha =1:110$. For a $A=1$, $\omega =0.0001$; b $A=1$, $\omega =0.001$; c $A=1$, $\omega =0.01$; d $A=1$, $\omega =0.1$; e $A=2$, $\omega =0.0001$; f $A=2$, $\omega =0.001$; g $A=2$, $\omega =0.01$; h $A=2$, $\omega =0.1$

Fig. 4.

Sampled time series from node (55, 2) (example for left boundary area of network) and node (55, 55) (example for central area of network) in the square array network for $I_{\mathit{ex}}=2$, $\beta =1.2$ and $\alpha =1:110$. For a $A=1$, $\omega =0.0001$; b $A=1$, $\omega =0.001$; c $A=1$, $\omega =0.01$; d $A=1$, $\omega =0.1$; e $A=2$, $\omega =0.0001$; f $A=2$, $\omega =0.001$; g $A=2$, $\omega =0.01$; h $A=2$, $\omega =0.1$

Fig. 5.

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network at $time=15000$ time units for $I_{\mathit{ex}}=2.3$, $\beta =1.2$ and $\alpha =1:110$. For a $A=1$, $\omega =0.0001$; b $A=1$, $\omega =0.001$; c $A=1$, $\omega =0.01$; d $A=1$, $\omega =0.1$; e $A=2$, $\omega =0.0001$; f $A=$ 2, $\omega =0.001$; g $A=2$, $\omega =0.01$; h $A=2$, $\omega =0.1$

Fig. 6.

Sampled time series from node (55, 2) (example for left boundary area of network) and node (55, 55) (example for central area of network) in the square array network for $I_{\mathit{ex}}=2.3$, $\beta =1.2$ and $\alpha =1:110$. For a $A=1$, $\omega =0.0001$; b $A=1$, $\omega =0.001$; c $A=1$, $\omega =0.01$; d $A=1$, $\omega =0.1$; e $A=2$, $\omega =0.0001$; f $A=2$, $\omega =0.001$; g $A=2$, $\omega =0.01$; h $A=2$, $\omega =0.1$

In Figs. 4 and 6, the node (55, 2) represents time behavior of neurons on the left boundary, and the node (55, 55) is an example to shows the time behavior of the center of the network. In this paper, the duration of sampled time series is chosen based on the appropriate time span needed to show the time behavior of that specific neuron until it gets to a kind of stable state with no significant change in its behavior. In this way, the overall behavior of that specific instance neuron can be detected through a suitable time span. For example, the displayed duration by $\mathit{time}$ = 15,000 time units for the node (55, 2) in Fig. 4b, f is to show variations that need a longer time span for demonstration, while the node (55, 2) in Fig. 4a needs only $\mathit{time}$ = 2000 time units.

It is found that, by level of excitability prepared from $I_{\mathit{ex}}=2$ and $I_{\mathit{ex}}=2.3$, the plane waves take emergence and move across the plane. However, in this case, these plane waves are not able to reach out the opposite boundary of the plane in the beginning. Actually, here the level of excitability of the tissue makes it send back some plane waves in the middle of the plane. Consequently, there will appear some secondary recursive waves right in front of the primary waves in the opposite direction. Therefore, before primary plane waves generated from the left boundary (induced by our stimulation) reach out the opposite boundary, they meet some secondary plane waves coming to them in the opposite direction. This causes destruction for both groups of the wave fronts. However, amazingly, we have noticed this coexistence of two traveling wave fronts is just a transition. Actually, by providing a longer run time, it is possible to see the very slow movement of area that the initiated plane waves from the left boundary face the opposite ones, and then this area completely disappears. It means that, after a longer time span, one of the two groups, primary plane waves initiated from the left boundary and secondary plane waves from the right boundary prevail and occupy the whole tissue. This can be a useful explanation for the normal wave propagation in an excitable tissue including cardiac tissue, since the propagation of electrical waves should reach out the desired parts of the cardiac tissue through a normal rhythm. Figures 3 and 5 show different states of the resulting spatial pattern formation under a variety of changes in $A$ and $\omega$ with $I_{\mathit{ex}}=2$ and $I_{\mathit{ex}}=2.3$, respectively.

As it is shown in Fig. 3, increasing the amplitude and frequency of the external periodical force to some levels can provide appropriate conditions for emergence of the spiral wave (Fig. 3f). But, after that, further increase of these two parameters can make the migrating plane waves faster and more powerful, which limits existence of spiral seed.

We found that a slight increase in $I_{\mathit{ex}}$ from 2 to 2.3 leads the tissue get in appropriate circumstances to hold development of spiral waves, more (readers can compare snapshots in Figs. 3, 5). By this increase, emergence of spiral seed gets more possible, even though its persistence suffers from low amplitude and frequency both together, as it is depicted in Fig. 5 (Fig. 5a). In fact, increasing the amplitude accompanied by higher amounts of $\omega$ will give spiral waves a better development. Noticing Fig. 5e, f, greater amount of parameter $A$ without any significant increase in parameter $\omega$ neutralizes the spiral seed. That is because this set of parameters gives the migrating re-excited waves enough power to extinguish the existed rotating spiral seed.

The time behavior of instance node (55, 55), which is in the middle of the plane is shown in Figs. 4 and 6. As it is noticeable in Fig. 4, the node (55, 55) switches from spiking state to bursting state after about 1000 time units for $I_{\mathit{ex}}=2$, while this switching occurs after about 600 time units for $I_{\mathit{ex}}=2.3$ (Fig. 6). Actually, this point is exactly when the neuron (node) receives the first propagated wave front, which makes it start to burst for the rest of the time span. In fact, the ability of neurons to represent bursting behavior is essential for these continuous wave propagations.

Moreover, the time series of node (55, 2) is drawn in Figs. 4 and 6, which represents the time behavior of the left boundary neurons. Interestingly, by $\omega =0.001$ the left boundary of the excitable tissue takes a slightly different behavior, as it is clear through the time series of node (55, 2) in Figs. 4b, f and 6b, f. In this case, bursting behavior of the neuron subsides and goes for a spiking state in an assignable period, and then it starts to grow again after the pass of a specific time span to get back to bursting state. This will cause the tissue represent a kind of different behavior, which cannot be regarded a transition only, since this sequence of manner repeats all over again.

It is interesting to discover the collective behavior and pattern formation under chaotic circumstances. As we know through previous studies, magnetic HR neuron can represent chaotic behaviors by further increase of external constant current ($I_{\mathit{ex}}$) (Lv et al. 2016). Therefore, we set $I_{\mathit{ex}}=3.5$ for all the neurons and show the results of two examples of parameter setting of $A$ and $\omega$ (Fig. 7). It is found that by setting $I_{\mathit{ex}}=3.5$, formation of spiral seed gets very soon. In fact, in this case, spiral wave is observable almost from the beginning, and it starts to grow and cover the entire tissue with the passage of time. Moreover, despite previous cases, in this case, as soon as the spiral seed finds existence (which is too soon comparatively), it gets to power without any opportunity of being destructed. Thus, every single neurons of the tissue will be reorganized based on the governing dynamics of the spiral pattern. To have a closer point of view about this distinct behavior, and see how this occupation occurs, we capture the unique exhibition of resulting spiral waves over the time in Fig. 7. Figure 7a–d shows emergence and propagation of spiral waves with different chiralities (Li et al. 2013, 2014; Pan et al. 2013) over the time by $I_{\mathit{ex}}=3.5$, $A=2$, $\omega =0.001$. This occupation takes place by a pair of high frequency spiral tips rotating in the opposite direction (Fig. 7a–d). It is also noticeable in Fig. 7e–h that four spiral seeds appear near each other (almost from the beginning), and generate fortified wave fronts. For Fig. 7e–h we set $I_{\mathit{ex}}=3.5$, $A=1$, $\omega =0.01$.

Fig. 7.

Development of spiral waves in chaotic excitable tissue over the time. The snapshots show spatial distribution of membrane potential with color scale for neurons in the square array network for $\beta =1.2$ and $\alpha =1:110$. The parameters setting for a–d is $I_{\mathit{ex}}=3.5$, $\text{A}=2$, $\mathrm{\omega }=0.001$ and for e–h is $I_{\mathit{ex}}=3.5$, $\text{A}=1$, $\mathrm{\omega }=0.01$. For a, e $t_1=750$ time units; b, f $t_2=1500$ time units; c, g $t_3=3000$ time units; d, h $t_4=7500$ time units

With the given explanations and results, this question arises: What happens with the collective behavior of the neurons when their local dynamic is in fast spiking state? In order to study this case, we increased $I_{\mathit{ex}}$ to 4.5. Then we induced the external periodical force on the left boundary, just as before. The resulting spatiotemporal patterns and calculated time series are shown in Figs. 8, 9 and 10. In this case, the effect of periodical force in the center of the network is investigated, as well (Fig. 9). Hence, it would be possible to understand the effect of external force location, and the different spatial pattern brought to the tissue due to this different location.

Fig. 8.

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network at $time=15000$ time units for $I_{\mathit{ex}}=4.5$, $\beta =1.2$ and $\alpha =1:110$. For a $A=1$, $\omega =0.0001$; b $A=1$, $\omega =0.001$; c $A=1$, $\omega =0.01$; d $A=1$, $\omega =0.1$; e $A=2$, $\omega =0.0001$; f $A=2$, $\omega =0.001$; g $A=2$, $\omega =0.01$; h $A=2$, $\omega =0.1$

Fig. 9.

The snapshots of spatial distribution of membrane potential with color scale for neurons in the square array network at $time=15000$ time units for $I_{\mathit{ex}}=4.5$, $\alpha =\beta =53:56$. For a $A=1$, $\omega =0.0001$; b $A=1$, $\omega =0.001$; c $A=1$, $\omega =0.01$; d $A=1$, $\omega =0.1$; e $A=2$, $\omega =0.0001$; f $A=2$, $\omega =0.001$; g $A=2$, $\omega =0.01$; h $A=2$, $\omega =0.1$

Fig. 10.

Sampled time series from node (55, 2) (example for left boundary area of network) and node (55, 55) (example for central area of network) in the square array network for $I_{\mathit{ex}}=4.5$, $\beta =1.2$ and $\alpha =1:110$. For a $A=1$, $\omega =0.0001$; b $A=1$, $\omega =0.001$; c $A=1$, $\omega =0.01$; d $A=1$, $\omega =0.1$; e $A=2$, $\omega =0.0001$; f $A=2$, $\omega =0.001$; g $A=2$, $\omega =0.01$; h $A=2$, $\omega =0.1$

Actually, appliance of $I_{\mathit{ex}}=4.5$ raises the excitability of the tissue to a high level. Therefore, the tissue exhibits a very turbulent behavior, the propagation is so powerful that there is no evidence of recursive wave fronts. Besides, the edges of wave fronts are so slippery (Fig. 8). To our knowledge, spiral seed finds existence when the wave front meet the wave back under appropriate conditions, which can cause phase singularity (Huang et al. 2004, 2010; Winfree 2001). Here singularity refers to a dynamically stable state that can organize the neurons based on its own dynamics without increasing the power (Huang et al. 2010). As a result, in this case, possibility of formation of spiral seed gets very high due to high slipperiness of the wave borders. Amazingly, here it is far more difficult for the spiral seed to be completely annihilated due to its very high possibility to join another nearby spiral seed or even breakup to more spiral seeds. These mechanisms bring the tissue a distinct appearance with scarce behaviors.

In Fig. 10, the node (55, 2) represents time behavior of neurons on the left boundary, and the node (55, 55) is an example to shows the time behavior of the central area of the network. The duration of sampled time series is chosen based on the appropriate time span needed to detect overall behavior of the neuron. As it is shown in Fig. 10, in this case, the transition of the beginning of the time series gets too long and the fluctuations become so fast. In addition, reminding the behavior of node (55, 2) in Figs. 4b, f and 6b, f, here in this case a kind of off and on behavior is recognizable in Fig. 10b, f, too. For further investigation, the frequency spectra of each time series in Fig. 10 are shown in Fig. 11, correspondingly.

Fig. 11.

Frequency spectra for node (55, 2) (example for left boundary area of network) and node (55, 55) (example for central area of network) in the square array network for $I_{\mathit{ex}}=4.5$, $\beta =1.2$ and $\alpha =1:110$. For a $A=1$, $\omega =0.0001$; b $A=1$, $\omega =0.001$; c $A=1$, $\omega =0.01$; d $A=1$, $\omega =0.1$; e $A=2$, $\omega =0.0001$; f $A=2$, $\omega =0.001$; g $A=2$, $\omega =0.01$; h $A=2$, $\omega =0.1$

Conclusion

In this study, the possible mechanism of defects in an excitable tissue due to abnormalities caused by any biological reasons is investigated. Some time series and respective frequency spectra are calculated to detect mode transition in electrical activities. In order to explain an excitable tissue, a square array network of four-variable magnetic Hindmarsh–Rose neurons with nearest-neighbor connection is designed. A variety of possible spatiotemporal patterns in the excitable tissue model is detected. It is found that both level of excitability of the tissue and the induced external force are responsible for initiation of spiral wave. The robustness of emerged spiral seed is examined through variation of some parameters, as well. The level of excitability of the tissue plays a decisive role in making the tissue confine emergence of spiral wave or hold its development.