Variations of the spontaneous electrical activities of the neuronal networks imposed by the exposure of electromagnetic radiations using computational map-based modeling

The interaction between neurons in a neuronal network develops spontaneous electrical activities. But the effects of electromagnetic radiation on these activities have not yet been well explored. In this study, a ring of three coupled 1-dimensional Rulkov neurons and the generated electromagnetic field (EMF) are considered to investigate how the spontaneous activities might change regarding the EMF exposure. By employing the bifurcation analysis and time series, a comprehensive view of neuronal behavioral changes due to electromagnetic inductions is provided. The main findings of this study are as follows: 1) When a neuronal network is showing a spontaneous chaotic firing manner (without any external stimuli), a generated magnetic field inhibits this type of behavior. In fact, EMF completely eliminated the chaotic intrinsic behaviors of the neuronal loop. 2) When the network is exhibiting regular period-3 spiking patterns, the generated magnetic field changes its firing pattern to chaotic spiking, which is similar to epileptic seizures. 3) With weak synaptic connections, electromagnetic radiation inhibits and suppresses neuronal activities. 4) If the external magnetic flux has a high amplitude, it can change the shape of the induction current according to its shape 5) when there are weak synaptic connections in the network, a high-frequency external magnetic flux engenders high-frequency fluctuations in the membrane voltages. On the whole, electromagnetic radiation changes the pattern of the spontaneous activities of neuronal networks in the brain according to synaptic strengths and initial states of the neurons.


Introduction
Many studies have shown that spontaneous electrical activities, namely human brain sleep patterns, respiration, heart rate generation and regulation, motor control, and sensory information are responsible for almost 80 percent of metabolic energy consumption in the brain (Alekseev et al., 2010;Bortolotto et al., 2019;Dipalo et al., 2017;Friesen, 1989;Gu et al., 1994;Mazzoni et al. 2007;Moshtagh-Khorasani et al., 2013;Napoli & Obeid, 2016). Some patterns of periodic spontaneous activity such as respiration and heart rate generation accompany us throughout life. So, investigating mechanisms and interactions underlying these activities are of great importance (Mazzoni et al., 2007).
Spontaneous electrical activity of neuronal networks is known to occur during the absence of external stimulation. These activities have complex origins and underlying mechanisms, but they are generated due to interactions of populations of neurons concerning synaptic couplings (Luhmann et al., 2016); thus, resembling every other function of biological systems, these natural events require cybernetic modeling approach leading us to more realistic and biologically plausible models. These synaptic couplings and the strength of them also have significant effects in synchronization mode transitions (Li et al., 2022b).
Computational neuron models are generally divided into two separate groups of ordinary differential equations (ODE)-based Action Editor: Geir Halnes. models and phenomenological models. The ODE-based models suffer from huge parameter space (so-called curse of dimensionality) caused by a high-dimensional system of nonlinear partial differential equations (Girardi-Schappo et al., 2013). It has been proven that phenomenological models can resemble aspects of neural activity, qualitatively. In the last decade, another subclass of phenomenological models known as map-based systems, have received much attention (Courbage et al., 2007).
Accordingly, map-based models are designed in order to simulate the collective dynamics of neuronal networks (Courbage et al., 2007). These models have many advantages over continuous-time ODE-based ones, including less complexity, numerical reliability, and computational costs while considering actual interactions in alive biological systems (Girardi-Schappo et al., 2013;Ibarz et al., 2011). Modified and confined logistic (MCL) model of Mesbah and Rulkov map are two notable examples of comprehensive map-based neuronal models (Mesbah et al., 2014;Shilnikov & Rulkov, 2003).
With the increasing use of electromagnetic devices such as mobile phones and therapeutic methods like transcranial magnetic stimulation (TMS), studying the possible effects of electromagnetic radiations on intrinsic functions of the nervous system has become an attractive field of research in recent decades, especially due to the inconsistency of previous studies (Bodewein et al., 2019).
Research showed that noise could inhibit the firing rate of neurons or enhance transmitting weak signals . If the noise is intensified, it can lead a loop of two neurons to a state of noise-induced chaos . Similarly, electric fields may weaken multiple vibrational resonances in neuronal systems .
There have been several studies analyzing the effects of electromagnetic induction and TMS on neuronal behavior and they have shown excitatory and inhibitory effects, as well as the rivalry between firing-capable axonal constituents for cellular excitability (Miyawaki et al., 2012;Wu et al., 2015;Yang et al., 2021a). For instance, in (Gramowski-Voß et al., 2015), increased activity and intensified burst structure in neuronal network activity are seen due to exposure to electromagnetic field (EMF). In another study, Zandi et al. used the Hindmarsh-Rose model to study different firing patterns of two coupled neurons with various synaptic couplings in the presence of electromagnetic induction (Zandi-Mehran et al., 2020). Some studies also investigated the behavioral change in an isolated Hindmarsh-Rose neuron with the presence of external electromagnetic field . Similarly, in , the Hodgkin-Huxley model is employed under the electromagnetic radiation by introducing an additional membrane current to investigate its impacts on neuronal activity, and suppressing effects on the firing activity of neurons were disclosed that were in agreement with the experimental findings. Although that study was on a single neuron and not neuronal networks, which are more biologically plausible (Qu et al., 2016).
In another recent study, Kafraj et al. modified the Izhikevich neuron model and investigated the effects of magnetic field and noise on neuron behavior (Kafraj et al., 2020). They found that external fields can increase the firing rate and change the firing patterns of the neuron. Their study is also based on an isolated neuron without considering the neuronal interactions and couplings. Interestingly, neuronal network motifs constructed by three neurons using the Izhikevich model showed chaotic resonance under electromagnetic induction .
To investigate the neuronal behavior in the presence of electromagnetic induction, conventional approaches like ODE-based methods are inefficient and hard to use due to high reductionism and the need for large sets of parameters as well, while map-based modeling is much simpler and more comprehensive alternative (Ibarz et al., 2011). In a study by Bortolotto et al., the logistic KTz model is used to represent one single neuron under electromagnetic radiation and results are then compared to Hindmarsh-Rose model (Bortolotto et al., 2019). Logistic KTz map is 3-dimensional, therefore it has a high computational cost. Besides, the effect of synaptic coupling and intrinsic changes in behavior are not declared in their study.
Some other studies focused on EMF effects on neuronal behavior from other perspectives. In (Ramakrishnan et al., 2022), a map neuron is combined with a discrete memristor model to investigate electromagnetic inductions. Some neuronal behaviors have been found, including spiking, periodic bursting, and chaotic bursting. Furthermore, they have examined the synchronization behavior of 100 neurons in a bidirectional ring network with both electrical and chemical synaptic couplings.
Modified Hindmarsh-Rose is frequently used in previous works. An examination used this model and proposed optimal primary-secondary approach for synchronization of two Hindmarsh-Rose neurons under magnetic flow effect, which can be used in controlling the behavior of neuron (secondary) using a pacemaker (primary) (Wouapi et al., 2021). Additionally, Others used this model to study discharge patterns and the effects of electromagnetic induction on modular neural networks (An & Qiao, 2021;Liu et al., 2021). Liu et al. concluded that the interaction between different subnetworks and electromagnetic induction parameter weaken the dynamical robustness, in contrast, the self-induction parameter could improve the dynamical robustness of modular network (Liu et al., 2021).
Li et al. employed a two-dimensional Rulkov map combined with a discrete memristor model to explore magnetic flux influence on the electrical behavior of a single neuron (Li et al., 2022a). Temperature and time delay of signal transmission also affect the behavior of neuronal networks significantly. Synchronization is improved with coupling strength when there is no time delay and low temperature in the network (Wu et al., 2022). Filtered signals also play important roles in neuronal excitability. When the harmonic amplitude drops, the neuron's response to such filtered signals becomes more insignificant (Yu et al., 2023).
Izhikevich and FitzHugh-Nagumo models were also used for neuronal activity mode transitions from an energy point of view and transmission of low-frequency signal driven by high-frequency stimulus under EMF influences, respectively (Ge et al., 2020;Yang et al., 2021b).
None of the previous studies focused on spontaneous electrical activities of a neuronal network. In addition, all of them suffer from at least one of the following flaws: 1) considering a single and isolated neuron, 2) using computationally high-cost or continuous-time models, 3) not including external magnetic flux.
Thus, in this article, our focus is to comprehensively explore variations in neuronal behaviors that emerge spontaneously in neuronal networks under EMFs exposure using a computationally efficient approach.
Comprehending the interactions between neurons is a challenging topic, and even a network constituted by only three neurons can exhibit rich and complex behavior (Bashkirtseva et al., 2020). So, a ring of three unidirectionally coupled mapbased neurons with the discretized model of electromagnetic induction is considered, and the effects of EMF exposure on their intrinsic activity are investigated through bifurcation and time series analysis. The model used in this study is the one-dimensional Rulkov map, therefore this approach is computationally efficient in studying neuronal network behavior.
The paper is organized as follows. In Sect. 2, we introduce our employed models, including one-dimensional Rulkov model, three unidirectionally coupled neuron model, and discrete model of electromagnetic radiation. In Sect. 3, we proceed to the bifurcation analysis of the model under electromagnetic radiation, and the results of our research are represented. Ultimately, the conclusion is provided in Sect. 4.

Models
Because of the interaction between neurons in the biological system, neuronal networks should be considered as open systems (Von Bertalanffy, 1950). Open systems maintain their structure through dissipating energy which permanently keeps them in states of non-equilibrium and uncertainty (Grande García, 2007). So, considering the fact that biological systems are far from equilibrium, it is imprecise to analyze the biological systems from a deterministic point of view.
It has been reported that the Rulkov model can produce spatiotemporal regimes similar to Hodgkin-Huxley-like models (Courbage et al., 2007), such as bursting and spiking behaviors (Wagemakers & Sanjuán, 2013) using completely transparent iteration algorithms with shorter computational time (Cao & Sanjuán, 2009). In this section, we introduce our neuronal networks based on one-dimensional Rulkov and electromagnetic radiation models.
While most of the parameters in the phenomenological models are not exact representatives of biophysical properties, they reproduce the behavior and dynamics of the neurons accurately by discretization and bifurcation analyses (Courbage & Nekorkin, 2010;Ibarz et al., 2011;Mesbah et al., 2014).

Single neuron model
Rulkov map is one of the best map-based models due to representing rich neuronal dynamics and low computational cost (Rulkov et al., 2004;Shilnikov & Rulkov, 2003). Here, onedimensional Rulkov map, represented in Eq. (1), is considered for the fast dynamics of every single neuron in the neuronal network.
where f(γ, x) is the Rulkov function, and γ is a control parameter associated with slow neural dynamics, and the variable x describes fast dynamics of the neuron (NF, 2001). A bifurcation diagram of a single neuron with respect to γ is given in Fig. 1. with different values of γ, the map presented in Eq. (1), mimics different neuronal behaviors. As γ decreases, the neuron bifurcates and switches its behavior to more chaotic behaviors. To have chaotic behaviors, 1 is set to 4.1 (Bashkirtseva et al., 2020).

Model of three coupled neurons
The ring configuration is one of the simplest network motifs, often found in neuronal networks. We used three non-identical Bifurcation diagram of a single Rulkov neuron with respect to control parameter, γ Rulkov neurons and coupled them unidirectionally with linear couplings, similar to the presented approach in (Bashkirtseva et al., 2020). The model equations are represented as follows.
To reveal the dynamics of complex neuronal networks, motifs are used as smaller network building blocks, and there are numerous networks containing motif elements, such as gene transcription networks, brain functional networks, and neuronal networks (Li et al. 2022b) where σ is the coupling coefficient and x, y and z are membrane potentials of three coupled neurons in the loop, respectively. 2 = 1 + Δ, 3 = 1 + 2Δ , where Δ is the parameter mismatch. To simplify the model and enrich its dynamics, linear and unidirectional couplings are considered, respectively.
An electrical synaptic connection is considered in this study. This type of synapse allows direct transmission of ions through a gap junction. Each neuron's membrane voltage affects the other one in an electrically coupled neuronal network, which can be considered as a feedback term. So, in order to imply such influences, a voltage-feedback term is included in the equation of the membrane voltages of each neuron in the loop, similar to (Bashkirtseva et al., 2020;Zandi-Mehran et al., 2020). It is also notable that increasing σ in the model (as the representative of synaptic strength) is usually interpreted as an increment in synaptic connections and higher ion transmissions (Bashkirtseva et al., 2020;Latham et al., 2000).
It has been shown that the increase in coupling strength of the neuronal loop will result in behavioral changes from chaos to order and back spontaneously (Bashkirtseva et al., 2020). In this study, we have added electromagnetic induction to the neuronal ring and investigated the effects of this field on intrinsic behavioral changes through bifurcation diagrams and time series analysis.
Manipulating synaptic connections in a network is a popular method for investigating its dynamics. While many studies (Abarbanel et al., 1996;Parastesh et al., 2019;Pinto et al., 2000;Shi & Lu, 2005;Shuai & Durand, 1999;Varona et al., 2001) considered a variable as a simple electrical synaptic connection, (Zandi-Mehran et al., 2020) employed it to particularly investigate the electromagnetic field effects with different levels of strength of electrical synapses. In addition, new studies demonstrated that memristive-based neuronal networks are able to represent rich dynamics, similar to the dynamics of a real brain (Takembo et al., 2019). Our study benefits from both of these frameworks, meaning that we employed a memristive-based neuronal network with electrical synaptic connections. (2)

Discrete model of electromagnetic radiation
The dynamics of the spontaneous activity depend on the balance of excitation and inhibition of the synapses, and we have modified this balance utilizing electromagnetic radiation (Mazzoni et al. 2007). To add the electromagnetic radiation effect, an induction current is applied to a neuron in the ring. Actually, as a means to find out how electromagnetic field changes, a magnetic flux is considered. The discretized formulation presented in (Bortolotto et al., 2019) is considered here, resulting in the induction current term below: where φ and φ ext represent magnetic flux and external electromagnetic radiation, respectively. (φ(t)) is the nonlinear relationship between electrical charge and magnetic flux (memductance), which is given in Eq. (5): where 2 , β, k 0 , k 1 , k 2 are constant parameters. φ ext is considered as a periodic type wave, which is given in Eq. (6): In which, A is the intensity of external magnetic flux in (6). Since the sinusoidal waveform is a fundamental waveform generated by our surrounding electrical machines and every other type of waveform can be decomposed to a sum of sinusoidal waves, the periodic external magnetic flux is utilized here. Moreover, choosing a periodic external flux allows us to study the effect of increasing the frequency.
Lastly, the induction current of Eq. 3 is applied to one of the neurons (x) of the network, as it is shown in Eq. 7.
In discrete-time models, variables are dimensionless; hence, they can be employed for qualitative imitations.
The electromagnetic radiation in this model is a popular model of the electromagnetic induction generated by neurons on themselves. To be more precise, based on Faraday's law of induction, the action potential propagation in neurons results in the production of a magnetic field, and feedback will be formed to regulate and change the electrical activity of neurons. Furthermore, based on Maxwell's electromagnetic induction theorem, some bioelectrical phenomena in the nervous system, such as changes in the concentration of ions inside and outside of a cell can trigger electromagnetic induction inside a cell. Thus, many studies suggested that the memristor model is beneficial for describing the memory effect by forming the connection between magnetic flux and membrane voltage Wu et al., 2019).

Results and discussion
Two cases of spontaneous behaviors in a neuronal network under an electromagnetic field are considered in this article. The first is when the network is exhibiting chaotic behavior with strong coupling between neurons, and the second is when the network is exhibiting order behavior under strong couplings. After that, the disruptive effects of electromagnetic field on these spontaneous behaviors are disclosed.

Intrinsic chaotic behavior
It has been realized that when every single neuron in a neuronal network is in a stable equilibrium state, chaotic regimes will appear spontaneously as the coupling coefficient increases (Bashkirtseva et al., 2020). It is observed here that in the presence of electromagnetic radiation, these spontaneous chaotic regimes will be suppressed. In other words, electromagnetic radiation has inhibitory effects on the spontaneous chaotic behaviors of a neuronal network. This is grasped through the bifurcation diagrams of Fig. 2 as the limit cycles and equilibrium points emerge as well as the negative Lyapunov exponents in Fig. 3a.
The parameter values for this scenario are chosen such that interactions between neurons through strong synaptic connections lead to intrinsic behavioral change from order to chaos. Parameters of the system are γ 1 = 0.55 and Δ = 0.01. The small value of Δ has been considered as the parameter mismatch, to reflect the real biological neural network motifs, and it was also concluded that parameter mismatch (Δ) does not affect system stability (Bashkirtseva et al., 2020). Other parameters were chosen as k 0 = 1, k 1 = β = 0.01, k 2 = 0.5, = 0.1 and 2 = 0.4 based on grid search.
As it can be seen in Figs. 2b and 3b, when neuronal network is functioning naturally with the absence of electromagnetic field, spontaneous chaotic regimes appear as coupling strength is increased (due to the disappearance of limit cycles and appearance of positive leading Lyapunov exponents), even though all three neurons are at stable regimes at the beginning (non-positive Lyapunov exponents). The critical change is seen in Fig. 2a, where the induced current due to the presence of electromagnetic field has disrupted chaotic window in spontaneous behavior of the neuronal network. In other words, we were expecting to see transitions from order to chaos in the neuronal ring as the coupling strength increased, while electromagnetic induction eliminated the spontaneous activity of the neuronal network. This effect is markedly visible through the analysis of the bifurcation diagram plotted in Figs. 2a and 3a, where, the emergence of the negative largest Lyapunov exponents indicates that as the coupling coefficient is increased, order behaviors are seen instead of chaotic behaviors.
Largest Lyapunov exponents are also plotted in Fig. 3. The positive and negative Lyapunov exponent indicates chaos and periodic spiking, respectively. Figure 3a clearly shows that there will be no chaotic behavior when the coupling strength (σ) is increased in the presence of the electromagnetic field, while Fig. 3b shows the expectation of a chaotic regime with increasing the coupling strength.
In order to reach a better understanding of this impact, it is helpful to observe the time series of neuron membrane potential and induction currents. According to Fig. 2b, if the Fig. 2 Bifurcation diagrams of x in the neuronal ring, with a the presence, and b the absence of electromagnetic induction. Considering the endogenous magnetic field, the spontaneous behaviors are corrupted with very high and very low synaptic strengths coupling coefficient (σ) is set to 0.07, the ring is supposed to change its behavior from quiescent to burst, spontaneously. Figure 4a shows the time series of x (a neuron in the ring), where electromagnetic induction is applied at t = 200. After the application of the magnetic field, the amplitude of oscillations is decreased significantly, which can be interpreted as subthreshold oscillations. Therefore, we can conclude that in lower values of coupling coefficient, electromagnetic field suppresses spontaneous burst activity of the neuronal network.
The induction current is also plotted in Fig. 4b, which is zero for t < 200, where there is no electromagnetic radiation. When electromagnetic radiation starts at t = 200, it forces membrane voltage and changes the behavior of the neuron in the ring structure according to the induced current.
Magnetic flux is shown in Fig. 4c, as well. It can be noticed from Fig. 4c that magnetic flux is significantly influenced by external magnetic flux (φ ext ).
Outcomes of Figs. 2 and 3 show the possible effect of electromagnetic radiation on the activity of neuronal networks. Bashkirtseva et al. (Bashkirtseva et al., 2020) showed spontaneous electrical activity of neuronal networks from order to chaos, and some experimental studies observed corruptive impacts of electromagnetic fields on spontaneous chaotic behaviors. In (Alekseev et al., 2010), the rat sural nerve was exposed to electromagnetic radiation, and inhibitory effects of the field on spontaneous activities of this nerve were observed. Regarding the bifurcation analysis in Fig. 2, we also found that the presence of electromagnetic If we increase the coupling coefficient further ( = 0.45 ), where chaotic behaviors are expected to occur, naturally, Fig. 5a shows that when electromagnetic induction starts at t = 600, intrinsic chaotic behavior changes to periodic spiking. This indicates that even at higher values of coupling strength, the electromagnetic field changes the activity patterns of neurons in the network. The induction current, in this case, is plotted in Fig 5b. Similar to Fig. 4, these results also demonstrate that the behavior of exposed neurons is highly dependent on the current induced by electromagnetic radiation. Magnetic flux is also severely dependent on external magnetic flux, which is a cosine-shaped wave.
To employ a stronger external magnetic flux which is introduced in Eq. (6), the amplitude (A) of a cosine-shaped wave is further increased from 0.1 to 2 in Fig. 6. As a result, magnetic flux (Fig. 6c) on the neuron has completely followed the form of external flux. In addition to magnetic flux, the induction current also started to shape like a cosineshaped wave. As the final consequence, the membrane voltage of the neuron has developed a background trend in accordance with the strong periodic external flux in Fig. 6a.

Intrinsic order behavior
For the second case, where the neuronal network changes its behavior through interactions from chaos to order, parameters are γ 1 = -2.2 and Δ = 0.01, and for electromagnetic field, the same parameters in the previous section are utilized and = 2 0.4. Fig. 5 Time series of a x (a neuron in the ring exposed to electromagnetic radiation) with γ 1 = 0.55, Δ = 0.01 and σ=0.45, b Induced current by electromagnetic field on neuron and c Magnetic flux. Electromagnetic induction starts at t = 600. Due to the generation of magnetic field, normal chaotic waves are changed to regular periodic waves Fig. 6 Stronger External field. Time series of a x, a neuron in the ring exposed to electromagnetic radiation, with γ 1 = 0.55, Δ = 0.01, σ=0.45 and A = 2, b Induced current by electromagnetic field on neuron and c Magnetic flux. Electromagnetic induction starts at t = 800. A sinusoidal external magnetic flux with large amplitude gives a sinusoidal-shaped trend to the induction current and firing pattern of neuron Figure 7 illustrates bifurcation diagrams of a neuron in the neuronal network concerning the coupling coefficient. In Fig. 7a, the bifurcation diagram is presented in the presence of electromagnetic radiation. Parameters are chosen to set every single neuron in a chaotic regime initially. As Fig. 7b illustrates, spontaneous transitions from chaos to order arise as the coupling coefficient increases. At ≈ 2 , bifurcation occurs, and transitions from chaos to order appear (Bashkirtseva et al., 2020). Figure 7a illustrates when the neuronal network is exposed to electromagnetic radiation, spontaneous transitions from chaos to order are disrupted. To be more precise, it was expected from natural behaviors of the neuronal network to exhibit periodic spiking behaviors as a result of increased coupling strength, but electromagnetic radiations eliminated these spontaneous activities, and periodic window in the bifurcation diagram is disappeared.
Largest Lyapunov exponents are also calculated in Fig. 8. The exponents are positive for every value of in Fig. 8a, which indicates that behavior is always chaotic and periodic spiking regimes never appear. Figure 9a is the time series of x in the neuronal network, in which σ = 0.25 in order to have 3-cycle regular spiking behavior (Fig. 7b). According to this figure, electromagnetic induction starts at t = 200 and transition from regular spiking to chaotic spiking is seen. Electromagnetic radiation has eliminated spontaneous regular spiking regime and changed it to a chaotic behavior. Figure 9b and c show the induced current and magnetic flux across the membrane, respectively. Similar to previous time series, magnetic flux is primarily dependent on external cosine flux. Minor perturbations on magnetic flux are due to interactions with neuron fields and mutual impacts.   Figure 10 illustrates the effects of a stronger external magnetic flux in which A in Eq. (6) is increased from 0.1 to 2. In this mode, when electromagnetic radiation has corrupted the order behavior of the neuronal loop, although a stronger external field still affects the induction current and imposes a periodic background trend to it compared to Fig. 9, it is still unable to set significant changes to neuronal membrane of external flux, whereas the pattern of membrane voltage is more significantly dependent on the intensity of external flux as shown in Fig. 6.
The biological plausibility of our study is rooted in corruptive neuronal activities in neurological disorders like epilepsy. While (Bashkirtseva et al., 2020) demonstrated the spontaneous electrical activities in a neuronal network, we illustrated how the additive intrinsic electromagnetic induction in such networks can interfere the normal spontaneous activity and produce epileptic-like patterns. According to (van Drongelen, 2013), many neurological disorders, including epilepsy, are poorly understood since there are no experimentally efficient tools to investigate neuronal activities at small scales. So, the significant role of modeling is clarified here. Additionally, Zhao & Wang, 2021) suggested using the memristor model in order to explore the mechanisms of an endogenous magnetic field in epilepsy, which is what we did in this study. Based on our results, we propose that endogenous electromagnetic induction and increased synaptic connection between the neurons of a neuronal network are the possible causes of abnormal spontaneous firing patterns.
Even though time is dimensionless in map-based models, and thus frequency cannot be defined, increasing the fluctuation Fig. 9 Time series of a x, a neuron in the ring exposed to electromagnetic radiation, with γ 1 = -2.2, Δ = 0.01 and σ=0.25, b Induced current by electromagnetic field on neuron and c Magnetic flux. Electromagnetic induction starts at t = 200. Regular spiking changed to chaotic spiking at the onset of electromagnetic induction Fig. 10 Strong External Flux. Time series of a x (a neuron in the ring exposed to electromagnetic radiation) with γ 1 = -2.2, Δ = 0.01, σ =0.25 and A = 2, b Induced current by electromagnetic field on neuron, and c Magnetic flux. Electromagnetic induction starts at t = 300. In case of transition from order to chaos, strong external flux has less influence on the induction current compared to transitions from chaos to order. But still, it gives a sinusoidal trend to the induction current rate of the external flux is beneficial to explore. With lower amplitudes of the outer flux, no significant change in the outcomes was observed with increasing frequency. Whereas increasing both the intensity and frequency of the external flux increased the firing frequency of a network with low synaptic couplings. Thus, in weak synaptic couplings where the network is supposed to be in a quiescent state, applying the EMF stimulates the network, and the firing frequency of the neurons rises with regard to the frequency of the external flux. This outcome is illustrated in Fig. 11.   Fig. 11 Effect of increasing the frequency of external magnetic flux by tenfold. a Time series, induction current, and magnetic flux of x, a neuron in the ring exposed to electromagnetic radiation with γ 1 = 0.55, Δ = 0.01, A = 2, σ=0.01, and high frequency external flux. b Same time series for a low frequency external magnetic flux. Electromagnetic induction starts at t = 300. High-frequency external magnetic flux engenders high-frequency oscillations in the membrane voltage when the synaptic strength is low a common way to validate the results of a map-based model is to change the parameters in a valid numerical range and see whether the results remain the same. We have chosen the parameters of the model of memristive electromagnetic induction based on the previous numerical and behavioral analysis studies (Bortolotto et al., 2019;Usha & Subha, 2019).
It is also noteworthy to point out that further decrease in induction parameters, including α 2 , means lower memductance, and hence, shows the effect of magnetic flux across the membrane of neurons is decreased. So, decreasing α 2 below 0.1 will gradually eradicate the disruptive effects of EMF radiation on the neuronal loop. Similarly, k 0 is the feedback gain of induction current on membrane voltage (switching factor of magnetic field) which is suggested to be considered 0.9 or 1 according to (Bortolotto et al., 2019;Wu et al., 2019). Therefore, the decrease in k 0 , as well as the decline in α 0 , resulted in less disruptive effects. Although setting α 2 between 0.1 to 0.4 does not change the results of our findings, we have set it to 0.4 according to (Usha & Subha, 2019).
The interaction between magnetic flux and membrane potential is characterized by k 1 and k 2 . Due to the fact that membrane potential has no significant effect on magnetic flux, k 1 is considered 0.01, but even increasing it does not change results. It will only intensify fluctuations of the magnetic flux. In a similar way, changes in k 2 and β only result in an increase or decrease in fluctuations of magnetic flux over a sinusoidal trend. Thus, changing k 2 also does not impose significant changes. In fact, regardless of the minor changes in the shape of magnetic flux, the mere presence of EMF radiation imposes changes in spiking patterns of the spontaneous electrical activity of a neuronal network, which was the main finding of this study.
Ultimately, the demonstrated effects depend on parameter values, and one cannot determine whether these effects are realistic or not before estimating the most accurate values for parameters and field strength. While the critical finding of our study on the emergence of demonstrated corruptive behaviors in the presence of EMF fields remains valid, like other simulation studies, finding the exact strength of harmful electromagnetic fields requires experimental studies.

Conclusion
In this paper, the spontaneous electrical activity of a ring of three coupled neurons is investigated under electromagnetic radiation, using computational simulations with the basis of the Rulkov one-dimensional map model, and a ring of three coupled neurons is considered a representative of a simple neuronal network in the brain. As previous studies illustrated spontaneous transitions from order to chaos in the electrical activity of neuronal networks, this study explored the possible effects of electromagnetic radiations on these spontaneous behavioral transitions.
Since a great proportion of neuronal activities in a biological system are spontaneous, we must delve into the analysis of such behaviors. We have found that the current induced by surrounding electromagnetic fields might have patternchanging effects on intrinsic neuronal activities. When the synaptic coupling between the neurons is strong and neurons are exhibiting chaotic behavior, magnetic induction eliminates these behaviors and forces the neurons to change their behavior according to the induced current. No chaotic regime is seen after the application of magnetic flux. On the other hand, when neurons are exhibiting regular spiking behavior in the neuronal ring, the generated magnetic flux also disrupts regular spiking and takes the neurons into chaotic regimes instead.
Generally, it is highlighted that in the first case, when neurons are in quiescent states, the generated magnetic flux suppresses the spontaneous behavior of the exposed neurons in the neuronal ring when the coupling coefficient is low, and changes chaotic behaviors to regular spiking or burst when coupling coefficient is high.
In the second case, when neurons are initially in chaotic states, magnetic flux changes the behavior of neurons from 3-cycle spiking to chaotic spiking when the synaptic connections are strengthened.
It is also notable that in the first scenario in which the behavior of the neuronal loop alters from chaos to order in the presence of electromagnetic radiation, the membrane voltage is highly dependent on the amplitude of external magnetic flux, while this is not the case when the neuronal ring changes from order to chaos under electromagnetic induction in the second scenario.
Results of this paper are suggesting that corruptive firing patterns arise due to increased synaptic connections and endogenous magnetic field. Abnormal spiking patterns of neuronal networks are common in neurological disorders such as epilepsy. These demonstrations provide novel insights both into pathology and clinical interventions.
To extend the current work, considering synapse types other than electrical for the network, like chemical synapses, to see whether only electrical synapses cause corruptive effects on the spontaneous electrical activities of the network or not. Furthermore, investigating the factors that can decrease the endogenous electromagnetic induction in a neuronal network via in-vitro or in-vivo experiments might be helpful in preventing epileptic seizures.