The Synchronization and Synchronization Transition of Coupled Fractional-order Neuronal Networks

Different from the previous researches on the synchronization and synchronization transition of neuronal networks constructed by integer-order neuronal models, the synchronization and synchronization transition of fractional-order neuronal network are investigated in this paper. The fractional-order ring neuronal network constructed by fractional-order Hindmarsh-Rose (HR) neuronal models without electromagnetic radiation are proposed, and it’s synchronization behaviors are investigated numerically. The synchronization behaviors of two coupled fractional-order neuronal models and ring neuronal network under electromagnetic radiation are studied numerically. By research results, several novel phenomena and conclusions can be drawn. First, for the fractional-order HR model’s ring neuronal network without electromagnetic radiation, if the fractional-order q is changed, the threshold of the coupling strength when the network is in perfect synchronization will change. Furthermore, the change of fractional-order can induce the transition of periodic synchronization and chaotic synchronization. Second, for the two coupled neurons under electromagnetic radiation, the synchronization degree is influenced by fractional-order and the feedback gain parameter k 1 . In addition, the fractional-order and parameter k 1 can induce the synchronization transition of bursting synchronization, perfect synchronization and phase synchronization. For the perfect synchronization, the synchronization transition of chaotic synchronization and periodic synchronization induced by q and parameter k 1 is also observed. Especially, When the fractional-order is small, like 0.6, the synchronization behavior will be more complex. Third, for the ring neuronal network under electromagnetic radiation, with the change of memory-conductance parameter β , parameter k 1 and fractional-order q of electromagnetic radiation, the synchronization behaviors are different. When β > 0 . 02 , the synchronization will be strengthened with the decreasing of fractional-order. The parameter k 1 can induce the synchronization transition of perfect periodic-10 synchronization, perfect periodic-7 synchronization, perfect periodic-5 synchronization and perfect periodic-4 synchronization. It is hard for the system to syn-chronize and q has little effect on the synchronization when − 0 . 06 < β < 0 . 02 . When β < − 0 . 06 , the network moves directly from asynchronization to perfect synchronization, and the synchronization factor goes from 0.1 to 1 with the small change of fractional-order. Larger the factional-order is, larger the range of synchronization is. The synchronization degree increases with the increasing of k 1 .


Introduction
The firing behavior of neuron is a nonlinear process, and the neuron is a complex nonlinear dynamic system. In 1952, Hodgkin and Huxley used the equivalent circuits and large amounts of data from experiments to modeled and analyzed the data, then they constructed the Hodgkin-Huxley(HH) neuron model through theoretical derivation [1]. Then, FItzHugu, Morris and Lecar, Hindmarsh and Rose proposed the FHN model [2], ML model [3], and HR model [4] respectively. Synchronization is an important phenomenon in neuronal system and operational mechanism of the brain. A number of researchers used the coupled neuronal models to try to explain some of the synchronization phenomena observed in the experiments.
For the synchronization behaviors of two coupled neurons and neuronal network, a lot of researches have been done by relevant scholars . For the synchronization behaviors of two coupled neurons, the synchronization transition [5,6] and the influence of different coupling way [7][8][9] are investigated. The synchronization transition of two coupled ML neurons is studied in [5], and it found that the bursting synchronization happens before spiking synchronization. The effects of memeristive synapse-coupled is investigated in [7]. When multiple neurons are coupled into a network, there will be more emergence phenomenon. Many researchers investigated the factors influencing network synchronization [10][11][12][13][14][15][16][17][18][19][20][21][22][23], such as the network topology [10][11][12][13][14], the time delay and partial time delay [15,16], the coupled way and initial value [17][18][19][20], and the layers of networks [21]. In the previous researches, most neuron models are integer-order models. Fractional-order model can describe the real phenomenon more completely. The dynamic characteristics of the fractionalorder Hindmarsh-Rose (HR) neuronal model was investigated in [24], and it was found that different fractionalorder can induce different dynamic behaviors. For the fractional-order coupled neurons and neuronal networks, there are few studies. For the fractional-order coupled neurons, the two neurons reach the perfect synchronization through the design of the controller in most of the literature [25][26][27]. In [28], it is found that the change of fractional-order can change the synchronization mode. But the synchronization behaviors of fractional-order neuronal network aren't investigated.
Over the past period of time, the dynamic behaviors of neuron under electromagnetic radiation have been paid widely scholars' attention [26,[29][30][31]. In fact, the neurons are usually exposed to electromagnetic radiation, and the external magnetic flux can affect the electromagnetic radiation of the neurons. In [30] and [31], the external stimulus current and external electromagnetic radiation are omitted, respectively, and the corresponding models are proposed. According to [30], the single fractional-order neuron model under electromagnetic radiation is proposed, and the dynamic behaviors are investigated in [26]. The effects of electromagnetic radiation on the dynamical characteristic and synchronization behaviors of integer-order coupled neuronal networks were investigated in [31][32][33][34]. The synchronization behavior of one main network coupled with some sub-networks is investigated in [32]. In [33], it is found that the rhythm synchronization will happen under appropriate coupling strength and electromagnetic radiation. In [26] and [27], the two coupled neurons with and without time-delay were in perfect synchronization by designing appropriate controller, but they didn't investigate the effects of parameter and the synchronization transition. From the above analysis, the synchronization and synchronization transition of coupled neurons and neuronal network under electromagnetic radiation are not investigated in the previous researches. In the researches mentioned above, most method to study the fractional-order system are the predict-corrector method [35]. Predict-corrector method has a high accuracy, but the calculation is large. The ADM method [36] used in this paper has a higher accuracy, and calculation amount is smaller than that of the predict-corrector method [37].
From this perspective, this paper investigates the synchronization transition of coupled fractional-order neuronal networks. First, the ring network for fractionalorder HR model without electromagnetic radiation is constructed, and the synchronization transition induced by the fractional-order is investigated. Then, different from [25][26][27], the synchronization transition of two coupled neurons under electromagnetic radiation induced by the fractional-order, coupling strength and parameter k 1 is studied. Finally, this paper investigates the synchronization behavior of the ring neuronal network under electromagnetic radiation.

Model description
In this paper, in order to analyze the dynamic behaviors of the fractional-order neuronal network, the HR model is adopted for single neuron model. The HR model is described as follows [24]: Where x is the membrane action potential, y is a recovery variable, z is a slow adaption current, D q t is The synchronization and synchronization transition of coupled fractional-order neuronal networks 3 the differential operator defined by Caputo, q is the fractional-order. I is the external stimulus current. In this paper, parameters except fractional-order and external stimulus current are fixed as a = 1, b = 3, c = 1, d = 5, r = 0.006, x = −1.56, s = 4 [28]. The fractionalorder ring neuronal network without electromagnetic radiation can be described as follows: Where i = 1, 2, ..., N , C is the coupling strength of the network. If node i and node j are connected, then a ij = a ji = 1. Each neuron is symmetrically coupled to its 2P nearest neighbors. It is important to notice that j = −m, (m = 1, 2, ..) implies node j coupled with node 101 − m . In this paper, the parameter P is set as 1.
The network structure is shown in Fig.1.
In this paper, the two coupled fractional-order neurons under electromagnetic radiation can be described as follows [29]: φ denotes the magnetic flux across the cell membrance. φ 0 is the external magnetic flux. k 1 is the feedback gain. The cubic flux-controlled memristor model W (φ) = dq(φ)/dφ = α+3βφ 3 is introduced in this model, and it is used to estimate the effect of feedback regulation on menberance potential when magnetic flux is changed. The two parameters, α and β, describe the memoryconductance, and they will vary with the environment and their own conditions. In this paper, α = 0.2, k 2 = 0.4, φ 0 = 1 .  The fractional-order neuronal ring network under electromagnetic radiation can be described as follows: 3 The synchronization behavior of fractional-order neuronal network without electromagnetic radiation In this paper, the synchronization factor R is adopted to describe the synchronization of network. R [34] is given by where ⟨·⟩ denotes the time averaging. The value of R is between 0 and 1, and it increases with decreasing average membrane potential errors. More precisely, perfect synchronization is expected when R is close to 1 and non-synchronization state may appear when R is close to 0. The ring network consists of 100 nodes. Numerical simulation is provided for the model (2) by utilizing the ADM method. As shown in Fig.2, the synchronous threshold of coupling strength will vary with the fractional-order. The change trend of threshold is to increase first and then decrease from Fig.2(b). In the previous researches, for the integer-order neuronal network, the threshold of coupling strength can change with other influence factors [10,15], such as the structure of network and time-delay. In this paper, the fractionalorder is proved to be an influence factor.
As shown in previous researches, for the integerorder neuronal network, time-delay can induce the synchronization transition of neuronal network [15,16]. In this paper, the results will show that fractional-order not only makes the synchronous system become asynchronization, but also changes the synchronization mode. When the neuronal network is in perfect synchronization, Fig. 3 shows the phase diagram of (z, x) when the fractional-order is 0.65, 0.75, 0.85, and 0.95. The neurons display periodic-6 bursting, periodic-10 bursting, and periodic-5 bursting when q = 0.65,q = 0.75, and q = 0.85 as shown in Fig.3(a)(b)(c). The network is in perfect periodic-6, perfect periodic-10 and perfect periodic-5 synchronization. When q = 0.95, as shown in Fig.3(d), neurons display chaotic bursting, so

The synchronization behavior of two fractional-order coupled neurons under electromagnetic radiation
The fractional-order HR neuron under electromagnetic radiation is investigated in [27], and two neurons reach perfect synchronization through adaptive control. But the synchronization behavior and synchronization transition induce by the fractional-order and other parameter are not reported in [27]. In this section, the synchronization behavior and synchronization transition induced by fractional-order and parameter k 1 are studied.
In this part, the similarity function is introduced to measure the synchronization of the system [28]. The model of similarity function can be described as follows: Where ⟨·⟩ stands for the time average. Obviously, the perfect synchronization can be observed when S is equal to 0, and bigger the S, worse the synchronization.
When the system is not in perfect synchronization, it may be in phase synchronization. The phase information can be calculated by detecting the time of sampled time series(t 1 , t 2 , ..., t n )across the Poincare section, and the phase is calculated by And the phase difference between two neurons is defined by When the system is not in perfect and phase synchronization, it may be in bursting synchronization. Bursting synchronization means rhythm synchronization of slow variables, so bursting synchronization can be measured by the slow variable's similarity function [38].
The synchronization and synchronization transition of coupled fractional-order neuronal networks 5 The model of similarity function can be described as follows [38]:  Fig.4, where the blue color region with S equal to 0 denote the system in perfect synchronization. As shown in Fig.4, when 0.55 < q < 0.67 and coupling strength around 0.367, the similarity function undergoes a complex change, which means that synchronous threshold of coupling strength changes great. It is different from [28] because of the electromagnetic radiation. When 0.67 < q < 1 , the synchronous threshold of coupling strength decreases with increasing fractionalorder. When 0.366 < C < 0.378, the S ∼ q curves are shown in Fig.5. As shown in Fig.4, the similarity function has striking changes when 0.366 < C < 0.378 , so the synchronization transition can be observed when 0.366 < C < 0.378 . The synchronization transition when C = 0.372 is taken as an example to observe the change of synchronization. When q = 0.56 , Fig.6(a) is the phase diagram of (x 1 , x 2 ) , and Fig.6(b) is the corresponding time series of x, so the system is in imperfect synchronization. As shown in Fig.6(c)(f), the system is in perfect synchronization when q = 0.61 and q = 0.9 , because the phase plane is located on a three quadrant angular bisector. The system is in imperfect synchronization when  From Fig.6, the two neurons are in imperfect synchronization and the phase differences are calculated as follow. As show in Fig.7(a), the phase different are al-ways around 0 and 2π , so the system gets in phase synchronization. From the Fig.7(b), the phase difference is around 0, so the system is also in phase synchronization when q = 0.7 .
As the analysis mentioned above, the synchronization transition mode induced by fractional-order is phase synchronization -perfect synchronization -phase synchronization -perfect synchronization. The transition mode is more complex than the coupled fractional-order neurons without electromagnetic radiation [28].
As shown in Fig.4, the two neurons are in perfect synchronization when C = 0.5 and 0.55 < q < 1 . Now, the ISI bifurcation of first neuron for C = 0.5 is shown in Fig.8. There exists double period bifurcation when q = 0.66 and q = 0.61 with the decreas-ing of fractional-order. The system is in chaotic perfect synchronization when 0.57 < q < 0.59 , but the system is in periodic perfect synchronization at other fractional-order. The Fig.9 show the phase diagram of (z 1 , x 1 ) and (z 2 , x 2 ) when q = 0.56 , q = 0.58 , q = 0.6 , q = 0.65 and q = 0.9 for C = 0.5 . The neurons display periodic-3 bursting, chaotic bursting, periodic-4 bursting, periodic-2 bursting, and spiking respectively. The synchronization transition, which is perfect periodic-3 synchronization -perfect chaotic synchronizationperfect periodic-4 synchronization -perfect periodic-2 synchronization -perfect spiking synchronization, is observed. When the system is in perfect synchronization, the transition mode is also more complex than the coupled neurons without electromagnetic radiation [28], in which the synchronization mode is only periodic synchronization in all fractional-order [28]. The results can be concluded that when the system is in perfect synchronization, fractional-order and electromagnetic radiation can change the synchronization mode. In this section, the influence of parameter k 1 and coupling strength on the synchronization behaviors, and the synchronization transition induced by k 1 will be investigated. The similarity function, phase difference, and slow variable's similarity function also are used to measure the synchronization degree of the system. Form the analysis mentioned above, different fractional-order will induce different dynamic behaviors. The influence of parameter k 1 and coupling strength on synchronization will be studied under different fractional-order.
Therefore, the value of the similarity function in the plane of (C, k 1 ) for several value of fractional-order is calculated, as shown in Fig.10. In Fig.10, the blue color region where S equals 0 denote the system in perfect synchronization. Generally, the conclusion that large parameter k 1 and coupling strength can make the system in perfect synchronization can be drawn. But there is an exception, when q = 0.8 and q = 0.95. There are existing that the smaller k 1 can make the system be in perfect synchronization but the larger k 1 can't. As shown in Fig.11(a), when q = 0.8, C = 0.34, 0.1 < k 1 < 0.27 , the value of similarity function decreases with increasing of k 1 , and then S = 0 when 0.27 < k 1 < 0.35. But with the increasing of k 1 sequentially, the S > 0 when 0.35 < k 1 < 0.39 . Then the S = 0 when k 1 > 0.39. To observe the above process more visually, some figures about phase diagram and the corresponding time series are shown in Fig.11(b)-(f) when q = 0.8, C = 0.34 . As shown in Fig.11(b)(c) , the phase diagram of (x 1 , x 2 ) and the corresponding time series of x 1 and x 2 show that the system is in imperfect synchronization when k 1 = 0.2. From Fig. 10(d)(f), the system is in perfect synchronization when k 1 = 0.3 and k 1 = 0.4. The system is in asynchronization when k 1 = 0.37, as shown in Fig.11(e).
The phase differences of x 1 and x 2 when k 1 = 0.2 and k 1 = 0.37 are plot in Fig.12. we can conclude that the two neurons are in phase synchronization when k 1 = 0.37 , because the phase different is small around 0. The two neurons are not in phase synchronization when k 1 = 0.2 but the phase is locked in some time periods, and the slow variable's similarity function is calculated as S z = 0.05, so the two neurons are in bursting synchronization. The synchronization transition induced by parameter k 1 , that is bursting synchronization -perfect synchronization -phase synchronization -perfect synchronization can be observed.
Meanwhile Fig.10 also shows that the range of synchronization when q = 0.6 is largest in three value of q. As shown in Fig.13, C = 0.26, k 1 = 0.38, the system is in asynchronization when q = 0.8 and q = 0.95, but system is in perfect synchronization when q = 0.6.
If we change the parameter k 1 when the system is in perfect synchronization(C = 0.5) for several value of q, the synchronization mode will change. The ISI bifurcation of first neuron is plot in Fig.14. As shown in Fig.14(a), when 0.1 < k 1 < 0.14 and 0.25 < k 1 < 0.33, the neurons display chaotic firing, but when 0.14 < k 1 < 0.24 and k 1 > 0.33, the neurons display periodic firing. When q = 0.6, C = 0.5, the parameter k 1 will induce the transition of chaotic perfect synchronization and periodic perfect synchronization. As shown in Fig.14(b), when q = 0.8, C = 0.5. There is double pe-  The phase diagrams of (z 1 , x 1 ) and (z 2 , x 2 ) when C = 0.5 are shown in Fig.15 . And the synchronization transition induced by the parameter k 1 can be explained. Fig.15(a)(c) shows that the neurons display  Fig.15(e)(f).
In [26], the effects of parameter k 1 on the synchronization are not investigated. And in [33], when the neuron model is integer-order model, the synchronization mode is perfect spiking synchronization for all k 1 when C = 0.5. The conclusion can be summarized that parameter k 1 can induce the synchronization transition and the synchronization behaviors are different when fractional-order varies. As shown in Fig.14 and Fig.15, the small fractional-order, such as 0.6, can induce more complex synchronization transition.

The synchronization behavior of fractional-order neuronal network under electromagnetic radiation
In this part, the effects of some parameters on synchronization behaviors of ring neuronal network under electromagnetic radiation will be investigated. In [33], the integer-order neuron's firing activity is influenced by the parameter k 1 . In this paper, the firing activity is also influenced by the parameter β except k 1 . In addition, in this paper, the results show that fractionalorder cannot influence the neurons' firing mode in the network. It is obviously different with the two coupled neurons under electromagnetic radiation, single neuron and the ring neuronal network without electromagnetic radiation. Although the fractional-order can't influence the neuronal firing mode, the fractional-order can influence the synchronization behaviors of the ring neuronal network under electromagnetic radiation at different conditions. For example, the ring neuronal network constructed by integer-order neuronal models is in perfect synchronization, but the perfect synchronization maybe destroyed in the same conditions when the integer-order neuronal models is instead of fractional-order neuronal models, vice versa. Therefore, this part focus on the effects of parameter β, k 1 and q on the synchronization behaviors and synchronization transition of ring neuronal network constructed by fractional-order neuronal models in the next. Now the firing mode with different parameter β when k 1 = 0.4, C = 10 will be plot in Fig.16, and the other parameters are set as mentioned in section 1. Fig.16 shows the firing mode of the first neuron when the parameter β is different. From numerical simulation, we can find that the neurons display chaotic bursting, spiking firing, and periodic bursting when −0.06 < β < 0.02, β < −0.06, and β > 0.02. In this section, the synchronization behaviors in the three regions of β will be studied numerically. The Bursting synchronization of neuronal network can be described by slow variable's synchronization factor R z . R z is given by where ⟨·⟩ denotes the time averaging. When R z is close to 1, the system reaches bursting synchronization.
5.1 the synchronization behavior of neuronal network when β > 0.02 In this part, β = 0.04 is set as an example. The curve of R ∼ C for different k 1 are shown in Fig.17. From  Fig.17, we will find that the effect of parameter k 1 and coupling strength on synchronization behaviors. The synchronization factor increases with the increase of coupling strength and k 1 . When the β = 0.04, k 1 plays a dominant role in network synchronization, because the network can't reach perfect synchronization when k 1 = 0.15, 0.2, 0.25, 0.3 and the coupling strength ranges from 1 to 16. The state of network can be transformed from asynchronization to perfect synchronization with minor change of k 1 when k 1 > 0.35. To observe the effect of fractional-order, k 1 is set as 0.4. Fractional-order cannot influence the neurons' firing mode, but it can change the synchronization degree. Fig.18 shows the R ∼ q curves for different coupling strength. We can conclude that the increase of fractional-order will weaken the synchronization of the neuronal network. The synchronization is robust against alterations of coupling strength when fractional-order is small. When the fractional-order is 0.7, the difference of synchronization factor between coupling strength 1 and 6 is only 0.05, but when the fractional-order is 1, the difference becomes 0.2. Fig.19 shows neuronal network's spatiotemporal patterns and corresponding snapshots when C = 4.5, k 1 = 0.35 at t = 2218. It is concluded that the strength of synchronization is higher when the fractional-order is 0.7 than the one when the fractional-order is 0.98. When C = 4.5, q = 0.7, the x i are uniform distributed value, so the network is in nearly perfect synchronization. When C = 4.5, q = 0.98, the x i are disorderly. But the slow variable's synchronization factor R z is calculated as 0.99, so the network is in bursting synchronization. The fractional-order can also induce the transition of perfect synchronization and bursting synchronization when the ring network under electromagnetic radiation. In addition, k 1 can also induce synchronization transition, that is to say, the firing mode of neurons will change for different k 1 . The parameters are set as C = 10, q = 0.8. Fig.20 show the phase diagram of (z i , x i ) when k 1 = 0.4, 0.5, 0.6, 0.7. As shown in Fig.20, the neurons of the network display periodic-10 bursting, periodic-7 bursting, periodic-5 bursting, and periodic-4 bursting when k 1 = 0.4, 0.5, 0.6, 0.7. The number of spiking in one bursting decreases with the increasing of k 1 , so the synchronization mode varies with k 1 . The synchronization transition induced by k 1 , which is perfect periodic-10 synchronization -perfect periodic-7 synchronization -perfect periodic-5 synchronizationperfect periodic-4 synchronization, is observed. In this part, β = 0.02 is set as an example. The synchronization behavior is different from the above analysis. As shown in Fig.21, the synchronization factor also increases with the increase of coupling strength and K 1 , but we can conclude that it is difficult to reach synchronization. The coupling strength ranges from 1 to 16, the synchronization factor just reaches around 0.3. Neuronal network's spatiotemporal patterns and corresponding snapshots at t = 2500 when K 1 = 0.4 are shown in Fig.22. It is found that the x i1 are disorderly when the coupling strength is 1 and 16. And the degree of synchronization is higher when the coupling strength is 16 than the one when the coupling strength is 1, but they are all not in perfect synchronization. After calculating the slow variable's synchronization factor, the network is not in bursting synchronization. This part of view is different from the case of network without electromagnetic radiation, as shown in Fig.3(d), under the same fractional-order and coupling strength, the network can be in perfect synchronization. In this condition, it is necessary to mention that the synchronization behaviors are minimally affected by fractional-order. In this part, β = −0.08 is set as an example. There will be some novel phenomenon. The R ∼ C curves for different fractional-order are plot in Fig.23. We can find that network is in perfect synchronization when C > 7, but the neurons' dynamic behaviors are different when 6.4 < C < 7 for different fractional-order. In addition, it can be found that the transition of synchronization and asynchronization is abrupt when the fractional-order and coupling strength changes slightly. The synchronization factor changes 0.1 to 1 directly. This phenomenon is not found in previous studies. Fig.24 shows the neuronal network's spatiotemporal patterns at different fractional-order and coupling strength. As shown in Fig.24, for each fractional-order, network will undergo several sudden transitions of asynchronization and perfect synchronization with the variation of coupling strength. The dynamic behavior of neurons in network is different at different fractionalorder and coupling strength. When C = 6.7, network Fig. 23 The curves of R ∼ C for different q is in asynchronization at q = 0.7 and q = 0.8, but network is in perfect synchronization at q = 0.9. When C = 0.648, network is in perfect synchronization at q = 0.7, but network is in asynchronization at q = 0.8 and q = 0.9. When β < −0.06, fractional-order can also induce synchronization transition. And we can find that larger the factional-order is, larger the range of asynchronization from Fig.23. From the value of slow variable's synchronization factor when the network is not in perfect synchronization, the network cannot get in bursting synchronization. In the neuronal network without electromagnetic radiation, we cannot find the abrupt change of synchronization factor and synchronization behaviors.
For the influence of k 1 , as shown in Fig.25, the synchronization degree increases when the k 1 increases, but the network is hard to get in perfect synchronization. The parameters are set as q = 0.8, C = 6.7.

Conclusion
This paper investigates the synchronization behavior and synchronization transition in three cases, which are the fractional-order HR model's ring neuronal network without electromagnetic radiation, two coupled fractional-order neuronal models under electromagnetic radiation and fractional-order ring neuronal network under electromagnetic radiation. First, for the fractionalorder model's ring neuronal network without electromagnetic radiation, like other influence factors reported in [21][22][23][24][25][26][27][28][29][30][31][32][33], the fractional-order can also change the threshold of the coupling strength. And the transition of periodic synchronization and chaotic synchronization induced by the fractional-order is found. Second, For the two coupled fractional-order neurons under electromagnetic radiation, the synchronous threshold of coupling strength will vary with the fractional-order. Compared with [28] (without electromagnetic radiation), the variation is more complex. The synchronization transition is also investigated in this situation. The transition of bursting synchronization, perfect synchronization, phase synchronization and perfect spiking synchronization is founded when the fractional-order changes. In addition, compared with [28], when the two coupled neurons are in perfect synchronization, the transition of perfect chaotic synchronization and perfect periodic synchronization will be founded when changing the fractionalorder and parameter k 1 . And when the fractional-order is small like q = 0.6, from the ISI bifurcation, the dynamics behavior will be more complex. Third, For the fractional-order ring neuronal network under electromagnetic radiation, it had been investigated that the parameter k 1 can change the firing activity in [33]. Different from the results in [33], this paper focus on the influence of parameter β, k 1 and fractional-order q on synchronization behaviors and synchronization transition. In network, fractional-order cannot change the firing activity of neurons. It is obviously different with the two coupled neurons and neuronal network without electromagnetic radiation. When β > 0.02, the synchronization degree decreases with fractional-order increasing. The parameter k 1 can induce the synchronization transition of perfect periodic-10 synchronization, perfect periodic-7 synchronization, perfect periodic-5 synchronization and perfect periodic-4 synchronization. When −0.06 < β < 0.02, the network is difficult to get into perfect synchronization, and the fractional-order is hard to change the synchronization degree. In particular, when β < −0.06, the network has sudden transition of asynchronization and perfect synchronization. The synchronization factor goes suddeny from 0.1 to 1. And larger the factional-order is, larger the range of asynchronization is. The synchronization degree increases with the increasing of k 1 .

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Conflict of Interest:
The authors declare that they have no conflict of interest.