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bi-resonance and wave propagation in FitzHugh–Nagumo neural systems under electromagnetic induction
Chaos, Solitons and Fractals 133 (2020) 109645
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Vibrational mono-/bi-resonance and wave propagation in FitzHugh–Nagumo neural systems under electromagnetic induction Mengyan Ge a, Lulu Lu a, Ying Xu a, Rozihajim Mamatimin b, Qiming Pei c, Ya Jia a,∗ a
Department of Physics, Central China Normal University, Wuhan 430079, China Institute of Civil Engineering, Kashi University, Kashi 844008, China c School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China b
a r t i c l e
i n f o
Article history: Received 28 May 2019 Revised 18 January 2020 Accepted 22 January 2020
Keywords: Vibrational resonance FitzHugh–Nagumo model Feed-forward feedback neural network Electromagnetic induction Chaos and bifurcation analysis
a b s t r a c t In this paper, an modified FitzHugh–Nagumo (FHN) neural model was employed to investigate the vibrational resonance (VR) phenomenon, the collective behaviors, and the transmission of weak low-frequency (LF) signal driven by high-frequency (HF) stimulus under the action of different electromagnetic induction in single FHN neuron and feed-forward feedback network (FFN) system, respectively. For the single FHN system, by increasing the amplitude of HF stimulus, the phenomena of vibrational mono-/bi-resonance are observed, and the input weak signal and output of system are synchronized, and the information of the weak LF signal is amplified. For the FFN system, the phenomena of vibrational mono-/bi-resonances are also occurred, both frequency and amplitude of the HF stimulus play an important role in the vibrational bi-resonances and transmission of weak LF signal in the FHN neural FFN.
1. Introduction Fluctuations are ubiquitous in various nonlinear dynamic processes (or signaling systems), for instance, the spikes of neurons are understood as random processes [1] where noise plays very important roles. In previous investigations, noise-induced complex dynamic behaviors were found. One of the most significant findings is the stochastic resonance (SR) phenomenon, which is expressed in a nonlinear system wherein the external information (such as weak signals) is input. It was found that noises are used to amplify and optimize information in the SR phenomenon [2– 4]. There are many most recently publications of SR phenomenon. For example, low stochastic resonance is observed by respectively varying noise strength and self-correlation time of sine-wiener noise [5]. The coherence resonance in one layer can not only be controlled by the network topology, intralayer and interlayer timedelayed couplings, but also by self-induced stochastic resonance in the other layer [6]. Recent investigations have revealed that the responses of nonlinear systems to a weak low-frequency (LF) signal are magnified by using a high-frequency (HF) stimulus, which is similar to the effect of noise on the SR phenomenon. In a nonlinear system, it is actually affected by two periodic stimulus, LF (as signal) and HF
∗
Corresponding author. E-mail address: [email protected] (Y. Jia).
https://doi.org/10.1016/j.chaos.2020.109645 0960-0779/© 2020 Elsevier Ltd. All rights reserved.
© 2020 Elsevier Ltd. All rights reserved.
(as carrier), where the HF stimulus can enhance the processing of weak LF signal. In the presence of additional white noise, vibrational resonance (VR) has been theoretically proven in the bistable system [7,8]. For the external fields, more than two frequency periodic stimulus could be often used in various signal systems, such as the commutation techniques [9], neuroscience [10], acoustics [11] and laser physics [12]. Vibration bi-resonances are a phenomenon in which a weak LF signal is amplified by different values of HF stimulus. There are many experimental researches on VR phenomenon and vibrational bi-resonances, such as excitable electronic circuit [13], overdamped duffing oscillator simulation [14], and bistable optical cavity laser [15]. In stochastic bistable systems, the VR was studied by experimental and numerical methods to increase the detection and recovery of weak subthreshold aperiodic binary signals [16]. The VR phenomenon and vibrational bi-resonances in FitzHugh–Nagumo (FHN) nervous system were studied in different levels of neurons and networks under diverse biophysical conditions [17]. For instance, the analysis of VR in a single FHN nervous system showed that the optimal amplitude of the HF stimulus does increase the neural response to the weak LF signal [18]. A single VR phenomenon was observed in coupled and networks of FHN model [19–21]. When the amplitude and frequency of the HF stimulus are simultaneously adjusted, a single VR and vibrational bi-resonances were observed in FHN neuron [22]. In the case of changes in potassium and sodium ion concentrations, Hodgkin-Huxley (HH) neuron significantly enhanced vibrational
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multi-resonances [23]. Moreover, The HF applied electric field externally changes the dynamics of the system, which leads to VR [24]. An optimal amplitude of the HF stimulus increases the response of coupled firing neurons to a subthreshold LF input signal, and the chemical synapse coupling is more obvious [25]. At the level of neural collective behavior, the phenomenon of VR, synchronization and information propagation have been studied extensively at different coupling topologies including feedforward [26,27], random [28], small-world networks [29,30], and so on. For instance, the moderated high temperatures induce a small variability of the intervals of bursting neurons leading to phase synchronization [31]. The algebraic connectivity is the important parameter for the synchronization, which links both the coupling intensity and the topology properties of network [32]. Moreover, some research results also showed that VR is subject to changes in demographic characteristics such as population size changes, topological complexity, and local neuronal dynamics [33]. The VR in a scale-free network has been also observed in many functional brain regions [34–37]. Actually, the coupled type for some networks can be unidirectional, and thus neural FFN is built to talk about the unidirectional coupling [38,39]. The pure feedforward network is multiple layers and the layer receives the input information from the upper-layer [40,41]. On the one hand, the memristor was firstly proposed by Chua [42] in 1971, which is the forth foundation circuit element, and connects the flux with charge. In 2008, HP Labs developed the first practical memristor device of TiO2 [43,44]. This component characteristic is very suitable for simulating neuron synapses, and making computer neural networks closer to the human brain [45–49]. Nowadays, memristors have been widely used in neuron systems [50]. In 2010, a flexible platform is described, different types of memristors can be simulated, and experiments had proved that memristors can be used to simulate neuronal synapses [51]. The simple way that a memristor is considered is to think of it as a potentiometer, that is, a resistor is adjusted. In an idealized memristor, because the resistance relies on the magnetic flux, a pulse with half the potential and twice the duration caused the same resistance change in 2012 [52]. In 2013, some researches in memristor-based neuronal systems were reviewed, and the excitement and rapid development were proven in the past few years [53]. Recently, the memristors have been considered as magnetic current through additional variables, which have improved many neural models of memristors, where the membrane voltage can be changed by additional variables [54–57]. The neural firing activities were observed under different feedback gains of induction current, and synchronization phenomena were investigated in improved Hindmarsh-Rose neuronal systems by memristors [54–56]. When each ion channel is regarded as a physical memristor, it was found that the shape of pinched hysteresis loop of memristor relies on both membrane patch temperature and input voltage in improved HH neural model [57]. On the other hand, although the improved FHN model was used to study synchronization in two coupled neurons and chaotic behavior of single neuron [58], the vibrational resonance phenomenon and transmission of weak LF signal have not been investigated in single improved FHN neuron and feed-forward improved FHN neural network. Therefore, an interesting question raises: how do strength of noise and electromagnetic induction affect VR phenomenon, collective behavior, and transmission of weak LF signal driven by HF stimulus in the improved FHN neural systems? This paper is organized as follows: Firstly, the effects of systemic parameter on the dynamics without LF signal and HF stimulus for single FHN neuron under electromagnetic induction are studied. Secondly, the vibrational mono-/bi-resonances are observed on excitable single FHN neural system. Then, we systematically analyze the bi-resonances and propagation of weak LF signal
in the FHN neural FFN with HF stimulus under different electromagnetic induction. Finally, we end with our conclusions. 2. Model description 2. 1. Single improved FHN neural model It is well known that the FHN neuronal model was derived from the nonlinear relaxation oscillator proposed by Vanderpol in 1926 [59], and FitzHugh and Nagumo independently proposed mathematical forms of excitable neurons, which were later called the FitzHugh–Nagumo neuron model [60,61]. The distribution and exchange of charged ions (Na+ , K+ , Ca2+ ) can induce complex electromagnetic field inside and outside the neuronal cell membrane, the membrane potential of neuron is modulated by the electromagnetic field. By using the Faraday’s law of electromagnetic induction, the effect of electromagnetic induction in cell must be considered. The transmembrane magnetic flow ϕ is regarded as an additional variable to describe the influence of electromagnetic induction. Therefore, the memristor was applied to improve coupling on neural membrane potential [42,62], and the memristor memductance is described by
β (ϕ ) =
dq(ϕ ) = α + 3βϕ 2 , dϕ
(1)
where ϕ is the magnetic flux across the memristor, α , β are parameters, q is the charge across the memristor. The improved FHN model [58] that is constituted by leading additive variable into magnetic flow which changes the membrane potential by a memristor and is a paradigmatic model representing firing spikes of neural activities. Here, the improved model under electromagnetic radiation is given by
⎧ dx ⎪ ⎨ε dt = x − ⎪ ⎩
dy dt dϕ dt
x3 3
− y + I − k1 ρ (ϕ )x,
= x + a − by,
(2)
= k2 x − 0.5ϕ ,
where x, y, and ϕ are the neural trans-membrane potential, conductivity of the potassium channels existing in neural membrane, and neural magnetic flow across membrane, respectively. The parameter k1 represents the feedback gain and the parameters k2 is the influence of membrane potential on magnetic flow. The electromagnetic induction is induced by time-varying change of intercellular and extracellular ion concentration [63], the induction current and electromagnetic induction are defined as follows
i =
dq(ϕ ) dq(ϕ ) dϕ = = ρ (ϕ )V = k1 ρ (ϕ )x, dt dϕ dt
(3)
where V is the induction voltage and holds the same physical units as variable x of membrane potential, and the parameter k1 describes the modulation strength of induction current on the membrane potential. The time scale ε = 0.08, is the inherent time scale that separate the fast and slow dynamics. I is a bi-chromatic signal consisting of a subthreshold LF signal and a HF stimulus modeled as Acos(ωt)+Bcos(Nωt), where Acos(ωt) is external LF signal and is subthreshold. Bcos(Nωt) is external HF stimulus with amplitude B, its frequency ω is N times of the LF signal (N ˃˃ 1). In the FHN neural model of Eq. (1), parameter a is a critical parameter which can affect the system’s dynamics. The parameters in Eq. (1) are set as α = 0.4, β = 0.02, b = 0.45, A = 0.32 and ω = 0.3. 2. 2. Feed-forward feedback neural network We consider that a FHN neural FFN system is constituted by 201 nodes, in which the dynamics for each node is described by
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the improved FHN model, and external LF signal and HF stimulus are only imposed on the central neuron (i = 101). The dynamics of improved FHN neural membrane potentials is rewritten as:
⎧ dx i ⎪ ⎨ε dt = xi − d yi
⎪ ⎩ ddtϕi dt
xi 3 3
− yi + Ii + DR xi −1 − xi + DL xi +1 − xi − k1 ρ (ϕi )xi ,
= xi + a − byi , = k2 xi − 0. 5ϕi ,
(4) where subscript i is the neural position in the FHN neural FFN system. Ii represents the two driving signals on the neural position i. The term k1 ρ (ϕ i )xi represents induction current of i neuron, and a feedback term k2 xi is introduced in the third formula. The memristor conductance of i neuron is often represented by ρ (ϕ i ) = α + 3βϕ i 2 . The system’s response of input LF signal is estimated by computing the Fourier coefficient Q of i neuron for the input frequency ω [7], which is defined as following
⎧ Q = ⎪ ⎪ ⎨ sin,i
ω
2π m
T0 +2π m/ω T0
2xi (t ) sin(ωt )dt,
T +2π m/ω Qcos,i = 2πωm T00 2xi (t ) cos(ωt )dt,
⎪ ⎪ ⎩Qi = Q 2 + Q 2 . cos,i sin,i
(5)
In neural system, the information is carried through large spikes instead of subthreshold oscillations, we are more interested in the frequency of spikes [19,20,22]. The threshold xs is set as 0 in the computation of Qi of i neuron. If the mean membrane potential ˂xi (t)˃ is less than xs , then ˂xi (t)˃ is displaced by the fixed point ˂xi (t)˃ = −1, otherwise, xi (t) keeps the same. The larger the Qi is, the more phase synchronization between input LF signal and output firing will be, and the much more information is transmitted by the excitable system. When the single neuron is considered, the mean membrane potential ˂xi (t)˃ is the x membrane potential of single neuron in Eq. (2). When the neuron i = 101, the I101 is Acos(ωt)+Bcos(Nωt), however, external forcing current of others neurons are 0. Parameters DL and DR ... are the coupling intensity between adjacent neurons: (i) DR ... = 0 and DL = D if when 1 ≤ i ≤ n; (ii) DR ... = D and DL = 0 if n + 1 ≤ i ≤ 2n+1. A statistical variable is applied to research collective behaviors by the mean field theory [61], expression of synchronization factor R is given by N 1
F= xi , R = N i=1
2 F
(1/N )
N
i=1
− F 2
xi 2 − xi 2
,
(6)
where N is the number of neurons (N = 201), and · is the mean of variable over time. According to previous researches, when the synchronization factor R is close to 1, the collective behaviors of neurons (the phase of the sequence of discharges) are basically the same. Otherwise, the non-perfect synchronization is reached when the R is close zero. 3. Results and discussions In the numerical simulations of this paper, the Euler algorithm is used and the time step h is set as 0.001. The initial variables are set as x0 = 0.1, y0 = 0.1 and ϕ 0 = 0.1. The time series of membrane potential and the inter-spike interval (ISI) are computed based on the selected parameters a, B, log10 N. 3.1. Single FHN neuron under electromagnetic induction According to [64], the nonlinear system’s orbit in the basin of attraction of a stable equilibrium or a limit cycle is similar to that
Fig. 1. Phase portraits (a)–(d) and bifurcation diagram (e) associated with different a (red solid line of the single FHN neuron without external LF signal and external HF stimulus under electromagnetic induction. (a) a = 0.64; (b) a = 0.65; (c) a = 0.655; (d) a > 0.657.
of a linear system. Similar to the bifurcation of fundamental cycles [65], in single FHN neuron model, parameter a plays a critical role in the dynamics of the system and the effects without LF signal and HF stimulus under electromagnetic induction are shown in Fig. 1. The Andronov-Hopf bifurcation occurs at a ≈ 0.6507. These results different from [41] in which the electromagnetic induction was not taken account into (the Andronov-Hopf bifurcation occurred at a ≈ 0.69). When the value of a is less than 0.6507, the neuron keeps in an excitable region and has a stable periodical solution as shown Fig. 1(a) and (b), respectively. Moreover, in a small vicinity of a = 0.6507, there are canard explosion that small oscillations near the unstable fixed point before rising abruptly the oscillation amplitude. When the parameter a is in the range of 0.6507 to 0.657, the neuron presents subthreshold oscillation. When the parameter a is bigger than 0.657, the neuron keeps in damp oscillation. These results correspond to Fig. 1(e) which shows the change of ISI with the a. When a is less than 0.6507, the value of ISI is greater than zero, and the neuron keeps excited oscillation. Because of being canard explosion, the value of ISI is rise gradually with the increasing of a. When a is equal to 0.6507, the value of ISI is zero, and the neuron keeps subthreshold oscillation or damped oscillation. Fig. 2 shows the different firing modes of a single FHN neuron with different parameters a under electromagnetic induction. From Fig. 2(b), the canard explosion keeps at a [0.6499, 0.6507]. For a < 0.6499, the FHN neuron exists periodic spiking state on the well known big excursion loop, see Fig. 2(a). In the parameter region of a (0.6507, 0.657], there is a subthreshold oscillatory behavior, see Fig. 2(c). When the parameter a is bigger than 0.657, the firing modes transform subthreshold oscillatory to damped oscillation. Even if the change in a is small, the changes of neuron oscillation are obvious. The canard explosion is quit sensitive to external perturbations, and has an important effects on the signal propagation. Therefore, in the following simulations, we take a = 0.78 to keep neuron in a stable region.
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Fig. 2. Time series of membrane potential x of single FHN neural model for different parameter a. (a) Periodic spiking when a < 0.6499; (b) 0.6499 ≤ a ≤ 0.6507; (c) subthreshold periodic oscillation when 0.6507 < a ≤ 0.657; (d) damp oscillation when a > 0.657.
3.2. Vibrational resonance in single FHN neuron system According to the Fig. 1, we take the parameter a of 0.78 in the following researches. Fig. 3 shows the time evolution of membrane potential x for different HF stimulus amplitudes B and frequency ratio log10 N under different electromagnetic inductions. For comparison, the LF signal is plotted in Fig. 3, indexed by red lines. When electromagnetic induction is weak, the amplitude of HF stimulus is not big (B = 0.4 as in Fig. 3(a1)), the total output of system is below threshold, and carries few information about the LF signal. With the increasing of amplitude of HF stimulus (B = 0.63 as in Fig. 3(b1)), the input LF signal and output of system are synchronized, and the information of the LF signal is amplified. However, when the amplitude of HF stimulus is large, the processing of signal is degraded again. Therefore, the VR is observed when the amplitude of HF stimulus chosen as a proper value. When electromagnetic induction is strong, the frequency ratio of HF stimulus is very small (log10 N = 0.71 as in Fig. 3(a2)], the total output of the system is intermittent spike discharge and carries some information about the LF signal. With the increasing of the frequency ratio of HF stimulus (log10 N = 1.35 as in Fig. 3(c2)), the output of system appears phenomenon of VR. However, when the frequency ratio of HF stimulus is big, the signal synchronization phenomenon is not well. From Fig. 3(e1), when the amplitude of HF stimulus is smaller than 0.53, the value of ISI is zero, and the neuron keeps subthreshold oscillatory or damped oscillation. With the increasing of the amplitude, the state of neuron changes from subthreshold oscillatory or damped oscillation to canard explosion and periodic oscillation. From Fig. 3(e2), when the frequency ratio of HF stimulus is smaller than 0.51, the value of ISI is zero, and the neuron keeps subthreshold oscillatory or damped oscillation. With the in-
creasing of the frequency ratio of HF stimulus, the state of neuron changes from subthreshold oscillatory or damped oscillation to excitability oscillation. However, when the frequency ratio is bigger than 1.42, the neuron is in subthreshold oscillatory or damped oscillation. In order to better understand the effects of the amplitude B and HF stimulus frequency ratio log10 N at different electromagnetic radiations on output of weak signal, the contour plot of the linear response Q for the excitable FHN model is plotted in Fig. 4. Fig. 4(a) shows the amplitude of the HF stimulus has a wider region under weak electromagnetic radiation. The red region represents the maximum value of Q, and the blue and dark areas are small value of Q. In the regions around (log10 N, B) = (1.52, 0.52) and (1.46, 0.33), the weak signal can get a lot of amplification. In the same way, Fig. 4(b) shows the amplitude of the HF stimulus has a wider region under strong electromagnetic radiation. Different from the Fig. 4(a), when the VR phenomenon is occurred, the both the amplitude and frequency ratio of the HF stimulus are smaller than those of weak electromagnetic radiation. Moreover, the linear response Q varies with HF stimulus frequency ratio log10 N when the amplitude is fixed, the plots of Q versus log10 N also exist a VR phenomenon. Likewise, the same is true of fixed frequencies. The results are shown in Fig. (5). In order to more clearly see the change of the corresponding factors with the change of the amplitude B and HF stimulus frequency ratio log10 N. Fig. 5 shows the dependencies of response measure Q on amplitude B (Fig. 5a, b) and different HF stimulus frequency under different electromagnetic inductions. Fig. 5 corresponds to Fig. 4 which fixes the amplitude B and HF stimulus frequency ratio log10 N. The Fig. 5(a, c) and Fig. 5(b, d) show the effects of weak electromagnetic induction (k1 = 0.4, k2 = 0.9) and
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Fig. 3. Time series of membrane potential x (a1)-(d2) and bifurcation diagrams (e1)-(e2) for different HF amplitude B and frequency ratio log10 N. The profiles of the LF modulation are also superimposed as red curves. (a1, b1, c1, d1) k1 = 0.4, k2 = 0.9, log10 N = 1.58, B = 0.4, 0.63, 1.32, 1.5; (a2, b2, c2, d2) k1 = 0.8, k2 = 2.0, B = 0.33, log10 N = 0.71, 0.89, 1.35, 2.13.
Fig. 4. Contour plot of Q for the excitable FHN model on the amplitude B and HF stimulus frequency ratio log10 N at different regions (k1 , k2 ). (a) k1 = 0.4, k2 = 0.9; (b) k1 = 0.8, k2 = 2.0. Red and orange indicate higher values of Q.
strong electromagnetic induction (k1 = 0.8, k2 = 2.0) on VR. According to Fig. 5(a, c), the weak LF signal is amplified by two values of amplitude B and HF stimulus frequency, called as vibrational bi-resonances. When the HF stimulus frequency ratio is fixed (log10 N = 1.542), the vibrational bi-resonances phenomenon
also is observed at strong electromagnetic induction in Fig. 5(a). In addition, Fig. 5(c) shows the amplitude is fixed (B = 0.558) under weak electromagnetic induction, the plots of Q versus log10 N have two peak value and exist a vibrational bi-resonances phenomenon. However, when the HF stimulus frequency ratio is fixed
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Fig. 5. Dependencies of response measure Q on amplitude B (a)-(b) and different HF stimulus frequency ratio (c)-(d). (a) log10 N = 1.542; (b) log10 N = 0.986; (c) B = 0.558; (d) B = 0.314. Fig. 5(b) and (d) show a single vibrational resonance, Fig. 5(a) and (c) show vibrational bi-resonances.
Fig. 6. Contour plot of Q for the excitable FHN model on the (k1 , k2 ) plane at different amplitude B. log10 N = 1.26. (a) B = 0.1; (b) B = 1.3. Red and orange indicate higher values of Q.
(log10 N = 0.986), a single VR phenomenon can be observed in Fig. 5(b). Moreover, Fig. 5(d) also shows that the linear response Q varies with HF stimulus frequency ratio log10 N when the amplitude is fixed (B = 0.314), the plots of Q versus log10 N have one peak value and exist a single VR phenomenon. However, To get a global view about effects of electromagnetic induction at small and big HF amplitude B on information transmission, a contour plot of the linear response Q for the excitable
FHN model is plotted in Fig. 6. Fig. 6(a and b) show the effects of linear responses on the small and large HF amplitude B. When the HF amplitude is small, the changes of linear responses with the feedback gain k1 are analogous, that is, regardless of the size of parameter k2 , the linear response Q decreases with the increasing of k1 . It reveals that a small value of the feedback gain k1, where the output of the excitable system is at best correlated with the weak LF signal, and the value of k2 has a few effects on the information
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Fig. 7. Time series of membrane potentials of all neurons (a1, b1, c1, d1) and mean-field activities (a2, b2, c2, d2) of the neural network obtained for different HF driving amplitude B, k1 = 0.4, k2 = 0.9, log10 N = 1.26. (a1, a2) B = 0.01; (b1, b2) B = 0.05; (c1, c2) B = 0.1; (d1, d2) B = 0.5.
transmission of weak signal. However, when the HF amplitude is large, the changes of linear responses with the feedback gain k1 and k2 are small, and the linear responses are small. Therefore, it indicates the synchronization phenomenon is not well, and signal processing is degraded at big HF amplitude, whether the electromagnetic induction is weak or strong.
3.3. Vibrational bi-resonances and propagation in the FHN neural FFN system In this section, we systematically analyze the wave emission and propagation of weak LF signal in the FHN neural FFN system with HF stimulus under different electromagnetic induction, i.e. the VR phenomenon. In this FHN neural FFN system, only the central neuron (i = 101) is imposed by external forcing current as I101 = Acos(ωt)+Bcos(Nωt), while Ii = 0 is imposed on the others neurons. Figs. 7 and 8 plot the time series of membrane potentials of all neurons and mean-field activity (black curves) of the neural network obtained for different HF driving amplitude B and different HF stimulus frequency ratio log10 N. Fig. 7 shows the effects of HF amplitude B on the wave emitting and propagation of weak LF signal in FFN neural system. When the amplitude B is small (B = 0.01 in Fig. 7(a1 and a2)), all neurons are at quiescent state and cannot carry any information about the LF signal (red curve). When the amplitude B is increased, waves will propagate from the central neuron to the neurons at both ends and the mean-field activities of neural network are related to the rhythm of LF signal, which means the appearance of VR. When the amplitude B keeps further increasing, more and more waves are emitted from
the central neuron to both ends. Nevertheless, the coherence between collective activities and subthreshold signal will disappear. Fig. 8 shows the effects of HF stimulus frequency ratio log10 N on the wave emission and propagation of weak LF signal in the FFN network. When the HF stimulus frequency ratio log10 N is small (log10 N = 0.26 in Fig. 8(a1 and a2)), all neurons keep in a quiescent state and cannot carry any information about the LF signal. When the HF stimulus frequency ratio log10 N is moderate, waves can propagate from the central neuron to the neurons at both ends and the mean-field activities of neural network are related to the rhythm of LF signal, which means the appearance of VR. However, with the increasing of the frequency ratio log10 N, no waves are emitted from the central neuron to both ends, because the central neuron keep also in the quiescent state. Moreover, the coherence between collective activities and LF signal also disappears. Fig. 9 shows the dependencies of synchronization factor R and response measure Q on the amplitude B (Fig. 9(a, c)) and different HF stimulus frequency (Fig. 9(b, d)) under different electromagnetic induction in FFN neural network. Fig. 9(a) shows the dependencies of synchronization factor R on the frequency ratio log10 N. When the frequency ratio is small, the synchronization factor is big, and the neurons keep quiescent state. When the frequency ratio is increased, the neurons are in excitability oscillation and the wave can be spread. However, when continue to grow the frequency ratio, the wave cannot be spread and most neurons keep in quiescent state. Fig. 9(c) shows that the linear response Q varies with HF stimulus frequency ratio log10 N when the amplitude is fixed (B = 0.1), the plots of Q versus log10 N have two peak value as a whole and occur vibrational bi-resonances phenomenon. With a quite high frequency of the HF stimulus, the coherence be-
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Fig. 8. Time series of membrane potentials of all neurons (a1, b1, c1, d1) and mean-field activities (a2, b2, c2, d2) of the neuronal network obtained for different HF stimulus frequency ratio log10 N, k1 = 0.4, k2 = 0.9, B = 0.1. (a1, a2) log10 N = 0.26; (b1, b2) log10 N = 0.5; (c1, c2) log10 N = 1.26; (d1, d2) log10 N = 2.5.
Fig. 9. Dependencies of synchronization factor R (a)-(b) and response measure Q (c)-(d) for the FFN neural system on amplitude B (a)-(c) and HF stimulus frequency ratio log10 N (b)-(d). k1 = 0.4, k2 = 0.9, (a)-(c) B = 0.1; (b)-(d) log10 N = 1.26.
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Fig. 10. Dependencies of synchronization factor R (a) and response measure Q (b) for the chain network in different regions (k1, k2 ). log10 N = 1.26, B = 0.1. Red and orange indicate higher values of Q.
tween collective behavior and LF signal cannot be observed. However, Fig. 9(b) shows the dependencies of synchronization factor R on the amplitude B. When the amplitude is quite small, the synchronization factor is big, and all the neurons keep quiescent state. When the amplitude is increased, the neurons are in subthreshold oscillatory or damped oscillation and the wave can be spread. When the HF stimulus frequency ratio is fixed (log10 N = 1.26), the vibrational bi-resonances phenomenon is observed in Fig. 9(d). The two peak value B = 0.15 and 0.98 are the highest and the second peak, respectively. Similar to Fig. 9(c), the coherence between collective behavior and LF signal cannot be observed by increasing the amplitude of LF signal. These results indicate that the electromagnetic induction and the HF stimulus play an important role in the VR and information transmission of FFN system. To get global views about how the electromagnetic induction affects transmission of weak LF signal, the contour plots of the synchronization factor R and linear response Q for the FFN neural system are drawn in Fig. 10. The Fig. 10(a), when the parameters k1 and k2 are small, the synchronization factors are small, and the neurons keep in excitability oscillation and the wave can be spread. When the parameters k1 and k2 are large, the synchronization factor is big, and the wave cannot be spread. In Fig. 10(b), the changes of linear responses with the feedback gain k1 are alike, that is, regardless of the size of parameter k2 , the linear response Q decreases with the increasing of k1 . It is shown that the small values of feedback gain k1 at which the output of weak LF signal in the FFN neural system are at best correlated, and the value of k2 has a few effects on the transmission of weak LF signal. When the value of the feedback gain k1 is large, the linear responses are small, and the coherence between collective behavior cannot be observed. Hence, it means that the transmission of weak LF signal is degraded at big feedback gain k1 .
vibrational mono-/bi-resonances are observed for excitable single FHN neuron system. We showed that the input weak signal and output of system keep well synchronized and the information of the LF signal is amplified by increasing the amplitude of HF stimulus. When the electromagnetic induction is weak, the amplitude of HF stimulus is not big, the total outputs of the system are below threshold and carry few information about the LF signal. In the FFN neural FHN system, with increasing the amplitude of HF stimulus, the synchronization factor becomes smaller, and more and more waves are emitted from the central neuron to both ends. Nevertheless, the coherence between collective activities and subthreshold signal disappear. The electromagnetic induction and HF stimulus have important effects on the VR phenomenon and weak signal transmission. In our future studies, the VR and transmission of weak LF signal by HF stimulus might be taken account into improved multilayer neural systems [66,67]. Based on this dynamic mechanism, HF stimulus also could be replaced by various noises [68–74].
4. Conclusions
Acknowledgment
In this paper, an improved FHN neural model was employed to study the VR phenomena, collective behavior, and transmission of weak LF signal by HF stimulus under different electromagnetic induction in the single excitable FHN neuron and the FHN neural FFN system, respectively. In the single FHN neural model, the systemic parameter a plays a critical role in the dynamics of system without LF signal and HF stimulus under electromagnetic induction. The phenomena of
The work was supported by the National Natural Science Foundation of China under Nos. 11775091(Y.J.) and 11605014(Q.P.).
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Mengyan Ge: Visualization, Supervision, Formal analysis, Writing - review & editing. Lulu Lu: Supervision, Formal analysis. Ying Xu: Supervision, Formal analysis. Rozihajim Mamatimin: Supervision. Qiming Pei: Formal analysis. Ya Jia: Visualization, Formal analysis, Writing - review & editing.
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Vibrational mono-/bi-resonance and wave propagation in FitzHugh–Nagumo neural systems under electromagnetic induction Mengyan Ge a, Lulu Lu a, Ying Xu a, Rozihajim Mamatimin b, Qiming Pei c, Ya Jia a,∗ a
Department of Physics, Central China Normal University, Wuhan 430079, China Institute of Civil Engineering, Kashi University, Kashi 844008, China c School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China b
a r t i c l e
i n f o
Article history: Received 28 May 2019 Revised 18 January 2020 Accepted 22 January 2020
Keywords: Vibrational resonance FitzHugh–Nagumo model Feed-forward feedback neural network Electromagnetic induction Chaos and bifurcation analysis
a b s t r a c t In this paper, an modified FitzHugh–Nagumo (FHN) neural model was employed to investigate the vibrational resonance (VR) phenomenon, the collective behaviors, and the transmission of weak low-frequency (LF) signal driven by high-frequency (HF) stimulus under the action of different electromagnetic induction in single FHN neuron and feed-forward feedback network (FFN) system, respectively. For the single FHN system, by increasing the amplitude of HF stimulus, the phenomena of vibrational mono-/bi-resonance are observed, and the input weak signal and output of system are synchronized, and the information of the weak LF signal is amplified. For the FFN system, the phenomena of vibrational mono-/bi-resonances are also occurred, both frequency and amplitude of the HF stimulus play an important role in the vibrational bi-resonances and transmission of weak LF signal in the FHN neural FFN.
1. Introduction Fluctuations are ubiquitous in various nonlinear dynamic processes (or signaling systems), for instance, the spikes of neurons are understood as random processes [1] where noise plays very important roles. In previous investigations, noise-induced complex dynamic behaviors were found. One of the most significant findings is the stochastic resonance (SR) phenomenon, which is expressed in a nonlinear system wherein the external information (such as weak signals) is input. It was found that noises are used to amplify and optimize information in the SR phenomenon [2– 4]. There are many most recently publications of SR phenomenon. For example, low stochastic resonance is observed by respectively varying noise strength and self-correlation time of sine-wiener noise [5]. The coherence resonance in one layer can not only be controlled by the network topology, intralayer and interlayer timedelayed couplings, but also by self-induced stochastic resonance in the other layer [6]. Recent investigations have revealed that the responses of nonlinear systems to a weak low-frequency (LF) signal are magnified by using a high-frequency (HF) stimulus, which is similar to the effect of noise on the SR phenomenon. In a nonlinear system, it is actually affected by two periodic stimulus, LF (as signal) and HF
∗
Corresponding author. E-mail address: [email protected] (Y. Jia).
https://doi.org/10.1016/j.chaos.2020.109645 0960-0779/© 2020 Elsevier Ltd. All rights reserved.
© 2020 Elsevier Ltd. All rights reserved.
(as carrier), where the HF stimulus can enhance the processing of weak LF signal. In the presence of additional white noise, vibrational resonance (VR) has been theoretically proven in the bistable system [7,8]. For the external fields, more than two frequency periodic stimulus could be often used in various signal systems, such as the commutation techniques [9], neuroscience [10], acoustics [11] and laser physics [12]. Vibration bi-resonances are a phenomenon in which a weak LF signal is amplified by different values of HF stimulus. There are many experimental researches on VR phenomenon and vibrational bi-resonances, such as excitable electronic circuit [13], overdamped duffing oscillator simulation [14], and bistable optical cavity laser [15]. In stochastic bistable systems, the VR was studied by experimental and numerical methods to increase the detection and recovery of weak subthreshold aperiodic binary signals [16]. The VR phenomenon and vibrational bi-resonances in FitzHugh–Nagumo (FHN) nervous system were studied in different levels of neurons and networks under diverse biophysical conditions [17]. For instance, the analysis of VR in a single FHN nervous system showed that the optimal amplitude of the HF stimulus does increase the neural response to the weak LF signal [18]. A single VR phenomenon was observed in coupled and networks of FHN model [19–21]. When the amplitude and frequency of the HF stimulus are simultaneously adjusted, a single VR and vibrational bi-resonances were observed in FHN neuron [22]. In the case of changes in potassium and sodium ion concentrations, Hodgkin-Huxley (HH) neuron significantly enhanced vibrational
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multi-resonances [23]. Moreover, The HF applied electric field externally changes the dynamics of the system, which leads to VR [24]. An optimal amplitude of the HF stimulus increases the response of coupled firing neurons to a subthreshold LF input signal, and the chemical synapse coupling is more obvious [25]. At the level of neural collective behavior, the phenomenon of VR, synchronization and information propagation have been studied extensively at different coupling topologies including feedforward [26,27], random [28], small-world networks [29,30], and so on. For instance, the moderated high temperatures induce a small variability of the intervals of bursting neurons leading to phase synchronization [31]. The algebraic connectivity is the important parameter for the synchronization, which links both the coupling intensity and the topology properties of network [32]. Moreover, some research results also showed that VR is subject to changes in demographic characteristics such as population size changes, topological complexity, and local neuronal dynamics [33]. The VR in a scale-free network has been also observed in many functional brain regions [34–37]. Actually, the coupled type for some networks can be unidirectional, and thus neural FFN is built to talk about the unidirectional coupling [38,39]. The pure feedforward network is multiple layers and the layer receives the input information from the upper-layer [40,41]. On the one hand, the memristor was firstly proposed by Chua [42] in 1971, which is the forth foundation circuit element, and connects the flux with charge. In 2008, HP Labs developed the first practical memristor device of TiO2 [43,44]. This component characteristic is very suitable for simulating neuron synapses, and making computer neural networks closer to the human brain [45–49]. Nowadays, memristors have been widely used in neuron systems [50]. In 2010, a flexible platform is described, different types of memristors can be simulated, and experiments had proved that memristors can be used to simulate neuronal synapses [51]. The simple way that a memristor is considered is to think of it as a potentiometer, that is, a resistor is adjusted. In an idealized memristor, because the resistance relies on the magnetic flux, a pulse with half the potential and twice the duration caused the same resistance change in 2012 [52]. In 2013, some researches in memristor-based neuronal systems were reviewed, and the excitement and rapid development were proven in the past few years [53]. Recently, the memristors have been considered as magnetic current through additional variables, which have improved many neural models of memristors, where the membrane voltage can be changed by additional variables [54–57]. The neural firing activities were observed under different feedback gains of induction current, and synchronization phenomena were investigated in improved Hindmarsh-Rose neuronal systems by memristors [54–56]. When each ion channel is regarded as a physical memristor, it was found that the shape of pinched hysteresis loop of memristor relies on both membrane patch temperature and input voltage in improved HH neural model [57]. On the other hand, although the improved FHN model was used to study synchronization in two coupled neurons and chaotic behavior of single neuron [58], the vibrational resonance phenomenon and transmission of weak LF signal have not been investigated in single improved FHN neuron and feed-forward improved FHN neural network. Therefore, an interesting question raises: how do strength of noise and electromagnetic induction affect VR phenomenon, collective behavior, and transmission of weak LF signal driven by HF stimulus in the improved FHN neural systems? This paper is organized as follows: Firstly, the effects of systemic parameter on the dynamics without LF signal and HF stimulus for single FHN neuron under electromagnetic induction are studied. Secondly, the vibrational mono-/bi-resonances are observed on excitable single FHN neural system. Then, we systematically analyze the bi-resonances and propagation of weak LF signal
in the FHN neural FFN with HF stimulus under different electromagnetic induction. Finally, we end with our conclusions. 2. Model description 2. 1. Single improved FHN neural model It is well known that the FHN neuronal model was derived from the nonlinear relaxation oscillator proposed by Vanderpol in 1926 [59], and FitzHugh and Nagumo independently proposed mathematical forms of excitable neurons, which were later called the FitzHugh–Nagumo neuron model [60,61]. The distribution and exchange of charged ions (Na+ , K+ , Ca2+ ) can induce complex electromagnetic field inside and outside the neuronal cell membrane, the membrane potential of neuron is modulated by the electromagnetic field. By using the Faraday’s law of electromagnetic induction, the effect of electromagnetic induction in cell must be considered. The transmembrane magnetic flow ϕ is regarded as an additional variable to describe the influence of electromagnetic induction. Therefore, the memristor was applied to improve coupling on neural membrane potential [42,62], and the memristor memductance is described by
β (ϕ ) =
dq(ϕ ) = α + 3βϕ 2 , dϕ
(1)
where ϕ is the magnetic flux across the memristor, α , β are parameters, q is the charge across the memristor. The improved FHN model [58] that is constituted by leading additive variable into magnetic flow which changes the membrane potential by a memristor and is a paradigmatic model representing firing spikes of neural activities. Here, the improved model under electromagnetic radiation is given by
⎧ dx ⎪ ⎨ε dt = x − ⎪ ⎩
dy dt dϕ dt
x3 3
− y + I − k1 ρ (ϕ )x,
= x + a − by,
(2)
= k2 x − 0.5ϕ ,
where x, y, and ϕ are the neural trans-membrane potential, conductivity of the potassium channels existing in neural membrane, and neural magnetic flow across membrane, respectively. The parameter k1 represents the feedback gain and the parameters k2 is the influence of membrane potential on magnetic flow. The electromagnetic induction is induced by time-varying change of intercellular and extracellular ion concentration [63], the induction current and electromagnetic induction are defined as follows
i =
dq(ϕ ) dq(ϕ ) dϕ = = ρ (ϕ )V = k1 ρ (ϕ )x, dt dϕ dt
(3)
where V is the induction voltage and holds the same physical units as variable x of membrane potential, and the parameter k1 describes the modulation strength of induction current on the membrane potential. The time scale ε = 0.08, is the inherent time scale that separate the fast and slow dynamics. I is a bi-chromatic signal consisting of a subthreshold LF signal and a HF stimulus modeled as Acos(ωt)+Bcos(Nωt), where Acos(ωt) is external LF signal and is subthreshold. Bcos(Nωt) is external HF stimulus with amplitude B, its frequency ω is N times of the LF signal (N ˃˃ 1). In the FHN neural model of Eq. (1), parameter a is a critical parameter which can affect the system’s dynamics. The parameters in Eq. (1) are set as α = 0.4, β = 0.02, b = 0.45, A = 0.32 and ω = 0.3. 2. 2. Feed-forward feedback neural network We consider that a FHN neural FFN system is constituted by 201 nodes, in which the dynamics for each node is described by
M. Ge, L. Lu and Y. Xu et al. / Chaos, Solitons and Fractals 133 (2020) 109645
3
the improved FHN model, and external LF signal and HF stimulus are only imposed on the central neuron (i = 101). The dynamics of improved FHN neural membrane potentials is rewritten as:
⎧ dx i ⎪ ⎨ε dt = xi − d yi
⎪ ⎩ ddtϕi dt
xi 3 3
− yi + Ii + DR xi −1 − xi + DL xi +1 − xi − k1 ρ (ϕi )xi ,
= xi + a − byi , = k2 xi − 0. 5ϕi ,
(4) where subscript i is the neural position in the FHN neural FFN system. Ii represents the two driving signals on the neural position i. The term k1 ρ (ϕ i )xi represents induction current of i neuron, and a feedback term k2 xi is introduced in the third formula. The memristor conductance of i neuron is often represented by ρ (ϕ i ) = α + 3βϕ i 2 . The system’s response of input LF signal is estimated by computing the Fourier coefficient Q of i neuron for the input frequency ω [7], which is defined as following
⎧ Q = ⎪ ⎪ ⎨ sin,i
ω
2π m
T0 +2π m/ω T0
2xi (t ) sin(ωt )dt,
T +2π m/ω Qcos,i = 2πωm T00 2xi (t ) cos(ωt )dt,
⎪ ⎪ ⎩Qi = Q 2 + Q 2 . cos,i sin,i
(5)
In neural system, the information is carried through large spikes instead of subthreshold oscillations, we are more interested in the frequency of spikes [19,20,22]. The threshold xs is set as 0 in the computation of Qi of i neuron. If the mean membrane potential ˂xi (t)˃ is less than xs , then ˂xi (t)˃ is displaced by the fixed point ˂xi (t)˃ = −1, otherwise, xi (t) keeps the same. The larger the Qi is, the more phase synchronization between input LF signal and output firing will be, and the much more information is transmitted by the excitable system. When the single neuron is considered, the mean membrane potential ˂xi (t)˃ is the x membrane potential of single neuron in Eq. (2). When the neuron i = 101, the I101 is Acos(ωt)+Bcos(Nωt), however, external forcing current of others neurons are 0. Parameters DL and DR ... are the coupling intensity between adjacent neurons: (i) DR ... = 0 and DL = D if when 1 ≤ i ≤ n; (ii) DR ... = D and DL = 0 if n + 1 ≤ i ≤ 2n+1. A statistical variable is applied to research collective behaviors by the mean field theory [61], expression of synchronization factor R is given by N 1
F= xi , R = N i=1
2 F
(1/N )
N
i=1
− F 2
xi 2 − xi 2
,
(6)
where N is the number of neurons (N = 201), and · is the mean of variable over time. According to previous researches, when the synchronization factor R is close to 1, the collective behaviors of neurons (the phase of the sequence of discharges) are basically the same. Otherwise, the non-perfect synchronization is reached when the R is close zero. 3. Results and discussions In the numerical simulations of this paper, the Euler algorithm is used and the time step h is set as 0.001. The initial variables are set as x0 = 0.1, y0 = 0.1 and ϕ 0 = 0.1. The time series of membrane potential and the inter-spike interval (ISI) are computed based on the selected parameters a, B, log10 N. 3.1. Single FHN neuron under electromagnetic induction According to [64], the nonlinear system’s orbit in the basin of attraction of a stable equilibrium or a limit cycle is similar to that
Fig. 1. Phase portraits (a)–(d) and bifurcation diagram (e) associated with different a (red solid line of the single FHN neuron without external LF signal and external HF stimulus under electromagnetic induction. (a) a = 0.64; (b) a = 0.65; (c) a = 0.655; (d) a > 0.657.
of a linear system. Similar to the bifurcation of fundamental cycles [65], in single FHN neuron model, parameter a plays a critical role in the dynamics of the system and the effects without LF signal and HF stimulus under electromagnetic induction are shown in Fig. 1. The Andronov-Hopf bifurcation occurs at a ≈ 0.6507. These results different from [41] in which the electromagnetic induction was not taken account into (the Andronov-Hopf bifurcation occurred at a ≈ 0.69). When the value of a is less than 0.6507, the neuron keeps in an excitable region and has a stable periodical solution as shown Fig. 1(a) and (b), respectively. Moreover, in a small vicinity of a = 0.6507, there are canard explosion that small oscillations near the unstable fixed point before rising abruptly the oscillation amplitude. When the parameter a is in the range of 0.6507 to 0.657, the neuron presents subthreshold oscillation. When the parameter a is bigger than 0.657, the neuron keeps in damp oscillation. These results correspond to Fig. 1(e) which shows the change of ISI with the a. When a is less than 0.6507, the value of ISI is greater than zero, and the neuron keeps excited oscillation. Because of being canard explosion, the value of ISI is rise gradually with the increasing of a. When a is equal to 0.6507, the value of ISI is zero, and the neuron keeps subthreshold oscillation or damped oscillation. Fig. 2 shows the different firing modes of a single FHN neuron with different parameters a under electromagnetic induction. From Fig. 2(b), the canard explosion keeps at a [0.6499, 0.6507]. For a < 0.6499, the FHN neuron exists periodic spiking state on the well known big excursion loop, see Fig. 2(a). In the parameter region of a (0.6507, 0.657], there is a subthreshold oscillatory behavior, see Fig. 2(c). When the parameter a is bigger than 0.657, the firing modes transform subthreshold oscillatory to damped oscillation. Even if the change in a is small, the changes of neuron oscillation are obvious. The canard explosion is quit sensitive to external perturbations, and has an important effects on the signal propagation. Therefore, in the following simulations, we take a = 0.78 to keep neuron in a stable region.
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Fig. 2. Time series of membrane potential x of single FHN neural model for different parameter a. (a) Periodic spiking when a < 0.6499; (b) 0.6499 ≤ a ≤ 0.6507; (c) subthreshold periodic oscillation when 0.6507 < a ≤ 0.657; (d) damp oscillation when a > 0.657.
3.2. Vibrational resonance in single FHN neuron system According to the Fig. 1, we take the parameter a of 0.78 in the following researches. Fig. 3 shows the time evolution of membrane potential x for different HF stimulus amplitudes B and frequency ratio log10 N under different electromagnetic inductions. For comparison, the LF signal is plotted in Fig. 3, indexed by red lines. When electromagnetic induction is weak, the amplitude of HF stimulus is not big (B = 0.4 as in Fig. 3(a1)), the total output of system is below threshold, and carries few information about the LF signal. With the increasing of amplitude of HF stimulus (B = 0.63 as in Fig. 3(b1)), the input LF signal and output of system are synchronized, and the information of the LF signal is amplified. However, when the amplitude of HF stimulus is large, the processing of signal is degraded again. Therefore, the VR is observed when the amplitude of HF stimulus chosen as a proper value. When electromagnetic induction is strong, the frequency ratio of HF stimulus is very small (log10 N = 0.71 as in Fig. 3(a2)], the total output of the system is intermittent spike discharge and carries some information about the LF signal. With the increasing of the frequency ratio of HF stimulus (log10 N = 1.35 as in Fig. 3(c2)), the output of system appears phenomenon of VR. However, when the frequency ratio of HF stimulus is big, the signal synchronization phenomenon is not well. From Fig. 3(e1), when the amplitude of HF stimulus is smaller than 0.53, the value of ISI is zero, and the neuron keeps subthreshold oscillatory or damped oscillation. With the increasing of the amplitude, the state of neuron changes from subthreshold oscillatory or damped oscillation to canard explosion and periodic oscillation. From Fig. 3(e2), when the frequency ratio of HF stimulus is smaller than 0.51, the value of ISI is zero, and the neuron keeps subthreshold oscillatory or damped oscillation. With the in-
creasing of the frequency ratio of HF stimulus, the state of neuron changes from subthreshold oscillatory or damped oscillation to excitability oscillation. However, when the frequency ratio is bigger than 1.42, the neuron is in subthreshold oscillatory or damped oscillation. In order to better understand the effects of the amplitude B and HF stimulus frequency ratio log10 N at different electromagnetic radiations on output of weak signal, the contour plot of the linear response Q for the excitable FHN model is plotted in Fig. 4. Fig. 4(a) shows the amplitude of the HF stimulus has a wider region under weak electromagnetic radiation. The red region represents the maximum value of Q, and the blue and dark areas are small value of Q. In the regions around (log10 N, B) = (1.52, 0.52) and (1.46, 0.33), the weak signal can get a lot of amplification. In the same way, Fig. 4(b) shows the amplitude of the HF stimulus has a wider region under strong electromagnetic radiation. Different from the Fig. 4(a), when the VR phenomenon is occurred, the both the amplitude and frequency ratio of the HF stimulus are smaller than those of weak electromagnetic radiation. Moreover, the linear response Q varies with HF stimulus frequency ratio log10 N when the amplitude is fixed, the plots of Q versus log10 N also exist a VR phenomenon. Likewise, the same is true of fixed frequencies. The results are shown in Fig. (5). In order to more clearly see the change of the corresponding factors with the change of the amplitude B and HF stimulus frequency ratio log10 N. Fig. 5 shows the dependencies of response measure Q on amplitude B (Fig. 5a, b) and different HF stimulus frequency under different electromagnetic inductions. Fig. 5 corresponds to Fig. 4 which fixes the amplitude B and HF stimulus frequency ratio log10 N. The Fig. 5(a, c) and Fig. 5(b, d) show the effects of weak electromagnetic induction (k1 = 0.4, k2 = 0.9) and
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Fig. 3. Time series of membrane potential x (a1)-(d2) and bifurcation diagrams (e1)-(e2) for different HF amplitude B and frequency ratio log10 N. The profiles of the LF modulation are also superimposed as red curves. (a1, b1, c1, d1) k1 = 0.4, k2 = 0.9, log10 N = 1.58, B = 0.4, 0.63, 1.32, 1.5; (a2, b2, c2, d2) k1 = 0.8, k2 = 2.0, B = 0.33, log10 N = 0.71, 0.89, 1.35, 2.13.
Fig. 4. Contour plot of Q for the excitable FHN model on the amplitude B and HF stimulus frequency ratio log10 N at different regions (k1 , k2 ). (a) k1 = 0.4, k2 = 0.9; (b) k1 = 0.8, k2 = 2.0. Red and orange indicate higher values of Q.
strong electromagnetic induction (k1 = 0.8, k2 = 2.0) on VR. According to Fig. 5(a, c), the weak LF signal is amplified by two values of amplitude B and HF stimulus frequency, called as vibrational bi-resonances. When the HF stimulus frequency ratio is fixed (log10 N = 1.542), the vibrational bi-resonances phenomenon
also is observed at strong electromagnetic induction in Fig. 5(a). In addition, Fig. 5(c) shows the amplitude is fixed (B = 0.558) under weak electromagnetic induction, the plots of Q versus log10 N have two peak value and exist a vibrational bi-resonances phenomenon. However, when the HF stimulus frequency ratio is fixed
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Fig. 5. Dependencies of response measure Q on amplitude B (a)-(b) and different HF stimulus frequency ratio (c)-(d). (a) log10 N = 1.542; (b) log10 N = 0.986; (c) B = 0.558; (d) B = 0.314. Fig. 5(b) and (d) show a single vibrational resonance, Fig. 5(a) and (c) show vibrational bi-resonances.
Fig. 6. Contour plot of Q for the excitable FHN model on the (k1 , k2 ) plane at different amplitude B. log10 N = 1.26. (a) B = 0.1; (b) B = 1.3. Red and orange indicate higher values of Q.
(log10 N = 0.986), a single VR phenomenon can be observed in Fig. 5(b). Moreover, Fig. 5(d) also shows that the linear response Q varies with HF stimulus frequency ratio log10 N when the amplitude is fixed (B = 0.314), the plots of Q versus log10 N have one peak value and exist a single VR phenomenon. However, To get a global view about effects of electromagnetic induction at small and big HF amplitude B on information transmission, a contour plot of the linear response Q for the excitable
FHN model is plotted in Fig. 6. Fig. 6(a and b) show the effects of linear responses on the small and large HF amplitude B. When the HF amplitude is small, the changes of linear responses with the feedback gain k1 are analogous, that is, regardless of the size of parameter k2 , the linear response Q decreases with the increasing of k1 . It reveals that a small value of the feedback gain k1, where the output of the excitable system is at best correlated with the weak LF signal, and the value of k2 has a few effects on the information
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Fig. 7. Time series of membrane potentials of all neurons (a1, b1, c1, d1) and mean-field activities (a2, b2, c2, d2) of the neural network obtained for different HF driving amplitude B, k1 = 0.4, k2 = 0.9, log10 N = 1.26. (a1, a2) B = 0.01; (b1, b2) B = 0.05; (c1, c2) B = 0.1; (d1, d2) B = 0.5.
transmission of weak signal. However, when the HF amplitude is large, the changes of linear responses with the feedback gain k1 and k2 are small, and the linear responses are small. Therefore, it indicates the synchronization phenomenon is not well, and signal processing is degraded at big HF amplitude, whether the electromagnetic induction is weak or strong.
3.3. Vibrational bi-resonances and propagation in the FHN neural FFN system In this section, we systematically analyze the wave emission and propagation of weak LF signal in the FHN neural FFN system with HF stimulus under different electromagnetic induction, i.e. the VR phenomenon. In this FHN neural FFN system, only the central neuron (i = 101) is imposed by external forcing current as I101 = Acos(ωt)+Bcos(Nωt), while Ii = 0 is imposed on the others neurons. Figs. 7 and 8 plot the time series of membrane potentials of all neurons and mean-field activity (black curves) of the neural network obtained for different HF driving amplitude B and different HF stimulus frequency ratio log10 N. Fig. 7 shows the effects of HF amplitude B on the wave emitting and propagation of weak LF signal in FFN neural system. When the amplitude B is small (B = 0.01 in Fig. 7(a1 and a2)), all neurons are at quiescent state and cannot carry any information about the LF signal (red curve). When the amplitude B is increased, waves will propagate from the central neuron to the neurons at both ends and the mean-field activities of neural network are related to the rhythm of LF signal, which means the appearance of VR. When the amplitude B keeps further increasing, more and more waves are emitted from
the central neuron to both ends. Nevertheless, the coherence between collective activities and subthreshold signal will disappear. Fig. 8 shows the effects of HF stimulus frequency ratio log10 N on the wave emission and propagation of weak LF signal in the FFN network. When the HF stimulus frequency ratio log10 N is small (log10 N = 0.26 in Fig. 8(a1 and a2)), all neurons keep in a quiescent state and cannot carry any information about the LF signal. When the HF stimulus frequency ratio log10 N is moderate, waves can propagate from the central neuron to the neurons at both ends and the mean-field activities of neural network are related to the rhythm of LF signal, which means the appearance of VR. However, with the increasing of the frequency ratio log10 N, no waves are emitted from the central neuron to both ends, because the central neuron keep also in the quiescent state. Moreover, the coherence between collective activities and LF signal also disappears. Fig. 9 shows the dependencies of synchronization factor R and response measure Q on the amplitude B (Fig. 9(a, c)) and different HF stimulus frequency (Fig. 9(b, d)) under different electromagnetic induction in FFN neural network. Fig. 9(a) shows the dependencies of synchronization factor R on the frequency ratio log10 N. When the frequency ratio is small, the synchronization factor is big, and the neurons keep quiescent state. When the frequency ratio is increased, the neurons are in excitability oscillation and the wave can be spread. However, when continue to grow the frequency ratio, the wave cannot be spread and most neurons keep in quiescent state. Fig. 9(c) shows that the linear response Q varies with HF stimulus frequency ratio log10 N when the amplitude is fixed (B = 0.1), the plots of Q versus log10 N have two peak value as a whole and occur vibrational bi-resonances phenomenon. With a quite high frequency of the HF stimulus, the coherence be-
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Fig. 8. Time series of membrane potentials of all neurons (a1, b1, c1, d1) and mean-field activities (a2, b2, c2, d2) of the neuronal network obtained for different HF stimulus frequency ratio log10 N, k1 = 0.4, k2 = 0.9, B = 0.1. (a1, a2) log10 N = 0.26; (b1, b2) log10 N = 0.5; (c1, c2) log10 N = 1.26; (d1, d2) log10 N = 2.5.
Fig. 9. Dependencies of synchronization factor R (a)-(b) and response measure Q (c)-(d) for the FFN neural system on amplitude B (a)-(c) and HF stimulus frequency ratio log10 N (b)-(d). k1 = 0.4, k2 = 0.9, (a)-(c) B = 0.1; (b)-(d) log10 N = 1.26.
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Fig. 10. Dependencies of synchronization factor R (a) and response measure Q (b) for the chain network in different regions (k1, k2 ). log10 N = 1.26, B = 0.1. Red and orange indicate higher values of Q.
tween collective behavior and LF signal cannot be observed. However, Fig. 9(b) shows the dependencies of synchronization factor R on the amplitude B. When the amplitude is quite small, the synchronization factor is big, and all the neurons keep quiescent state. When the amplitude is increased, the neurons are in subthreshold oscillatory or damped oscillation and the wave can be spread. When the HF stimulus frequency ratio is fixed (log10 N = 1.26), the vibrational bi-resonances phenomenon is observed in Fig. 9(d). The two peak value B = 0.15 and 0.98 are the highest and the second peak, respectively. Similar to Fig. 9(c), the coherence between collective behavior and LF signal cannot be observed by increasing the amplitude of LF signal. These results indicate that the electromagnetic induction and the HF stimulus play an important role in the VR and information transmission of FFN system. To get global views about how the electromagnetic induction affects transmission of weak LF signal, the contour plots of the synchronization factor R and linear response Q for the FFN neural system are drawn in Fig. 10. The Fig. 10(a), when the parameters k1 and k2 are small, the synchronization factors are small, and the neurons keep in excitability oscillation and the wave can be spread. When the parameters k1 and k2 are large, the synchronization factor is big, and the wave cannot be spread. In Fig. 10(b), the changes of linear responses with the feedback gain k1 are alike, that is, regardless of the size of parameter k2 , the linear response Q decreases with the increasing of k1 . It is shown that the small values of feedback gain k1 at which the output of weak LF signal in the FFN neural system are at best correlated, and the value of k2 has a few effects on the transmission of weak LF signal. When the value of the feedback gain k1 is large, the linear responses are small, and the coherence between collective behavior cannot be observed. Hence, it means that the transmission of weak LF signal is degraded at big feedback gain k1 .
vibrational mono-/bi-resonances are observed for excitable single FHN neuron system. We showed that the input weak signal and output of system keep well synchronized and the information of the LF signal is amplified by increasing the amplitude of HF stimulus. When the electromagnetic induction is weak, the amplitude of HF stimulus is not big, the total outputs of the system are below threshold and carry few information about the LF signal. In the FFN neural FHN system, with increasing the amplitude of HF stimulus, the synchronization factor becomes smaller, and more and more waves are emitted from the central neuron to both ends. Nevertheless, the coherence between collective activities and subthreshold signal disappear. The electromagnetic induction and HF stimulus have important effects on the VR phenomenon and weak signal transmission. In our future studies, the VR and transmission of weak LF signal by HF stimulus might be taken account into improved multilayer neural systems [66,67]. Based on this dynamic mechanism, HF stimulus also could be replaced by various noises [68–74].
4. Conclusions
Acknowledgment
In this paper, an improved FHN neural model was employed to study the VR phenomena, collective behavior, and transmission of weak LF signal by HF stimulus under different electromagnetic induction in the single excitable FHN neuron and the FHN neural FFN system, respectively. In the single FHN neural model, the systemic parameter a plays a critical role in the dynamics of system without LF signal and HF stimulus under electromagnetic induction. The phenomena of
The work was supported by the National Natural Science Foundation of China under Nos. 11775091(Y.J.) and 11605014(Q.P.).
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Mengyan Ge: Visualization, Supervision, Formal analysis, Writing - review & editing. Lulu Lu: Supervision, Formal analysis. Ying Xu: Supervision, Formal analysis. Rozihajim Mamatimin: Supervision. Qiming Pei: Formal analysis. Ya Jia: Visualization, Formal analysis, Writing - review & editing.
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