Electrical activity and synchronization of HR-tabu neuron network coupled by Chua Corsage Memristor

The processing and transmission of biological neural information are jointly completed by the electromagnetic activities of neurons in different brain regions. And memristor is the appropriate candidate for mimicking synapse due to its unique memory function and synapse-like plasticity. Therefore, it is of great significance to explore the electrical behavior of heterogeneous neuron network coupled by memristor. This paper focuses on the electrical activity and synchronization of a bi-neuron network (HR-tabu neuron network) built by coupling Hindmarsh–Rose and tabu learning models with Chua Corsage Memristor (CCM). The electrical activities of HR-tabu network, such as spiking discharge and bursting discharge, are revealed under appropriate external stimuli and coupling strength. Interestingly, we find that the initial value-related state switching of HR-tabu network is associated with the equilibrium states of CCM. In addition, the synchronization behavior of HR-tabu network depending on the coupling strength, external stimuli and system parameters is investigated in detail by analyzing the phase difference and synchronization factor. It is shown that phase synchronization of HR-tabu neuron network can be achieved under small coupling strength, and that complete synchronization can be achieved when the coupling strength is large enough. The obtained results provide possible guidance for regulating the firing activity and synchronous behavior of artificial neurons, and therefore have potential applications in brain science and biomimetics.


Introduction
Heterogeneous neurons are the neurons in different brain regions that display different functions and structures.The signal processing and transmission of the nervous system are completed through the firing activities of neurons, and the function discrimination of neural signal is jointly achieved by different brain neurons [1][2][3].Therefore, the electrical activity and synchronization of heterogeneous neurons play an important role in the study of neural signal processing [4][5][6] and also help to explore the neural function of the brain [7][8][9][10][11].
In order to facilitate the research of the electrical activity and action potential mechanism of the nervous system, various neuron models, such as Hodgkin-Huxley (HH) [12], Rulkov [13], Morris-Lecar (ML) [14], FitzHugh-Nagumo (FN) [15], Hindmarsh-Rose (HR) [16], Hopfield neuron (HN) [17] and tabu learning neuron [18], were constructed.Among these models, HR neuron is increasingly employed to qualitatively describe the complex dynamics of neural electrical activities.For example, the discharge mode and the Hamiltonian energy of the memristive HR model were studied in [19], it is revealed that the HR network exhibits faster spiking behavior when the value of the external current increases, and that faster spiking behavior needs more energy.In [20], the discharge modes of chaos, bursting, spiking and cluster state in two coupled HR neurons were investigated, and various spatiotemporal patterns in ringstar HR networks were explored.In [21], the phenomena of coexisting hyperchaotic and periodic bursting of 3D HR neuron injected by external alternating current were reported.Tabu learning neuron is evolved from Hopfield neural network for better optimization efficiency.And accordingly, the discharge dynamics of tabu neuron is highly regarded.For example, the authors in [22] investigated the dynamical behaviors of a tabu two-neuron network with linear proximity function based on Hopf bifurcation theory.Also, the dynamical behavior of a tabu leaning neuron model with two delays is explored by choosing the sum of the two delays as the bifurcation parameter [23].The authors in [24] studied the bifurcation behavior of a tabu neuron model and found that the model will produce rich neural discharge behaviors under the stimulation of external forced current.Recently, three kinds of tabu neuron models were, respectively, built under forced current stimulus, electromagnetic radiation stimulus, and both the stimuli of forced current and electromagnetic radiation.The firing activities of these neurons were then studied by analyzing bifurcation diagram, time sequence, attraction basin and Hamilton energy, showing that the behavior of absorbing or dissipating energy in the discharge process depends on the imposed stimulation [25].
Memristor has remarkable bionic properties such as plasticity, memorability, nonlinearity and nanometer size, and can therefore be taken to mimic biological synapse for reproducing the electrical activity in neurons [26][27][28][29][30][31].Ding investigated the discharge dynamics of fractional-order memristor-coupled HR neurons considering synaptic cross talk [32].Fida presented a rate-coded memristive spiking neural network, which was built with an active memristor neuron based on vanadium dioxide coupled with a nonvolatile memristor synapse [33].Sun constructed a HR-FN-HR neural network coupled by a bistable locally active memristor, and the complex dynamical behavior was discovered by estimating the energy released during the transition between various electrical activities [34].Bao presented a memristive neuron model by considering the flux-controlled memristor to imitate the electromagnetic induction effect of adapting synapses [35].Li constructed a HR-FN neural network by considering a bistable active memristor as the synapse and investigated its coexisting firing patterns [36].Based on the 2D HR neuron model, a memristor-coupled 3D HR neuron model was introduced and it is displayed with hidden chaotic and periodic bursting firing patterns [21].The authors in [37] introduced a novel FitzHugh-Nagumo neuron model with locally memristive synapse to reproduce biophysical firing patterns from chemical synapse.However, the effect of the multistability of the memristor on the discharge behavior of the nervous system is rarely considered in the existed works.In fact, the multistability feature of the memristor can realize the multistability of nonlinear dynamic system, which provides an important pathway for achieving multi-mode discharge of neural system.
As an important device for simulating biological synapse, memristor simultaneously plays an energetic role in neuron synchronization [38][39][40][41].By coupling two identical single-neuron models with a memristor, a memristor-coupled network is constructed.It is discovered that complete synchronization can be achieved at large coupling strength, and parallel-offset synchronization appears when the memristor initial conditions of two neurons are mismatched [42].Bao introduces a bi-neuron network by bidirectionally coupling two Morris-Lecar neurons with a memristor synapse.It is investigated that the synchronous firing activities are associated with the induction coefficient and the initial values of memristor synapse and coupling neurons [43].Zhou studied the collective dynamics of HR network under synapse coupling, and standard deviation and synchronization factor were referred to explore the synchronization behavior.It is found that the memristor synapse can induce synchronization more effectively for lower magnetic coupling strength and cell size [44].Shang constructed a ring Chialvo neural network coupled by memristor, and the synchronization and chimera state in the network are analyzed, showing that memristor plays the role of synapse well and successfully realizes the neuron synchronization [45].Wu designed an equivalent circuit using two memristive Josephson junctions coupled by a resistor and two inductors.By identifying the peak intervals of the phase difference and the average Hamiltonian energy, it is found that in-phase and inverse-phase synchronization can be achieved by the interaction of resistive and magnetic coupling [46].However, the presented works mainly focused on studying the synchronization behavior of memristorcoupled homogeneous neurons [47].In fact, the biological neural functions are jointly realized by the electrical activity of neurons in different brain regions.Thus, it is necessary to explore the dynamics and synchronization behavior in the heterogeneous memristive neural network [48].
In this paper, we build a bi-neuron network by coupling HR neuron and tabu leaning neuron with 6-lobe CCM.It is revealed that under appropriate external stimuli and coupling strength, the HR-tabu network exhibits rich electrical activities such as spiking discharge and bursting discharge.And the multistability of CCM can induce the multimodal discharge of HR-tabu network, which means that the initial value-related state switching of HR-tabu network is associated with the equilibrium states of CCM.Furthermore, the synchronous activity of HR-tabu network is investigated in detail by analyzing the phase difference and synchronization factor.The results show that the phase synchronization of HRtabu network can be obtained under small coupling strength, and that the complete synchronization can be obtained when the coupling strength is large enough.
The structure of this paper is listed as follows.In Sect.2, CCM is used to couple HR neuron and tabu leaning neuron to form a heterogeneous neural network, and the stability of the equilibrium point is analyzed.In Sect.3, the electrical activity of HR-tabu neuron network is explored by bifurcation diagrams and time series.In Sect.4, the switching of discharge behavior induced by the initial value of the memristor is studied.Then, the synchronization activity of the heterogeneous neural network is analyzed in Sect. 5. Finally, the conclusion is drawn in Sect.6.

Description of HR-tabu network
The 2D HR neuron is modeled by the following mathematical expression [49] where x is the membrane potential, y is the recovery variable of the neuron, and a, b, c and d are positive parameters.
The tabu learning neuron is developed from Hopfield neural network, which can be described by [25] where x and y denote the membrane potential and tabu learning state variable of the neuron, respectively.a, b, c and d are four positive parameters; f(x) is the activation function, which disrupts the balance of membrane potential difference and stimulates the action potential to achieve neuronal firing.The activation function required in the tabu learning neuron should be bounded and differentiable.Therefore, the following activation function is considered [25] f ðxÞ¼2tanhðwxÞÀtanhðwxþ1:5wÞÀtanhðwxÀ1:5wÞ CCM means that the DC V-I Plot of the memristor holds the shape of shoelace.The multifunctional CCM family has three general-purpose memristors, namely, 2-lobe CCM, 4-lobe CCM and 6-lobe CCM, described by [50] i where U, v and i denote the internal state, input voltage and output current, respectively.The intrinsic memductance is fitted with the scaling constant G 0 so that the current of CCM is not excessive.For the sake of simplicity, G 0 is chosen to be 1.The state function f m (U) (m = {2,4,6}) of the three types of generalpurpose CCMs is different from each other.This paper only considers the 6-lobe CCM, the corresponding state function is [50] The DC V-I plot is a powerful visualization tool to determine whether a memristor is locally active, which occurs when the plot has a negative slope.The DC V-I relation of the 6-lobe CCM is expressed by Eq. ( 6) when _ U = 0 in Eq. ( 4).
According to Eq. ( 6), the DC V-I plot of CCM can be plotted in Fig. 1.Each point on the DC V-I curve corresponds to an equilibrium state U, which may be stable (solid line) or unstable (dotted line).When the voltage is -7 V B V B 7 V, the DC V-I curve exhibits multiple values, which is hence known as multivalued DC V-I curve.The CCM is locally active when the voltage belongs to the interval of (-3 V, -1 V) since the slope of the CCM is always negative at any point of this interval.
In order to explore the electrical activity and synchronization behavior of heterogeneous neural network, the 6-lobe CCM is employed to couple two different neurons (HR neurons and tabu neurons) to form a HR-tabu heterogeneous neural network, as shown in Fig. 2. When there emerges a difference of membrane potential between the two neurons, the electromagnetic induced current will be generated by the memristor synapse.The coupled network is described by where k is the coupling strength of the memristor, and the parameters are determined as

Stability analysis of equilibrium point
Letting the left side of Eq. ( 7) be zero, the equilibrium points can be obtained by and the equilibrium points is then expressed as And the values of x 1 and U can be obtained by solving the following two equations When k = 0.1 and 1, the two function curves in Eqs. ( 10) and ( 11) can be numerically drawn in Fig. 3a and b.As shown in Fig. 3, there are always 11 intersections under different conditions, which corresponds to the 11 equilibrium points of this neural network.
According to Eq. ( 7), the Jacobian matrix is yielded as For the two given parameters k = 0.1 and 1, the values of (U, x 1) for the 11 equilibrium states and their corresponding eigenvalues and stabilities are tabulated in Tables 1 and 2, respectively.The tables reveal that all equilibrium points are unstable saddle focus.The bifurcation diagram in Fig. 4 illustrates that different discharge patterns can be induced by adjusting the coupling strength k.The sampled time series are plotted in Fig. 5 for better observation.It is displayed from Fig. 5 that neuron 1 exhibits periodic single spike discharge when k = 0, periodic double spike discharge when k = 0.0025, periodic spike discharge with alternating double spike and single spike when k = 0.0064, periodic spike discharge with double spike plus single spike when k = 0.11, periodic spike discharge with triple spike plus single spike when k = 0.124.It is found from the variation of the spike number and discharge period that the type of spike discharge is influenced by the coupling strength k.

Firing pattern affected by DC stimulus
We know that external stimuli can modulate the excitability of neurons, and the coupling strength affects the signal exchange between neurons.Therefore, in order to simulate the realistic cellular environment, we applied different external DC stimuli I 1 = I 2 = I to explore the firing pattern of heterogeneous neural networks.
When  6.In this case, neuron 2 remained in a periodic discharge state, so we will focus on neuron 1 to analyze its firing pattern.As shown in Fig. 7, we know that neuron 1 exhibits periodic single spike discharge when I = 0.03, periodic spike discharge with double spike plus single spike when I = 0.23, periodic double spike discharge when I = 0.43, periodic triple spike discharge when I = 0.76.As we find that the DC stimuli I mainly influences the frequency of spike discharge.

Firing pattern affected by periodic current stimulus
Bursting discharge is a complex dynamic behavior characterized by the alternate large-amplitude and small-amplitude discharges [51].In this section, the sinusoidal excitation I = Asin(xt) is taken as the bifurcation parameter for investigating the bursting discharge of HR-tabu network.The analysis shows that the amplitude A has little effect on bursting discharge, thus we fix parameter A = 1 and coupling strength k = 0.01.To simulate the realistic environment of neurons, sinusoidal excitation is, respectively, applied to the single-neuron model and the coupled neuron model.When the sinusoidal excitation I is only exerted on neuron 1, the bifurcation diagram by varying the frequency x is obtain in Fig. 8.In this case, neuron 1 generates a bursting discharge, while neuron 2 remains in a periodic discharge, as shown in Fig. 9a-d.And we know that the discharge period and the number of spikes gradually decrease as x increases.When x [ 0.13, the HR-tabu network initiates a protective suppression mechanism so that the triggered discharge period does not change periodically with the increase of x, as shown in Fig. 9e for x = 0.33.
When the sinusoidal excitation I is exerted on both neurons of HR-tabu network, the corresponding bifurcation diagram versus x is shown in Fig. 10.The firing states of neuron 1 and neuron 2 are similar to Fig. 9a-c, and the discharge period and spike number of each burst decrease when increasing x, as shown in Fig. 11a-c.When x = 0.32, HR-tabu network initiates a protective suppression mechanism, which makes the bursting discharge change irregularly, and mostly in the periodic discharge state, as shown in Fig. 11d.Until x = 0.52, the network reaches phase  synchronization after a brief period of alternating spike and period discharge, as depicted in Fig. 11e.However, when x = 0.8, the system enters a periodic discharge after a brief spike discharge, as shown in Fig. 11f.

State switching of HR-tabu network
The state switching of dynamical system refers to the fact that a tiny perturbation at the critical point may lead to a qualitative change of the dynamical behavior [52].The power-off plot (POP) of the 6-lobe CCM used in HR-tabu network is plotted in Fig. 12.It is observed that there exist four intersections Q 1 (3, 0), Q 3 (15, 0), Q 5 (35,0) and Q 7 (63, 0) with negative slope, which indicates that the memristor is nonvolatile and stable in the neighborhood of the four equilibrium points.
When the system parameters and initial values are, respectively, assigned as 83, k = 0.001 and x(0) = (0.01 0.05 0.02 0.1 U(0)), the localized basin of attraction representing the dynamical state of neuron 1 is obtained in Fig. 13.It can be found by comparison that the boundaries of different state switching emerge in the vicinity of the unstable equilibrium point, and even some of them correspond to each other.The red, light blue, blue, magenta and yellow regions in Fig. 13 represent the period-1 discharge, resting discharge, period-3 discharge, chaos discharge and period-1 discharge, respectively.
The corresponding time waveforms and phase diagrams for different initial conditions are shown in Fig. 14.It can be observed that in the light blue region, neuron 1 and neuron 2 exhibit period-1 discharge, as shown in Fig. 14a.But in the red region, it can be found that neuron 1 exhibits resting discharge, while neuron 2 exhibits period-1 discharge, as shown in Fig. 14b.In the blue region, neuron 1 exhibits period-3 discharge and neuron 2 exhibits period-6 discharge, as shown in Fig. 14c.In the magenta region, neuron 1  and neuron 2 exhibit chaos discharge, as shown in Fig. 14d.And in the yellow region, neuron 1 and neuron 2 exhibit phase-locked period-1 discharge, as shown in Fig. 14e.Therefore, the discharge behavior of the memristor-coupled heterogeneous neural network can be regulated by adjusting the initial states of the memristors in different regions, which is easier to achieve state switching than those multiple discharges that depend on some specific initial values.

Synchronization of HR-tabu network
It's evidenced that the synchronization of neurons in different brain regions is essential for neural signal processing.Consequently, obtaining the synchronization of the membrane potential of the coupled HR-tabu network is of great importance.
Phase synchronization can be determined by observing whether the phase difference between the two neurons is close to the phase-locked state, and the instantaneous phase of the membrane potential can be calculated by sampling time series and the corresponding Hilbert transform [16].The Hilbert transform x1 ðtÞ and x2 ðtÞ of the sampling series x 1 (t) and x 2 (t) are determined by  x 1 ðsÞ t À s ds where PV is the integral in the Cauchy sense.Hence, the phase difference D/ between membrane potentials x 1 and x 2 can be given by When the absolute value of D/ is less than 2p, the two coupled neurons will achieve phase synchronization [36].
The synchronization factor R characterizes the correlation of time sequences from the perspective of mean-field theory, which is defined as [16] where N denotes the number of sequence, for the bineuron model, N = 2, x i h i ¼ 1=Dt P t f t¼t i x i , t i and t f represent the start and end times of the calculation, Dt = t ft i .Obviously, the larger the value of R, the higher of sequences correlation.Especially, the  The time evolution, phase difference and sequence error h(t) = x 1 (t)x 2 (t) of the membrane potentials are obtained in Fig. 16 for different k.It is found that with the increase of k, the state of the membrane potentials x 1 and x 2 evolves from phase synchronization to complete synchronization.

Synchronization affected by DC stimulus
The effect of DC excitation I (I 1 = I 2 = I) on the synchronization behavior of HR-tabu neuron network is displayed in Fig. 17  more effectively, as depicted in Fig. 17b and d.But the opposite is true for parameter c 2 , and the dependence of the synchronization on the parameter value c 2 will be greatly reduced if the external stimulus I is too large, as depicted in Fig. 17c.
What is more, it is observed that the effect of parameter variation on phase difference D/ is slightly different from the effect on synchronization factor R, and the dependence of the phase difference on the parameters is relatively high.For parameter a 1 , the phase difference D/ tends to 0 as I increases But the change trend is completely different for a 1 = 0.7 and I \ 0, and it decreases substantially with a sudden peak near I = -0.2.When I [ 0, the phase difference gradually increases again with the increase of the external current I.When b 1 = 1, the phase difference decreases gradually when I is greater than 0.4.And when b 1 = 2, the phase difference decreases gradually when I is greater than 0.8.As for the parameters c 2 and d 1 , the phase difference gradually increases overall with the external current I.The stimulus frequency x is first employed to explore the influence of periodic current on the synchronization behavior, as shown in Fig. 18.From Fig. 18a, it can be seen that when the coupling strength is smaller, the dependence of the synchronization behavior on the stimulus frequency x is strong and the synchronization rate is relatively low.However, if the coupling strength k is large enough (k = 10), the stimulus frequency x has small influence on synchronization and the synchronization factor R is stable at 1.When k is fixed at 0.1, the synchronization factor R decreases slightly with the increase of x, as depicted in Fig. 18b.And the larger amplitude A will bring out the higher synchronization rate.
The influence of different system parameters on synchronization by increasing the stimulation frequency x is then explored.It can be observed from Fig. 19a that with the increase of stimulation frequency x, the synchronization factor R fluctuates within [0.9, 1] when a 1 = 0.

Conclusion
The function of biological neural information is jointly completed by the electromagnetic activities of neurons in different brain regions that hold different biophysical functions.In addition, as the fourth basic component, the nanometer-sized memristor has the important features of memory and synapse-like plasticity, which make it the most suitable electronic device for constructing artificial neural synapses.Therefore, it is of great significance to explore the electrical behavior of heterogeneous neuron network coupled by memristor.This paper presented a heterogeneous neural network by coupling HR neuron and tabu neuron with Chua Corsage Memristor synapses.It is found that the initial value-related state switching of HR-tabu network is associated with the equilibrium states of CCM rather than the distributions of equilibrium points in the system.Furthermore, the analysis of time-evolution waveform shows that the heterogeneous neural network can exhibit complex electrical activities such as spike discharge and bursting discharge under appropriate external stimuli and coupling strength.Subsequently, the synchronization behavior of the heterogeneous neural network was investigated by analyzing the phase difference, synchronization factor and sequence error of the neurons in HR-tabu network, and the dependence of the synchronization behavior on the coupling strength, external stimuli and system parameters was also explored.These results revealed the rich dynamical behavior and synchronization regulation of HR-tabu network, which provides an important reference for investigating the electrical activity of heterogeneous neural network with a more complex topology.

Fig. 1
Fig. 1 DC V -I plot of the 6-lobe CCM

Fig. 2
Fig.2The connection topology of non-homologous neurons model

3
Firing pattern of HR-tabu network under different system parameters This section will analyze the influence of coupling strength, DC stimulus and periodic current stimulus on the firing behavior of HR-tabu network by employing phase diagram, bifurcation diagram and time waveform diagram.

Fig. 5
Fig. 5 Time waveforms of neuron 1 under different coupling strengths

Fig. 10
Fig. 10 Bifurcation diagram versus x when sinusoidal excitation I exerts on both neurons

Fig. 11 1 . 5 . 1
Fig. 11 Time waveforms when sinusoidal excitation I exerts on both neurons , by setting the system parameters as a 1 = 1, b 1 = 3, c 1 = 1, d 1 = 5, a 2 = 0.2, b 2 = 0.3, c 2 = 0.5, d 2 = 1, w = 5.2, b = 4 and k = 0.1.It is found from Fig. 17 that the effect of different system parameters on the synchronization behavior of HR-tabu network is different as the external current I increases.For parameter a 1 , both too large and too small values can promote the synchronization behavior more effectively, as depicted in Fig. 17a.While for parameters b 1 and d 1 , smaller parameter values can enhance the synchronization ability of the network
4, and the synchronization factor is consistently high.While the synchronization factor induced by other values of a 1 is prone to be influenced by the stimulation frequency x.It is noted for parameter b 1 that the synchronization factor fluctuates in the range of [0.7, 0.8] except b 1 = 1, and smaller parameter b 1 can enhance the synchronization ability of the network more effectively.

Fig. 15
Fig. 15 a 1 -d 1 Dependence of the phase difference on coupling strength k; a 2 -d 2 Dependence of synchronization factor on coupling strength k

Fig. 16
Fig. 16 Time waveforms, phase difference and sequence error for a k = 1; b k = 50

Fig. 17
Fig. 17a 1 -d 1 Dependence of the phase difference on DC external excitation; a 2 -d 2 Dependence of synchronization factor on DC external excitation

a 1 -d 1
Fig. 17a 1 -d 1 Dependence of the phase difference on DC external excitation; a 2 -d 2 Dependence of synchronization factor on DC external excitation

Fig. 18
Fig. 18 Dependence of synchronization on excitation frequency x

Table 1
Stability of equilibrium points for k = 0.1

Table 2
Stability of equilibrium points for k = 1