Three-Dimensional Memristive Hindmarsh – Rose Neuron Model with Hidden Coexisting Asymmetric Behaviors

Since the electrical activities of neurons are closely related to complex electrophysiological environment in neuronal system, a novel three-dimensional memristive Hindmarsh–Rose (HR) neuron model is presented in this paper to describe complex dynamics of neuronal activities with electromagnetic induction. The proposed memristive HR neuron model has no equilibrium point but can show hidden dynamical behaviors of coexisting asymmetric attractors, which has not been reported in the previous references for the HR neuron model. Mathematical model based numerical simulations for hidden coexisting asymmetric attractors are performed by bifurcation analyses, phase portraits, attraction basins, and dynamical maps, which just demonstrate the occurrence of complex dynamical behaviors of electrical activities in neuron with electromagnetic induction. Additionally, circuit breadboard based experimental results well confirm the numerical simulations.

Inspired by the constructing approach of the threedimensional HR neuron model [3,24], a novel threedimensional memristive HR neuron model is presented in this paper, which could be used to better describe complex dynamical characteristics of neuronal electrical activities with electromagnetic induction or further exhibit some undiscovered complex dynamical behaviors in neuronal electrical activities.Interestingly, our proposed memristive HR neuron model has no equilibrium point, which can be classified as a particular dynamical system with hidden oscillating patterns [28][29][30][31].Furthermore, coexisting asymmetric attractors' behavior can also be observed in such memristive HR neuron model as well, indicating the emergence of bistability dynamics, which has been found in some specified neuron models [32][33][34][35].However, the phenomenon of hidden coexisting asymmetric attractors has not been previously reported for the HR neuron model.This paper is organized as follows.In Section 2, based on the brief reviews on the HR neuron model, a threedimensional memristive HR neuron model is presented, upon which hidden coexisting asymmetric attractors are numerically revealed by phase portraits and time series and its bistability dynamics are confirmed by the attraction basins related to the initial values.In Section 3, hidden coexisting asymmetric attractors' behaviors are demonstrated by bifurcation diagrams, Lyapunov exponents, and dynamical maps, from which numerous types of coexisting asymmetric

Three-Dimensional Memristive HR Neuron Model
2.1.Brief Reviews on the HR Neuron Model.Through simplifying the classical Hodgkin-Huxley model [1], the twodimensional Hindmarsh-Rose (HR) neuron model was proposed by Hindmarsh and Rose [2] in 1982, which is described by two first-order ordinary differential equations as in which two variables  and  are the membrane potential and recovery variable (also called spiking variable), respectively, and a term  is the externally applied current.The parameters , , , and  are four positive constants, which are often assumed as  = 1,  = 3,  = 1, and  = 5, respectively [2,[11][12][13][14].
To permit numerous dynamical behaviors, for example, chaotic dynamics, for the membrane potential, an extra third equation was introduced by Hindmarsh and Rose [3] in 1984 to improve the two-dimensional neuron model (1), which is expressed by three first-order ordinary differential equations as where the variable  is the bursting variable and the constant  1 is the resting potential of the model.The newly added parameters  and  are two positive constants but  is very small.Thus, a new variable , a slowly evolving current, is coupled into the first equation of the two-dimensional model (1) to tune the externally applied current .If the threedimensional neuron model ( 2) is in its firing state, the value of  increases [24].

Constructed Memristive HR Neuron
Model.Motivated by the above constructing approach of the model (2), a threedimensional memristive HR neuron model with electromagnetic induction effect is proposed through introducing a flux-controlled ideal memristor into the first equation of the model (1), which can be mathematically modeled by where the new variable  is the magnetic flux indicating the time integral of the membrane potential .The newly adding term  represents the externally applied electromagnetic induction and  is the strength of the electromagnetic induction.
It is important to stress that the introduced memristor in (3) is ideal and flux-controlled.According to the definitions of ideal memristor by state-dependent Ohm's laws between the terminal voltage V and terminal current  [36,37], a fluxcontrolled ideal memristor  is thereby given as where the memductance () can be interpreted as the flux-dependent rate of change of charge.Therefore, the memductance () utilized in (3) can be written by in which the coefficient  is positive.
To exhibit three characteristic fingerprints of pinched hysteresis loop of the memristor modeled by ( 4) and ( 5) [38], a sinusoidal voltage source V =  sin(2) is connected at the input terminals of the memristor, where  and  are the amplitude and frequency, respectively.Let  = 1.When  = 4 is maintained unchanged and  is assigned as 0.1, 0.2, and 0.5, respectively, the V −  plots are shown in Figure 1(a), while when  = 0.1 is fixed and  is determined as 3, 4, and 5, respectively, the V −  plots are shown in Figure 1(b).It is seen from Figure 1 that the V −  plots are the hysteresis loops pinched at the origin.The hysteresis loop is pinched regardless of the stimulus amplitudes but shrinks into a linear function at infinite frequency and its lobe area decreases with increasing the frequency.The numerical results in Figure 1 indicate that the memristor modeled by (4) can behave three fingerprints for distinguishing memristors [38].
In the next work, the three-dimensional memristive HR neuron model given in (3) is considered.It should be remarked that the adjustable parameters of interest are  and , and their regions are correspondent to the first quadrant of the parameter space ( > 0 and  > 0).For any uncertain parameter , the existence of any equilibrium point is not allowed in the three-dimensional memristive HR neuron model, neither stable nor unstable.Only if the applied current  = −1 will the model show an equilibrium point, which is not in the considered parameter region.This case is often encountered in various kinds of nonlinear dynamical systems that are known to generate the specified hidden attractors [28][29][30][31].

Coexisting Asymmetric Attractors.
When the original parameters are selected as  = 1,  = 3,  = 1, and  = 5, respectively [2], an example for model (3) with  = 1 and  = 0.9 is given as shown in Figure 2, where the orbits marked by the red and blue colors emerge from the initial values (0, 0, −2) and (0, 0, 2), respectively.In Figure 2(a), the phase portraits in the  −  plane display the bistability phenomenon of hidden coexisting asymmetric x Particularly, the emerging coexisting asymmetric attractors do not associate with any equilibrium point, indicating that the memristive HR neuron model always operates in hidden oscillating patterns [28][29][30][31].Additionally, it is interesting to note that, just like the self-excited coexisting asymmetric attractors in hyperbolic-type memristor based Hopfield neural network [32], such hidden coexisting asymmetric attractors in the memristive HR neuron model are induced by electromagnetic induction also, which illustrates the occurrence of complex dynamical behaviors of electrical activities in neuron with electromagnetic induction.

Hidden Coexisting Asymmetric Attractors' Behavior
When the applied current  and electromagnetic induction strength  are considered as two bifurcation parameters, hidden coexisting asymmetric behaviors of the memristive HR neuron model are numerically studied by MATLAB ODE45 algorithm under two sets of the initial values (0, 0, −2) and (0, 0, 2). the initial values (0, 0, −2) and (0, 0, 2), respectively, and in Figure 4(b) the Lyapunov exponents marked by the red and fuchsia colors correspond to the initial values (0, 0, −2) and those marked by blue and green colors correspond to (0, 0, 2).It can be seen from Figure 4 that complex dynamics appear in the memristive HR neuron model, in which chaotic attractors with different topologies, limit cycles with different periodicities, period-doubling bifurcation routes, tangent bifurcation routes, crisis scenarios, coexisting bifurcation modes, and so on can be found.Therefore, the electromagnetic induction by the introduced memristor induces numerous complex dynamics for the membrane potential , especially including hidden coexisting asymmetric behaviors.

Bifurcation Behaviors with Increasing 𝑘.
The concernedly coexisting behaviors of asymmetric attractors mainly locate in two parameter regions [0.718, 0.736] and [0.754, 0.909], in which some different types of hidden coexisting asymmetric attractors occur.When  = 1 and  = 0.735 and 0.81, respectively, the phase portraits in the - plane for the other two types of hidden coexisting asymmetric attractors are depicted in Figure 5.In detail, Figure 5(a) exhibits the coexistence of hidden chaotic attractor and hidden limit cycle, and Figure 5(b) displays the coexistence of two hidden limit cycles with different periodicities.

Bifurcation
Behaviors with Increasing . Figure 6 demonstrates the bifurcation diagrams of  and the first two Lyapunov exponents as  = 0∼2.4and  = 0.9, where the initial values for different colored orbits in Figure 6(a) and different colored Lyapunov exponents in Figure 6(b) are consistent with those used in Figures 4(a) and 4(b).In the same way, it can be observed from Figure 6 that complex dynamics are coined in the memristive HR neuron model, reflecting the dynamical effect of the externally applied current  in neurons.
The parameter region [0.98, 1.65] has the benefit for the coexisting behaviors of asymmetric attractors, in which several different types of hidden coexisting asymmetric attractors can be clearly found.When  = 0.9 is fixed and  is set to 1.15 and to 1.62, respectively, the phase portraits in the - plane for the two types of hidden coexisting asymmetric attractors are plotted in Figure 7, where in Figure 7(a) the coexistence of hidden chaotic attractor and hidden period 1 limit cycle is displayed, and in Figure 7(b) the coexistence of hidden period 2 limit cycle and hidden chaotic attractor with large size is presented.

Coexisting Asymmetric Behaviors in the Parameter Space.
For intuitively manifesting the coexisting behaviors of asymmetric attractors in the memristive HR neuron model, dynamical maps depicted by the largest Lyapunov exponent under two sets of the initial values are numerically plotted in the - parameter space [40], as shown in Figures 8(a) and 8(b), where the luminous yellow, red, and black colored regions stand for the chaotic, periodic, and divergent behaviors, respectively.Figure 8 indicates how dynamical evolution in the electromagnetic induction strength  and applied current  affects the coexisting behaviors under different initial values being considered.When the two parameters  and  are evolved, some chaotic regions are embedded in the periodic regions; however, different chaotic regions appear on the parameter spaces of Figures 8(a) and 8(b), which are caused by the coexisting asymmetric attractors' behaviors under different initial values.The numerical results in Figure 8 illustrate that the dynamical behaviors depicted by the largest Lyapunov exponent based dynamical maps are well agreed with those revealed by the bifurcation behaviors in Figures 4 and 6.
It should be remarkable for the dynamical maps in Figure 8 that the chaotic attractors in different locations of the chaotic region have different topologies and the limit cycles in different locations of the periodic region have different periodicities.Specifically, except for several types of coexisting asymmetric behaviors shown in Figures 2, 5, and  7, another type of coexisting asymmetric behaviors of chaotic attractor and divergent orbit can also be uncovered, which means that another form of bistability exists in the memristive HR neuron model.4) and ( 5) and its constructing three-dimensional memristive HR neuron model expressed by ( 3) can be physically realized by using an electronic circuit via analog multipliers and operational amplifiers connected with resistors and/or capacitors [41][42][43], as drawn in Figures 9(a course, this three-dimensional memristive HR neuron model also can be digitally implemented in field-programmable gate arrays (FPGA) as well [44,45].The implementation circuit of the flux-controlled ideal memristor in Figure 9(a) contains an integrator with time constant , an inverter, a multiplier  0 , and a resistor   .For the input voltage V and output current , the mathematical model for the memristor emulator can be easily given as

Circuit Design and Breadboard Experiments
where V  is the inner variable of the memristor emulator and  0 is the gain of the multipliers  0 ,  =  0 /  , and (V  ) = V  /.The main circuit of the memristive HR neuron model in Figure 9(b) has two integrating channels for implementing the first and second equations of (3).According to Kirchhoff 's circuit laws and electrical properties of the circuit components, the circuit equations of Figure 9(b) are written as where V  and V  are two circuit variables,   and  0 are two applied voltages, and  1 ,  2 , and  3 are the gains of the multipliers  1 ,  2 , and  3 , respectively.Considering that the dynamic amplitude of the recovery variable  in the numerical simulations exceeds the linear operation ranges of operational amplifier and multiplier, the following linear transformation Let the time constant  = 10 kΩ × 33 nF = 330 s; that is,  = 10 kΩ and  = 33 nF and the multiplier gains  0 =  1 = 0.1 and  2 =  3 = 1.According to (9), the circuit parameters for the breadboard experiments of the memristive HR neuron model can be calculated, as listed in Table 1.

Results
Captured from Breadboard Experiments.According to the circuit diagrams in Figure 9 and circuit parameters in Table 1, a hardware circuit using commercially discrete components can be welded on a breadboard.The operational amplifiers AD711JN and analog multipliers AD633JN supplied by ±15 V voltage modules are chosen.The DC voltages   and  0 are provided by Tektronix PWS 2326 DC Power Supply and the experimental results are measured by Tektronix TDS 3054C Digital Phosphor Oscilloscope.The photograph of the connectedly experimental prototype for the memristive HR neuron model is displayed in Figure 10.
For experimentally measuring the pinched hysteresis loops of the memristor emulator given in Figure 9(a),  The circuit parameters listed in Table 1 are used and the different initial voltages of three capacitors are randomly sensed by repeatedly switching on and off the experimental power supply [46].For the typical circuit parameters in Table 1, two adjustable circuit parameters of  1 and   correspond to the adjustable model parameters of  = 1 and  = 0.9.Corresponding to Figure 2, the phase portraits in the V  -V  plane and time series of the variable V  that emerged from different initial voltages are experimentally obtained, as shown in Figure 12.The experimental results indicate that hidden coexisting asymmetric attractors also can be measured from the breadboard experiments of the memristive HR neuron model as well.
When the applied current  = 1, that is, the resistance  1 is fixed as 10 kΩ and the resistance   is set to 1.36 kΩ and to 1.23 kΩ, respectively, the phase portraits in the V  -V  plane are captured, as shown in Figures 13(a) and 13(b).Furthermore, when the electromagnetic induction strength  = 0.9, that is,   = 1.11 kΩ and  1 is set to 8.70 kΩ and to 6.17 kΩ, respectively, the phase portraits in the V  -V  plane are captured, as shown in Figures 13(c) and 13(d).Ignoring some tiny differences between numerical simulations and breadboard experiments due to the computational errors and parasitic circuit parameters, the experimental results are almost the same as the numerical simulations, which imply that the coexisting asymmetric attractors' behaviors that emerged from the memristive HR neuron model can be validated experimentally.

Conclusions
This paper presents a novel three-dimensional memristive HR neuron model to describe complex dynamics of neuronal activities with electromagnetic induction.The most prominent feature of this neuron model is that it does not contain any equilibrium point but can exhibit hidden coexisting behaviors of asymmetric attractors.Through executing bifurcation analyses, phase portraits, attraction basins, and dynamical maps, hidden coexisting asymmetric attractors are uncovered from the mathematical model and verified from the corresponding breadboard experiments.Thus, the proposed memristive HR neuron model can imitate the complex dynamical behaviors of electrical activities in neuron with electromagnetic induction.Further investigations will be performed in our future works.

4. 1 .
Physical Circuit Designs and Parameter Selections.The flux-controlled ideal memristor  characterized by (

Figure 9 :
Figure 9: Physical electronic circuit implementation for the memristive HR neuron model.(a) Implementation circuit of the flux-controlled ideal memristor.(b) Main circuit of the memristive HR neuron model.

Figure 10 :Figure 11 :Figure 12 :
Figure 10: Photograph of the experimental breadboard and typical chaotic attractor captured by the digital oscilloscope.

Table 1 :
Circuit parameters of the memristive HR neuron model for breadboard experiments.