THE MEMRISTOR AS AN ELECTRIC SYNAPSE - IEEE Xplore

THE MEMRISTOR AS AN ELECTRIC SYNAPSE – SYNCHRONIZATION PHENOMENA Ch. K. Volos1, I. M. Kyprianidis2, and I. N. Stouboulos2 1

Department of Mathematics and Engineering Sciences, University of Military Education, Greece 2 Department of Physics, Aristotle University of Thessaloniki, Greece ABSTRACT

Today many scientists see nonlinear science as the most important frontier for the fundamental understanding of Nature. Especially, the recent implementation of the memristor has led to the interpretation of phenomena not only in electronic devices but also in biological systems. Many research teams work on projects, which use memristor to simulate the behavior of biological synapses. Based on this research approach, we have studied using computer simulations, the dynamic behavior of two coupled, via a memristor, identical nonlinear circuits, which play the role of “neurons”. The proposed memristor is a flux-controlled memristor where the relation between charge and magnetic flux is a smooth continuous cubic function. Very interesting synchronization phenomena such as inverse π-lag synchronization and complete chaotic synchronization were observed. Index Terms— Memristor, biological synapses, double scroll circuit, complete chaotic synchronization, inverse π-lag synchronization 1. INTRODUCTION In 1971, Leon Chua from the University of California at Berkley, reasoned that there should be a fourth fundamental electronic element, except from the three well known, resistor, capacitor and inductor. These circuit elements reflect relationships between pairs of the four electromagnetic quantities of charge (q), current (i), voltage (v) and magnetic flux (φ). But a link between charge and flux was missing. Chua dubbed this missing link by introducing the memristor (short for memory resistor) [1] and created a crude example to demonstrate its key property that it becomes more or less resistive (less or more conductive) depending on the amount of charge that has flowed through it. Such an electronic device would not be reported until 2008, when a physical model of a two-terminal device behaving as a memristor was announced in Nature [2]. Scientists at the Laboratories of Hewlett-Packard, reported the realization of a new nanometer-scale electric switch, which “remembers” whether it is “on” or “off” after its

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power is turned off. This discovery has attracted a great attention. The features of memristor explain several phenomena in nanoscale systems, such as in thermistor [3] and spintronic devices [4]. Memristors can also be used to design nonlinear oscillators by letting them be the nonlinear device in Chua’s circuit family [5-8]. Also, electronic circuits with memory circuit elements (memristors, memcapcitors and meminductors) can simulate processes typical of biological systems such as the adaptive behavior of unicellular organisms [9], learning and associative memory [10]. Nowadays, neuromorphic computing circuits are designed by borrowing principles of operation typical of the human (or animal) brain and, therefore, due to their intrinsic analog capabilities they can potentially solve problems that are cumbersome (or outright intractable) by digital computation. Therefore, certain realizations of memristors can be very useful in such circuits because of their intrinsic properties which mimic to some extent the behavior of biological synapses. Just like a synapse, which is essentially a programmable wire used to connect groups of neurons together, the memristor changes its resistance in varying levels. Many research teams [11-13] found that memristors can simulate synapses because electrical synaptic connections between two neurons can seemingly strengthen or weaken depending on when the neurons fire. Based on the above mentioned fact that memristors mimic the behavior of biological synapses, we have studied, via computer simulations the dynamical behavior of two identical nonlinear circuits coupled via a memristor. The circuit, which is used, has a very interesting dynamic behavior as for certain values of its elements exhibits double scroll chaotic attractors. The proposed memristor is a flux controlled memristor, which has a cubic relation between flux (φ) and charge (q). So, in the next sections the memristor and the coupling system are described followed by the simulation analysis. In this procedure we have used various tools of nonlinear systems’ theory, such as, bifurcation diagrams, phase portraits and Lyapunov exponents. The great sensitivity on initial conditions, the phenomenon of complete chaotic synchronization and inverse π-lag synchronization, are observed. Finally, conclusions remarks are included in the last section.

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Fig. 1. The coupled nonlinear circuits via the memristor M and its q – φ characteristic curve.

2. THE PROPOSED MEMRISTOR

W(ϕ) =

The memristor is an electronic element which satisfies the Eq. (1) between charge (q) and flux (φ). W(q) is called memductance and associates current (iM) and voltage (υM) of the memristor with the Eq. (2). dq(ϕ) (1) W(ϕ) = dϕ i Μ = W(ϕ) ⋅ υM (2)

In the case of a linear element, W is a constant, and the memristor is identical to resistor. However, in the case of W being itself a function of φ, produced by a nonlinear circuit element, then no combination of the fundamental circuit elements reproduces the same results as the memristor. Nowadays, many scientists see the memristor as a nonlinear circuit element with specific characteristics. For this reason various forms of meductances, such as cubic or piecewise linear [5, 6, 8], are proposed, except of the HP’s memristor. The proposed memristor is a flux-controlled memristor described by the function W(φ(t)), where q(φ) in Eq. (3) is a smooth continuous cubic function of the form: q(ϕ) = −k1 ⋅ ϕ + k 3 ⋅ ϕ3

(3)

with k1, k3 > 0. As a result, in this case the memductance W(φ) is provided by the following expression:

dq(ϕ) = − k1 + k 3 ⋅ ϕ 2 dϕ

(4)

3. THE COUPLED SYSTEM

In Fig. 1 two identical nonlinear circuits, which produce double scroll chaotic attractors [14], are coupled, via a memristor M. The state equations, describing the normalized system, are: ⎧ dx1 ⎪ dτ = y1 ⎪ dy R ⎪ 1 = z1 + ⋅ ( k1 + 3 ⋅ k 3 ⋅ w 2 ) ⋅ (y 2 − y1 ) 2 ⎪ dτ ⎪ dz1 = − x1 − y1 − z1 + R ⋅ f (x ) 1 ⎪ dτ 2 2 2 2⋅ Rx ⎪⎪ dx 2 (5) = y2 ⎨ dτ ⎪ dy ⎪ 2 = z 2 + R ⋅ ( k1 + 3 ⋅ k 3 ⋅ w 2 ) ⋅ (y1 − y 2 ) 2 ⎪ dτ ⎪ dz 2 = − x 2 − y 2 − z 2 + R ⋅ f (x ) 2 ⎪ dτ 2 2 2 2⋅ Rx ⎪ dw R ⋅ C ⎪ = ( y 2 − y1 ) 2 ⎪⎩ dτ State variables x1,2, y1,2, and z1,2, represent the voltages at the outputs of the operational amplifiers numbered as ‘‘1’’, ‘‘2’’

(a) (b) Fig. 2. The bifurcation diagrams of x2 – x1 versus k3, with k1 = 0.5 ⋅ 10−4 C/Wb and initial values, (a) (x1,y1, z1, x2, y2, z2, w) = (–2, 1, 0.08, 0.8, 1.5, –0.03, 0.00001) and (b) (x1,y1, z1, x2, y2, z2, w) = (–1, 0.5, 0.08, 1, -1.5, –0.03, 0).

and ‘‘3’’, respectively (Fig. 1). The first three equations of system (5) describe the first of the two coupled identical double scroll circuits, while the other three describe the second one. Also, the state variable w represents the magnetic flux (φ) of the memristor M. The saturation functions f(x1,2), used in Eq. (5) are defined by the following expression: if x1,2 ≥ n ⎧ 1, ⎪1 (6) f (x1,2 ) = ⎨ ⋅ x1,2 , if − n ≤ x1,2 < n ⎪−n1, if x1,2 < −n ⎩ where n=R2/R3. This implementation demonstrates an i-v characteristic with two saturation plateaus at ±1, as well as, an intermediate linear part with slope 1/n. The values of the circuit elements were: R = 20.0 kΩ, R1 = 1.0 kΩ, R2 = 14.3 kΩ, R3 = 20.4 kΩ, RX = 12.5 kΩ, and C = 1.0 nF. Also, time scaling: τ = (1/RC1)t, was included. All the operational amplifiers were of the type LF411. The voltages of the positive and negative power supplies were set ±15V. The outputs of the saturated voltages were ±14.3V. For this specific implementation, each circuit operated independently in a chaotic mode, demonstrating a double scroll chaotic attractor. This was further confirmed numerically calculating the related Lyapunov exponents [15], which were found to possess the following values: LE1 = 0.13271, LE2 = 0.00000, and LE3 = –0.85410. Since, according to the numerical analysis, one of the Lyapunov exponents is positive, we consider that each one of the coupled nonlinear circuits operates in a chaotic mode. 4. DYNAMICAL BEHAVIOR OF THE SYSTEM

In our recent works the bidirectional coupling via a linear resistor between two identical double scroll circuits of this type, was studied [16, 17]. Very interesting synchronization phenomena, such as complete chaotic synchronization, and inverse π-lag synchronization, depending on the coupling

factor and the initial conditions, were raised. The complex behavior of this linearly coupled system was the reason to proceed to the study of the coupling system via a memristor. Our aim was to study the complex dynamic behavior exhibited by coupled nonlinear circuits. In this work, we have used the system (5) and we have observed a very interesting dynamical behavior. We have chosen the factor k1 = 0.5 ⋅10 −4 C/Wb of the Eq. (3), while the other factor k3 played the role of the control parameter. The system was studied by employing a fourth order RungeKutta algorithm, which is a numerical tool for solving ordinary differential equations. In Fig. 2 the bifurcation diagrams of x2 – x1 versus k3, are shown. These diagrams are produced by increasing the factor k3, while the initial conditions of the system in each iteration are the same. The two coupled nonlinear circuits described by the equation system (5) exhibit a variety of dynamical behaviors, including various types of synchronization and regions of desynchronization, depending on the factor k3 and initial conditions. Desynchronization is commonly used when the coupling system is out of synchronization (complete or inverse π-lag) and exhibits either a chaotic or periodic behavior. In the next paragraphs, the above mentioned dynamical behaviors are studied thoroughly by using the phase portraits and the signals waveforms. Firstly, it must been mentioned that the coupled system was at the saturation for low values of the factor k3. For that reason, the two bifurcation diagrams of Fig. 2 begin at the values of k3 = 3 ⋅104 C/Wb3 and k3 = 2.5 ⋅10 4 C/Wb3, respectively. Secondly, in general the morphology is the same in both bifurcation diagrams of Fig. 2, with a difference at the width of regions of dynamical behaviors. In details, the system after the saturation region appears a region of chaotic desynchronization as it shown in the phase portrait of x2 versus x1, for k3 = 2.85 ⋅104 C/Wb3 in the case of initial conditions of Fig. 2(b). In this region each one

Fig. 3. Phase plot in x2 versus x1 plane, for k3 = 2.85 ⋅ 10 4 C/Wb3, in the case of initial conditions of Fig. 2(b) (chaotic desynchronization).

Fig. 5. Phase plot in x2 versus x1 plane, for k3 = 3.6 ⋅ 104 C/Wb3, in the case of initial conditions of Fig. 2(a) (stable chaotic desynchronization).

Fig. 4. Phase plot in x2 versus x1 plane, for k3 = 2.97 ⋅ 10 4 C/Wb3, in the case of initial conditions of Fig. 2(b) (periodic behavior).

Fig. 6. Phase plot in x2 versus x1 plane, for k3 = 3.55 ⋅ 104 C/Wb3, in the case of initial conditions of Fig. 2(b) (periodic behavior).

of the two coupled circuits produces different spiral attractors with a tendency for expansion. In the bifurcation diagram of Fig. 2(b) we can observe that the chaotic desynchronization region appears a small discontinuity in the ranges of values 2.96 ⋅104 C/Wb3 ≤ k3 < 2.99 ⋅10 4 C/Wb3. In this region the system is also desynchronized appearing a periodic steady state (Fig. 4). The first region of desynchronization is followed by a compressed region of chaotic behavior with expanded periodic windows. In this region the chaotic spiral attractor of the system become stable without any further expansions (Fig. 5). Also, the observed periodic behavior (Fig. 6) in this interrupted chaotic region has the same form with this of Fig. 4. For k3 = 3.89 ⋅10 4 C/Wb3 in the case of Fig. 2(a) and for k3 = 3.58 ⋅104 C/Wb3 in the case of Fig. 2(b) the system enters into a region of period-1 steady state. Fig. 7 shows the periodic attractor, in x2 versus x1 phase diagram, for

Fig. 7. Phase plot in x2 versus x1 plane, for k3 = 4 ⋅ 10 4 C/Wb3, in the case of initial conditions of Fig. 2(a) (period-1 steady state).

k3 = 4 ⋅10 4 C/Wb3 (for Fig. 2(a)), which appears to be a line but in reality it is a closed curve. In this periodic region

S(τ) =

⎡⎣ − x 2 ( t + τ ) − x1 ( t ) ⎤⎦

2

(8) 1 ⎡ x 2 ( t ) ⋅ ( − x ( t ) )2 ⎤ 2 2 ⎢⎣ 1 ⎥⎦ As shown in Fig. 10 the time lag τmin = 0.9 ms is equal to x1, x2 half-period (T/2) in this region. Consequently, the sum of x1(t) and x2(t + τmin) = −x1(t) is

Fig. 8. Phase portrait in y versus x plane of each circuit, for k3 = 4 ⋅ 10 4 C/Wb3, in the case of initial conditions of Fig. 2(a) (period-1 steady state).

Fig. 10. The similarity function (S) versus time (t), for k3 = 4 ⋅ 10 4 C/Wb3, in the case of initial conditions of Fig. 2(a). S(τmin) = 0 means lag with time shift of τmin = T/2 = 0.9 ms.

Fig. 9. Time-series of x1 (upper side) and x2 (lower side), for k3 = 4 ⋅ 10 4 C/Wb3, in the case of initial conditions of Fig. 2(a). Inverse π-lag synchronization is observed.

each coupled circuit appears a symmetrical period-1 limit cycle in the x-y plane, as to (x1, x2) = (0, 0) (Fig. 8). As a result in this case the coupled system demonstrates a phenomenon that we term as “Inverse π-lag Synchronization” [17]. This phenomenon results to a synchronized mode of operation between two coupled circuits in such a way that the signal of the first circuit (x1) is opposite (mirrored) to the signal of the second one (x2) with a time lag τmin. This time lag is equal to T/2, where T is the period of x1 and x2: x1 (t) = − x 2 (t + τmin ), τ min = T / 2 (7) Time-series x1, x2 are presented in Fig. 9 confirming the inverse π-lag synchronization behavior of the system (for the case of Fig. 2(a)), since signals x1 and x2 are mirrored with a time lag τmin = T/2. To compute this time lag we have used the similarity function (Eq. 8), defined with respect to the state variables, x1 and −x2, of the coupled chaotic oscillators.

Fig. 11. Time-series of x1(t) + x2(t + τmin), for k3 = 4 ⋅ 10 4 C/Wb3, in the case of initial conditions of Fig. 2(a). Inverse π-lag synchronization is observed.

zero as shown in Fig. 11. The system remains in the region of inverse π-lag synchronization as the factor k3 increases, but step-by-step enters into the complete chaotic synchronization mode, which happens when the difference x2 – x1 becomes equal to zero. As a result, in the phase portrait of x2 versus x1 the trajectory remains strictly on the diagonal (Fig.12). In the synchronization region, each one of the coupled circuits appears double scroll chaotic attractors. Also, in the case of Fig. 2a the system’s behavior switches, depending on the value of the factor k3, between the inverse π-lag and

complete chaotic synchronization. This phenomenon is also observed in the second bifurcation diagram of Fig. 2b, but not in so obvious way, as in the first one.

[3] M. Sapoff and R. M. Oppenheim, “Theory and Application of Self-heated Thermistors,” Proc. IEEE, vol. 51, pp. 1292-1305, 1963. [4] Y. V. Pershin and M. Di Ventra, “Spin Memristive Systems: Spin Memory Effects in Semiconductor Spintronics,” Phys. Rev. B, Condens. Matter, vol. 78, pp. 113309/1-4, 2008. [5] M. Itoh, and L. O. Chua, “Memristor Oscillators,” Int. J. Bifurcation and Chaos, vol. 18, pp. 3183-3206, 2008. [6] I. M. Kyprianidis, Ch. K. Volos, and I. N. Stouboulos, “Chaotic Dynamics from a Nonlinear Circuit Based on Memristor with Cubic Nonlinearity,” Proc. 7th Conference of the Balkan Physical Union, AIP Proc., vol. 1203, pp. 626-631, 2010. [7] B. Muthuswamy and P. P. Kokate, “Memristor Based Chaotic Circuits,” IETE Techn. Rev., vol. 26, pp. 415-426, 2009.

Fig. 12. Phase plot in x2 versus x1 plane, for k3 = 6 ⋅ 104 C/Wb3, in the case of initial conditions of Fig. 2(b) (chaotic synchronization).

5. CONCLUSIONS

It is already known that memristors can mimic the behavior of biological synapses. So, for the shake of simplicity, in this report we have studied, for the first time, the coupling scheme between two identical nonlinear circuits via a memristor. The conclusions of this work are very useful especially in the field of neuromorphic systems. The study confirms that the coupled system maintains the essential characteristic of dynamic systems, such as the great sensitivity of the system on initial conditions. This phenomenon is confirmed by the bifurcation diagrams of Fig. 2, in which it is stated that a small variation in initial conditions can cause a different dynamical behavior. Also, the coupled system via the memristor, shows similar but more interesting behavior in relation to that observed in the coupling between the same circuits via a linear resistor [17]. In general, as the coupling parameter k3 is increased, the system undergoes a transition from chaotic desynchronization to inverse π-lag synchronization and finally to complete chaotic synchronization. Inverse π-lag synchronization in coupled nonlinear systems is a recent observed special case of lag synchronization, which is found when the coupled system is in phase locked state. 6. REFERENCES [1] L. O. Chua, “Memristor-The Missing Circuit Element,” IEEE Trans. on Circuit Theory, vol. CT-18, pp. 507-519, 1971. [2] D. Strukov, G. Snider, G. Stewart, and R. Williams, “The Missing Memristor Found,” Nature, vol. 453, pp. 80-83, 2008.

[8] B. Muthuswamy, “Implementing Memristor Based Chaotic Circuits,” Int. J. Bifurcation and Chaos, vol. 20, pp. 1335-1350, 2010. [9] Y. V. Pershin, S. La Fontaine, and M. Di Ventra, “Memristive Model of Amoeba Learning,” Phys. Rev. E, vol. 80, pp. 021 926– 1–021 926–6, 2009. [10] Y. V. Pershin and M. Di Ventra, “Experimental Demonstration of Associative Memory with Memristive Neural Networks,” Neural Networks, vol. 23, p. 881, 2010. [11] S. H. Jo, T. Chang, I. Ebong, B. B. Bhadviya, P. Mazumder, and W. Lu, “Nanoscale Memristor Device as Synapse in Neuromorphic Systems,” Nano Lett., vol. 10, pp. 1297-1301, 2010. [12] M. Laiho and E. Lehtonen, “Cellular Nanoscale Network Cell with Memristors for Local Implication Logic and Synapses,” Proc. IEEE International Symposium on Circuits and Systems (ISCAS 2010), pp. 2051-2054, 2010. [13] B. Linares-Barranco and T. Serrano-Gotarredona, “Memristance Can Explain Spike-Time-Dependent-Plasticity in Neural Synapses,” Proc. Nature, pp. 1-4, 2009. [14] Ch. K. Volos, I. M. Kyprianidis, I. N. Stouboulos, and A. N. Anagnostopoulos, “Experimental Study of the Dynamic Behavior of a Double Scroll Circuit,” J. Applied Functional Analysis, vol. 4, pp. 703-711, 2009. [15] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov Exponents from a Time Series,” Physica D, vol. 16, pp. 285-317, 1985. [16] Ch. K. Volos, I. M. Kyprianidis, S. G. Stavrinides, I. N. Stouboulos, and A. N. Anagnostopoulos, “Inverse Lag Synchronization in Mutually Coupled Nonlinear Circuits,” Proc. Latest Trends on Communications, pp. 31-36, 2010. [17] Ch. K. Volos, I. M. Kyprianidis, and I. N. Stouboulos, “Various Synchronization Phenomena in Bidirectionally Coupled Double Scroll Circuits,” Commun. Nonlinear Sci. Numer. Simulat., vol. 16, pp. 3356–3366, 2011.