Antimonotonicity in a FitzHugh Nagumo Type Circuit

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1 Antimonotonicity in a FitzHugh Nagumo Type Circuit I. Μ. KYPRIANIDI, A. T. MAKRI, I. N. TOUBOUO Physics Department Aristotle University of Thessaloniki Thessaloniki, 5414 GREECE imkypr@auth.gr, aimakri@physics.auth.gr, stouboulos@physics.auth.gr Ch. K. VOO Faculty of Mathematics Engineering tudies Department of Military cience Hellenic Army Academy Vari, GR-1667 GREECE chvolos@gmail.com Abstract: - Antimonotonicity, i.e. forward period-doubling bifurcation sequences followed by reverse perioddoubling sequences, as a parameter is varied in a monotone way, is observed in a FitzHugh Nagumo type circuit, with a nonlinear resistor, which has a smooth cubic characteristic. The FitzHugh Nagumo type electrical nonlinear oscillators simulate neuron cells, so the study of their dynamics is very important. Key-Words: - FitzHugh Nagumo, nonlinear circuits, chaos, antimonotonicity, bubbles. 1 Introduction Electric circuits with a nonlinear resistor, which is characterized by a smooth cubic i υ characteristic, have emerged as a simple, yet powerful experimental analytical tool in studying chaotic behavior in nonlinear dynamics. Among the electrical oscillators that have been studied, the FitzHugh Nagumo type oscillator [1,] is very important, because can simulate neuron cells. The system of two FitzHugh-Nagumo cells coupled with gap junctions is the simplest possible system simulating two coupled neuron cells via an electric synapse []. As introduced by Fitzhugh [1], his model for a spiking neuron is a two dimensional reduction of the Hodgkin Huxley equations [4]. A qualitative description of the single neuron activity is given, according to FitzHugh, by the system of coupled nonlinear differential equations dx 1 γx x yz dt dy 1 x α βy dt γ (1) The variable x describes the potential difference across the neural membrane y can be considered as a combination of the different ion channel conductivities, present in the Hodgkin-Huxley model. The control parameter z of the FitzHugh system describes the intensity of the stimulating current. Nagumo et al. [] proposed an electronic simulator of the model of FitzHugh using a tunnel diode as the nonlinear element (Fig.1). Fig.1. The electronic simulator of the FitzHugh model proposed by Nagumo et al. []. The FitzHugh model of nonlinear differential equations (1) can be simulated by a different nonlinear electric circuit, using a nonlinear resistor with a smooth cubic i υ characteristic (Fig.). IBN:

2 The nonlinear differential equations (7) (8) are the FitzHugh equations (1). Fig.. The electronic simulator of the model of FitzHugh, proposed by Kyprianidis et al [5]. Analysis of the FitzHugh Model The smooth cubic i υ characteristic of the nonlinear resistor of the circuit of Fig. is given by the following equation 1 1 υ in g(υ) υ ρ V () 0 where ρ V 0 are normalization parameters. From Kirchhoff s laws i i i i () C N di (4) υ E Ri dt or dυ υ υ i i (5) dt C ρ di 1 υ Ri E (6) dt By introducing new, normalized t υ ρi variables, τ, x, y C ρi z, equations (5) (6) are reduced to equations (7) (8), dx 1 γx x yz (7) dτ dy 1 x βy α (8) dτ γ where, E R 1 α, β γ (9) V ρ ρ C 0.1 The FitzHugh Type Model Proposed by Rajasekar akshmanan Rajasekar akshmanan proposed a slightly different form of FitzHugh model [6,7] given by the following state equations, which are of Bonhoeffer van der Pol type, dx 1 x x yz dt (10) dy cxaby dτ The study of Eqs.(10) revealed the existence of chaotic behavior, following the period doubling route to chaos, devil s staircases. The nonlinear differential equations (10) can be also simulated by a nonlinear electric circuit, using a nonlinear resistor with a smooth cubic i-v characteristic. The nonlinear electric circuit is shown in Fig.. The smooth cubic i υ characteristic of the nonlinear resistor of the circuit of Fig. is given by the same equation, as before, 1 1 υ in g(υ) υ ρ V (11) 0 where ρ V 0 are normalization parameters. From Kirchhoff s laws i i i i (1) C N υ E Ri dt di (1) t By introducing new, normalized variables, τ, ρc υ ρi ρi x, y, z, equations (1) (1) are reduced to equations (10), where, E R ρ C a, b c (14) ρ In the general case, the driving current source has the following form i I Icosπf t (15) DC 0 including a DC plus a sinusoidal term of frequency f, so z BDC B0 cosπfτ (16) where the normalized frequency f will be f ρcf. IBN:

3 In the bifurcation diagrams of Figs.4-7 chaotic states are observed (chaotic bubbles, [8]), while in the bifurcation diagrams of Figs.8-9, only periodic states (periodic bubbles). For B 0 = 0., the system undergoes the sequence: p 1 p p 1. This bifurcation diagram is the primer bubble [8]. Fig.. The nonlinear electric circuit simulating Eqs.(10). The topology of the circuits of Fig. Fig. is exactly the same, proving the equivalence of equations (1) (10). We have studied the dynamics of the circuit keeping constant the following parameters: a = 0.7, b = 0.8, c = 0.1 f = For B DC = 0.0, the bifurcation diagram y vs. B 0 is shown in Fig.4. The system follows a period-doubling route to chaos, for 0.6 < B 0 < 0.8, then a period-4 window is observed, followed by a chaotic regime, 1.1 < B 0 < 1.6. Then, the system undergoes a reverse period-doubling sequence a period-1 steady state is observed for B 0 > 1.7. These forward reverse period doubling sequences, as a parameter of the system increases in a monotone way, is called antimonotonicity [8-1]. By varying the value of the DC component of the input current, the bifurcation diagrams of the system are shown in Figs.5-9. The period-4 window in Fig.4 has been disappeared the period-1 bubbles are clearly observed. Fig.5. Bifurcation diagram y vs. B 0 of the nonlinear circuit of Fig., for f = B DC = A chaotic bubble of period-1. Fig.6. Bifurcation diagram y vs. B 0 of the nonlinear circuit of Fig., for f = B DC = A chaotic bubble. Fig.4. Bifurcation diagram y vs. B 0 of the nonlinear circuit of Fig., for f = B DC = 0.0. The FitzHugh-Nagumo Type Circuit Driven by a Voltage ource In the circuits of Figs., the driving source is a current source. But in most cases, circuits are driven by voltage sources. In this section, we will study the circuit of Fig. driven by a voltage source, as it is shown in Fig.10. IBN:

4 Fig.7. Bifurcation diagram y vs. B 0 of the nonlinear circuit of Fig., for f = B DC = A chaotic bubble. Fig.8. Bifurcation diagram y vs. B 0 of the nonlinear circuit of Fig., for f = B DC = A periodic bubble. Fig.9. Bifurcation diagram y vs. B 0 of the nonlinear circuit of Fig., for f = B DC = The primer bubble. Fig.10. The circuit of Fig. driven by a voltage source. The smooth cubic i υ characteristic of the nonlinear resistor of the circuit of Fig.10 remains the same as before 1 1 υ in g(υ) υ ρ V (17) 0 applying Kirhhoff s laws we have i i i i (18) where, C i N υ υ (19) R di (0) υ E Ri dt t By introducing the normalized time, τ ρc the normalized variables υ ρi ρυ x, y, u (1) RV 0 the normalized state equations are the following: dx 1 x1 ε x yu dτ () dy cxaby dτ where, 0 E R ρ C ρ a, b, c ε () V ρ R In the general case, the driving voltage source has the following form: υ V V cosπf t (4) DC 0 including a DC plus a sinusoidal term of frequency f s, so u UDC U0 cosπfτ (5) IBN:

5 where the normalized frequency f will be f ρcf..1 Dynamics of the Circuit Bifurcation Diagrams We have studied the dynamics of the circuit keeping constant the following parameters: a = 0.7, b= 0.8, c = 0.1, f = U DC = 0.0. The bifurcation diagrams, y vs. U 0, for different values of factor ε, are shown in the following figures antimonotonicity is observed. In Figs chaotic bubbles of period-1 are observed, while periodic bubbles are shown in Figs Fig.1. Bifurcation diagram, y vs. U 0, for ε = Fig.11. Bifurcation diagram, y vs. U 0, for ε = U DC = 0.0. A chaotic bubble of period-1. Fig.14. Bifurcation diagram, y vs. U 0, for ε = Fig.1. Bifurcation diagram, y vs. U 0, for ε = Fig.15. Bifurcation diagram, y vs. U 0, for ε = Conclusion Cascades of period-doubling bifurcations have long been recognized to be one of the most common routes to chaos, as exemplified e.g. by the IBN:

6 Fig.16. Bifurcation diagram, y vs. U 0, for ε = 0.00 U DC = 0.0. A periodic bubble. Fig.17. Bifurcation diagram, y vs. U 0, for ε = 0.50 U DC = 0.0. The primer bubble. one-dimensional (1D) logistic map x n+1 = λx n (1 x n ). As the parameter λ in such a map is increased, it is known that periodic orbits are only created but never destroyed. Unlike the monotone bifurcation behavior of the logistic map, however it has been shown that, in many common nonlinear dynamical systems, forward period-doubling bifurcation sequences are followed by reverse period-doubling sequences, as a parameter is varied in a monotone way. This type of creation annihilation of periodic orbits is called antimonotonicity, it is very important to know this type of dynamics in nonlinear systems, because we can work in different parameter regimes having the same dynamics. Also, when antimonotonicity is present, phenomena of hysteresis are not observed [10]. References: [1] R. FitzHugh, Impulses physiological states in theoretical models of nerve membrane, Biophys. Journal, Vol.1, 1961, pp [] J. Nagumo,. Arimoto,.Yoshizawa, An Active Pulse Transmission ine imulating Nerve Axon, Proc. IRE, Vol.50, 196, pp [] M. Aqil, K-. Hong, M-Y. Jeong, ynchronization of coupled chaotic FitzHugh- Nagumo systems, Commun Nonlinear ci Numer imlat., Vol.17, 01, pp [4] A.. Hodgkin A. F. Huxley, A quantitative description of membrane current its application to conduction excitation in nerve, J. Physiol., Vol.117, 195, pp [5] I. M. Kyprianidis, V. Papachristou, I. N. touboulos Ch. Volos, Dynamics of Coupled Chaotic Bonhoeffer van der Pol Oscillators, WEA Trans. yst., 01, pp [6]. Rajasekar M. akshmanan, Period- Doubling Bifurcations, Chaos, Phase-ocking Devil s taircase in a Bonhoeffer van der Pol oscillator, Physica D, Vol., 1988, pp [7]. Rajasekar M. akshmanan, Algorithms for Controlling Chaotic Motion: Application for the BVP oscillator, Physica D, Vol.67, 199, pp [8] M. Bier T. C. Bountis, Remerging Feigenbaum Trees in Dynamical ystems, Phys. ett. A, Vol.104, 1984, pp [9] I. M. Kyprianidis, P. Haralabidis, I. N. touboulos, T. Bountis, Antimonotonicity Chaotic Dynamics in a Fourth Order Autonomous Nonlinear Electric Circuit, Int. J. Bifurcation & Chaos, Vol.10, 000, pp [10] I. M. Kyprianidis M. E. Fotiadou, Complex Dynamics in Chua s Canonical Circuit with a Cubic Nonlinearity, WEA Trans. Circuits yst., Vol.5, 006, pp [11] I. M. Kyprianidis, Antimonotonicity in Chua s Canonical Circuit, Proc. of the 5th WEA Int. Conf. on Non-inear Analysis, Non-inear ystems Chaos, Bucarest, Romania, 006, pp [1] I. N. touboulos, I. M. Kyprianidis, M.. Papadopoulou, Experimental tudy of Antimonotonicity in a 4th Order Nonlinear Autonomous Electric Circuit, WEA Trans. Circuits yst., Vol.5, 006, pp [1] I. M. Kyprianidis, New Chaotic Dynamics in Chua s Canonical Circuit, WEA Trans. Circuits yst., Vol.5, 006, pp IBN: