^{1,a)}, Alexey Feoktistov

^{1}, Vadim S. Anishchenko

^{1}and Jürgen Kurths

^{2,3}

### Abstract

Using a model system of FitzHugh-Nagumo type in the excitable regime, the similarity between synchronization of self-sustained and noise-induced oscillations is studied for the case of more than one main frequency in the spectrum. It is shown that this excitable system undergoes the same frequency lockings as a self-sustained quasiperiodic oscillator. The presence of noise-induced both stable and unstable limit cycles and tori, as well as their tangential bifurcations, are discussed. As the FitzHugh-Nagumo oscillator represents one of the basic neural models, the obtained results are of high importance for neuroscience.

One of the basic models in neuroscience is the FitzHugh-Nagumo oscillator, which describes the excitable dynamics of a single neuron. In the excitable regime, under the influence of noise the model demonstrates the phenomenon of Coherence Resonance: the oscillations become coherent at certain noise intensity. It is well known that such oscillations can be synchronized by an external harmonic force and even mutual synchronization may appear when a pair of non-identical oscillators is coupled. Using numerical simulations and electronic experiments, we show that the noise-induced oscillations with few main frequencies in excitable systems of the FitzHugh-Nagumo type demonstrate a kind of synchronization which obeys the same scenario as the synchronization of deterministic self-sustained quasiperiodic oscillations. This enables us to predict the existence of stable and unstable noise-induced limit cycles and tori which should possess similar tangential bifurcations as in the case of quasiperiodic oscillations.

S.A. and V.A. are grateful for the financial support of the Alexander von Humboldt Foundation and Russian Ministry of Education and Sciences (Grant No. 2.2.2.2/11514) and J.K. for the support by the federal ministry of education and research Germany (BCCN2, Grant No. 01GQ1001A). The authors dedicate this paper to the genuine memory of Frank Moss.

I. INTRODUCTION

II. THE ROLE OF SADDLE-NODE BIFURCATION IN SYNCHRONIZATION OF PERIODIC AND QUASI-PERIODIC OSCILLATIONS THROUGH THE FREQUENCY LOCKING

III. SYNCHRONIZATION OF NOISE-INDUCED COHERENT OSCILLATIONS IN AN EXCITABLE MODEL SYSTEM

IV. EXPERIMENT

V. ENSEMBLE OF EXCITABLE MODEL SYSTEMS

VI. CONCLUSION

### Key Topics

- Bifurcations
- 30.0
- Oscillators
- 28.0
- Synchronization
- 18.0
- Neuroscience
- 6.0
- Coherence
- 4.0

## Figures

(Color online) The qualitative phase space structure of a quasi-harmonic oscillator under external harmonic forcing (1): (a) outside synchronization region—ergodic two-dimensional torus; (b) inside synchronization region—stable (light, green online) and unstable (dark, red online) limit cycles on the surface of the former two-dimensional torus.

(Color online) The qualitative phase space structure of a quasi-harmonic oscillator under external harmonic forcing (1): (a) outside synchronization region—ergodic two-dimensional torus; (b) inside synchronization region—stable (light, green online) and unstable (dark, red online) limit cycles on the surface of the former two-dimensional torus.

The phase space structure of the reduced system (2): (a) inside the synchronization region—two fixed points (a saddle and a stable node); (b) outside the synchronization region (no fixed points).

The phase space structure of the reduced system (2): (a) inside the synchronization region—two fixed points (a saddle and a stable node); (b) outside the synchronization region (no fixed points).

The phase space structure of the reduced system (4) in the case of two-frequency quasi-periodic self-sustained oscillations under a harmonic force: (a) all frequencies are locked, four fixed points exist on the phase plane; (b) after saddle-node bifurcations of fixed points, only one frequency is locked, two invariant closed curves exist: stable (*l* _{1}) and unstable (*l* _{2}); (c) after tangential bifurcation of the invariant closed curves, no frequency locking.

The phase space structure of the reduced system (4) in the case of two-frequency quasi-periodic self-sustained oscillations under a harmonic force: (a) all frequencies are locked, four fixed points exist on the phase plane; (b) after saddle-node bifurcations of fixed points, only one frequency is locked, two invariant closed curves exist: stable (*l* _{1}) and unstable (*l* _{2}); (c) after tangential bifurcation of the invariant closed curves, no frequency locking.

Mean frequencies versus coupling coefficient in system (6). The value of 〈*ω _{i} * 〉 is given by the right-hand side of Eq. (6) averaged over the integration time interval.

Mean frequencies versus coupling coefficient in system (6). The value of 〈*ω _{i} * 〉 is given by the right-hand side of Eq. (6) averaged over the integration time interval.

(Color online) The Fourier spectrum evolution for system (8) calculated for variables *x* _{1} (dark grey, blue in color) and *x* _{2} (light grey, red in color). Here, *ω* _{1,2} are mean frequencies of the first and second subsystems, respectively; *ω _{ex} * is frequency of the external forcing;

*A*is amplitude of the external forcing: (a)

_{ex}*A*= 10

_{ex}^{−5}(lower region

**T**in Fig. 6); (b)

^{2}*A*= 4 · 10

_{ex}^{−5}(region

**T**in Fig. 6); (c)

^{3}*A*= 6 · 10

_{ex}^{−5}(region

**T**in Fig. 6); (d)

^{3}*A*= 8 · 10

_{ex}^{−5}(upper region

**T**in Fig. 6).

^{2}(Color online) The Fourier spectrum evolution for system (8) calculated for variables *x* _{1} (dark grey, blue in color) and *x* _{2} (light grey, red in color). Here, *ω* _{1,2} are mean frequencies of the first and second subsystems, respectively; *ω _{ex} * is frequency of the external forcing;

*A*is amplitude of the external forcing: (a)

_{ex}*A*= 10

_{ex}^{−5}(lower region

**T**in Fig. 6); (b)

^{2}*A*= 4 · 10

_{ex}^{−5}(region

**T**in Fig. 6); (c)

^{3}*A*= 6 · 10

_{ex}^{−5}(region

**T**in Fig. 6); (d)

^{3}*A*= 8 · 10

_{ex}^{−5}(upper region

**T**in Fig. 6).

^{2}(Color online) The bifurcation diagram of system (8). Region **C**: one main frequency in the spectrum, both frequencies of noise-induced oscillations entrained by an external harmonic force. Regions **T ^{2} **: two main frequencies in the spectrum, either one oscillator is synchronized by the external force at a frequency which differs from that of the other oscillator, or both oscillators are mutually synchronized at a frequency different from external forcing frequency. Regions

**T**: three main frequencies in the spectrum, no synchronization in the system.

^{3}(Color online) The bifurcation diagram of system (8). Region **C**: one main frequency in the spectrum, both frequencies of noise-induced oscillations entrained by an external harmonic force. Regions **T ^{2} **: two main frequencies in the spectrum, either one oscillator is synchronized by the external force at a frequency which differs from that of the other oscillator, or both oscillators are mutually synchronized at a frequency different from external forcing frequency. Regions

**T**: three main frequencies in the spectrum, no synchronization in the system.

^{3}Electronic setup modeling the dynamics of system (8).

Electronic setup modeling the dynamics of system (8).

(Color online) The bifurcation diagram obtained from the electronic experiment. Region **C**: one main frequency in the spectrum, both frequencies of noise-induced oscillations entrained by an external harmonic force. Regions **T ^{2} **: two main frequencies in the spectrum—either one oscillator is synchronized by the external force at a frequency which differs from the frequency of the other oscillator or both oscillators are mutually synchronized on the frequency different from external force frequency. Regions

**T**: three main frequencies in the spectrum, no synchronization in the system.

^{3}(Color online) The bifurcation diagram obtained from the electronic experiment. Region **C**: one main frequency in the spectrum, both frequencies of noise-induced oscillations entrained by an external harmonic force. Regions **T ^{2} **: two main frequencies in the spectrum—either one oscillator is synchronized by the external force at a frequency which differs from the frequency of the other oscillator or both oscillators are mutually synchronized on the frequency different from external force frequency. Regions

**T**: three main frequencies in the spectrum, no synchronization in the system.

^{3}(Color online) Frequencies versus coupling coefficient in system (9).

(Color online) Frequencies versus coupling coefficient in system (9).

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