We show that a wide class of uncoupled limit-cycle oscillators can be in-phase synchronized by common weak additive noise. An expression of the Lyapunov exponent is analytically derived to study the stability of the noise-driven synchronizing state. The result shows that such a synchronization can be achieved in a broad class of oscillators with little constraint on their intrinsic property. On the other hand, the leaky integrate-and-fire neuron oscillators do not belong to this class, generating intermittent phase slips according to a power law distribution of their intervals.
The connectivity of complex networks and functional implications has been attracting much interest in many physical, biological and social systems. However, the significance of the weight distributions of network links remains largely unknown except for uniformly- or Gaussian-weighted links. Here, we show analytically and numerically, that recurrent neural networks can robustly generate internal noise optimal for spike transmission between neurons with the help of a long-tailed distribution in the weights of recurrent connections. The structure of spontaneous activity in such networks involves weak-dense connections that redistribute excitatory activity over the network as noise sources to optimally enhance the responses of individual neurons to input at sparse-strong connections, thus opening multiple signal transmission pathways. Electrophysiological experiments confirm the importance of a highly broad connectivity spectrum supported by the model. Our results identify a simple network mechanism for internal noise generation by highly inhomogeneous connection strengths supporting both stability and optimal communication.
We formulate a phase-reduction method for a general class of noisy limit cycle oscillators and find that the phase equation is parametrized by the ratio between time scales of the noise correlation and amplitude relaxation of the limit cycle. The equation naturally includes previously proposed and mutually exclusive phase equations as special cases. The validity of the theory is numerically confirmed. Using the method, we reveal how noise and its correlation time affect limit cycle oscillations.
The phase description is a powerful tool for analyzing noisy limit cycle oscillators. The method, however, has found only limited applications so far, because the present theory is applicable only to the Gaussian noise while noise in the real world often has non-Gaussian statistics. Here, we provide the phase reduction method for limit cycle oscillators subject to general, colored and non-Gaussian, noise including heavy-tailed one. We derive quantifiers like mean frequency, diffusion constant, and the Lyapunov exponent to confirm consistency of the results. Applying our results, we additionally study a resonance between the phase and noise. Limit cycle oscillators effectively model various sustained oscillations in many fields of science and technology including chemical reactions, biology, electric circuits, and lasers [1][2][3][4]. The phase reduction method is a powerful analytical tool which approximates highdimensional dynamics of limit cycle oscillators with single phase variable that characterizes timing of oscillation [1,5]. Since the phase is neutrally stable, phase perturbations persist in time and result in various remarkable phenomena where weak action leads to significant effects, such as those addressed in the theory of synchronization [6,7]. While the theory of the phase reduction had been developed for deterministic oscillators, recent studies successfully extended the theory to limit cycle oscillators subject to noise [4,8,9] and revealed that interplay between nonlinearity and noise results in fascinating noise-induced phenomena including frequency modulation and noise-induced synchronization [12,13]. This extended phase reduction method, however, has found limited applications so far, since the method is applicable only to Gaussian noise. While the noise in the real world often has non-Gaussian statistics, few theories have considered nonlinear systems subject to general non-Gaussian noise, which has forced people to use the Gaussian approximation. In particular, whether the phase description is still valid for oscillators subject to non-Gaussian noise and how quantifiers of the phase dynamics should be amended remains unknown. In this paper, we develop the phase reduction method for limit cycle oscillators subject to general, colored and nonGaussian noise. By correctly evaluating the influence of amplitude perturbations up to second order in the noise strength, we derive the stochastic differential equation of phase, which allows us to study nonlinear oscillations in the real world without the Gaussian approximation. To confirm consistency of the result, we derive closed expressions of quantifiers of the phase dynamics such as mean frequency, phase diffusion constant, and the Lyapunov exponent. The only limitation we impose is the weakness of the noise. Thus, the obtained results are applicable even when higher order moments of the noise diverge as long as the second order moment is finite and we confirm this fact numerically. As an application of the results, we study a limit cycle oscillato...
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