Many systems in physics, chemistry and biology exhibit oscillations

with a pronounced random component. Such stochastic oscillations can emerge via

different mechanisms, for example linear dynamics of a stable focus with

fluctuations, limit-cycle systems perturbed by noise, or excitable systems in

which random inputs lead to a train of pulses. Despite their diverse origins,

the phenomenology of random oscillations can be strikingly similar. In joint

work with Alberto Perez, Benjamin Lindner, and Boris Gutkin, we introduce a

nonlinear transformation of stochastic oscillators to a new complex-valued

function $Q^*_1(\mbx)$ that greatly simplifies and unifies the mathematical

description of the oscillator's spontaneous activity, its response to an

external time-dependent perturbation, and the correlation statistics of

different oscillators that are weakly coupled. The function $Q^*_1(\mbx)$ is the

eigenfunction of the Kolmogorov backward operator with the least negative (but

non-vanishing) eigenvalue $\lambda_1=\mu_1+i\omega_1$. The resulting power

spectrum of the complex-valued function is exactly given by a Lorentz spectrum

with peak frequency $\omega_1$ and half-width $\mu_1$; its susceptibility with

respect to a weak external forcing is given by a simple one-pole filter,

centered around $\omega_1$; and the cross-spectrum between two coupled

oscillators can be easily expressed by a combination of the spontaneous power

spectra of the uncoupled systems and their susceptibilities. Our approach makes

qualitatively different stochastic oscillators comparable, provides simple

characteristics for the coherence of the random oscillation, and gives a

framework for the description of weakly coupled stochastic oscillators.

Joint work with:

Alberto Perez-Cervera (Universitat Politècnica de Catalunya, Barcelona)

Boris Gutkin (Ecole Normale Supérieure, Paris)

Benjamin Lindner (Humboldt University, Berlin)