@article{SchwabedalPikovskij2010,
  author    = {Schwabedal, Justus T. C. and Pikovskij, Arkadij},
  title     = {Effective phase description of noise-perturbed and noise-induced oscillations},
  issn      = {1951-6355},
  doi       = {10.1140/epjst/e2010-01271-6},
  year      = {2010},
  abstract  = {An effective dynamical description of a general class of stochastic phase oscillators is presented. For this, the effective phase velocity is defined either by the stochastic phase oscillators invariant probability density or its first passage times. Using the first approach the effective phase exhibits the correct frequency and invariant distribution density, whereas the second approach models the proper phase resetting curve. The discrepancy of the effective models is most pronounced for noise-induced oscillations and is related to non-monotonicity of the stochastic phase variable due to fluctuations.},
  language  = {en}
}
@article{SchwabedalPikovskij2010,
  author    = {Schwabedal, Justus T. C. and Pikovskij, Arkadij},
  title     = {Effective phase dynamics of noise-induced oscillations in excitable systems},
  issn      = {1539-3755},
  doi       = {10.1103/Physreve.81.046218},
  year      = {2010},
  abstract  = {We develop an effective description of noise-induced oscillations based on deterministic phase dynamics. The phase equation is constructed to exhibit correct frequency and distribution density of noise-induced oscillations. In the simplest one-dimensional case the effective phase equation is obtained analytically, whereas for more complex situations a simple method of data processing is suggested. As an application an effective coupling function is constructed that quantitatively describes periodically forced noise-induced oscillations.},
  language  = {en}
}
@article{SchwabedalPikovskijKralemannetal.2012,
  author    = {Schwabedal, Justus T. C. and Pikovskij, Arkadij and Kralemann, Bj{\"o}rn and Rosenblum, Michael},
  title     = {Optimal phase description of chaotic oscillators},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {85},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {2},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {1539-3755},
  doi       = {10.1103/PhysRevE.85.026216},
  pages     = {9},
  year      = {2012},
  abstract  = {We introduce an optimal phase description of chaotic oscillations by generalizing the concept of isochrones. On chaotic attractors possessing a general phase description, we define the optimal isophases as Poincare surfaces showing return times as constant as possible. The dynamics of the resultant optimal phase is maximally decoupled from the amplitude dynamics and provides a proper description of the phase response of chaotic oscillations. The method is illustrated with the Rossler and Lorenz systems.},
  language  = {en}
}
@article{SchwabedalPikovskij2013,
  author    = {Schwabedal, Justus T. C. and Pikovskij, Arkadij},
  title     = {Phase description of stochastic oscillations},
  series = {Physical review letters},
  volume    = {110},
  journal   = {Physical review letters},
  number    = {20},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {0031-9007},
  doi       = {10.1103/PhysRevLett.110.204102},
  pages     = {5},
  year      = {2013},
  abstract  = {We introduce an invariant phase description of stochastic oscillations by generalizing the concept of standard isophases. The average isophases are constructed as sections in the state space, having a constant mean first return time. The approach allows us to obtain a global phase variable of noisy oscillations, even in the cases where the phase is ill defined in the deterministic limit. A simple numerical method for finding the isophases is illustrated for noise-induced switching between two coexisting limit cycles, and for noise-induced oscillation in an excitable system. We also discuss how to determine isophases of observed irregular oscillations, providing a basis for a refined phase description in data analysis.},
  language  = {en}
}