Collective phase description of globally coupled excitable elements

Abstract

We develop a theory of collective phase description for globally coupled noisy excitable elements exhibiting macroscopic oscillations. Collective phase equations describing macroscopic rhythms of the system are derived from Langevin-type equations of globally coupled active rotators via a nonlinear Fokker-Planck equation. The theory is an extension of the conventional phase reduction method for ordinary limit cycles to limit-cycle solutions in infinite-dimensional dynamical systems, such as the time-periodic solutions to nonlinear Fokker-Planck equations representing macroscopic rhythms. We demonstrate that the type of the collective phase sensitivity function near the onset of collective oscillations crucially depends on the type of the bifurcation, namely, it is type-I for the saddle-node bifurcation and type-II for the Hopf bifurcation.

Population dynamics of the modified theta model: macroscopic phase reduction and bifurcation analysis link microscopic neuronal interactions to macroscopic gamma oscillation

“…where u(u) ¼ c 1 (1 þ cosu)/C is the sensitivity of a single neuron to the phase u [20]. This equation provides unique insights into how the external perturbations and/or changes of internal states affect macroscopic properties of gamma oscillations and will be discussed in following sections.…”

Response of traveling waves to transient inputs in neural fields

“…Our analytical framework extends previous numerical studies of transient stimuli in neural fields [23,24] and makes use of the zero-eigenmode of the adjoint as a response function for spatially extended systems [20,38,39]. In particular, we study how the speed and location of traveling waves are affected, since such analyses could likely be experimentally testable [11].…”

“…By analogy, our wave response functions seem to be type II for pulses in networks with inhibition and type I otherwise, as in the case of fronts. In a related context, using a nonlinear Fokker-Planck equation, [39] recently found that the bifurcation structure of a coupled oscillator population determined whether its sensitivity function was type I or type II.…”

Optimal entrainment with smooth, pulse, and square signals in weakly forced nonlinear oscillators

“…In addition to the example of (36), the weakly stimulated Hodgkin-Huxley neuron model, whose phase response function is given in [17,28], is also considered: Z (θ) = 0.176116 + 0.371736 cos θ − 0.740283 sin θ − 0.819478 cos 2θ + 0.00225226 sin 2θ + 0.181875 cos 3θ + 0.403816 sin 3θ + 0.111446 cos 4θ − 0.0892503 sin 4θ − 0.0127103 cos 5θ − 0.0165083 sin 5θ . (37) As the procedure for obtaining all (local and global) optimal forcings is the same for both examples (36) and (37), we explain the case of (36) in detail, and for the case of (37), we omit detailed numerical data here and just mention the numerical results.…”

“…Case of 1 < p < ∞ for the example of (36) For the example of (36), here we numerically identify the optimal forcings for the case of 1 < p < ∞, as follows. The case of p = ∞ and the case of p = 1 are respectively considered in Sections 6.2 and 6.3.…”