Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators

Synchronization and coherence of multiple oscillators is of general interest to many scientific and engineering disciplines, notably in biology and neurophysiology where chemical and cellular processes are commonly modeled by dynamical systems [20, 33]. These phenomena have been studied in various contexts, such as weak coupling [32], coupling with delays [7, 8] and oscillators subject to noisy forcing [34], and are often considered in the setting of electrophysiological behaviours of individual cells using mathematical models like the Hodgkin-Huxley (HH) [15] and Morris-Lecar (ML) models [28]. More recently there has been a great deal of work that highlights the constructive role that noise can play in dynamical systems, for example, in the context of stochastic resonance, spike time reliability, and noise induced synchrony (see, e.g. [11, 23, 24, 29, 31] and references therein ). For the last of these, there have been a number of studies showing that identical uncoupled oscillators subject to a weak common noise source exhibit primarily synchronous dynamics. At first such a result appeared counter-intuitive, given the expectation that noise should increase disorder. However, it is not completely unexpected from the perspective that forcing two identical oscillating systems with the same smooth forcing function leads to entrainment of their dynamics. Noise-induced synchronization has been studied in a wide array of experimental contexts, for example, in olfactory bulb neurons and electroreceptors [10], [31]. There has also been a great deal of interest in noise-induced oscillator synchronization as it pertains to the phenomenon of spike time reliability (STR) of neuronal cells. Experiments involving prepared slices of neuronal tissue have exhibited this phenomenon [5, 25], with complementary theoretical work in [4] [9]. STR is demonstrated by the reliable reproduction of a train of spikes or oscillatory responses of an oscillator over a series of repeated trials. In each trial an identical noisy stimulus applied to a neuron or oscillator reproduces a train of spikes or oscillatory responses with reliable timing, while a constant current fails to do so. The same phenomenon can be demonstrated by simultaneously subjecting similar uncoupled neuronal oscillators to the same noisy forcing simultaneously, as considered in this paper and references discussed below. Then these oscillators exhibit a response of synchronized spikes or in-phase oscillations. Such a response, in the absence of any coupling besides the noisy forcing, is analogous to the reliable replication over repeated trials observed in STR. Recently, the phenomenon of STR has been studied in the context where the oscillator is a conditional oscillator, that is, in parameter ranges where the system is quiescent in the absence of noise [40]. There the nearly regular oscillations are driven by a noise-induced subthreshold activity, in contrast to other contexts for STR where the underlying deterministic system exhibits spiking, large amplitude oscillations, or a limit cycle, as in, for example, [9]. Oscillator synchronization via common noisy input has been studied using the Fokker-Planck equation (FPE) for the probability density function for the phase difference [13]. In that setting, each of the oscillators has an attracting limit cycle in the absence of noise. Goldobin and Pikovsky studied the case