Stochastic entrainment of a stochastic oscillator

Forward entrainment: Psychophysics, neural correlates, and function

We define forward entrainment as that part of behavioral or neural entrainment that outlasts the entraining stimulus. In this review, we examine conditions under which one may optimally observe forward entrainment. In Part 1, we review and evaluate studies that have observed forward entrainment using a variety of psychophysical methods (detection, discrimination, and reaction times), different target stimuli (tones, noise, and gaps), different entraining sequences (sinusoidal, rectangular, or sawtooth waveforms), a variety of physiological measures (MEG, EEG, ECoG, CSD), in different modalities (auditory and visual), across modalities (audiovisual and auditory-motor), and in different species. In Part 2, we describe those experimental conditions that place constraints on the magnitude of forward entrainment, including an evaluation of the effects of signal uncertainty and attention, temporal envelope complexity, signal-to-noise ratio (SNR), rhythmic rate, prior experience, and intersubject variability. In Part 3 we theorize on potential mechanisms and propose that forward entrainment may instantiate a dynamic auditory afterimage that lasts a fraction of a second to minimize prediction error in signal processing.

Entrainment of a cellular circadian oscillator by light in the presence of molecular noise

In this paper, we consider a stochastic molecular circadian oscillator described by a sequence of biological reactions and its deterministic kinetics governed by a system of ordinary differential equations in the limit of large numbers of molecules. The oscillations in the model are generated by negative feedback regulation of a gene. The focus of this paper is the entrainment of the oscillator by a periodic light signal that affects the maximal transcription rate of the gene. We introduce two scalings of the model parameters that provide independent control over the natural frequency of the oscillator and the relative noise level. We study entrainment in two ways: by visualizing the stochastic limit cycle in various projections of the discrete phase space of the system and by evaluating the maximum of the normalized cross correlation of the light signal with the number of protein molecules in the cell. The visualization method ignores the phase of the oscillator, and we find in this way that entrainment has a subtle organizing effect on the limit cycle as a whole. The cross correlation results reveal an interval of natural frequencies of the oscillator surrounding the frequency of the light signal within which maximal entrainment occurs with rather sharp drops in entrainment at the edges of this interval. The width of the interval of maximal entrainment increases with the amplitude of the light signal. These statements are applicable both to the stochastic oscillator and to its deterministic limit, but the results are most clear-cut in the deterministic case and degrade from there as the relative noise level increases.