Universality in Systems with Power-Law Memory and Fractional Dynamics

There are a few different ways to extend regular nonlinear dynamical systems by introducing power-law memory or considering fractional differential/difference equations instead of integer ones. This extension allows the introduction of families of nonlinear dynamical systems converging to regular systems in the case of an integer power-law memory or an integer order of derivatives/differences. The examples considered in this review include the logistic family of maps (converging in the case of the first order difference to the regular logistic map), the universal family of maps, and the standard family of maps (the latter two converging, in the case of the second difference, to the regular universal and standard maps). Correspondingly, the phenomenon of transition to chaos through a period doubling cascade of bifurcations in regular nonlinear systems, known as "universality", can be extended to fractional maps, which are maps with power-/asymptotically power-law memory. The new features of universality, including cascades of bifurcations on single trajectories, which appear in fractional (with memory) nonlinear dynamical systems are the main subject of this review.Comment: 23 pages 7 Figures, to appear Oct 28 201

Evaluating Adaptation and Olivocochlear Efferent Feedback as Potential Explanations of Psychophysical Overshoot

Masked detection threshold for a short tone in noise improves as the tone’s onset is delayed from the masker’s onset. This improvement, known as “overshoot,” is maximal at mid-masker levels and is reduced by temporary and permanent cochlear hearing loss. Computational modeling was used in the present study to evaluate proposed physiological mechanisms of overshoot, including classic firing rate adaptation and medial olivocochlear (MOC) feedback, for both normal hearing and cochlear hearing loss conditions. These theories were tested using an established model of the auditory periphery and signal detection theory techniques. The influence of several analysis variables on predicted tone-pip detection in broadband noise was evaluated, including: auditory nerve fiber spontaneous-rate (SR) pooling, range of characteristic frequencies, number of synapses per characteristic frequency, analysis window duration, and detection rule. The results revealed that overshoot similar to perceptual data in terms of both magnitude and level dependence could be predicted when the effects of MOC efferent feedback were included in the auditory nerve model. Conversely, simulations without MOC feedback effects never produced overshoot despite the model’s ability to account for classic firing rate adaptation and dynamic range adaptation in auditory nerve responses. Cochlear hearing loss was predicted to reduce the size of overshoot only for model versions that included the effects of MOC efferent feedback. These findings suggest that overshoot in normal and hearing-impaired listeners is mediated by some form of dynamic range adaptation other than what is observed in the auditory nerve of anesthetized animals. Mechanisms for this adaptation may occur at several levels along the auditory pathway. Among these mechanisms, the MOC reflex may play a leading role