Nonlinear dynamics and chaos in a vocal-ventricular fold system.

In humans, ventricular folds are located superiorly to the vocal folds. Under special circumstances such as voice pathology or singing, they vibrate together with the vocal folds to contribute to the production of voice. In the present study, experimental data measured from physical models of the vocal and ventricular folds were analyzed in the light of nonlinear dynamics. The physical models provide a useful experimental framework to study the biomechanics of human vocalizations. Of particular interest in this experiment are co-oscillations of the vocal and ventricular folds, occasionally accompanied by irregular dynamics. We show that such a system can be regarded as two coupled oscillators, which give rise to various cooperative behaviors such as synchronized oscillations with a 1:1 or 1:2 frequency ratio and desynchronized oscillations with torus or chaos. The insight gained from the view of nonlinear dynamics should be of significant use for the diagnosis of voice pathologies, such as ventricular fold dysphonia.

Potential negative effect of total parenteral nutrition on the human circadian clock.

When patients cannot eat on their own, total parenteral nutrition (TPN) is a clinically beneficial method of maintaining nutrition. However, many animal studies have demonstrated that circadian rhythms are strongly affected by feeding time, raising the concern that continuous TPN around the clock may have an unexpected negative impact on the circadian clock of patients. To investigate this concern, we compared clock gene expression of aged subjects with or without TPN using hair follicle cells and found that while none of the non-TPN subjects showed any obvious defects in circadian rhythms of peripheral clock gene expression, a portion of aged subjects receiving continuous TPN showed abnormal circadian rhythms in peripheral clocks. Continuous TPN around the clock may therefore potentially perturb peripheral circadian rhythms, giving rise to the proposal that TPN needs to be administered with consideration to time factors.

A spatial model of the plant circadian clock reveals design principles for coordinated timing.

Individual plant cells possess a genetic network, the circadian clock, that times internal processes to the day-night cycle. Mathematical models of the clock are typically either "whole-plant" that ignore tissue or cell type-specific clock behavior, or "phase-only" that do not include molecular components. To address the complex spatial coordination observed in experiments, here we implemented a clock network model on a template of a seedling. In our model, the sensitivity to light varies across the plant, and cells communicate their timing via local or long-distance sharing of clock components, causing their rhythms to couple. We found that both varied light sensitivity and long-distance coupling could generate period differences between organs, while local coupling was required to generate the spatial waves of clock gene expression observed experimentally. We then examined our model under noisy light-dark cycles and found that local coupling minimized timing errors caused by the noise while allowing each plant region to maintain a different clock phase. Thus, local sensitivity to environmental inputs combined with local coupling enables flexible yet robust circadian timing.

Ventricular fold oscillations lower the vocal pitch in rhesus macaques.

We carried out ex vivo and in vivo experiments to explore the functional role of the ventricular folds in sound production in macaques. In the ex vivo experiments, 29 recordings out of 67 showed that the ventricular folds co-oscillated with the vocal folds. Transitions from normal vocal fold oscillations to vocal-ventricular fold co-oscillations as well as chaotic irregular oscillations were also observed. The in vivo experiments indicated that the vocal-ventricular fold co-oscillations were also observed in two macaque individuals. In both ex vivo and in vivo experiments, the vocal-ventricular fold co-oscillations significantly lowered the fundamental frequency. A mathematical model revealed that the lowering of the fundamental frequency was caused by a low oscillation frequency inherent in the ventricular folds, which entrained the vocal folds to their low-frequency oscillations. From a physiological standpoint, the macaques may utilize the ventricular fold oscillations more frequently than humans. The advantages as well as disadvantages of using the ventricular folds as an additional vocal repertory are discussed.

Experimental study on the quasi-steady approximation of glottal flows.

To examine the quasi-steady approximation of the glottal flow, widely used in the modeling of vocal fold oscillations, intraglottal pressure distributions were measured in a scaled-up static vocal fold model under time-varying flow conditions. The left and right vocal folds were slightly open and set to a symmetric and oblique configuration with a divergence angle. To realize time-varying flow conditions, the flow rate was sinusoidally modulated with a frequency of 2 and 10 Hz, which correspond to 112.5 and 562.5 Hz, respectively, in real life. Measurements of the intraglottal pressures under both steady and time-varying flows revealed that the pressure profiles of the time-varying flow conditions are non-distinguishable from those of the steady flow conditions as far as they have the same subglottal pressure as an input pressure. The air-jet separation point was also non-distinguishable between the steady and the time-varying flow conditions. Our study therefore suggests that the time-varying glottal flow can be approximated as a series of steady flow states with a matching subglottal pressure in the range of normal vocalization frequencies. Since the glottal closure was not taken into account in the present experiment, our argument is valid except for such a critical situation.

Electrical coupling controls dimensionality and chaotic firing of inferior olive neurons.

We previously proposed, on theoretical grounds, that the cerebellum must regulate the dimensionality of its neuronal activity during motor learning and control to cope with the low firing frequency of inferior olive neurons, which form one of two major inputs to the cerebellar cortex. Such dimensionality regulation is possible via modulation of electrical coupling through the gap junctions between inferior olive neurons by inhibitory GABAergic synapses. In addition, we previously showed in simulations that intermediate coupling strengths induce chaotic firing of inferior olive neurons and increase their information carrying capacity. However, there is no in vivo experimental data supporting these two theoretical predictions. Here, we computed the levels of synchrony, dimensionality, and chaos of the inferior olive code by analyzing in vivo recordings of Purkinje cell complex spike activity in three different coupling conditions: carbenoxolone (gap junctions blocker), control, and picrotoxin (GABA-A receptor antagonist). To examine the effect of electrical coupling on dimensionality and chaotic dynamics, we first determined the physiological range of effective coupling strengths between inferior olive neurons in the three conditions using a combination of a biophysical network model of the inferior olive and a novel Bayesian model averaging approach. We found that effective coupling co-varied with synchrony and was inversely related to the dimensionality of inferior olive firing dynamics, as measured via a principal component analysis of the spike trains in each condition. Furthermore, for both the model and the data, we found an inverted U-shaped relationship between coupling strengths and complexity entropy, a measure of chaos for spiking neural data. These results are consistent with our hypothesis according to which electrical coupling regulates the dimensionality and the complexity in the inferior olive neurons in order to optimize both motor learning and control of high dimensional motor systems by the cerebellum.

Experimental study of vocal-ventricular fold oscillations in voice production.

Ventricular folds are located in the supraglottal region above the vocal folds. Although the ventricular folds do not vibrate under normal vocalizations, they vibrate under certain conditions, e.g., throat singing or ventricular fold dysphonia. In throat singing, the ventricular folds vibrate at the same frequency as (or at integer ratios of) the vocal fold vibration frequency. In ventricular fold dysphonia, on the other hand, the ventricular folds interfere with the vocal folds, giving rise to a hoarse voice. In the present study, the synthetic larynx model was utilized to examine the vocal-ventricular fold oscillations. Our experiments revealed that the vocal and ventricular folds can co-oscillate at the same frequency with an out-of-phase relation. Compared to the control condition, under which no ventricular folds exist, the phonation threshold pressure was increased in the presence of the ventricular folds. Acoustic analysis indicated that jitter was reduced and vocal efficiency was increased by the ventricular folds. Distance between the vocal and ventricular folds did not alter these oscillation properties. A computational model was further simulated to elucidate the mechanism underlying the observed vocal-ventricular fold oscillations. It has been suggested that out-of-phase oscillations of the vocal and ventricular folds are important for sustaining periodic laryngeal vibrations.

Weak coupling between intracellular feedback loops explains dissociation of clock gene dynamics.

Circadian rhythms are generated by interlocked transcriptional-translational negative feedback loops (TTFLs), the molecular process implemented within a cell. The contributions, weighting and balancing between the multiple feedback loops remain debated. Dissociated, free-running dynamics in the expression of distinct clock genes has been described in recent experimental studies that applied various perturbations such as slice preparations, light pulses, jet-lag, and culture medium exchange. In this paper, we provide evidence that this "presumably transient" dissociation of circadian gene expression oscillations may occur at the single-cell level. Conceptual and detailed mechanistic mathematical modeling suggests that such dissociation is due to a weak interaction between multiple feedback loops present within a single cell. The dissociable loops provide insights into underlying mechanisms and general design principles of the molecular circadian clock.

Reducing the complexity of mathematical models for the plant circadian clock by distributed delays.

A major bottleneck in the modelling of biological networks is the parameter explosion problem - the exponential increase in the number of parameters that need to be optimised to data as the size of the model increases. Here, we address this problem in the context of the plant circadian clock by applying the method of distributed delays. We show that using this approach, the system architecture can be simplified efficiently - reducing the number of parameters - whilst still preserving the core mechanistic dynamics of the gene regulatory network. Compared to models with discrete time-delays, which are governed by functional differential equations, the distributed delay models can be converted into sets of equivalent ordinary differential equations, enabling the use of standard methods for numerical integration, and for stability and bifurcation analyses. We demonstrate the efficiency of our modelling approach by applying it to three exemplar mathematical models of the Arabidopsis circadian clock of varying complexity, obtaining significant reductions in complexity in each case. Moreover, we revise one of the most up-to-date Arabidopsis models, updating the regulation of the PRR9 and PRR7 genes by LHY in accordance with recent experimental data. The revised model more accurately reproduces the LHY-induction experiments of core clock genes, compared with the original model. Our work thus shows that the method of distributed delays facilitates the optimisation and reformulation of genetic network models.

Coherency of circadian rhythms in the SCN is governed by the interplay of two coupling factors.

Circadian clocks are autonomous oscillators driving daily rhythms in physiology and behavior. In mammals, a network of coupled neurons in the suprachiasmatic nucleus (SCN) is entrained to environmental light-dark cycles and orchestrates the timing of peripheral organs. In each neuron, transcriptional feedbacks generate noisy oscillations. Coupling mediated by neuropeptides such as VIP and AVP lends precision and robustness to circadian rhythms. The detailed coupling mechanisms between SCN neurons are debated. We analyze organotypic SCN slices from neonatal and adult mice in wild-type and multiple knockout conditions. Different degrees of rhythmicity are quantified by pixel-level analysis of bioluminescence data. We use empirical orthogonal functions (EOFs) to characterize spatio-temporal patterns. Simulations of coupled stochastic single cell oscillators can reproduce the diversity of observed patterns. Our combination of data analysis and modeling provides deeper insight into the enormous complexity of the data: (1) Neonatal slices are typically stronger oscillators than adult slices pointing to developmental changes of coupling. (2) Wild-type slices are completely synchronized and exhibit specific spatio-temporal patterns of phases. (3) Some slices of Cry double knockouts obey impaired synchrony that can lead to co-existing rhythms ("splitting"). (4) The loss of VIP-coupling leads to desynchronized rhythms with few residual local clusters. Additional information was extracted from co-culturing slices with rhythmic neonatal wild-type SCNs. These co-culturing experiments were simulated using external forcing terms representing VIP and AVP signaling. The rescue of rhythmicity via co-culturing lead to surprising results, since a cocktail of AVP-antagonists improved synchrony. Our modeling suggests that these counter-intuitive observations are pointing to an antagonistic action of VIP and AVP coupling. Our systematic theoretical and experimental study shows that dual coupling mechanisms can explain the astonishing complexity of spatio-temporal patterns in SCN slices.

A practical method for estimating coupling functions in complex dynamical systems.

A foremost challenge in modern network science is the inverse problem of reconstruction (inference) of coupling equations and network topology from the measurements of the network dynamics. Of particular interest are the methods that can operate on real (empirical) data without interfering with the system. One such earlier attempt (Tokuda et al. 2007 Phys. Rev. Lett. 99, 064101. (doi:10.1103/PhysRevLett.99.064101)) was a method suited for general limit-cycle oscillators, yielding both oscillators' natural frequencies and coupling functions between them (phase equations) from empirically measured time series. The present paper reviews the above method in a way comprehensive to domain-scientists other than physics. It also presents applications of the method to (i) detection of the network connectivity, (ii) inference of the phase sensitivity function, (iii) approximation of the interaction among phase-coherent chaotic oscillators, and (iv) experimental data from a forced Van der Pol electric circuit. This reaffirms the range of applicability of the method for reconstructing coupling functions and makes it accessible to a much wider scientific community. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Experimental study on nonlinear source-filter interaction using synthetic vocal fold models.

Under certain conditions, e.g., singing voice, the fundamental frequency of the vocal folds can go up and interfere with the formant frequencies. Acoustic feedback from the vocal tract filter to the vocal fold source then becomes strong and non-negligible. An experimental study was presented on such source-filter interaction using three types of synthetic vocal fold models. Asymmetry was also created between the left and right vocal folds. The experiment reproduced various nonlinear phenomena, such as frequency jump and quenching, as reported in humans. Increase in phonation threshold pressure was also observed when resonant frequency of the vocal tract and fundamental frequency of the vocal folds crossed each other. As a combined effect, the phonation threshold pressure was further increased by the left-right asymmetry. Simulation of the asymmetric two-mass model reproduced the experiments to some extent. One of the intriguing findings of this study is the variable strength of the source-filter interaction over different model types. Among the three models, two models were strongly influenced by the vocal tract, while no clear effect of the vocal tract was observed in the other model. This implies that the level of source-filter interaction may vary considerably from one subject to another in humans.

Photic phase-response curves for cycling female mice.

The photic entrainment system is critical for the internal circadian clock to be synchronized by external time cues. In nocturnal rodents, exposure to light during the early subjective night causes a phase delay, whereas it causes a phase advance during the late subjective night. This is represented by a phase-response curve (PRC). The PRC of females has not been well studied due to their estrous cycles. Our aim in this study was to understand the characteristics of photic entrainment in female cycling rodents and identify differences in photic entrainment among the stages of the estrous cycle. To establish two types of PRC, immediate PRC (iPRC) and steady state PRC (ssPRC), in each stage of the estrous cycle, we recorded circadian rhythms of wheel running activity, applying a 15-min light pulse to cycling female mice in constant darkness. In the iPRC, which was evaluated on the next day of the light pulse, the amount of phase shift in the diestrus was larger than that in the metestrus stage at circadian time (CT) 2. Similarly, the amount of phase shift in metestrus was larger than that in proestrus at CT 10. In the ssPRC, which was evaluated after completion of a new steady state, no significant estrous variations in the amount of photic phase shifts were detected for any CTs. Although these results indicate that the intrinsic photic entrainment system is not influenced by the estrous cycle, it may affect photoreception and cause sudden behavioral changes.

Synchronization of two coupled turbulent fires.

We numerically study the scale-free nature of a buoyancy-induced turbulent fire and synchronization of two coupled turbulent fires. A scale-free structure is detected in weighted networks between vortices, while its lifetime obeys a clear power law, indicating intermittent appearances, disappearances, and reappearances of the scale-free property. A significant decrease in the distance between the two fire sources gives rise to a synchronized state in the near field dominated by the unstable motion of transverse vortex rings. The synchronized state vanishes in the far field forming well-developed turbulent plumes, regardless of the distance between the two fire sources.

Grasshopper mice employ distinct vocal production mechanisms in different social contexts.

Functional changes in vocal organ morphology and motor control facilitate the evolution of acoustic signal diversity. Although many rodents produce vocalizations in a variety of social contexts, few studies have explored the underlying production mechanisms. Here, we describe mechanisms of audible and ultrasonic vocalizations (USVs) produced by grasshopper mice (genus Onychomys). Grasshopper mice are predatory rodents of the desert that produce both loud, long-distance advertisement calls and USVs in close-distance mating contexts. Using live-animal recording in normal air and heliox, laryngeal and vocal tract morphological investigations, and biomechanical modelling, we found that grasshopper mice employ two distinct vocal production mechanisms. In heliox, changes in higher-harmonic amplitudes of long-distance calls indicate an airflow-induced tissue vibration mechanism, whereas changes in fundamental frequency of USVs support a whistle mechanism. Vocal membranes and a thin lamina propria aid in the production of long-distance calls by increasing glottal efficiency and permitting high frequencies, respectively. In addition, tuning of fundamental frequency to the second resonance of a bell-shaped vocal tract increases call amplitude. Our findings indicate that grasshopper mice can dynamically adjust motor control to suit the social context and have novel morphological adaptations that facilitate long-distance communication.

Effect of level difference between left and right vocal folds on phonation: Physical experiment and theoretical study.

As an alternative factor to produce asymmetry between left and right vocal folds, the present study focuses on level difference, which is defined as the distance between the upper surfaces of the bilateral vocal folds in the inferior-superior direction. Physical models of the vocal folds were utilized to study the effect of the level difference on the phonation threshold pressure. A vocal tract model was also attached to the vocal fold model. For two types of different models, experiments revealed that the phonation threshold pressure tended to increase as the level difference was extended. Based upon a small amplitude approximation of the vocal fold oscillations, a theoretical formula was derived for the phonation threshold pressure. This theory agrees with the experiments, especially when the phase difference between the left and right vocal folds is not extensive. Furthermore, an asymmetric two-mass model was simulated with a level difference to validate the experiments as well as the theory. The primary conclusion is that the level difference has a potential effect on voice production especially for patients with an extended level of vertical difference in the vocal folds, which might be taken into account for the diagnosis of voice disorders.

Coupling Controls the Synchrony of Clock Cells in Development and Knockouts.

In mammals, a network of coupled neurons within the hypothalamus coordinates physiological rhythms with daily changes in the environment. In each neuron, delayed negative transcriptional feedbacks generate oscillations, albeit noisy and unreliable ones. Coupling mediated by diffusible neuropeptides lends precision and robustness to circadian rhythms. The double knockout of Cryptochrome Cry turns adult mice arrhythmic. But, remarkably, double knockout neonates continue to show robust oscillation much like wild-type neonates and appear to lose rhythmicity with development. We study quantitatively dispersed neurons and brain slices from wild-type and Cry double knockout mice to understand the links between single cell rhythmicity and intercellular coupling. We quantify oscillator properties of dispersed cells using nonlinear regression and study bifurcations diagrams of network models. We find that varying just three parameters-oscillator strength, strength of coupling, and timing of coupling-can reproduce experimentally observed features. In particular, modeling reveals that minor changes in timing of coupling can destroy synchronization as observed in adult slices from knockout mice.

Functional morphology of the Alligator mississippiensis larynx with implications for vocal production.

Sauropsid vocalization is mediated by the syrinx in birds and the larynx in extant reptiles; but whereas avian vocal production has received much attention, the vocal mechanism of basal reptilians is poorly understood. The American alligator (Alligator mississippiensis) displays a large vocal repertoire during mating and in parent-offspring interactions. Although vocal outputs of these behaviors have received some attention, the underlying mechanism of sound production remains speculative. Here, we investigate the laryngeal anatomy of juvenile and adult animals by macroscopic and histological methods. Observations of the cartilaginous framework and associated muscles largely corroborate earlier findings, but one muscle, the cricoarytenoideus, exhibits a heretofore unknown extrinsic insertion that has important implications for effective regulation of vocal fold length and tension. Histological investigation of the larynx revealed a layered vocal fold morphology. The thick lamina propria consists of non-homogenous extracellular matrix containing collagen fibers that are tightly packed below the epithelium but loosely organized deep inside the vocal fold. We found few elastic fibers but comparatively high proportions of hyaluronan. Similar organizational complexity is also seen in mammalian vocal folds and the labia of the avian syrinx: convergent morphologies that suggest analogous mechanisms for sound production. In tensile tests, alligator vocal folds demonstrated a linear stress-strain behavior in the low strain region and nonlinear stress responses at strains larger than 15%, which is similar to mammalian vocal fold tissue. We have integrated morphological and physiological data in a two-mass vocal fold model, providing a systematic description of the possible acoustic space that could be available to an alligator larynx. Mapping actual call production onto possible acoustic space validates the model's predictions.

An extended mathematical model for reproducing the phase response of Arabidopsis thaliana under various light conditions.

Experimental studies showed that light qualities such as color and strength influence the phase response properties of plant circadian systems. These effects, however, have yet to be properly addressed in theoretical models of plant circadian systems. To fill this gap, the present paper develops a mathematical model of a plant circadian clock that takes into account the intensity and wavelength of the input light. Based on experimental knowledge, we model three photoreceptors, Phytochrome A, Phytochrome B, and Cryptochrome 1, which respond to red and/or blue light, in Arabidopsis thaliana. The three photoreceptors are incorporated into a standard mathematical model of the plant system, in which activator and repressor genes form a single feedback loop. The model capability is examined by a phase response curve (PRC), which plots the phase shifts elicited by the light perturbation as a function of the perturbation phase. Numerical experiments demonstrate that the extended model reproduces the essential features of the PRCs measured experimentally under various light conditions. Particularly, unlike conventional models, the model generates the inherent shape of the PRC under dark pulse stimuli. The outcome of our modeling approach may motivate future theoretical and experimental studies of plant circadian rhythms.

The source-filter theory of whistle-like calls in marmosets: Acoustic analysis and simulation of helium-modulated voices.

Whistle-like high-pitched "phee" calls are often used as long-distance vocal advertisements by small-bodied marmosets and tamarins in the dense forests of South America. While the source-filter theory proposes that vibration of the vocal fold is modified independently from the resonance of the supralaryngeal vocal tract (SVT) in human speech, a source-filter coupling that constrains the vibration frequency to SVT resonance effectively produces loud tonal sounds in some musical instruments. Here, a combined approach of acoustic analyses and simulation with helium-modulated voices was used to show that phee calls are produced principally with the same mechanism as in human speech. The animal keeps the fundamental frequency (f0) close to the first formant (F1) of the SVT, to amplify f0. Although f0 and F1 are primarily independent, the degree of their tuning can be strengthened further by a flexible source-filter interaction, the variable strength of which depends upon the cross-sectional area of the laryngeal cavity. The results highlight the evolutionary antiquity and universality of the source-filter model in primates, but the study can also explore the diversification of vocal physiology, including source-filter interaction and its anatomical basis in non-human primates.