On the origin and evolution of the dual oscillator model underlying the photoperiodic clockwork in the suprachiasmatic nucleus

Decades have now passed since Colin Pittendrigh first proposed a model of a circadian clock composed of two coupled oscillators, individually responsive to the rising and setting sun, as a flexible solution to the challenge of behavioral and physiological adaptation to the changing seasons. The elegance and predictive power of this postulation has stimulated laboratories around the world in searches to identify and localize such hypothesized evening and morning oscillators, or sets of oscillators, in insects, rodents, and humans, with experimental designs and approaches keeping pace over the years with technological advances in biology and neuroscience. Here, we recount the conceptual origin and highlight the subsequent evolution of this dual oscillator model for the circadian clock in the mammalian suprachiasmatic nucleus; and how, despite our increasingly sophisticated view of this multicellular pacemaker, Pittendrigh's binary conception has remained influential in our clock models and metaphors.

Frequency Shifts in a Local Oscillator Model for the Generation of Spontaneous Otoacoustic Emissions by the Lizard Ear

INTRODUCTION

In order to understand human hearing, it helps to understand how the ears of lower vertebrates, like, for instance, lizards, function. A key feature in common is that the ears of both humans and lizards emit faint, pure tones known as spontaneous otoacoustic emissions (SOAEs). More than four decades after their discovery, the mechanism underlying these emissions is still imperfectly understood, although it is known that they are important for improving the sensitivity and sharpness of hearing. In both humans and lizards, the frequencies of SOAEs change by a few percent when static pressure is applied to the tympanic membrane. For the human ear, this observation is normally explained by a so-called global oscillator model (such as with Shera's coherent reflection model), in which the emissions result from standing waves, and external pressure changes the boundary conditions - the stiffness of the oval and round windows - which then has a global effect on the SOAE frequencies.

METHODS

Here we investigate how changing parameters of an earlier developed local oscillator model for the lizard ear can change the frequencies of the SOAEs. A major feature of the model is that each oscillator is coupled only to its immediate neighbours. The oscillators then cluster into groups of identical frequency, and each of these so-called frequency plateaus can be taken to represent an SOAE.

RESULTS

Even though the natural (unperturbed) frequencies of all the oscillators remain fixed, here we find for several model parameters that by slightly changing their value the frequency plateaus - the SOAEs - shift by a few percent. Plots of how these changes alter SOAE frequencies are given, and their magnitude corresponds well with observations of SOAE changes in lizards.

DISCUSSION

Investigation of the influence of the change of parameters in an earlier developed local oscillator model for the lizard ear shows that a local oscillator model can explain small SOAE frequency changes as well as a global oscillator model.

Patterns and Stability of Coupled Multi-Stable Nonlinear Oscillators

Nonlinear isolated and coupled oscillators are extensively studied as prototypical nonlinear dynamics models. Much attention has been devoted to oscillator synchronization or the lack thereof. Here, we study the synchronization and stability of coupled driven-damped Helmholtz-Duffing oscillators in bi-stability regimes. We find that despite the fact that the system parameters and the driving force are identical, the stability of the two states to spatially non-uniform perturbations is very different. Moreover, the final stable states, resulting from these spatial perturbations, are not solely dictated by the wavelength of the perturbing mode and take different spatial configurations in terms of the coupled oscillator phases.

Temporal variations in the pattern of breathing: techniques, sources, and applications to translational sciences

The breathing process possesses a complex variability caused in part by the respiratory central pattern generator in the brainstem; however, it also arises from chemical and mechanical feedback control loops, network reorganization and network sharing with nonrespiratory motor acts, as well as inputs from cortical and subcortical systems. The notion that respiratory fluctuations contain hidden information has prompted scientists to decipher respiratory signals to better understand the fundamental mechanisms of respiratory pattern generation, interactions with emotion, influences on the cortical neuronal networks associated with cognition, and changes in variability in healthy and disease-carrying individuals. Respiration can be used to express and control emotion. Furthermore, respiration appears to organize brain-wide network oscillations via cross-frequency coupling, optimizing cognitive performance. With the aid of information theory-based techniques and machine learning, the hidden information can be translated into a form usable in clinical practice for diagnosis, emotion recognition, and mental conditioning.

Phase response approaches to neural activity models with distributed delay

In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson-Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks.

Mathematical Modeling in Circadian Rhythmicity

Circadian clocks are autonomous systems able to oscillate in a self-sustained manner in the absence of external cues, although such Zeitgebers are typically present. At the cellular level, the molecular clockwork consists of a complex network of interlocked feedback loops. This chapter discusses self-sustained circadian oscillators in the context of nonlinear dynamics theory. We suggest basic steps that can help in constructing a mathematical model and introduce how self-sustained generations can be modeled using ordinary differential equations. Moreover, we discuss how coupled oscillators synchronize among themselves or entrain to periodic signals. The development of mathematical models over the last years has helped to understand such complex network systems and to highlight the basic building blocks in which oscillating systems are built upon. We argue that, through theoretical predictions, the use of simple models can guide experimental research and is thus suitable to model biological systems qualitatively.

Phase portraits as movement primitives for fast humanoid robot control

Currently, usual approaches for fast robot control are largely reliant on solving online optimal control problems. Such methods are known to be computationally intensive and sensitive to model accuracy. On the other hand, animals plan complex motor actions not only fast but seemingly with little effort even on unseen tasks. This natural sense to infer temporal dynamics and coordination motivates us to approach robot control from a motor skill learning perspective to design fast and computationally light controllers that can be learned autonomously by the robot under mild modeling assumptions. This article introduces Phase Portrait Movement Primitives (PPMP), a primitive that predicts dynamics on a low dimensional phase space which in turn is used to govern the high dimensional kinematics of the task. The stark difference with other primitive formulations is a built-in mechanism for phase prediction in the form of coupled oscillators that replaces model-based state estimators such as Kalman filters. The policy is trained by optimizing the parameters of the oscillators whose output is connected to a kinematic distribution in the form of a phase portrait. The drastic reduction in dimensionality allows us to efficiently train and execute PPMPs on a real human-sized, dual-arm humanoid upper body on a task involving 20 degrees-of-freedom. We demonstrate PPMPs in interactions requiring fast reactions times while generating anticipative pose adaptation in both discrete and cyclic tasks.

Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review

Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott-Antonsen and Watanabe-Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.

Sensorimotor synchronization during gait is altered by the addition of variability to an external cue

Sensorimotor synchronization has been used in the rehabilitation of gait, yet much remains unknown regarding the optimal use of this technique. The purpose of this study was to test the hypothesis that adding small amounts of variability to the motion of a vertically oscillating treadmill would affect the behavior of healthy walkers. Sixteen young adults walked on a treadmill and pneumatically actuated platform for one control trial (no oscillation) and eight trials in which the walking surface oscillated in the vertical direction under different conditions of variability. During the oscillation trials, the mean frequency of oscillation was equal to the preferred step frequency of the participant, but each individual cycle period was allowed to vary within a pre-determined range from 0% (no variability) to ±25% (high variability) of the mean cycle period. The amount of variance of each cycle period within each condition was drawn randomly from a white noise generator. Synchronization was improved when a small amount of noise was added to the platform motion but synchronization significantly decreased at higher levels of noise. Coefficient of variation of stride duration was relatively unchanged at lower levels of variability, but increased significantly at higher levels of variability. Statistical persistence of stride duration was significantly reduced during all trials with vertical oscillation relative to normal walking, but was not significantly altered by variability in the treadmill oscillation. These results suggest that the addition of a small amount of random variability to the cycle period of an oscillator may enhance sensorimotor synchronization of gait to an external signal. These data may have implications for the use of synchronization in a therapeutic setting.

Predicting Pattern Formation in Multilayer Networks

We investigate how the structure of interactions between coupled oscillators influences the formation of asynchronous patterns in a multilayer network by formulating a simple, general multilayer oscillator model. We demonstrate the analysis of this model in three-oscillator systems, illustrating the role of interactions among oscillators in sustaining differences in both the phase and amplitude of oscillations leading to the formation of asynchronous patterns. Finally, we demonstrate the generalizability of our model's predictions through comparison with a more realistic multilayer model. Overall, our model provides a useful approach for predicting the types of asynchronous patterns that multilayer networks of coupled oscillators which cannot be achieved by the existing methods which focus on characterizing the synchronous state.

Cochlear impulse responses resolved into sets of gammatones: the case for beating of closely spaced local resonances

Gammatones have had a long history in auditory studies, and recent theoretical work suggests they may play an important role in cochlear mechanics as well. Following this lead, the present paper takes five examples of basilar membrane impulse responses and uses a curve-fitting algorithm to decompose them into a number of discrete gammatones. The limits of this ‘sum of gammatones’ (SOG) method to accurately represent the impulse response waveforms were tested and it was found that at least two and up to six gammatones could be isolated from each example. Their frequencies were stable and largely independent of stimulus parameters. The gammatones typically formed a regular series in which the frequency ratio between successive members was about 1.1. Adding together the first few gammatones in a set produced beating-like waveforms which mimicked waxing and waning, and the instantaneous frequencies of the waveforms were also well reproduced, providing an explanation for frequency glides. Consideration was also given to the impulse response of a pair of elastically coupled masses—the basis of two-degree-of-freedom models comprised of coupled basilar and tectorial membranes—and the resulting waveform was similar to a pair of beating gammatones, perhaps explaining why the SOG method seems to work well in describing cochlear impulse responses. A major limitation of the SOG method is that it cannot distinguish a waveform resulting from an actual physical resonance from one derived from overfitting, but taken together the method points to the presence of a series of closely spaced local resonances in the cochlea.

Coupled Oscillators Model of Hyperexcitable Neuroglial Networks

Glial populations within neuronal networks of the brain have recently gained much interest in the context of hyperexcitability and epilepsy. In this paper, we present an oscillator-based neuroglial model capable of generating Spontaneous Electrical Discharges (SEDs) in hyperexcitable conditions. The network is composed of 16 coupled Cognitive Rhythm Generators (CRGs), which are oscillator-based mathematical constructs previously described by our research team. CRGs are well-suited for modeling assemblies of excitable cells, and in this network, each represents one of the following populations: excitatory pyramidal cells, inhibitory interneurons, astrocytes, and microglia. We investigated various pathways leading to hyperexcitability, and our results suggest an important role for astrocytes and microglia in the generation of SEDs of various durations. Analysis of the resultant SEDs revealed two underlying duration distributions with differing properties. Particularly, short and long SEDs are associated with deterministic and random underlying processes, respectively. The mesoscale of this model makes it well-suited for (a) the elucidation of glia-related hypotheses in hyperexcitable conditions, (b) use as a testing platform for neuromodulation purposes, and (c) a hardware implementation for closed-loop neuromodulation.

Capacitive coupling synchronizes autonomous microfluidic oscillators

Even identically designed autonomous microfluidic oscillators have device-to-device oscillation variability that arises due to inconsistencies in fabrication, materials, and operation conditions. This work demonstrates, experimentally and theoretically, that with appropriate capacitive coupling these microfluidic oscillators can be synchronized. The size and characteristics of the capacitive coupling needed and the range of input flow rate differences that can be synchronized are also characterized. In addition to device-to-device variability, there is also within-device oscillation noise that arises. An additional advantage of coupling multiple fluidic oscillators together is that the oscillation noise decreases. The ability to synchronize multiple autonomous oscillators is also a first step towards enhancing their usefulness as tools for biochemical research applications where multiplicate experiments with identical temporal-stimulation conditions are required.

Entrainment of Arteriole Vasomotor Fluctuations by Neural Activity Is a Basis of Blood-Oxygenation-Level-Dependent "Resting-State" Connectivity

Resting-state signals in blood-oxygenation-level-dependent (BOLD) imaging are used to parcellate brain regions and define "functional connections" between regions. Yet a physiological link between fluctuations in blood oxygenation with those in neuronal signaling pathways is missing. We present evidence from studies on mouse cortex that modulation of vasomotion, i.e., intrinsic ultra-slow (0.1 Hz) fluctuations in arteriole diameter, provides this link. First, ultra-slow fluctuations in neuronal signaling, which occur as an envelope over γ-band activity, entrains vasomotion. Second, optogenetic manipulations confirm that entrainment is unidirectional. Third, co-fluctuations in the diameter of pairs of arterioles within the same hemisphere diminish to chance for separations >1.4 mm. Yet the diameters of arterioles in distant (>5 mm), mirrored transhemispheric sites strongly co-fluctuate; these correlations are diminished in acallosal mice. Fourth, fluctuations in arteriole diameter coherently drive fluctuations in blood oxygenation. Thus, entrainment of vasomotion links neuronal pathways to functional connections.

A framework for quantification and physical modeling of cell mixing applied to oscillator synchronization in vertebrate somitogenesis

In development and disease, cells move as they exchange signals. One example is found in vertebrate development, during which the timing of segment formation is set by a ‘segmentation clock’, in which oscillating gene expression is synchronized across a population of cells by Delta-Notch signaling. Delta-Notch signaling requires local cell-cell contact, but in the zebrafish embryonic tailbud, oscillating cells move rapidly, exchanging neighbors. Previous theoretical studies proposed that this relative movement or cell mixing might alter signaling and thereby enhance synchronization. However, it remains unclear whether the mixing timescale in the tissue is in the right range for this effect, because a framework to reliably measure the mixing timescale and compare it with signaling timescale is lacking. Here, we develop such a framework using a quantitative description of cell mixing without the need for an external reference frame and constructing a physical model of cell movement based on the data. Numerical simulations show that mixing with experimentally observed statistics enhances synchronization of coupled phase oscillators, suggesting that mixing in the tailbud is fast enough to affect the coherence of rhythmic gene expression. Our approach will find general application in analyzing the relative movements of communicating cells during development and disease.

Summary: We develop a framework to quantify and model cell mixing independent of a choice of reference frames, and apply this to study oscillator synchronization in the zebrafish segmentation clock.

Oscillating systems with cointegrated phase processes

We present cointegration analysis as a method to infer the network structure of a linearly phase coupled oscillating system. By defining a class of oscillating systems with interacting phases, we derive a data generating process where we can specify the coupling structure of a network that resembles biological processes. In particular we study a network of Winfree oscillators, for which we present a statistical analysis of various simulated networks, where we conclude on the coupling structure: the direction of feedback in the phase processes and proportional coupling strength between individual components of the system. We show that we can correctly classify the network structure for such a system by cointegration analysis, for various types of coupling, including uni-/bi-directional and all-to-all coupling. Finally, we analyze a set of EEG recordings and discuss the current applicability of cointegration analysis in the field of neuroscience.

Wave Generation in Unidirectional Chains of Idealized Neural Oscillators

We investigate the dynamics of unidirectional semi-infinite chains of type-I oscillators that are periodically forced at their root node, as an archetype of wave generation in neural networks. In previous studies, numerical simulations based on uniform forcing have revealed that trajectories approach a traveling wave in the far-downstream, large time limit. While this phenomenon seems typical, it is hardly anticipated because the system does not exhibit any of the crucial properties employed in available proofs of existence of traveling waves in lattice dynamical systems. Here, we give a full mathematical proof of generation under uniform forcing in a simple piecewise affine setting for which the dynamics can be solved explicitly. In particular, our analysis proves existence, global stability, and robustness with respect to perturbations of the forcing, of families of waves with arbitrary period/wave number in some range, for every value of the parameters in the system.

Beyond in-phase and anti-phase coordination in a model of joint action

In 1985, Haken, Kelso and Bunz proposed a system of coupled nonlinear oscillators as a model of rhythmic movement patterns in human bimanual coordination. Since then, the Haken-Kelso-Bunz (HKB) model has become a modelling paradigm applied extensively in all areas of movement science, including interpersonal motor coordination. However, all previous studies have followed a line of analysis based on slowly varying amplitudes and rotating wave approximations. These approximations lead to a reduced system, consisting of a single differential equation representing the evolution of the relative phase of the two coupled oscillators: the HKB model of the relative phase. Here we take a different approach and systematically investigate the behaviour of the HKB model in the full four-dimensional state space and for general coupling strengths. We perform detailed numerical bifurcation analyses and reveal that the HKB model supports previously unreported dynamical regimes as well as bistability between a variety of coordination patterns. Furthermore, we identify the stability boundaries of distinct coordination regimes in the model and discuss the applicability of our findings to interpersonal coordination and other joint action tasks.

The vibrating reed frequency meter: digital investigation of an early cochlear model

The vibrating reed frequency meter, originally employed by Békésy and later by Wilson as a cochlear model, uses a set of tuned reeds to represent the cochlea’s graded bank of resonant elements and an elastic band threaded between them to provide nearest-neighbour coupling. Here the system, constructed of 21 reeds progressively tuned from 45 to 55 Hz, is simulated numerically as an elastically coupled bank of passive harmonic oscillators driven simultaneously by an external sinusoidal force. To uncover more detail, simulations were extended to 201 oscillators covering the range 1–2 kHz. Calculations mirror the results reported by Wilson and show expected characteristics such as traveling waves, phase plateaus, and a response with a broad peak at a forcing frequency just above the natural frequency. The system also displays additional fine-grain features that resemble those which have only recently been recognised in the cochlea. Thus, detailed analysis brings to light a secondary peak beyond the main peak, a set of closely spaced low-amplitude ripples, rapid rotation of phase as the driving frequency is swept, frequency plateaus, clustering, and waxing and waning of impulse responses. Further investigation shows that each reed’s vibrations are strongly localised, with small energy flow along the chain. The distinctive set of equally spaced ripples is an inherent feature which is found to be largely independent of boundary conditions. Although the vibrating reed model is functionally different to the standard transmission line, its cochlea-like properties make it an intriguing local oscillator model whose relevance to cochlear mechanics needs further investigation.

Respiratory calcium fluctuations in low-frequency oscillating astrocytes in the pre-Bötzinger complex

Astrocytes have been found to modulate neuronal activity through calcium-dependent signaling in various brain regions. However, whether astrocytes of the pre-Bötzinger complex (preBötC) exhibit respiratory rhythmic fluctuations is still controversial. Here we evaluated calcium-imaging experiments within preBötC in rhythmically active medullary slices from TgN(hGFAP-EGFP) mice using advanced analyses. 13.8% of EGFP-negative cells, putative neurons, showed rhythmic fluorescent changes that were highly correlated to the respiratory rhythmic fluctuation (cross-correlation coefficient>0.5 and dF/F>0.2%). In contrast, a considerable number of astrocyte somata exhibited synchronized low-frequency (<0.03Hz) calcium oscillations. After band-pass filtering, signals that irregularly preceded the calcium signal of EGFP-negative cells were observed in 10.2% of astrocytes, indicating a functional coupling between astrocytes and neurons in preBötC. A model simulation confirmed that such preinspiratory astrocytic signals can arise from coupled neuronal and astrocytic oscillators, supporting a concept that slow oscillatory changes of astrocytic functions modulate neighboring neuronal activity to add variability in respiratory rhythm.