Short Term Frequency Instability Effects in Networks of Coupled Oscillators

Coupled oscillator networks (CONs) occur in various domains such as biology and electronics. The nonlinear phenomenon of “injection locking” leads to frequency synchronization among the oscillators in a CON. In this chapter, we first obtain simplified tools to analyze the phenomenon of injection locking and present the Generalized Adler’s equation. The Generalized Adler’s equation is applicable for analyzing the injection locking in any type of oscillator. We demonstrate the use of Generalized Adler’s equation for injection locking analysis on a ring oscillator and a Hodgkin-Huxley neuronal oscillator. Next, we describe efficient system-level simulation techniques (synchronized steady state and transient) for coupled oscillator networks using the Perturbation Projection Vector (PPV) phase macromodel of the oscillator. The coupled oscillator simulation techniques are generic in nature and can be applied for the simulation of any coupled oscillator network, like the ones found in biological or electronic systems. We apply our system-level techniques on a −G m LC (harmonic) and a Hodgkin-Huxley (non-harmonic) CON.

Coupled oscillator networks occur in various domains such as biology, astrophysics and electronics. In this paper, we present a comprehensive procedure for rapid and accurate simulation of large coupled oscillator networks using widely accepted, fully-nonlinear perturbation projection vector (PPV) phase macromodels. We validate our method against full simulation of 20times20 coupled network of Brusselator biochemical oscillator and obtain computational speedups of 170x over full simulation. Furthermore, we apply the method to study self-organization phenomenon of Brusselator under asymmetric coupling and time period variations.

The dynamical tolerance of coupled oscillator networks against local failures is studied. As the fraction of failed oscillator nodes gradually increases, the mean oscillation amplitude in the entire network decreases and then suddenly vanishes at a critical fraction as a phase transition. This critical fraction, widely used as a measure of the network robustness, was analytically derived for random failures but not for targeted attacks so far. Here we derive the general formula for the critical fraction, which can be applied to both random failures and targeted attacks. We consider the effects of targeting oscillator nodes based on their degrees. First we deal with coupled identical oscillators with homogeneous edge weights. Then our theory is applied to networks with heterogeneous edge weights and to those with nonidentical oscillators. The analytical results are validated by numerical experiments. Our results reveal the key factors governing the robustness and fragility of oscillator networks.

In this chapter we describe phenomenon of ragged synchronizability. First we give a theoretical introduction about detecting of the complete synchronization in the network of coupled systems. As an example we choose a network of non-diagonally coupled forced van der Pol oscillators.We present construction of the single electrical van der Pol oscillator and array of such systems. We show numerical and experimental results of the existence of the complete synchronization in the array and we discuss it in the very detailed way. We also consider the synchronization in clusters, experimentally in the small array and numerically in the larger network. Our numerical simulations show good agreement with the experimental observations.

The synchronization problems of networks of coupled harmonic oscillators are addressed by the edge event-triggered approach in this paper. The network dynamics with respect to edge states are presented and a new edge event-triggered control protocol is designed. Combined with the periodic event-detecting and edge event-triggered approach, sufficient conditions that guarantee the synchronization of coupled harmonic oscillators are presented. Two event-detecting rules are given to achieve the synchronization of coupled harmonic oscillators with low resource consumption. Finally, simulations are conducted to illustrate the effectiveness of the edge event-triggered control algorithm.

In modern times, satellite-based global positioning and navigation systems, such as the GPS, include precise time-keeping devices, e.g. atomic clocks, which are crucial for navigation and for a wide range of economic and industrial applications. However, precise timing might not be available when the environment renders satellite equipment inoperable. In response to this critical need, we have been carrying out, over the past three years, theory and preliminary experiments [Buono et al., 2018a; Buono et al., 2018b; Palacios et al., 2020], towards developing a novel and inexpensive precision timing device that can function independently of GPS availability. The fundamental idea is to exploit collective behavior generated by networks of coupled nonlinear oscillators. Common sense may suggest that synchronized oscillations may lead to higher accuracy. Previous works show, however, that it is not synchronization but rather, traveling wave patterns, in which consecutive oscillators are out of phase by a constant amount, that can better reduce the negative effects of noise and material imperfections which cause phase drift. In this work we advance the state-of-art in the network-based concept by studying, mainly computationally, collective behavior in networks of Colpitts oscillators. These type of oscillators are chosen because they offer a wide range of advantages (such as the ability to tune up the oscillations over a broad frequency range). The results highlight the regions of parameter space, including coupling strength, where traveling wave patterns have the largest basins of attraction and the ability to reduce phase drift by a [Formula: see text] scaling law, where [Formula: see text] is the number of oscillators in the network. The results should also provide guidelines for follow-up design and fabrication tasks of a network-based technology for precision timing.

High output power density millimeterwave (mm-wave) signal sources are demonstrated at 60 and 140 GHz by a magnetic plasmon waveguide (MPW) based in-phase coupled oscillator network (CON) in 65-nm CMOS. Each oscillator unit-cell is designed by an MPW-based low-loss inter-digital zero-phase coupler (ZPC). With multiple in-phase coupled ZPC unit-cells, a high output power mm-wave signal is generated at the center of the CON by an in-phase power combination. A 60-GHz CON-based signal source is implemented with four in-phase CON unit-cells at 60 GHz in a compact core chip area of 0.11 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . It is measured with 2-mW peak differential output power, 2.2% power efficiency, 18.2-mW/mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> output power density, -113.9-dBc/Hz phase noise at 10-MHz offset, and 15.8% frequency tuning range (FTR) centered at 63.1 GHz. Moreover, a 140-GHz signal source is implemented by the combination of second-order harmonics of four in-phase coupled ZPC unit-cells at 70 GHz in a compact core chip area of 0.13 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . It is measured with 3.5-mW peak output power, 2.4% power efficiency, 26.9-mW/mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> output power density, -104.9-dBc/Hz phase noise at 25-MHz offset, and 9.7% FTR centered at 133.5 GHz.

A metamaterial-based zero-phase coupler (ZPC) is proposed for coupled oscillator network (CON) to build high output power density and low phase noise 60GHz signal source in CMOS. The ZPC itself is both high-Q resonator and low-loss zero-phase shifter. The unit-cell oscillator of CON can be implemented by the ZPC, which has minimized loss for in-phase coupling when connecting with other ZPC-based unit-cells. The frequency tuning range is realized by inductive tuning at each ZPC unit-cell. A 4-unit-cell ZPC-based CON is demonstrated at 60GHz in 65nm CMOS process with measured maximum output power of 2mW, power efficiency of 2.2%, phase noise of -116.7dBc/Hz at 10MHz offset and frequency tuning range of 15.8% centered at 63.5GHz.

Neuromorphic computing is a wide research field aimed to the realization of brain-inspired hardware, apt to tackle computation of unstructured data more efficiently than currently done with standard computational units. Oscillatory neural networks are known for their associative memory capability, which enables to retrieve the information stored in the system from noisy or incomplete data. The development of phase-transition materials such as vanadiumdioxide (VO2) allows to design compact relaxation oscillator units which can be coupled in frequency and phase to realize an oscillatory neural network in hardware. In this thesis, we investigate the oscillatory neural network technology from the realization of the basic oscillator components with VO2 to the exploitation of the coupled oscillators as analog filters in convolutional neural networks applications. VO2 phase-transition devices are realized in a CMOS compatible process in two geometries, a planar and a crossbar configuration. The impact of the polycrystallinity of the VO2 film on the insulator-to-metal transition of the device is analyzed; through the contacting of a single grain we demonstrate the realization of a VO2 device with a single, sharp phase transition. The VO2 devices are connected in circuits to build networks of coupled oscillators. Through coupling with resistive and capacitive elements, experimental demonstrations of a 4-VO2 coupled oscillator network is shown. The network encodes the input and output information in the relative phase of the oscillators. The associative memory capability of the system is used to extract features from hand-written digits. By expanding the network to a 3×3 coupled oscillator system, we demonstrate in simulations how an oscillatory neural network can replace up to five digital filters in a convolutional neural network, retaining the same image processing capabilities.

In this chapter, we analyze multicluster states in a network of adaptively coupled phase oscillators. Multicluster states are composed of several one-clusters with distinct frequencies. Starting from random initial conditions, our numerical simulations show two different types of states. These are the splay and the antipodal type multicluster states. A third mixed type multicluster state is found by specially prepared initial conditions. For all these states the collective motion of oscillators, the shape of the coupling structure, and the interaction between the frequency clusters are presented in detail. It turns out that due to the adaptive dynamics of the coupling weights, the oscillators are able to form groups of strongly connected units. The interaction between the groups is weak compared to the interaction within the groups. We provide an analytical description for the existence of multicluster states by using methods from asymptotic analysis. Employing the fact of weak inter-cluster coupling, we find stability conditions for multicluster states.

We devised a new method of data mining for a large-scale database. In the method, a network of locally coupled phase oscillators subject to Kuramoto's model substitutes for given multivariate data to generate major features through phase locking of the oscillators, i.e., phase transition of the data set. We applied the method to the national database of care needs certification for the Japanese public long-term care insurance program, and found three major patterns in the aging process of the frail elderly. This work revealed the latent utility of Kuramoto's model for data processing.

This study presents three EEG/MEG applications in which the modeling of oscillatory signal components offers complementary analysis and an improved explanation of the underlying generator structures. Coupled oscillator networks were used for modeling. Parameters of the corresponding ordinary coupled differential equation (ODE) system are identified using EEG/MEG data and the resulting solution yields the modeled signals. This model-related analysis strategy provides information about the coupling quantity and quality between signal components (example 1, neonatal EEG during quiet sleep), allows identification of the possible contribution of hidden generator structures (example 2, 600-Hz MEG oscillations in somatosensory evoked magnetic fields), and can explain complex signal characteristics such as amplitude-frequency coupling and frequency entrainment (example 3, EEG burst patterns in sedated patients).

Coupled oscillator networks often display transitions between qualitatively different phase-locked solutions—such as synchrony and rotating wave solutions—following perturbation or parameter variation. In the limit of weak coupling, these transitions can be understood in terms of commonly studied phase approximations. As the coupling strength increases, however, predicting the location and criticality of transition, whether continuous or discontinuous, from the phase dynamics may depend on the order of the phase approximation—or a phase description of the network dynamics that neglects amplitudes may become impossible altogether. Here we analyze synchronization transitions and their criticality systematically for varying coupling strength in theory and experiments with coupled electrochemical oscillators. First, we analyze bifurcations analysis of synchrony and splay states in an abstract phase model and discuss conditions under which synchronization transitions with different criticalities are possible. In particular, we show that such conditions can be understood by considering the relative contributions of higher harmonics to the phase dynamics. Second, we illustrate that transitions with different criticality indeed occur in experimental systems. Third, we highlight that the amplitude dynamics observed in the experiments can be captured in a numerical bifurcation analysis of delay-coupled oscillators. Our results showcase that reduced order phase models may miss important features that one would expect in the dynamics of the full system. Published by the American Physical Society 2024

In this letter, we derive an exact and general closed-form expression for the bit error rate (BER) of Gray coded M-ary phase shift keying (MPSK) in the presence of a fixed phase error. We also propose a very simple and tight approximate expression from the derived exact expression. To validate the exactness of the proposed exact and general expression and the tightness of the proposed approximation, we compare the results obtained from the proposed expressions with those from computer simulations. We also present the average BER results of MPSK for random phase errors caused by phase-locked loop or imperfect channel estimation by using the derived expressions.

A recurrent hippocampal network model was constructed using mapped clock oscillators (MCOs) and shown to reliably generate spontaneous epileptiform activity under simulated hyperexcitable conditions. Such a network may prove useful as a platform for launching investigations into seizure control, and for complementing in vitro and animal models of epilepsy. More importantly, the therapeutic potential of MCO networks in the treatment of neurological disorders like epilepsy is only beginning to be realized