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Journal articles on the topic 'Biological nonlinear oscillators'
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1
Murugesh, T. S., and M. Senthilkumar. "Control Techniques for Synchronizing the States of Two Coupled Van der Pol Oscillators." Asian Journal of Engineering and Applied Technology 5, no. 2 (2016): 44–50. http://dx.doi.org/10.51983/ajeat-2016.5.2.799.
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Mathematical models of nonlinear oscillators are used to describe a wide variety of physical and biological phenomena that exhibit self-sustained oscillatory behavior. When these oscillators are strongly driven by forces that are periodic in time, they often exhibit a remarkable ‘‘mode-locking’’ that synchronizes the nonlinear oscillations to the driving force. Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states and is characterized by their amplitude and their phase. Their interactions can result in a systematic process of synchronization which is the adjustment of rhythms of oscillating objects due to an interaction and is quite distinct from a simple stimulus response pattern. Oscillators respond to stimuli at some times in their cycle and may not respond at others. Many important physical, chemical and biological systems are composed of coupled nonlinear oscillators. The Van der Pol equation has been used to model a number of biological processes such as the heartbeat, circadian rhythms, biochemical oscillators, and pacemaker neurons. Two such resistively coupled Van der Pol oscillators are analyzed and the phenomenon of synchronization between the states of the coupled oscillators is explored. Several control techniques to achieve synchronization are designed, implemented and performance evaluation carried out by simulation using MATLAB Software.
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BODE, M., D. RUWISCH, P. SCHÜTZ, et al. "PARALLEL ANALOG COMPUTATION OF COUPLED BIOLOGICAL OSCILLATORS." Journal of Biological Systems 03, no. 01 (1995): 81–93. http://dx.doi.org/10.1142/s0218339095000083.
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In this work the dynamics of coupled nonlinear oscillators, which are ubiquitous in biology, is experimentally studied by using electrical relaxation oscillators. The results of this analog computation obtained with two and three coupled oscillators are in agreement with the results known from numerical approaches. Phase death, which is a mutual annihilation of oscillations, is a generic phenomenon. All modes known from approaches using identical oscillators have been found. Additionally we observed new generic modes that are caused by inhomogeneities of the oscillators, such differences being typical for biological cells. Simulations of excitable electrical oscillators yield similar results.
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Shcherbak, Volodymyr, and Iryna Dmytryshyn. "Estimation of oscillation velocities of oscillator network." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 32 (December 28, 2018): 182–89. http://dx.doi.org/10.37069/1683-4720-2018-32-17.
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The study of the collective behavior of multiscale dynamic processes is currently one of the most urgent problems of nonlinear dynamics. Such systems arise on modelling of many cyclical biological or physical processes. It is of fundamental importance for understanding the basic laws of synchronous dynamics of distributed active subsystems with oscillations, such as neural ensembles, biomechanical models of cardiac or locomotor activity, models of turbulent media, etc. Since the nonlinear oscillations that are observed in such systems have a stable limit cycle , which does not depend on the initial conditions, then a system of interconnected nonlinear oscillators is usually used as a model of multiscale processes. The equations of Lienar type are often used as the main dynamic model of each of these oscillators. In a number of practical control problems of such interconnected oscillators it is necessary to determine the oscillation velocities by known data. This problem is considered as observation problem for nonlinear dynamical system. A new method – a synthesis of invariant relations is used to design a nonlinear observer. The method allows us to represent unknowns as a function of known quantities. The scheme of the construction of invariant relations consists in the expansion of the original dynamical system by equations of some controlled subsystem (integrator). Control in the additional system is used for the synthesis of some relations that are invariant for the extended system and have the attraction property for all of its trajectories. Such relations are considered in observation problems as additional equations for unknown state vector of initial oscillators ensemble. To design the observer, first we introduce a observer for unique oscillator of Lienar type and prove its exponential convergence. This observer is then extended on several coupled Lienar type oscillators. The performance of the proposed method is investigated by numerical simulations.
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Indic, Premananda, William J. Schwartz, and David Paydarfar. "Design principles for phase-splitting behaviour of coupled cellular oscillators: clues from hamsters with ‘split’ circadian rhythms." Journal of The Royal Society Interface 5, no. 25 (2007): 873–83. http://dx.doi.org/10.1098/rsif.2007.1248.
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Nonlinear interactions among coupled cellular oscillators are likely to underlie a variety of complex rhythmic behaviours. Here we consider the case of one such behaviour, a doubling of rhythm frequency caused by the spontaneous splitting of a population of synchronized oscillators into two subgroups each oscillating in anti-phase ( phase-splitting ). An example of biological phase-splitting is the frequency doubling of the circadian locomotor rhythm in hamsters housed in constant light, in which the pacemaker in the suprachiasmatic nucleus (SCN) is reconfigured with its left and right halves oscillating in anti-phase. We apply the theory of coupled phase oscillators to show that stable phase-splitting requires the presence of negative coupling terms, through delayed and/or inhibitory interactions. We also find that the inclusion of real biological constraints (that the SCN contains a finite number of non-identical noisy oscillators) implies the existence of an underlying non-uniform network architecture, in which the population of oscillators must interact through at least two types of connections. We propose that a key design principle for the frequency doubling of a population of biological oscillators is inhomogeneity of oscillator coupling.
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Rosenblum, Michael, and Arkady Pikovsky. "Nonlinear phase coupling functions: a numerical study." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2160 (2019): 20190093. http://dx.doi.org/10.1098/rsta.2019.0093.
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Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here, we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the periodically forced Stuart–Landau oscillator as the paradigmatic model, we determine and numerically analyse the coupling functions up to the fourth order in the force strength. We show that the found nonlinear phase coupling functions can be used for predicting synchronization regions of the forced oscillator. This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences’.
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Padoan, Alberto, Fulvio Forni, and Rodolphe Sepulchre. "Balanced truncation for model reduction of biological oscillators." Biological Cybernetics 115, no. 4 (2021): 383–95. http://dx.doi.org/10.1007/s00422-021-00888-4.
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AbstractModel reduction is a central problem in mathematical biology. Reduced order models enable modeling of a biological system at different levels of complexity and the quantitative analysis of its properties, like sensitivity to parameter variations and resilience to exogenous perturbations. However, available model reduction methods often fail to capture a diverse range of nonlinear behaviors observed in biology, such as multistability and limit cycle oscillations. The paper addresses this need using differential analysis. This approach leads to a nonlinear enhancement of classical balanced truncation for biological systems whose behavior is not restricted to the stability of a single equilibrium. Numerical results suggest that the proposed framework may be relevant to the approximation of classical models of biological systems.
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NEIMAN, ALEXANDER B., and DAVID F. RUSSELL. "STOCHASTIC DYNAMICS OF ELECTRORECEPTORS IN PADDLEFISH." Fluctuation and Noise Letters 04, no. 01 (2004): L139—L149. http://dx.doi.org/10.1142/s0219477504001744.
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Electroreceptors in paddlefish serve as accessible and well-defined biological models for studying the functional roles in sensory nervous systems of noisy oscillations and the nonlinear phenomena associated with them, including synchronization, noise-induced transitions, and noise-induced bursting. The spontaneous dynamics of paddlefish electroreceptors show two oscillatory modes: one associated with 26 Hz oscillations in the sensory epithelia, and another with 30-65 Hz periodicities of afferent terminals. This novel type of organization of peripheral sensory receptors, with two distinct types of embedded oscillators, results in stochastic biperiodic firing patterns of primary afferents. The biperiodicity can be explained qualitatively in terms of a simple model based on a stochastic circle map. Stimulation with broadband Gaussian noise changes the tonic firing pattern of electroreceptors to a bursting mode, indicating a noise-induced transition. This qualitative change in dynamics leads to burst synchronization among different electroreceptors.
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Alanazy, Asma, Galal Moatimid, Tarek Amer, and Abdallah Galal. "Non-Perturbative Approach in Scrutinizing Nonlinear Time-Delay of Van der Pol-Duffing Oscillator." European Journal of Pure and Applied Mathematics 18, no. 1 (2025): 5495. https://doi.org/10.29020/nybg.ejpam.v18i1.5495.
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The time-delayed (TD) of velocity and position are employed throughout this investigation to lessen the nonlinear vibration of an exciting Van der Pol-Duffing oscillator (VdPD). The issue encompasses multiple real-world elements such as feedback lags, signal transmission delays, and delayed responses in mechanical, electrical, or biological systems. Actually, examining this oscillator facilitates the investigation of complex dynamics, including chaos, bifurcations, and stability alterations, rendering it essential for disciplines like control theory, engineering, and neuroscience. The current oscillator is analyzed using the NPA. This methodology is based mainly on the He’s frequency formula (HFF). Simply, this approach transforms the nonlinear ordinary differential equation (ODE) into a linear one. Accordingly, the stability standards are constructed, depicted, and sketched. Contrasting the analytical solution (AS) with the associated numerical data, which reveals high nonlinearity, the numerical estimation is validated via the Mathematica Software (MS). In contrast to other traditional perturbation methods, the NPA exhibits high convenience, accessibility, and great precision in analyzing the behavior of strong nonlinear oscillators. Subsequently, this technique enables the analysis of issues related to other oscillators in dynamical systems. It is an effective and promising method for addressing similar dynamic system challenges, providing a qualitative assessment of theoretical outcomes. The study describes time histories of solutions for different natural frequencies and TD parameters and discusses the main findings based on displayed curves. It also examines how various regulatory limits impact the vibrating system. The performance is applicable in engineering and other domains owing to its flexibility in various nonlinear systems. Consequently, the NPA can be considered significant, effective, and intriguing, with potential for use in more categories within the domain of coupled dynamical systems.
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Wu, Hui, and Dongwook Kim. "Distribution of Order Parameter for Kuramoto Model." International Journal for Innovation Education and Research 3, no. 9 (2015): 52–68. http://dx.doi.org/10.31686/ijier.vol3.iss9.432.
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The synchronization in large populations of interacting oscillators has been observed abundantly in nature, emergining in fields such as physical, biological and chemical system. For this reason, many scientists are seeking to understand the underlying mechansim of the generation of synchronous patterns in oscillatory system. The synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. The Kuramoto model can be used to understand the emergence of synchronization in nextworks of coupled, nonlinear oscillators. In particular, this model presents a phase transition from incoherence to synchronization. In this paper, we investigated the distribution of order parameter γ which describes the strength of synchrony of these oscillators. The larger the order parameter γ is, the more extent the oscillators are synchronized together. This order parameter γ is a critical parameter in the Kuramoto model. Kuramoto gave a initial estimate equation for the value of the order parameter by giving the value of the coupling constant. But our numerical results show that the distribution of the order parameter is slightly different from Kuramoto’s estimation. We gave an estimation for the distribution of order parameter for different values of initial conditions. We discussed how the numerical result will be distributed around Kuramoto’s analytical equation.
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olde Scheper, Tjeerd V. "Criticality Analysis: Bio-Inspired Nonlinear Data Representation." Entropy 25, no. 12 (2023): 1660. http://dx.doi.org/10.3390/e25121660.
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The representation of arbitrary data in a biological system is one of the most elusive elements of biological information processing. The often logarithmic nature of information in amplitude and frequency presented to biosystems prevents simple encapsulation of the information contained in the input. Criticality Analysis (CA) is a bio-inspired method of information representation within a controlled Self-Organised Critical system that allows scale-free representation. This is based on the concept of a reservoir of dynamic behaviour in which self-similar data will create dynamic nonlinear representations. This unique projection of data preserves the similarity of data within a multidimensional neighbourhood. The input can be reduced dimensionally to a projection output that retains the features of the overall data, yet has a much simpler dynamic response. The method depends only on the Rate Control of Chaos applied to the underlying controlled models, which allows the encoding of arbitrary data and promises optimal encoding of data given biologically relevant networks of oscillators. The CA method allows for a biologically relevant encoding mechanism of arbitrary input to biosystems, creating a suitable model for information processing in varying complexity of organisms and scale-free data representation for machine learning.
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Rapin, V. V. "Theoretical investigation of injection-locked differential oscillator." Radiotekhnika, no. 211 (December 30, 2022): 143–47. http://dx.doi.org/10.30837/rt.2022.4.211.11.
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A preliminary analysis of published works on this topic showed that at present there is no sufficiently substantiated theory of such devices, and the approximate approaches used are rough and do not always meet the requirements of practice. The proposed transition from a differential self-oscillator to an equivalent single-circuit oscillator has not received a convincing justification.
 This article presents a methodology for studying a synchronized differential oscillator using rigorous methods. A mathematical model of such oscillator is presented in the form of nonlinear differential equations obtained on the basis of Kirchhoff's laws. Their analysis made it possible to substantiate the transition to the model of a single circuit LC oscillator, equivalent to a differential one. A technique for such a transition is proposed, including the procedure for determining the nonlinear characteristics of the amplifying element of this self-oscillator, based on the nonlinear characteristics of two amplifying elements of the differential oscillator.
 The mathematical model of an equivalent oscillator is represented by a non-linear differential Van der Pol equation in a dimensionless form, it is simple and accurate. This form of representation made it possible to single out a small parameter and estimate its value. In the case of small values of the small parameter, as is known, traditional methods can be used for its analysis. Thus, the task of studying the synchronization process of a differential oscillator is reduced to the study of the synchronization process of a Van der Pol oscillator. The presented results can be useful in the development of various devices based on synchronized differential oscillators.
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Saavedra, Martín, Juan Pérez-Mercader, and Alberto P. Muñuzuri. "Intermittent regimes as a synchronization phenomenon in two sets of nonlinear chemical oscillators." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 11 (2022): 113125. http://dx.doi.org/10.1063/5.0104610.
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Systems of nonlinear chemical oscillators can exhibit a large diversity of non-trivial states depending on the parameters that characterize them. Among these, a synchronization phenomenon is of special interest due to its direct link with chemical and biological processes in nature. We carry out numerical experiments for two different sets of chemical oscillators with different properties and immersed in a Belousov–Zhabotinsky solution. We document the emergence of different states of synchronization that depend on the parameters characterizing the solution. We also show that, in the interface regions, this system generates a stable dynamics of intermittency between the different synchronization states where interesting phenomena, such as the “devil's staircase,” emerge. In general, the added complexity introduced with the additional set of oscillators results in more complex non-trivial synchronization states.
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Ernst, U., K. Pawelzik, and T. Geisel. "Delay-induced multistable synchronization of biological oscillators." Physical Review E 57, no. 2 (1998): 2150–62. http://dx.doi.org/10.1103/physreve.57.2150.
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Pedergnana, Tiemo, and Nicolas Noiray. "Steady-state statistics, emergent patterns and intermittent energy transfer in a ring of oscillators." Nonlinear Dynamics 108, no. 2 (2022): 1133–63. http://dx.doi.org/10.1007/s11071-022-07275-z.
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AbstractNetworks of coupled nonlinear oscillators model a broad class of physical, chemical and biological systems. Understanding emergent patterns in such networks is an ongoing effort with profound implications for different fields. In this work, we analytically and numerically study a symmetric ring of N coupled self-oscillators of van der Pol type under external stochastic forcing. The system is proposed as a model of the thermo- and aeroacoustic interactions of sound fields in rigid enclosures with compact source regions in a can-annular combustor. The oscillators are connected via linear resistive coupling with nonlinear saturation. After transforming the system to amplitude-phase coordinates, deterministic and stochastic averaging is performed to eliminate the fast oscillating terms. By projecting the potential of the slow-flow dynamics onto the phase-locked quasi-limit cycle solutions, we obtain a compact, low-order description of the (de-)synchronization transition for an arbitrary number of oscillators. The stationary probability density function of the state variables is derived from the Fokker–Planck equation, studied for varying parameter values and compared to time series simulations. We leverage our analysis to offer explanations for the intermittent energy transfer between Bloch waves observed in acoustic pressure spectrograms observed of real-world gas turbines.
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Bi, Hongjie, and Tomoki Fukai. "Amplitude-mediated chimera states in nonlocally coupled Stuart–Landau oscillators." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 8 (2022): 083125. http://dx.doi.org/10.1063/5.0096284.
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Chimera states achieve the coexistence of coherent and incoherent subgroups through symmetry breaking and emerge in physical, chemical, and biological systems. We show the presence of amplitude-mediated multicluster chimera states in nonlocally coupled Stuart–Landau oscillators. We clarify the prerequisites for having different types of chimera states by analytically and numerically studying how phase transitions occur between these states. Our results demonstrate how the oscillation amplitudes interact with the phase degrees of freedom in chimera states and significantly advance our understanding of the generation mechanisms of such states in coupled oscillator systems.
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Williams, T. L., K. A. Sigvardt, N. Kopell, G. B. Ermentrout, and M. P. Remler. "Forcing of coupled nonlinear oscillators: studies of intersegmental coordination in the lamprey locomotor central pattern generator." Journal of Neurophysiology 64, no. 3 (1990): 862–71. http://dx.doi.org/10.1152/jn.1990.64.3.862.
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1. This paper reports the results of an investigation of the basic mechanisms underlying intersegmental coordination in lamprey locomotion, by the use of a combined mathematical and biological approach. 2. Mathematically, the lamprey central pattern generator (CPG) is described as a chain of coupled nonlinear oscillators; experimentally, entrainment of fictive locomotion by imposed movement has been investigated. Interpretation of the results in the context of the theory has allowed conclusions to be drawn about the nature of ascending and descending coupling in the lamprey spinal CPG. 3. Theory predicts and data show that 1) the greater the number of oscillators in the chain, the smaller is the entrainment frequency range and 2) it is possible to entrain both above and below the rest frequency at one end but only above or below at the other end. 4. In the context of the experimental results, the theory indicates the following: 1) ascending coupling sets the intersegmental phase lags, whereas descending coupling changes the frequency of the coupled oscillators; 2) there are differences in the ascending and descending coupling other than strength; and it also suggests that 3) coupling slows down the oscillators.
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Borg, Yanika, Ekkehard Ullner, Afnan Alagha, Ahmed Alsaedi, Darren Nesbeth, and Alexey Zaikin. "Complex and unexpected dynamics in simple genetic regulatory networks." International Journal of Modern Physics B 28, no. 14 (2014): 1430006. http://dx.doi.org/10.1142/s0217979214300060.
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One aim of synthetic biology is to construct increasingly complex genetic networks from interconnected simpler ones to address challenges in medicine and biotechnology. However, as systems increase in size and complexity, emergent properties lead to unexpected and complex dynamics due to nonlinear and nonequilibrium properties from component interactions. We focus on four different studies of biological systems which exhibit complex and unexpected dynamics. Using simple synthetic genetic networks, small and large populations of phase-coupled quorum sensing repressilators, Goodwin oscillators, and bistable switches, we review how coupled and stochastic components can result in clustering, chaos, noise-induced coherence and speed-dependent decision making. A system of repressilators exhibits oscillations, limit cycles, steady states or chaos depending on the nature and strength of the coupling mechanism. In large repressilator networks, rich dynamics can also be exhibited, such as clustering and chaos. In populations of Goodwin oscillators, noise can induce coherent oscillations. In bistable systems, the speed with which incoming external signals reach steady state can bias the network towards particular attractors. These studies showcase the range of dynamical behavior that simple synthetic genetic networks can exhibit. In addition, they demonstrate the ability of mathematical modeling to analyze nonlinearity and inhomogeneity within these systems.
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Carley, D. W., E. Onal, R. Aronson, and M. Lopata. "Breath-by-breath interactions between inspiratory and expiratory duration in occlusive sleep apnea." Journal of Applied Physiology 66, no. 5 (1989): 2312–19. http://dx.doi.org/10.1152/jappl.1989.66.5.2312.
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We examined interactions between inspiratory duration (TI), expiratory duration (TE), and inspiratory (esophageal) pressure (Pes) generation in seven subjects with confirmed occlusive sleep apnea. Breath-by-breath values of TI, TE, and Pes were identified by digital computer during 21 260-s epochs of repetitive occlusive apnea during non-rapid-eye-movement sleep. The control theory of interacting nonlinear oscillators was used to categorize the interaction between TI and TE for each epoch as either 1) synchronization, the strongest possible interaction between biological oscillators; 2) relative entrainment, a moderate interaction between oscillators; or 3) relative coordination, a weak interaction. The latter two interactions were characterized by systemic oscillations in the moving cross-correlation between TI and TE. The relationship between TI and Pes was analyzed in a similar fashion. Significant oscillations were present in all three parameters (P less than 0.0001 for each). We observed significant negative correlations between TI and TE and between TI and Pes (P less than 0.001 for each) when all breaths for all epochs were pooled. In no epoch was there a significant positive correlation between TI and TE or Pes. All three interactions were observed between TI and TE: five epochs of synchronization, nine of relative entrainment, and seven of relative coordination. In contrast, 19 of 21 epochs exhibited synchronization between TI and Pes, with 2 epochs of relative entrainment. The relative frequency of TI vs. Pes synchronization was significantly greater than TI vs. TE synchronization (P less than 0.005).(ABSTRACT TRUNCATED AT 250 WORDS)
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SORRENTINO, FRANCESCO, MARIO DI BERNARDO, and FRANCO GAROFALO. "SYNCHRONIZABILITY AND SYNCHRONIZATION DYNAMICS OF WEIGHED AND UNWEIGHED SCALE FREE NETWORKS WITH DEGREE MIXING." International Journal of Bifurcation and Chaos 17, no. 07 (2007): 2419–34. http://dx.doi.org/10.1142/s021812740701849x.
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We study the synchronizability and the synchronization dynamics of networks of nonlinear oscillators. We investigate how the synchronization of the network is influenced by some of its topological features such as variations of the power law degree distribution exponent γ and the degree correlation coefficient r. Using an appropriate construction algorithm based on clustering the network vertices in p classes according to their degrees, we construct networks with an assigned power law distribution but changing degree correlation properties. We find that the network synchronizability improves when the network becomes disassortative, i.e. when nodes with low degree are more likely to be connected to nodes with higher degree. We consider the case of both weighed and unweighed networks. The analytical results reported in the paper are then confirmed by a set of numerical observations obtained on weighed and unweighed networks of nonlinear Rössler oscillators. Using a nonlinear optimization strategy we also show that negative degree correlation is an emerging property of networks when synchronizability is to be optimized. This suggests that negative degree correlation observed experimentally in a number of physical and biological networks might be motivated by their need to synchronize better.
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Mchedlova, Elena, and Aleksej Trubeckov. "Self-organization as a result of nonlinear interactions in large groups of biological oscillators." Izvestiya VUZ. Applied Nonlinear Dynamics 3, no. 1 (1995): 73–81. https://doi.org/10.18500/0869-6632-1995-3-1-73-81.
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Interactions in the system of biological thythms under external influence are investigated numerically. Each of rhythms is represented by autooscillator. Mathematical -index, showing degree of desyncronization under different breaks in external influence is suggested. The model allows to study the influence of shift work on the human organism at the computer experiment.
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Yan, Jie, Xiaxia Kang, and Ling Yang. "The Trade-Off Mechanism in Mammalian Circadian Clock Model with Two Time Delays." International Journal of Bifurcation and Chaos 27, no. 09 (2017): 1750147. http://dx.doi.org/10.1142/s0218127417501474.
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Circadian clock is an autonomous oscillator which orchestrates the daily rhythms of physiology and behaviors. This study is devoted to explore how a positive feedback loop affects the dynamics of mammalian circadian clock. We simplify an experimentally validated mathematical model in our previous work, to a nonlinear differential equation with two time delays. This simplified mathematical model incorporates the pacemaker of mammalian circadian clock, a negative primary feedback loop, and a critical positive auxiliary feedback loop, [Formula: see text] loop. We perform analytical studies of the system. Delay-dependent conditions for the asymptotic stability of the nontrivial positive steady state of the model are investigated. We also prove the existence of Hopf bifurcation, which leads to self-sustained oscillation of mammalian circadian clock. Our theoretical analyses show that the oscillatory regime is reduced upon the participation of the delayed positive auxiliary loop. However, further simulations reveal that the auxiliary loop can enable the circadian clock gain widely adjustable amplitudes and robust period. Thus, the positive auxiliary feedback loop may provide a trade-off mechanism, to use the small loss in the robustness of oscillation in exchange for adaptable flexibility in mammalian circadian clock. The results obtained from the model may gain new insights into the dynamics of biological oscillators with interlocked feedback loops.
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Hennig, D., J. F. R. Archilla, and J. M. Romero. "Modelling the thermal evolution of enzyme-created bubbles in DNA." Journal of The Royal Society Interface 2, no. 2 (2005): 89–95. http://dx.doi.org/10.1098/rsif.2004.0024.
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The formation of bubbles in nucleic acids (NAs) is fundamental in many biological processes such as DNA replication, recombination, telomere formation and nucleotide excision repair, as well as RNA transcription and splicing. These processes are carried out by assembled complexes with enzymes that separate selected regions of NAs. Within the frame of a nonlinear dynamics approach, we model the structure of the DNA duplex by a nonlinear network of coupled oscillators. We show that, in fact, from certain local structural distortions, there originate oscillating localized patterns, that is, radial and torsional breathers, which are associated with localized H-bond deformations, reminiscent of the replication bubble. We further study the temperature dependence of these oscillating bubbles. To this aim, the underlying nonlinear oscillator network of the DNA duplex is brought into contact with a heat bath using the Nosé–Hoover method. Special attention is paid to the stability of the oscillating bubbles under the imposed thermal perturbations. It is demonstrated that the radial and torsional breathers sustain the impact of thermal perturbations even at temperatures as high as room temperature. Generally, for non-zero temperature, the H-bond breathers move coherently along the double chain, whereas at T =0 standing radial and torsional breathers result.
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Signorini, Maria G., and Diego di Bernardo. "Simulation of Heartbeat Dynamics: A Nonlinear Model." International Journal of Bifurcation and Chaos 08, no. 08 (1998): 1725–31. http://dx.doi.org/10.1142/s0218127498001418.
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The mathematical modeling of biological systems has proven to be a valuable tool by allowing experiments which would otherwise be unfeasible in a real situation. In this work we propose a system of nonlinear differential equations describing the macroscopic behavior of the cardiac conduction system. The model describes the interactoin between the SinoAtrial and AtrioVentricular node. Its very simple structure consists of two nonlinear oscillators resistively coupled. The numerical analysis detects different kinds of bifurcations whose pathophysiological meanings are discussed. Moreover, the model is able to classify different pathologies, such as several classes of arrhythmic events, as well as to suggest hypothesis on the mechanisms that induce them. These results also show that the mechanisms generating the heartbeat obey complex laws. The model provides a wuite complete description of different pathological phenomena and its simplicity can be exploited for further studies on the control of cardiac dynamics.
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DOI, Shinji. "Response Characteristics of Nonlinear Models to External Stimuli: Neuron Models and Biological Oscillators as an Example." IEICE ESS Fundamentals Review 13, no. 3 (2020): 187–96. http://dx.doi.org/10.1587/essfr.13.3_187.
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Ascoli, Alon, Martin Weiher, Melanie Herzig, Stefan Slesazeck, Thomas Mikolajick, and Ronald Tetzlaff. "Graph Coloring via Locally-Active Memristor Oscillatory Networks." Journal of Low Power Electronics and Applications 12, no. 2 (2022): 22. http://dx.doi.org/10.3390/jlpea12020022.
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This manuscript provides a comprehensive tutorial on the operating principles of a bio-inspired Cellular Nonlinear Network, leveraging the local activity of NbOx memristors to apply a spike-based computing paradigm, which is expected to deliver such a separation between the steady-state phases of its capacitively-coupled oscillators, relative to a reference cell, as to unveal the classification of the nodes of the associated graphs into the least number of groups, according to the rules of a non-deterministic polynomial-hard combinatorial optimization problem, known as vertex coloring. Besides providing the theoretical foundations of the bio-inspired signal-processing paradigm, implemented by the proposed Memristor Oscillatory Network, and presenting pedagogical examples, illustrating how the phase dynamics of the memristive computing engine enables to solve the graph coloring problem, the paper further presents strategies to compensate for an imbalance in the number of couplings per oscillator, to counteract the intrinsic variability observed in the electrical behaviours of memristor samples from the same batch, and to prevent the impasse appearing when the array attains a steady-state corresponding to a local minimum of the optimization goal. The proposed Memristor Cellular Nonlinear Network, endowed with ad hoc circuitry for the implementation of these control strategies, is found to classify the vertices of a wide set of graphs in a number of color groups lower than the cardinality of the set of colors identified by traditional either software or hardware competitor systems. Given that, under nominal operating conditions, a biological system, such as the brain, is naturally capable to optimise energy consumption in problem-solving activities, the capability of locally-active memristor nanotechnologies to enable the circuit implementation of bio-inspired signal processing paradigms is expected to pave the way toward electronics with higher time and energy efficiency than state-of-the-art purely-CMOS hardware.
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Zhogoleva, Nadiya, and Volodymyr Shcherbak. "Asymptotic evaluation of the state and stiffness of the van der Paul oscillator." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 33 (December 27, 2019): 91–99. http://dx.doi.org/10.37069/1683-4720-2019-33-7.
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In many applications of physics, biology, and other sciences, an approach based on the concept of model equations is used as an approximate model of complex nonlinear processes. The basis of this concept is the provision that a small number of characteristic types movements of simple mathematical models inherent in systems gives the key to understanding and exploring a huge number of different phenomena. With this approach it is a priori assumed that the entire physical diverseness can be represented in the form of fairly simple model equations. It is contributes to a qualitative study of complex systems for various physical nature since basic models individually are well studied, their parameters have a physical interpretation. In particular, it is well known that oscillatory motion of various systems with a stable limit cycle can be modeled by a system consisting of one or more coupled van der Pol oscillators. Such systems are widely represented in various technical devices and in the study and modeling of some biological functions of the body, such as cardiac activity, respiration, locomotor activity, etc. It is considered a typical situation for many practical applications of control theory when the complete state vector of the system is unknown and only some of the functions of the state variables -- the outputs of the system are accessible to measurement. Therefore, the problem of determining in real time the state and parameters of such systems based on the results of measuring the output signals are relevant. One of these inverse control problems, namely, the problem of observability and parameter identification of an model oscillatory system is considered in this article. For observation and identification scheme design the method of invariant relations developed in analytical mechanics is used. Its modification in control problems allows us to synthesize additional relationships between known and unknown quantities of a dynamical system that arise during the observed motion. The method does not involve linearization of the original system and is essentially non-linear. The constructed nonlinear observer provides an asymptotic estimation of unknown parameter and velocity of oscillations.
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Huang, Xiaojun, and Zigen Song. "Generation of stochastic mixed-mode oscillations in a pair of VDP oscillators with direct-indirect coupling." Mathematical Biosciences and Engineering 21, no. 1 (2023): 765–77. http://dx.doi.org/10.3934/mbe.2024032.
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<abstract> <p>Environmental noise can lead to complex stochastic dynamical behavior in nonlinear systems. In this paper, we studied the phenomenon of a pair of Van der Pol (VDP) oscillators with direct-indirect coupling affected by Gaussian white noise. That is to say, a noise-induced equilibrium transition oscillation was observed in three types of different parameter regions, where the deterministic system had two kinds of stable equilibrium points. Meanwhile, with the noise intensity increasing, we found that the stochastic system will constantly switch between two stable equilibrium points. To analyze the stochastic behavior, we used the stochastic sensitivity equation and confidence ellipse method. When the confidence ellipsoid crossed the boundary of the attraction basin of the equilibrium point, the system entered into the state of stochastic mixed-mode oscillations, which was consistent with the simulation results.</p> </abstract>
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Razzak, Md Abdur. "A simple harmonic balance method for solving strongly nonlinear oscillators." Journal of the Association of Arab Universities for Basic and Applied Sciences 21, no. 1 (2016): 68–76. http://dx.doi.org/10.1016/j.jaubas.2015.10.002.
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Ijspeert, Auke Jan, Jun Nakanishi, Heiko Hoffmann, Peter Pastor, and Stefan Schaal. "Dynamical Movement Primitives: Learning Attractor Models for Motor Behaviors." Neural Computation 25, no. 2 (2013): 328–73. http://dx.doi.org/10.1162/neco_a_00393.
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Nonlinear dynamical systems have been used in many disciplines to model complex behaviors, including biological motor control, robotics, perception, economics, traffic prediction, and neuroscience. While often the unexpected emergent behavior of nonlinear systems is the focus of investigations, it is of equal importance to create goal-directed behavior (e.g., stable locomotion from a system of coupled oscillators under perceptual guidance). Modeling goal-directed behavior with nonlinear systems is, however, rather difficult due to the parameter sensitivity of these systems, their complex phase transitions in response to subtle parameter changes, and the difficulty of analyzing and predicting their long-term behavior; intuition and time-consuming parameter tuning play a major role. This letter presents and reviews dynamical movement primitives, a line of research for modeling attractor behaviors of autonomous nonlinear dynamical systems with the help of statistical learning techniques. The essence of our approach is to start with a simple dynamical system, such as a set of linear differential equations, and transform those into a weakly nonlinear system with prescribed attractor dynamics by means of a learnable autonomous forcing term. Both point attractors and limit cycle attractors of almost arbitrary complexity can be generated. We explain the design principle of our approach and evaluate its properties in several example applications in motor control and robotics.
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Wang, Shuo, Erik D. Herzog, István Z. Kiss, et al. "Inferring dynamic topology for decoding spatiotemporal structures in complex heterogeneous networks." Proceedings of the National Academy of Sciences 115, no. 37 (2018): 9300–9305. http://dx.doi.org/10.1073/pnas.1721286115.
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Extracting complex interactions (i.e., dynamic topologies) has been an essential, but difficult, step toward understanding large, complex, and diverse systems including biological, financial, and electrical networks. However, reliable and efficient methods for the recovery or estimation of network topology remain a challenge due to the tremendous scale of emerging systems (e.g., brain and social networks) and the inherent nonlinearity within and between individual units. We develop a unified, data-driven approach to efficiently infer connections of networks (ICON). We apply ICON to determine topology of networks of oscillators with different periodicities, degree nodes, coupling functions, and time scales, arising in silico, and in electrochemistry, neuronal networks, and groups of mice. This method enables the formulation of these large-scale, nonlinear estimation problems as a linear inverse problem that can be solved using parallel computing. Working with data from networks, ICON is robust and versatile enough to reliably reveal full and partial resonance among fast chemical oscillators, coherent circadian rhythms among hundreds of cells, and functional connectivity mediating social synchronization of circadian rhythmicity among mice over weeks.
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Abrevaya, Germán, Guillaume Dumas, Aleksandr Y. Aravkin, et al. "Learning Brain Dynamics With Coupled Low-Dimensional Nonlinear Oscillators and Deep Recurrent Networks." Neural Computation 33, no. 8 (2021): 2087–127. http://dx.doi.org/10.1162/neco_a_01401.
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Many natural systems, especially biological ones, exhibit complex multivariate nonlinear dynamical behaviors that can be hard to capture by linear autoregressive models. On the other hand, generic nonlinear models such as deep recurrent neural networks often require large amounts of training data, not always available in domains such as brain imaging; also, they often lack interpretability. Domain knowledge about the types of dynamics typically observed in such systems, such as a certain type of dynamical systems models, could complement purely data-driven techniques by providing a good prior. In this work, we consider a class of ordinary differential equation (ODE) models known as van der Pol (VDP) oscil lators and evaluate their ability to capture a low-dimensional representation of neural activity measured by different brain imaging modalities, such as calcium imaging (CaI) and fMRI, in different living organisms: larval zebrafish, rat, and human. We develop a novel and efficient approach to the nontrivial problem of parameters estimation for a network of coupled dynamical systems from multivariate data and demonstrate that the resulting VDP models are both accurate and interpretable, as VDP's coupling matrix reveals anatomically meaningful excitatory and inhibitory interactions across different brain subsystems. VDP outperforms linear autoregressive models (VAR) in terms of both the data fit accuracy and the quality of insight provided by the coupling matrices and often tends to generalize better to unseen data when predicting future brain activity, being comparable to and sometimes better than the recurrent neural networks (LSTMs). Finally, we demonstrate that our (generative) VDP model can also serve as a data-augmentation tool leading to marked improvements in predictive accuracy of recurrent neural networks. Thus, our work contributes to both basic and applied dimensions of neuroimaging: gaining scientific insights and improving brain-based predictive models, an area of potentially high practical importance in clinical diagnosis and neurotechnology.
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olde Scheper, Tjeerd V. "Controlled bio-inspired self-organised criticality." PLOS ONE 17, no. 1 (2022): e0260016. http://dx.doi.org/10.1371/journal.pone.0260016.
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Complex biological systems are considered to be controlled using feedback mechanisms. Reduced systems modelling has been effective to describe these mechanisms, but this approach does not sufficiently encompass the required complexity that is needed to understand how localised control in a biological system can provide global stable states. Self-Organised Criticality (SOC) is a characteristic property of locally interacting physical systems, which readily emerges from changes to its dynamic state due to small nonlinear perturbations. These small changes in the local states, or in local interactions, can greatly affect the total system state of critical systems. It has long been conjectured that SOC is cardinal to biological systems, that show similar critical dynamics, and also may exhibit near power-law relations. Rate Control of Chaos (RCC) provides a suitable robust mechanism to generate SOC systems, which operates at the edge of chaos. The bio-inspired RCC method requires only local instantaneous knowledge of some of the variables of the system, and is capable of adapting to local perturbations. Importantly, connected RCC controlled oscillators can maintain global multi-stable states, and domains where power-law relations may emerge. The network of oscillators deterministically stabilises into different orbits for different perturbations, and the relation between the perturbation and amplitude can show exponential and power-law correlations. This can be considered to be representative of a basic mechanism of protein production and control, that underlies complex processes such as homeostasis. Providing feedback from the global state, the total system dynamic behaviour can be boosted or reduced. Controlled SOC can provide much greater understanding of biological control mechanisms, that are based on distributed local producers, with remote consumers of biological resources, and globally defined control.
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Chen, Bor-Sen, Chih-Yuan Hsu, and Jing-Jia Liou. "Robust Design of Biological Circuits: Evolutionary Systems Biology Approach." Journal of Biomedicine and Biotechnology 2011 (2011): 1–14. http://dx.doi.org/10.1155/2011/304236.
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Artificial gene circuits have been proposed to be embedded into microbial cells that function as switches, timers, oscillators, and the Boolean logic gates. Building more complex systems from these basic gene circuit components is one key advance for biologic circuit design and synthetic biology. However, the behavior of bioengineered gene circuits remains unstable and uncertain. In this study, a nonlinear stochastic system is proposed to model the biological systems with intrinsic parameter fluctuations and environmental molecular noise from the cellular context in the host cell. Based on evolutionary systems biology algorithm, the design parameters of target gene circuits can evolve to specific values in order to robustly track a desired biologic function in spite of intrinsic and environmental noise. The fitness function is selected to be inversely proportional to the tracking error so that the evolutionary biological circuit can achieve the optimal tracking mimicking the evolutionary process of a gene circuit. Finally, several design examples are givenin silicowith the Monte Carlo simulation to illustrate the design procedure and to confirm the robust performance of the proposed design method. The result shows that the designed gene circuits can robustly track desired behaviors with minimal errors even with nontrivial intrinsic and external noise.
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Brea, Jorge, David F. Russell, and Alexander B. Neiman. "Measuring direction in the coupling of biological oscillators: A case study for electroreceptors of paddlefish." Chaos: An Interdisciplinary Journal of Nonlinear Science 16, no. 2 (2006): 026111. http://dx.doi.org/10.1063/1.2201466.
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Minati, Ludovico, Hiroyuki Ito, Alessio Perinelli, et al. "Connectivity Influences on Nonlinear Dynamics in Weakly-Synchronized Networks: Insights From Rössler Systems, Electronic Chaotic Oscillators, Model and Biological Neurons." IEEE Access 7 (2019): 174793–821. http://dx.doi.org/10.1109/access.2019.2957014.
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Taylor, Mark A., Stavros Pavlou, and Ioannis G. Kevrekidis. "Microbial predation in coupled chemostats: A global study of two coupled nonlinear oscillators." Mathematical Biosciences 122, no. 1 (1994): 25–66. http://dx.doi.org/10.1016/0025-5564(94)90081-7.
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Ekeberg, Örjan, and Sten Grillner. "Simulations of neuromuscular control in lamprey swimming." Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences 354, no. 1385 (1999): 895–902. http://dx.doi.org/10.1098/rstb.1999.0441.
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The neuronal generation of vertebrate locomotion has been extensively studied in the lamprey. Models at different levels of abstraction are being used to describe this system, from abstract nonlinear oscillators to interconnected model neurons comprising multiple compartments and a Hodgkin–Huxley representation of the most relevant ion channels. To study the role of sensory feedback by simulation, it eventually also becomes necessary to incorporate the mechanical movements in the models. By using simplifying models of muscle activation, body mechanics, counteracting water forces, and sensory feedback through stretch receptors and vestibular organs, we have been able to close the feedback loop to enable studies of the interaction between the neuronal and the mechanical systems. The neuromechanical simulations reveal that the currently known network is sufficient for generating a whole repertoire of swimming patterns. Swimming at different speeds and with different wavelengths, together with the performance of lateral turns can all be achieved by simply varying the brainstem input. The neuronal mechanisms behind pitch and roll manoeuvres are less clear. We have put forward a ‘crossed–oscillators’ hypothesis where partly separate dorsal and ventral circuits are postulated. Neuromechanical simulations of this system show that it is also capable of generating realistic pitch turns and rolls, and that vestibular signals can stabilize the posture during swimming.
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Aldonin, Gennady, and Vasiliy Cherepanov. "Model of the Process of Self-Organization of the Heart Rhythm." Infocommunications and Radio Technologies 5, no. 4 (2022): 472–83. http://dx.doi.org/10.29039/2587-9936.2022.05.4.35.
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The paper considers a synergistic analysis of the physical and physiological nature of electrical processes in the human heart, namely in the most important biosystem – the conduction nervous system of the heart (CNSH), in particular, the heart pacemaker. Currently, promising methods for studying CNSH as an active medium are being actively developed, using the foundations of nonlinear dynamics. Methods for describing active media are widely used in the study of the phenomena of the work of the heart pacemaker, where the active medium is represented as an ensemble of some elements that locally interact with each other. Self-organization in biological systems can be represented on the basis of a non-linear dynamic approach to the description of mechanisms in CNSH, namely, the consideration of P-cells of the pacemaker as a system of coupled non-linear oscillators. Such a synergistic method provides a real basis for modeling the processes of generation and propagation of nerve excitation in the heart using the Fermi–Pasta–Ulam (FPU) “return” theorem and the Kolmogorov–Arnold–Moser (KAM) theorem.
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Wang, Binrui, Yixuan Liu, Zhongwen Li, Dijian Chen, Ruizi Ma, and Ling Wang. "Parallel Spine Design and CPG Motion Test of Quadruped Robot." International Journal of Pattern Recognition and Artificial Intelligence 34, no. 05 (2019): 2059013. http://dx.doi.org/10.1142/s0218001420590132.
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The spine of mammals aids in the stability of locomotion. Central Pattern Generators (CPGs) located in spinal cord can rapidly provide a rhythmic output signal during loss of sensory feedback on the basis of a simulated quadruped agent. In this paper, active spine of quadruped robot is shown to be extremely effective in motion. An active spine model based on the Parallel Kinematic Mechanism (PKM) system and biological phenomena is described. The general principles involved in constructing a neural network coupled with limbs and spine to solve specific problems are discussed. A CPG mathematical model based on Hopf nonlinear oscillators produces rhythmic signal during locomotion is described, where many parameters to be solved must be formulated in terms of desired stability, often subject to vertical stability analysis. Our simulations demonstrate that active spine with setting reasonable CPG parameters can reduce unnecessary lateral displacement during trot gait, improving the stability of quadruped robot. In addition, we demonstrate that physical prototype mechanism provides a framework which shows correctness of simulation, and stability can thus be easily embodied within locomotion.
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Koch, J., Z. Chen, A. Tuor, J. Drgona, and D. Vrabie. "Structural inference of networked dynamical systems with universal differential equations." Chaos: An Interdisciplinary Journal of Nonlinear Science 33, no. 2 (2023): 023103. http://dx.doi.org/10.1063/5.0109093.
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Networked dynamical systems are common throughout science in engineering; e.g., biological networks, reaction networks, power systems, and the like. For many such systems, nonlinearity drives populations of identical (or near-identical) units to exhibit a wide range of nontrivial behaviors, such as the emergence of coherent structures (e.g., waves and patterns) or otherwise notable dynamics (e.g., synchrony and chaos). In this work, we seek to infer (i) the intrinsic physics of a base unit of a population, (ii) the underlying graphical structure shared between units, and (iii) the coupling physics of a given networked dynamical system given observations of nodal states. These tasks are formulated around the notion of the Universal Differential Equation, whereby unknown dynamical systems can be approximated with neural networks, mathematical terms known a priori (albeit with unknown parameterizations), or combinations of the two. We demonstrate the value of these inference tasks by investigating not only future state predictions but also the inference of system behavior on varied network topologies. The effectiveness and utility of these methods are shown with their application to canonical networked nonlinear coupled oscillators.
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BILOTTA, ELEONORA, STEFANIA GERVASI, and PIETRO PANTANO. "READING COMPLEXITY IN CHUA'S OSCILLATOR THROUGH MUSIC. PART I: A NEW WAY OF UNDERSTANDING CHAOS." International Journal of Bifurcation and Chaos 15, no. 02 (2005): 253–382. http://dx.doi.org/10.1142/s0218127405012156.
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Modern Science is finding new methods of looking at biological, physical or social phenomena. Traditional methods of quantification are no longer sufficient and new approaches are emerging. These approaches make it apparent that the phenomena the observer is looking at are not classifiable by conventional methods. These phenomena are complex. A complex system, as Chua's oscillator, is a nonlinear configuration whose dynamical behavior is chaotic. Chua's oscillator equations allow to define the basic behavior of a dynamical system and to detect the changes in the qualitative behavior of a system when bifurcation occurs, as parameters are varied. The typical set of behavior of a dynamical system can be detailed as equilibrium points, limit cycles, strange attractors. The concepts, methods and paradigms of Dynamical Systems Theory can be applied to understand human behavior. Human behavior is emergent and behavior patterns emerge thanks to the way the parts or the processes are coordinated among themselves. In fact, the listening process in humans is complex and it develops over time as well. Sound and music can be both inside and outside humans. This tutorial concerns the translation of Chua's oscillators into music, in order to find a new way of understanding complexity by using music. By building up many computational models which allow the translation of some quantitative features of Chua's oscillator into sound and music, we have created many acoustical and musical compositions, which in turn present the characteristics of dynamical systems from a perceptual point of view. We have found interesting relationships between dynamical systems behavior and their musical translation since, in the process of listening, human subjects perceive many of the structures as possible to perceive in the behavior of Chua's oscillator. In other words, human cognitive abilities can analyze the large and complicated patterns produced by Chua's systems translated into music, achieving the cognitive economy and the coordination and synthesis of countless data at our disposal that occur in the perception of dynamic events in the real world. Music can be considered the semantics of dynamical systems, which gives us a powerful method for interpreting complexity.
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Belykh, Vladimir, Igor Belykh, and Martin Hasler. "Small-world networks: dynamical models and synchronization." Izvestiya VUZ. Applied Nonlinear Dynamics 11, no. 3 (2003): 67–76. http://dx.doi.org/10.18500/0869-6632-2003-11-3-67-76.
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This paper provides а short review оf recent results оn synchronization in small-world dynamical networks of coupled oscillators. We also propose a new model of small-world networks of cells with a time-varying coupling and study its synchronization properties. It is shown that such а time-varying structure of the network can ensure more reliable synchronization than the conventional small-worlds. The term «small world» refers to a network of locally connected nodes having a few additional long-range shortcuts chosen at random. The addition оf thе shortcuts sharply reduces the average distance between the nodes and therefore provides the so-called small-world effect. Discovered first in social networks, the small-world effect appeared to be а characteristic оf many real-world structure both human-generated ог of biological origin. For social networks, this property implies that almost any pair of people in the world can be connected to one another by a short chain of intermediate acquaintances, of typical length about six. However, the structure оf social networks is not homogeneous, there are always key persons аn provide distant out-local world connections between people. This paper is written in honor оf the 60th birthday оf our friend and colleague, Wadim S. Anishchenko, who is one of such key persons in the Nonlinear Dynamics community.
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Breakspear, Michael, and Cornelis J. Stam. "Dynamics of a neural system with a multiscale architecture." Philosophical Transactions of the Royal Society B: Biological Sciences 360, no. 1457 (2005): 1051–74. http://dx.doi.org/10.1098/rstb.2005.1643.
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The architecture of the brain is characterized by a modular organization repeated across a hierarchy of spatial scales—neurons, minicolumns, cortical columns, functional brain regions, and so on. It is important to consider that the processes governing neural dynamics at any given scale are not only determined by the behaviour of other neural structures at that scale, but also by the emergent behaviour of smaller scales, and the constraining influence of activity at larger scales. In this paper, we introduce a theoretical framework for neural systems in which the dynamics are nested within a multiscale architecture. In essence, the dynamics at each scale are determined by a coupled ensemble of nonlinear oscillators, which embody the principle scale-specific neurobiological processes. The dynamics at larger scales are ‘slaved’ to the emergent behaviour of smaller scales through a coupling function that depends on a multiscale wavelet decomposition. The approach is first explicated mathematically. Numerical examples are then given to illustrate phenomena such as between-scale bifurcations, and how synchronization in small-scale structures influences the dynamics in larger structures in an intuitive manner that cannot be captured by existing modelling approaches. A framework for relating the dynamical behaviour of the system to measured observables is presented and further extensions to capture wave phenomena and mode coupling are suggested.
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Yidi, Zhang, Guo Shan, and Sun Mingzhu. "Front Waves of Chemical Reactions and Travelling Waves of Neural Activity." Journal of NeuroPhilosophy 1, no. 2 (2022): 222–39. https://doi.org/10.5281/zenodo.7254050.
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Travelling waves crossing the nervous networks at mesoscopic/macroscopic scales have been correlated with different brain functions, from long-term memory to visual stimuli. Here we investigate a feasible relationship between wave generation/propagation in recurrent nervous networks and a physical/chemical model, namely the Belousov–Zhabotinsky reaction (BZ). Since BZ’s nonlinear, chaotic chemical process generates concentric/intersecting waves that closely resemble the diffusive nonlinear/chaotic oscillatory patterns crossing the nervous tissue, we aimed to investigate whether wave propagation of brain oscillations could be described in terms of BZ features. We compared experimentally detected oscillations during the spontaneous activity of the brain with BZ-like concentric waves simulated by a recently introduced artificial network. The observed overlap and agreement between simulated and measured oscillatory patterns suggests that changes in cortical areas’ neural activity might be described in terms of a recognizable diffusion pattern. We describe biological plausibility, benefits and limits of our approach and discuss the relationship among BZ-like networks, Pandemonium-like architectures and the spontaneous activity of the brain.
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Gkogkas, Marios Antonios, Benjamin Jüttner, Christian Kuehn, and Erik Andreas Martens. "Graphop mean-field limits and synchronization for the stochastic Kuramoto model." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 11 (2022): 113120. http://dx.doi.org/10.1063/5.0094009.
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Models of coupled oscillator networks play an important role in describing collective synchronization dynamics in biological and technological systems. The Kuramoto model describes oscillator’s phase evolution and explains the transition from incoherent to coherent oscillations under simplifying assumptions, including all-to-all coupling with uniform strength. Real world networks, however, often display heterogeneous connectivity and coupling weights that influence the critical threshold for this transition. We formulate a general mean-field theory (Vlasov–Focker Planck equation) for stochastic Kuramoto-type phase oscillator models, valid for coupling graphs/networks with heterogeneous connectivity and coupling strengths, using graphop theory in the mean-field limit. Considering symmetric odd-valued coupling functions, we mathematically prove an exact formula for the critical threshold for the incoherence–coherence transition. We numerically test the predicted threshold using large finite-size representations of the network model. For a large class of graph models, we find that the numerical tests agree very well with the predicted threshold obtained from mean-field theory. However, the prediction is more difficult in practice for graph structures that are sufficiently sparse. Our findings open future research avenues toward a deeper understanding of mean-field theories for heterogeneous systems.
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Zhao, Wenchuan, Yu Zhang, Kian Meng Lim, Lijian Yang, Ning Wang, and Linghui Peng. "Research on control strategy of pneumatic soft bionic robot based on improved CPG." PLOS ONE 19, no. 7 (2024): e0306320. http://dx.doi.org/10.1371/journal.pone.0306320.
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To achieve the accuracy and anti-interference of the motion control of the soft robot more effectively, the motion control strategy of the pneumatic soft bionic robot based on the improved Central Pattern Generator (CPG) is proposed. According to the structure and motion characteristics of the robot, a two-layer neural network topology model for the robot is constructed by coupling 22 Hopfield neuron nonlinear oscillators. Then, based on the Adaptive Neuro-Fuzzy Inference System (ANFIS), the membership functions are offline learned and trained to construct the CPG-ANFIS-PID motion control strategy for the robot. Through simulation research on the impact of CPG-ANFIS-PID input parameters on the swimming performance of the robot, it is verified that the control strategy can quickly respond to input parameter changes between different swimming modes, and stably output smooth and continuous dynamic position signals, which has certain advantages. Then, the motion performance of the robot prototype is analyzed experimentally and compared with the simulation results. The results show that the CPG-ANFIS-PID motion control strategy can output coupled waveform signals stably, and control the executing mechanisms of the pneumatic soft bionic robot to achieve biological rhythms motion propulsion waveforms, confirming that the control strategy has accuracy and anti-interference characteristics, and enable the robot have certain maneuverability, flexibility, and environmental adaptability. The significance of this work lies in establishing a CPG-ANFIS-PID control strategy applicable to pneumatic soft bionic robot and proposing a rhythmic motion control method applicable to pneumatic soft bionic robot.
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Doelling, Keith B., Luc H. Arnal, and M. Florencia Assaneo. "Adaptive oscillators support Bayesian prediction in temporal processing." PLOS Computational Biology 19, no. 11 (2023): e1011669. http://dx.doi.org/10.1371/journal.pcbi.1011669.
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Humans excel at predictively synchronizing their behavior with external rhythms, as in dance or music performance. The neural processes underlying rhythmic inferences are debated: whether predictive perception relies on high-level generative models or whether it can readily be implemented locally by hard-coded intrinsic oscillators synchronizing to rhythmic input remains unclear and different underlying computational mechanisms have been proposed. Here we explore human perception for tone sequences with some temporal regularity at varying rates, but with considerable variability. Next, using a dynamical systems perspective, we successfully model the participants behavior using an adaptive frequency oscillator which adjusts its spontaneous frequency based on the rate of stimuli. This model better reflects human behavior than a canonical nonlinear oscillator and a predictive ramping model–both widely used for temporal estimation and prediction–and demonstrate that the classical distinction between absolute and relative computational mechanisms can be unified under this framework. In addition, we show that neural oscillators may constitute hard-coded physiological priors–in a Bayesian sense–that reduce temporal uncertainty and facilitate the predictive processing of noisy rhythms. Together, the results show that adaptive oscillators provide an elegant and biologically plausible means to subserve rhythmic inference, reconciling previously incompatible frameworks for temporal inferential processes.
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Zippo, Antonio. "Matsuoka Nonlinear Oscillator Integration for Investigating Parkinsonian Tremor Dynamics through Multibody Simulation." Proceedings of the International Conference on Condition Monitoring and Asset Management 2024, no. 1 (2024): 7–13. https://doi.org/10.1784/cm2024.2b2.
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Parkinson's disease (PD) stands as a relentless neurodegenerative condition marked by a cascade of motor symptoms, such as tremors, rigidity, and bradykinesia. Among these manifestations, tremor emerges as a central feature, exerting a profound impact on the daily lives of those afflicted. Unraveling the complexities of Parkinsonian tremor dynamics holds importance in shaping innovative treatment modalities and interventions aimed at enhancing the well-being of individuals grappling with PD. In recent years, computational modeling techniques have emerged as powerful tools for investigating the complex dynamics of Parkinsonian tremor. Among these techniques, multibody simulation offers a versatile framework for analyzing the biomechanical interactions within the musculoskeletal system. By incorporating realistic anatomical structures and physiological parameters, multibody simulations enable researchers to mimic the dynamics of tremor generation and propagation in the upper limb with high fidelity. The integration of the Matsuoka nonlinear oscillator into the multibody simulation framework presents a novel approach for investigating Parkinsonian tremor dynamics. Inspired by neural circuitry underlying rhythmic movements in biological systems, the Matsuoka oscillator captures the nonlinear dynamics of neuromuscular interactions, producing oscillatory patterns closely resembling tremor characteristics observed in individuals with PD. By leveraging this integration, researchers can replicate pathological tremor phenomena and investigate the underlying mechanisms driving tremor generation. This paper provides a comprehensive overview of the methodology employed for modeling the upper limb and implementing the Matsuoka nonlinear oscillator within a multibody simulation framework. Through the synergistic combination of multibody simulation and the Matsuoka nonlinear oscillator, this study contributes to advancing our understanding of Parkinsonian tremor dynamics and lays the foundation for developing personalized therapeutic interventions for individuals with PD.
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KURRER, CHRISTIAN, and KLAUS SCHULTEN. "NEURONAL OSCILLATIONS AND STOCHASTIC LIMIT CYCLES." International Journal of Neural Systems 07, no. 04 (1996): 399–402. http://dx.doi.org/10.1142/s0129065796000373.
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We investigate a model for synchronous neural activity in networks of coupled neurons. The individual systems are governed by nonlinear dynamics and can continuously vary between excitable and oscillatory behavior. Analytical calculations and computer simulations show that coupled excitable systems can undergo two different phase transitions from synchronous to asynchronous firing behavior. One of the transitions is akin to the synchronization transitions in coupled oscillator systems, while the second transition can only be found in coupled excitable systems. Using the concept of Stochastic Limit Cycles, we present an analytical derivation of the two transitions and discuss implications for synchronization transitions in biological neural networks.
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مخزوم, هيثم, ابتسام دايرة, and اميرة بن فايد. "PERIODIC SOLUTIONS FOR IMPULSIVE NEUTRAL DYNAMIC EQUATIONS WITH INFINITE DELAY ON TIME SCALES SPACE." مجلة كلية التربية العلمية, no. 18 (June 15, 2025): 329–49. https://doi.org/10.37376/fesj.vi18.7293.
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This study addresses the problem of determining the existence and uniqueness of periodic solutions to a class of impulsive neutral dynamic equations that incorporate infinite delay, defined over a periodic time scale denoted by . The focal point of this work is a complex dynamic system that integrates multiple mathematical features: neutral terms, impulsive discontinuities at discrete instances, and an integral representation of the system’s historical behavior extending indefinitely into the past. The dynamic model under consideration involves a delta derivative, multiplicative operator terms, and delayed functional components, and is governed by impulsive effects at specified time points. The analysis is grounded in a general framework that accommodates both discrete and continuous behavior through the unifying language of time scale calculus. To establish the existence of periodic solutions, we utilize Krasnoselskii’s fixed point theorem—an essential tool in nonlinear operator theory known for its effectiveness in handling non-compact and non-linear mappings in Banach spaces. In contrast, the uniqueness of the solution is ensured by applying the Banach contraction principle, which demands more restrictive structural conditions on the system’s parameters but provides strong guarantees of solution distinctiveness. The theoretical contributions presented herein not only address the inherent analytical challenges posed by the neutral and impulsive dynamics but also offer valuable insights into systems exhibiting long-term memory. Such systems are prevalent in various scientific domains, including automatic control mechanisms with feedback delays, macroeconomic models driven by historical trends, and biological oscillators subject to abrupt environmental perturbations. By integrating advanced methods from time scale calculus, infinite-dimensional functional analysis, and fixed point theory, this work offers a comprehensive approach that enhances both the theoretical understanding and practical applicability of periodic solutions in delay-dominated dynamic systems.