Engineering entrainment and adaptation in limit cycle systems

Periodic behavior is key to life and is observed in multiple instances and at multiple time scales in our metabolism, our natural environment, and our engineered environment. A natural way of modeling or generating periodic behavior is done by using oscillators, i.e., dynamical systems that exhibit limit cycle behavior. While there is extensive literature on methods to analyze such dynamical systems, much less work has been done on methods to synthesize an oscillator to exhibit some specific desired characteristics. The goal of this article is twofold: (1) to provide a framework for characterizing and designing oscillators and (2) to review how classes of well-known oscillators can be understood and related to this framework. The basis of the framework is to characterize oscillators in terms of their fundamental temporal and spatial behavior and in terms of properties that these two behaviors can be designed to exhibit. This focus on fundamental properties is important because it allows us to systematically compare a large variety of oscillators that might at first sight appear very different from each other. We identify several specifications that are useful for design, such as frequency-locking behavior, phase-locking behavior, and specific output signal shape. We also identify two classes of design methods by which these specifications can be met, namely offline methods and online methods. By relating these specifications to our framework and by presenting several examples of how oscillators have been designed in the literature, this article provides a useful methodology and toolbox for designing oscillators for a wide range of purposes. In particular, the focus on synthesis of limit cycle dynamical systems should be useful both for engineering and for computational modeling of physical or biological phenomena.     

Hexapodal gaits and coupled nonlinear oscillator models

The general, model-independent features of different networks of six symmetrically coupled nonlinear oscillators are investigated. These networks are considered as possible models for locomotor central pattern generators (CPGs) in insects. Numerical experiments with a specific oscillator network model are briefly described. It is shown that some generic phase-locked oscillation-patterns for various systems of six symmetrically coupled nonlinear oscillators correspond to the common forward-walking gaits adopted by insects. It is also demonstrated that transitions observed in insect gaits can be modelled as standard symmetry-breaking bifurcations occurring in such systems. The present analysis, which leads to a natural classification of hexapodal gaits by symmetry and to natural sequences of gait bifurcations, relates observed gaits to the overall organizational structure of the underlying CPG. The implications of the present results for the development of simplified control systems for hexapodal walking robots are discussed.     

Systems and synthetic biology approaches in understanding biological oscillators

Background: Self-sustained oscillations are a ubiquitous and vital phenomenon in living systems. From primitive single-cellular bacteria to the most sophisticated organisms, periodicities have been observed in a broad spectrum of biological processes such as neuron firing, heart beats, cell cycles, circadian rhythms, etc. Defects in these oscillators can cause diseases from insomnia to cancer. Elucidating their fundamental mechanisms is of great significance to diseases, and yet challenging, due to the complexity and diversity of these oscillators. Results: Approaches in quantitative systems biology and synthetic biology have been most effective by simplifying the systems to contain only the most essential regulators. Here, we will review major progress that has been made in understanding biological oscillators using these approaches. The quantitative systems biology approach allows for identification of the essential components of an oscillator in an endogenous system. The synthetic biology approach makes use of the knowledge to design the simplest, de novo oscillators in both live cells and cell-free systems. These synthetic oscillators are tractable to further detailed analysis and manipulations. Conclusion: With the recent development of biological and computational tools, both approaches have made significant achievements.     

Sensitivity of basic oscillatory mechanisms for pattern generation and detection

 Intrinsic oscillators are the basic building blocks of central pattern generators, which model the neural circuits underlying pattern generation. Coupled intrinsic oscillators have been shown to synchronize their oscillatory frequencies and to maintain a characteristic pattern of phase relationships. Recently, oscillatory neurons have also been identified in sensory systems that are involved in decoding phase information. It has been hypothesized that the neural oscillators are part of neural circuits that implement phase-locked loops (PLLs), which are well-known electrical circuits for temporal decoding. Thus, there is evidence that intrinsic neural oscillators participate in both temporal pattern generation and temporal pattern decoding. The present paper investigates the dynamics underlying forced oscillators and forced PLLs, using a single framework, and compares both their stability and sensitivity characteristics. In particular, a method for assessing whether an oscillatory neuron is forced directly or indirectly, as part of a PLL, is developed and applied to published data. Received: 17 July 2000 / Accepted in revised form: 14 March 2001     

Analytical approximations for the amplitude and period of a relaxation oscillator

Background  

Analysis and design of complex systems benefit from mathematically tractable models, which are often derived by approximating a nonlinear system with an effective equivalent linear system. Biological oscillators with coupled positive and negative feedback loops, termed hysteresis or relaxation oscillators, are an important class of nonlinear systems and have been the subject of comprehensive computational studies. Analytical approximations have identified criteria for sustained oscillations, but have not linked the observed period and phase to compact formulas involving underlying molecular parameters.     

Temporal and spatial oscillations in bacteria

Oscillations pervade biological systems at all scales. In bacteria, oscillations control fundamental processes, including gene expression, cell cycle progression, cell division, DNA segregation and cell polarity. Oscillations are generated by biochemical oscillators that incorporate the periodic variation in a parameter over time to generate an oscillatory output. Temporal oscillators incorporate the periodic accumulation or activity of a protein to drive temporal cycles such as the cell and circadian cycles. Spatial oscillators incorporate the periodic variation in the localization of a protein to define subcellular positions such as the site of cell division and the localization of DNA. In this Review, we focus on the mechanisms of oscillators and discuss the design principles of temporal and spatial oscillatory systems.     

Mode coupling in living systems: implications for biology and medicine

Complex systems, and in particular biological ones, are characterized by large numbers of oscillations of widely differing frequencies. Various prejudices tend to lead to the assumption that such oscillators should generically be very weakly interacting. This paper reviews the basic ideas of linearity and nonlinearity as seen by a physicist, but with a view to biological systems. In particular, it is argued that large couplings between different oscillators of disparate frequencies are common, being present even in rather simple systems which are well-known in physics, although this issue is often glossed over. This suggests new experiments and investigations, as well as new approaches to therapies and human-environment interactions which, without the concepts described here, may otherwise seem unlikely to be interesting. The style of the paper is conversational with a minimum of mathematics, and no attempt at a complete list of references.     

A group-theoretic approach to rings of coupled biological oscillators

In this paper, a general approach for studying rings of coupled biological oscillators is presented. This approach, which is group-theoretic in nature, is based on the finding that symmetric ring networks of coupled non-linear oscillators possess generic patterns of phaselocked oscillations. The associated analysis is independent of the mathematical details of the oscillators' intrinsic dynamics and the nature of the coupling between them. The present approach thus provides a framework for distinguishing universal dynamic behaviour from that which depends upon further structure. In this study, the typical oscillation patterns for the general case of a symmetric ring of n coupled non-linear oscillators and the specific cases of three- and five-membered rings are considered. Transitions between different patterns of activity are modelled as symmetry-breaking bifurcations. The effects of one-way coupling in a ring network and the differences between discrete and continuous systems are discussed. The theoretical predictions for symmetric ring networks are compared with physiological observations and numerical simulations. This comparison is limited to two examples: neuronal networks and mammalian intestinal activity. The implications of the present approach for the development of physiologically meaningful oscillator models are discussed.     

Using variability to regulate long term biological rhythms

We present a model for the generation of precise, long term rhythms from a collection of imprecise, short term oscillators. The model uses variability between oscillators in conjunction with simple coupling rules to produce long term rhythms that are independent of rate equations (e.g. Arrhenius). The rhythms generated by the model are controlled by only two independent parameters and exhibit several physiologically interesting properties, including ready entrainment to external signals and splitting in response to strong constant signals. The model provides several predictions that can be tested in future experiments.     

Mutually synchronized relaxation oscillators as prototypes of oscillating systems in biology

Summary This paper discusses the analogy between phenomena in populations of coupled biological oscillators and the behaviour of systems of synchronized mathematical oscillators. Frequency entrainment in a set of coupled relaxation oscillators is investigated with perturbation methods. This analysis leads to quantitative results for entrainment and explains phenomena such as travelling waves in systems of spatially distributed oscillators.     

Synchronization of coupled nonidentical genetic oscillators

The study of the collective dynamics of synchronization among genetic oscillators is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. Genetic oscillators are biochemical networks, which can generally be modelled as nonlinear dynamic systems. We show in this paper that many genetic oscillators can be transformed into Lur'e form by exploiting the special structure of biological systems. By using a control theory approach, we provide a theoretical method for analysing the synchronization of coupled nonidentical genetic oscillators. Sufficient conditions for the synchronization as well as the estimation of the bound of the synchronization error are also obtained. To demonstrate the effectiveness of our theoretical results, a population of genetic oscillators based on the Goodwin model are adopted as numerical examples.     

Transient resetting: a novel mechanism for synchrony and its biological examples

The study of synchronization in biological systems is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. In this paper, by using simple dynamical systems theory, we present a novel mechanism, named transient resetting, for the synchronization of uncoupled biological oscillators with stimuli. This mechanism not only can unify and extend many existing results on (deterministic and stochastic) stimulus-induced synchrony, but also may actually play an important role in biological rhythms. We argue that transient resetting is a possible mechanism for the synchronization in many biological organisms, which might also be further used in the medical therapy of rhythmic disorders. Examples of the synchronization of neural and circadian oscillators as well as a chaotic neuron model are presented to verify our hypothesis.     

Multiple pulse interactions and averaging in systems of coupled neural oscillators

Oscillators coupled strongly are capable of complicated behavior which may be pathological for biological control systems. Nevertheless, strong coupling may be needed to prevent asynchrony. We discuss how some neural networks may be designed to achieve only simple locking behavior when the coupling is strong. The design is based on the fact that the method of averaging produces equations that are capable only of locking or drift, not pathological complexity. Furthermore, it is shown that oscillators that interact by means of multiple pulses per cycle, dispersed around the cycle, behave like averaged equations, even if the number of pulses is small. We discuss the biological intuition behind this scheme, and show numerically that it works when the oscillators are taken to be composites, each unit of which is governed by a well-known model of a neural oscillator. Finally, we describe numerical methods for computing from equations for coupled limit cycle oscillators the averaged coupling functions of our theory.Research partially supported by the National Science Foundation under grants DMS 8796235 and DMS 8701405 and the Air Force Office of Scientific Research under University Research Contract F 49620-C-0131 to Northeastern University     

Spatially Organized Dynamical States in Chemical Oscillator Networks: Synchronization,Dynamical Differentiation,and Chimera Patterns

Dynamical processes in many engineered and living systems take place on complex networks of discrete dynamical units. We present laboratory experiments with a networked chemical system of nickel electrodissolution in which synchronization patterns are recorded in systems with smooth periodic, relaxation periodic, and chaotic oscillators organized in networks composed of up to twenty dynamical units and 140 connections. The reaction system formed domains of synchronization patterns that are strongly affected by the architecture of the network. Spatially organized partial synchronization could be observed either due to densely connected network nodes or through the ‘chimera’ symmetry breaking mechanism. Relaxation periodic and chaotic oscillators formed structures by dynamical differentiation. We have identified effects of network structure on pattern selection (through permutation symmetry and coupling directness) and on formation of hierarchical and ‘fuzzy’ clusters. With chaotic oscillators we provide experimental evidence that critical coupling strengths at which transition to identical synchronization occurs can be interpreted by experiments with a pair of oscillators and analysis of the eigenvalues of the Laplacian connectivity matrix. The experiments thus provide an insight into the extent of the impact of the architecture of a network on self-organized synchronization patterns.     

Modelling periodic oscillation in gene regulatory networks by cyclic feedback systems

In this paper, we develop a new methodology to analyze and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using negative cyclic feedback systems. We show that negative cyclic feedback networks have no stable equilibria but stable periodic orbits when certain conditions are satisfied. Specifically, we first prove the basic properties of the biological networks composed of cyclic feedback loops, and then extend our results to general cyclic feedback network with less restriction, thereby making our theoretical analysis and design of oscillators easy to implement, even for large-scale systems. Finally, we use one circadian network formed by a period protein (PER) and per mRNA, and one biologically plausible synthetic gene network, to demonstrate the theoretical results. Since there is less restriction on the network structure, the results of this paper can be expected to apply to a wide variety of areas on modelling, analyzing and designing of biological systems.     

A new model for simple neural nets and its application in the design of a neural oscillator

The results of a previous theoretical study of a class of systems are applied for the design of neural nets which try to simulate biological behavior. Besides the models for single aperiodic and periodic neurons, a “neural oscillator” is developed which consists of two cross-excited neurons. Its response is similar to the firing pattern of certain biological neural oscillators, like the flying system of the locust. Also, by proper change of its parameters, it can be made highly irregular, providing a deterministic model for the spontaneous neural activity.     

Design principles for robust oscillatory behavior

Oscillatory responses are ubiquitous in regulatory networks of living organisms, a fact that has led to extensive efforts to study and replicate the circuits involved. However, to date, design principles that underlie the robustness of natural oscillators are not completely known. Here we study a three-component enzymatic network model in order to determine the topological requirements for robust oscillation. First, by simulating every possible topological arrangement and varying their parameter values, we demonstrate that robust oscillators can be obtained by augmenting the number of both negative feedback loops and positive autoregulations while maintaining an appropriate balance of positive and negative interactions. We then identify network motifs, whose presence in more complex topologies is a necessary condition for obtaining oscillatory responses. Finally, we pinpoint a series of simple architectural patterns that progressively render more robust oscillators. Together, these findings can help in the design of more reliable synthetic biomolecular networks and may also have implications in the understanding of other oscillatory systems.

Electronic supplementary material

The online version of this article (doi:10.1007/s11693-015-9178-6) contains supplementary material, which is available to authorized users.     

A Structural Classification of Candidate Oscillatory and Multistationary Biochemical Systems

Molecular systems are uncertain: The variability of reaction parameters and the presence of unknown interactions can weaken the predictive capacity of solid mathematical models. However, strong conclusions on the admissible dynamic behaviors of a model can often be achieved without detailed knowledge of its specific parameters. In systems with a sign-definite Jacobian, for instance, cycle-based criteria related to the famous Thomas’ conjectures have been largely used to characterize oscillatory and multistationary dynamic outcomes. We build on the rich literature focused on the identification of potential oscillatory and multistationary behaviors using parameter-free criteria. We propose a classification for sign-definite non-autocatalytic biochemical networks, which summarizes several existing results in the literature. We call weak (strong) candidate oscillators systems which can possibly (exclusively) transition to instability due to the presence of a complex pair of eigenvalues, while we call weak (strong) candidate multistationary systems those which can possibly (exclusively) transition to instability due to the presence of a real eigenvalue. For each category, we provide a characterization based on the exclusive or simultaneous presence of positive and negative cycles in the associated sign graph. Most realistic examples of biochemical networks fall in the gray area of systems in which both positive and negative cycles are present: Therefore, both oscillatory and bistable behaviors are in principle possible. However, many canonical example circuits exhibiting oscillations or bistability fall in the categories of strong candidate oscillators/multistationary systems, in agreement with our results.     

Local network parameters can affect inter-network phase lags in central pattern generators

Weakly coupled phase oscillators and strongly coupled relaxation oscillators have different mechanisms for creating stable phase lags. Many oscillations in central pattern generators combine features of each type of coupling: local networks composed of strongly coupled relaxation oscillators are weakly coupled to similar local networks. This paper analyzes the phase lags produced by this combination of mechanisms and shows how the parameters of a local network, such as the decay time of inhibition, can affect the phase lags between the local networks. The analysis is motivated by the crayfish central pattern generator used for swimming, and uses techniques from geometrical singular perturbation theory.     

Populations of biochemical oscillators as circadian clocks

Various types of populations of interacting oscillators were analyzed and their synchronization states were determined. One of the systems involving biochemical oscillators was simulated on the computer and the occurence of rhythm splitting was observed. A comparison of its attributes with experimental results on circadian ryhythms showed good agreement. This allows us to distinguish between types of mechanisms held responsible for the splitting phenomenon in the past. The present model also offers a new explanation about the differences of light action on diurnal and nocturnal organisms.