Are ecological systems chaotic - And if not, why not?

Chaotic dynamics emerge when positive/negative feedback systems are dominated much of the time by positive feedback growth processes. Although all ecological systems contain the seeds of chaos (positive feedback), empirical evidence and evolutionary/ecological reasoning support the view that ecosystems do not normally behave chaotically. They may, however, be driven to chaos by human actions that increase growth rates or induce delays in the regulatory (negative feedback) processes.     

Feedback theory and Darwinian evolution.

Feedback loops can have a significant impact on biological systems that are evolving under Darwinian natural selection. Many of the striking and sometimes bizarre patterns that characterize the evolution of such systems have simple, natural explanations that involve the effects of feedback loops. The two fundamental types of feedback loops, positive and negative, have effects that are radically different: negative feedback tends to produce stability and resistance to change; positive feedback produces instability and even catastrophe. Both types of feedback loops are important in biological systems, and both can produce chaos, whose mathematical complexity often produces strange, beautiful and totally unexpected patterns that have only begun to be explored using the computational capabilities of modern electronic computers. An understanding of the patterns that can result from the effects of feedback loops can produce important new insights into the patterns that mark the evolutionary development of biological systems.     

On the dynamics of controlled metabolic network and cellular behavior

The existence of elaborate control mechanisms for the various biochemical processes inside and within living cells is responsible for the coherent behaviour observed in its spatio-temporal organisation. Stability and sensitivity are both necessary properties of living systems and these are achieved through negetive and positive feedback loops as in other control systems. We have studied a three-step reaction scheme involving a negative and a positive feedback loop in the form of end-product inhibition and allosteric activation. The variety of behaviour exhibited by this system, under different conditions, includes steady state, simple limit cycle oscillations, complex oscillations and period bifurcations leading to random oscillations or chaos. The system also shows the existence of two distinct chaotic regimes under the variation of a single parameter. These results, in comparison with single biochemical control loops, show that new behaviours can be exhibited in a more complex network which are not seen in the single control loops. The results are discussed in the light of a diverse variety of cellular functions in normal and altered cells indicating the role of controlled metabolic network as the underlying basis for cellular behaviour.     

Dynamic Processes in Regulation and Some Implications for Biofeedback and Biobehavioral Interventions

Systems theory has long been used in psychology, biology, and sociology. This paper applies newer methods of control systems modeling for assessing system stability in health and disease. Control systems can be characterized as open or closed systems with feedback loops. Feedback produces oscillatory activity, and the complexity of naturally occurring oscillatory patterns reflects the multiplicity of feedback mechanisms, such that many mechanisms operate simultaneously to control the system. Unstable systems, often associated with poor health, are characterized by absence of oscillation, random noise, or a very simple pattern of oscillation. This modeling approach can be applied to a diverse range of phenomena, including cardiovascular and brain activity, mood and thermal regulation, and social system stability. External system stressors such as disease, psychological stress, injury, or interpersonal conflict may perturb a system, yet simultaneously stimulate oscillatory processes and exercise control mechanisms. Resonance can occur in systems with negative feedback loops, causing high-amplitude oscillations at a single frequency. Resonance effects can be used to strengthen modulatory oscillations, but may obscure other information and control mechanisms, and weaken system stability. Positive as well as negative feedback loops are important for system function and stability. Examples are presented of oscillatory processes in heart rate variability, and regulation of autonomic, thermal, pancreatic and central nervous system processes, as well as in social/organizational systems such as marriages and business organizations. Resonance in negative feedback loops can help stimulate oscillations and exercise control reflexes, but also can deprive the system of important information. Empirical hypotheses derived from this approach are presented, including that moderate stress may enhance health and functioning.     

The principle of homeostasis in the hypothalamus-pituitary-adrenal system: new insight from positive feedback

Feedback control, both negative and positive, is a fundamental feature of biological systems. Some of these systems strive to achieve a state of equilibrium or "homeostasis". The major endocrine systems are regulated by negative feedback, a process believed to maintain hormonal levels within a relatively narrow range. Positive feedback is often thought to have a destabilizing effect. Here, we present a "principle of homeostasis," which makes use of both positive and negative feedback loops. To test the hypothesis that this homeostatic concept is valid for the regulation of cortisol, we assessed experimental data in humans with different conditions (gender, obesity, endocrine disorders, medication) and analyzed these data by a novel computational approach. We showed that all obtained data sets were in agreement with the presented concept of homeostasis in the hypothalamus-pituitary-adrenal axis. According to this concept, a homeostatic system can stabilize itself with the help of a positive feedback loop. The brain mineralocorticoid and glucocorticoid receptors-with their known characteristics-fulfill the key functions in the homeostatic concept: binding cortisol with high and low affinities, acting in opposing manners, and mediating feedback effects on cortisol. This study supports the interaction between positive and negative feedback loops in the hypothalamus-pituitary-adrenal system and in this way sheds new light on the function of dual receptor regulation. Current knowledge suggests that this principle of homeostasis could also apply to other biological systems.     

The dynamics of production and destruction: Analytic insight into complex behavior

This paper analytically explores the properties of simple differential-difference equations that represent dynamic processes with feedback dependent on prior states of the system. Systems with pure negative and positive feedback are examined, as well as those with mixed (positive/negative) feedback characteristics. Very complex time dependent behaviors may arise from these processes. Indeed, the same mechanism may, depending on system parameters and initial conditions, produce simple, regular, repetitive patterns and completely irregular random-like fluctuations.For the differential-delay equations considered here we prove the existence of: (i) stable and unstable limit cycles, where the stable cycles may have an arbitrary number of extrema per period; and (ii) chaos, meaning the presence of infinitely many periodic solutions of different period and of infinitely many irregular and mixing solutions.     

Information processing in echo state networks at the edge of chaos

We investigate information processing in randomly connected recurrent neural networks. It has been shown previously that the computational capabilities of these networks are maximized when the recurrent layer is close to the border between a stable and an unstable dynamics regime, the so called edge of chaos. The reasons, however, for this maximized performance are not completely understood. We adopt an information-theoretical framework and are for the first time able to quantify the computational capabilities between elements of these networks directly as they undergo the phase transition to chaos. Specifically, we present evidence that both information transfer and storage in the recurrent layer are maximized close to this phase transition, providing an explanation for why guiding the recurrent layer toward the edge of chaos is computationally useful. As a consequence, our study suggests self-organized ways of improving performance in recurrent neural networks, driven by input data. Moreover, the networks we study share important features with biological systems such as feedback connections and online computation on input streams. A key example is the cerebral cortex, which was shown to also operate close to the edge of chaos. Consequently, the behavior of model systems as studied here is likely to shed light on reasons why biological systems are tuned into this specific regime.     

Feedback regulation of EGFR signalling: decision making by early and delayed loops

Human-made information relay systems invariably incorporate central regulatory components, which are mirrored in biological systems by dense feedback and feedforward loops. This type of system control is exemplified by positive and negative feedback loops (for example, receptor endocytosis and dephosphorylation) that enable growth factors and receptor Tyr kinases of the epidermal growth factor receptor (EGFR)/ERBB family to regulate cellular function. Recent studies show that the collection of feedback regulatory loops can perform computational tasks - such as decoding ligand specificity, transforming graded input signals into a digital output and regulating response kinetics. Aberrant signal processing and feedback regulation can lead to defects associated with pathologies such as cancer.     

Roles of positive and negative feedback in biological systems

We discuss the influence of positive and negative feedback on the stability of a system, which is not clear-cut, and involves complex, mathematical problems. We show in particular that positive feedback can have a stabilising effect on some systems. We also point out the role that positive feedback plays in the digital treatment of signals required by cellular signalling, drawing on analogies from electronics, and the role that negative feedback plays in making a system robust against alteration of its parameters. Both positive and negative feedback can be seen as important enhancers of the properties of biological systems.     

Tuning genetic clocks employing DNA binding sites

Periodic oscillations play a key role in cell physiology from the cell cycle to circadian clocks. The interplay of positive and negative feedback loops among genes and proteins is ubiquitous in these networks. Often, delays in a negative feedback loop and/or degradation rates are a crucial mechanism to obtain sustained oscillations. How does nature control delays and kinetic rates in feedback networks? Known mechanisms include proper selection of the number of steps composing a feedback loop and alteration of protease activity, respectively. Here, we show that a remarkably simple means to control both delays and effective kinetic rates is the employment of DNA binding sites. We illustrate this design principle on a widely studied activator-repressor clock motif, which is ubiquitous in natural systems. By suitably employing DNA target sites for the activator and/or the repressor, one can switch the clock “on” and “off” and precisely tune its period to a desired value. Our study reveals a design principle to engineer dynamic behavior in biomolecular networks, which may be largely exploited by natural systems and employed for the rational design of synthetic circuits.     

A dichotomy for a class of cyclic delay systems

Two complementary analyses of a cyclic negative feedback system with delay are considered in this paper. The first analysis applies the work by Sontag, Angeli, Enciso and others regarding monotone control systems under negative feedback, and it implies the global attractiveness towards an equilibrium for arbitrary delays. The second one concerns the existence of a Hopf bifurcation with respect to the delay parameter, and it implies the existence of nonconstant periodic solutions for special delay values. A key idea is the use of the Schwarzian derivative, and its application for the study of Hill function nonlinearities. The positive feedback case is also addressed.     

On a criterion of instability for positive feedback control systems

Perturbation of the input in open systems triggers a response which displaces the stationary state. It is shown that the gradients of this stationary displacement are comprised in the inverse matrix of the system. Hence the “trace criterion” of instability is examined in the inverse matrices of some positive and negative feedback oscillatory models. For positive feedback systems the results provide a phenomenological interpretation of instability, in terms of the control properties of the system. An objection to Clarke's conjecture is made.     

Cellular control models with linked positive and negative feedback and delays. II. Linear analysis and local stability

An analysis of local behavior is made of two nonlinear models which incorporate both an induction or positive feedback control mechanism and a repression or negative feedback control mechanism. The systems of differential equations with delays are linearized about their equilibria. The related characteristic equations which are exponential polynomials are studied to determine the local stability of the models. Computer studies are included to show the range of stability for different parameter values, and the biological significance is discussed briefly.     

Coupled positive and negative feedback circuits form an essential building block of cellular signaling pathways

Cellular circuits have positive and negative feedback loops that allow them to respond properly to noisy external stimuli. It is intriguing that such feedback loops exist in many cases in a particular form of coupled positive and negative feedback loops with different time delays. As a result of our mathematical simulations and investigations into various experimental evidences, we found that such coupled feedback circuits can rapidly turn on a reaction to a proper stimulus, robustly maintain its status, and immediately turn off the reaction when the stimulus disappears. In other words, coupled feedback loops enable cellular systems to produce perfect responses to noisy stimuli with respect to signal duration and amplitude. This suggests that coupled positive and negative feedback loops form essential signal transduction motifs in cellular signaling systems.     

Hindmarsh-Rose神经模型的混沌控制

应用稳定性准则的混沌控制方法控制单个Hindmarsh-Rose神经元模型的混沌发放峰序列和混沌爆发运动。通过对膜电压的非线性连续-时间反馈干扰的输入,将混沌运动控制到5峰／爆发（5spikes/burst）轨道上,该轨道嵌入在混沌吸引子内。数值模拟结果显示该方法在控制HR神经元模型方面是有效的。     

Biology using engineering tools

Negative feedback is an ubiquitous feature of biological networks. Recent work from Sturm and colleaguespresents experimental evidence that biological negative feedback can serve the same function as it does for engineered systems: robustness to perturbations within the feedback loop. Such behavior has important implications for how to attack deregulated signaling networks containing negative feedback in diseases such as cancer.     

Can noise induce chaos?

An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that "chaos" should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.