Consistent dynamics suggests tight regulation of biophysical parameters in a small network of bursting neurons

The neuronal firing patterns in the pyloric network of crustaceans are remarkably consistent among animals. Although this characteristic of the pyloric network is well-known, the biophysical mechanisms underlying the regulation of the systems output are receiving renewed attention. Computer simulations of the pyloric network recently demonstrated that consistent motor output can be achieved from neurons with disparate biophysical parameters among animals. Here we address this hypothesis by pharmacologically manipulating the pyloric network and analyzing the emerging voltage oscillations and firing patterns. Our results show that the pyloric network of the lobster stomatogastric ganglion maintains consistent and regular firing patterns even when entire populations of specific voltage-gated channels and synaptic receptors are blocked. The variations of temporal parameters used to characterize the burst patterns of the neurons as well as their intraburst spike dynamics do not display statistically significant increase after blocking the transient K-currents (with 4-aminopyridine), the glutamatergic inhibitory synapses (with picrotoxin), or the cholinergic synapses (with atropine) in pyloric networks from different animals. These data suggest that in this very compact circuit, the biophysical parameters are cell-specific and tightly regulated.     

Transient responses of a modeled bursting neuron: analysis with equilibrium and averaged nullclines

We utilized a state-space approach to study the dynamics of a modeled bursting neuron consisting of 11 state variables. Such an approach may be used on a high-order system when a small number of variables are rate-limiting and dominate the dynamics of the model. Calculation of equilibrium and averaged nullclines and saddle-node bifurcations of the full and reduced models provided measures that indicated the transition between silence and spiking and the dynamics of the system during both the silent and spiking phases of the burst cycle. The relative stability of tonic beating solutions in the presence and absence of 5-HT was calculated in the state-space of the slow variables and related to specific biophysical mechanisms. The results were compared with similar simulations performed in Butera et al. (1995) which utilized a current-voltage (I-V)-based method for analysis. While the state-space method is sometimes more difficult to link to specific biophysical mechanisms, it offers a wider portrait of the dynamics of the system. In contrast, the use of I-V plots offers a direct relationship to biophysical processes, but provides no information on the dynamics of non-voltage-dependent processes such as Ca. Received: 6 December 1996 / Accepted in revised form: 1 July 1997     

Resilience indicators: prospects and limitations for early warnings of regime shifts

In the vicinity of tipping points—or more precisely bifurcation points—ecosystems recover slowly from small perturbations. Such slowness may be interpreted as a sign of low resilience in the sense that the ecosystem could easily be tipped through a critical transition into a contrasting state. Indicators of this phenomenon of ‘critical slowing down (CSD)’ include a rise in temporal correlation and variance. Such indicators of CSD can provide an early warning signal of a nearby tipping point. Or, they may offer a possibility to rank reefs, lakes or other ecosystems according to their resilience. The fact that CSD may happen across a wide range of complex ecosystems close to tipping points implies a powerful generality. However, indicators of CSD are not manifested in all cases where regime shifts occur. This is because not all regime shifts are associated with tipping points. Here, we review the exploding literature about this issue to provide guidance on what to expect and what not to expect when it comes to the CSD-based early warning signals for critical transitions.     

Simulation of circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators

In mammals, circadian rhythms are driven by a pacemaker located in the suprachiasmatic nuclei (SCN) of the anterior hypothalamus. The firing rate of neurons within the SCN exhibits a circadian rhythm. There is evidence that individual neurons within the SCN act as circadian oscillators. Rhythm generation in the SCN was therefore modeled by a system of self-sustained oscillators. The model is composed of up to 10000 oscillatory elements arranged in a square array. Each oscillator has its own (randomly determined) intrinsic period reflecting the widely dispersed periods observed in the SCN. The model behavior was investigated mainly in the absence of synchronizing zeitgebers. Due to local coupling the oscillators synchronized and an overall rhythm emerged. This indicates that a locally coupled system is capable of integrating the output of individual clock cells with widely dispersed periods. The period of the global output (average of all oscillators) corresponded to the average of the intrinsic periods and was stable even for small amplitudes and during transients. Noise, reflecting biological fluctuations at the cellular level, distorted the global rhythm in small arrays. The period of the rhythm could be stabilized by increasing the array size, which thus increased the robustness against noise. Since different regions of the SCN have separate output pathways, the array of oscillators was subdivided into four quadrants. Sudden deviations of periodicity sometimes appeared in one quadrant, while the periods of the other quadrants were largely unaffected. This result could represent a model for splitting, which has been observed in animal experiments. In summary, the multi-oscillator model of the SCN showed a broad repertoire of dynamic patterns, revealed a stable period (even during transients) with robustness against noise, and was able to account for such a complex physiological behavior as splitting.     

Building Resilience Against Climate-Driven Shifts in a Temperate Reef System: Staying Away from Context-Dependent Ecological Thresholds

As climate-driven environmental changes and anthropogenic perturbations increasingly affect ecological systems, the number of abrupt phase shifts in ecosystem dynamics is rising, with far-reaching ecological, economic and social effects. These shifts are notoriously difficult to study, anticipate and manage. Although indicators of impending phase shifts in ecosystems have been described theoretically, they have only been observed empirically either after the fact or under controlled experiments. Here we demonstrate the usefulness of case-specific simulation models to estimate tipping points in the dynamics of real ecological systems, characterise how these thresholds may vary depending on local conditions and derive safe management targets associated with low risk of undesirable phase shifts. Under the combined effects of ocean changes and fishing, inshore rocky reefs in eastern Tasmania can transition from dense seaweed beds to sea urchin ‘barrens’ habitat, realising severe local loss of habitat, productivity and valuable fisheries. Using Monte-Carlo simulations with a validated model that realistically captures reef dynamics, we characterise the hysteresis in community dynamics and the variability in ecological thresholds along the gradient of environmental conditions. Simulation suggests that prevention of ongoing sea urchin destructive grazing of macroalgal beds is achievable but the yet-to-be-observed restoration of seaweed beds from extensive sea urchin barrens is highly unlikely. To guide management against undesirable phase shifts, we define target points associated with low risk of widespread barrens formation and show that, along with sea urchin culling, recognising the role of lobsters in mitigating sea urchin destructive grazing through predation is key to maintain reef productivity.     

Functional links and robustness in food webs

The robustness of ecosystems to species losses is a central question in ecology, given the current pace of extinctions and the many species threatened by human impacts, including habitat destruction and climate change. Robustness from the perspective of secondary extinctions has been addressed in the context of food webs to consider the complex network of species interactions that underlie responses to perturbations. In-silico removal experiments have examined the structural properties of food webs that enhance or hamper the robustness of ecosystems to species losses, with a focus on the role of hubs, the most connected species. Here we take a different approach and focus on the role of the connections themselves. We show that trophic links can be divided into functional and redundant based on their contribution to robustness. The analysis of empirical webs shows that hubs are not necessarily the most important species as they may hold many redundant links. Furthermore, the fraction of functional connections is high and constant across systems regardless of size and interconnectedness. The main consequence of this scaling pattern is that ecosystem robustness can be considerably reduced by species extinctions even when these do not result in any secondary extinctions. This introduces the possibility of tipping points in the collapse of ecosystems.     

Lack of Critical Slowing Down Suggests that Financial Meltdowns Are Not Critical Transitions,yet Rising Variability Could Signal Systemic Risk

Complex systems inspired analysis suggests a hypothesis that financial meltdowns are abrupt critical transitions that occur when the system reaches a tipping point. Theoretical and empirical studies on climatic and ecological dynamical systems have shown that approach to tipping points is preceded by a generic phenomenon called critical slowing down, i.e. an increasingly slow response of the system to perturbations. Therefore, it has been suggested that critical slowing down may be used as an early warning signal of imminent critical transitions. Whether financial markets exhibit critical slowing down prior to meltdowns remains unclear. Here, our analysis reveals that three major US (Dow Jones Index, S&P 500 and NASDAQ) and two European markets (DAX and FTSE) did not exhibit critical slowing down prior to major financial crashes over the last century. However, all markets showed strong trends of rising variability, quantified by time series variance and spectral function at low frequencies, prior to crashes. These results suggest that financial crashes are not critical transitions that occur in the vicinity of a tipping point. Using a simple model, we argue that financial crashes are likely to be stochastic transitions which can occur even when the system is far away from the tipping point. Specifically, we show that a gradually increasing strength of stochastic perturbations may have caused to abrupt transitions in the financial markets. Broadly, our results highlight the importance of stochastically driven abrupt transitions in real world scenarios. Our study offers rising variability as a precursor of financial meltdowns albeit with a limitation that they may signal false alarms.     

Phenotypic variability promotes diversity and stability in competitive communities

Intraspecific variation is at the core of evolutionary theory, and yet, from an ecological perspective, we have few robust expectations for how this variation should affect the dynamics of large communities. Here, by adapting an approach from evolutionary game theory, we show that the incorporation of phenotypic variability into competitive networks dramatically alters the dynamics across ecological timescales, stabilising the systems and buffering the communities against demographic perturbations. The beneficial effects of phenotypic variability are strongest when there are substantial differences among phenotypes and when phenotypes are inherited with moderately high fidelity; yet even low levels of variation lead to significant increases in diversity, stability, and robustness. By identifying a simple and ubiquitous stabilising force in competitive communities, this work contributes to our core understanding of how biological diversity is maintained in natural systems.     

Circadian organization inAplysia: internal desynchronization and amplitude of locomotor rhythm

Summary We have tested the hypothesis that the circadian oscillators in the eyes ofAplysia are coequal driver oscillators for the circadian locomotor rhythm. Three predictions based on this hypothesis were tested. Prediction 1: at a time when the phase difference between the eye rhythms is small, the amplitude of the locomotor rhythm in two eyed animals will be as great or greater than the amplitude in one eyed animals. Prediction 2: the amplitude of the locomotor rhythm of two eyed animals will decline under conditions in which the two eye rhythms become out of phase with each other. Prediction 3: the form of the locomotor rhythm will broaden or become biphasic in two eyed animals when the two eye rhythms become out of phase with each other.None of the predictions was confirmed. One eyedAplysia had higher amplitude locomotor rhythms than two eyedAplysia, even under conditions in which the two eye rhythms were probably not far out of phase with each other. There was no tendency for the amplitude of the locomotor rhythm of two eyed animals to decline under circumstances in which the phase difference between the two eye rhythms changes from less than 4 h to as much as 11.5 h. There was no tendency in two eyed animals for the locomotor rhythm to broaden or become biphasic as the eye rhythms became more out of phase with each other.The results led us to reject the hypothesis that the eyes are co-equal drivers for the locomotor rhythm. The ocular influence on locomotion is more likely to be mediated via mechanisms in the central nervous system that do not faithfully conserve the phase of the eye rhythms. One possibility is that the driver is a third circadian oscillator that interacts with the two eye oscillators.Abbreviations CAP compound action potentials - CC constant conditions - CT circadian time - DO driver oscillator - EO eye oscillator - RSD relative standard deviations (see Methods)     

生态系统的多稳态与突变

许多生态系统可能在短时间内发生难以预料的状态突变, 其中一些生态系统突变的机理可以用多稳态理论进行解释。近年来生态系统的多稳态和突变现象及其机理吸引了研究者和管理者的广泛关注。本文重点对生态系统多稳态的理论基础、识别方法及稳态转换发生的早期预警信号进行综述, 并基于典型生态系统过程对现实世界中可能观测到的稳态转换进行实例分析, 最后对多稳态概念框架和理论应用中的潜在争议进行讨论, 以期为非线性生态系统动态的理论研究、管理实践和生物多样性保护等提供参考。     

Lactation curve model with explicit representation of perturbations as a phenotyping tool for dairy livestock precision farming

In the context of dairy farming, ruminant females often face challenges inducing perturbations that affect their performance and welfare. A key issue is how to assess the effect of perturbations and provide metrics to quantify how animals cope with their environment. Milk production dynamics are good candidates to address this issue: i) they are easily accessible, ii) overall dynamics throughout lactation process are well described and iii) perturbations are visible through milk losses. In this study, a perturbed lactation model (PLM) with explicit representation of perturbations was developed. The model combines two components: i) the unperturbed lactation model that describes a theoretical lactation curve, assumed to reflect female production potential and ii) the perturbation model that describes all the deviations from the unperturbed lactation model with four parameters: starting date, intensity and shape (collapse and recovery). To illustrate the use of the PLM as a phenotyping tool, it was fitted on a data set of 319 complete lactations from 181 individual dairy goats. A total of 2 354 perturbations were detected, with an average of 7.40 perturbations per lactation. Loss of milk production for the whole lactation due to perturbations varied between 2 and 19% of the milk production predicted by the unperturbed lactation model. The number of perturbations was not the major factor explaining the loss of milk production, suggesting that there are different types of animal response to challenges. By incorporating explicit representation of perturbations in a lactation model, it was possible to determine for each female the potential milk production, characteristics of each perturbation and milk losses induced by perturbations. Further, it was possible to compare animals and analyze individual variability. The indicators produced by the PLM are likely to be useful to move from raw data to decision support tools in dairy production.     

A phase dynamics model of human circadian rhythms

Nonphotic entrainment of an overt sleep-wake rhythm and a circadian pacemaker-driving temperature/melatonin rhythm suggests existence of feedback mechanisms in the human circadian system. In this study, the authors constructed a phase dynamics model that consisted of two oscillators driving temperature/melatonin and sleep-wake rhythms, and an additional oscillator generating an overt sleep-wake rhythm. The feedback mechanism was implemented by modifying couplings between the constituent oscillators according to the history of correlations between them. The model successfully simulated the behavior of human circadian rhythms in response to forced rest-activity schedules under free-run situations: the sleep-wake rhythm is reentrained with the circadian pacemaker after release from the schedule, there is a critical period for the schedule to fully entrain the sleep-wake rhythm, and the forced rest-activity schedule can entrain the circadian pacemaker with the aid of exercise. The behavior of human circadian rhythms was reproduced with variations in only a few model parameters. Because conventional models are unable to reproduce the experimental results concerned here, it was suggested that the feedback mechanisms included in this model underlie nonphotic entrainment of human circadian rhythms.     

Lung and buccal ventilation in the frog: uncoupling coupled oscillators

The frog, with two distinct ventilatory acts, provides a useful model to investigate the prospective interaction of two oscillators in generating the respiratory rhythm. Building on evidence supporting the existence of separate oscillators generating buccal and lung ventilation, we have attempted to uncouple the two rhythms in the isolated brain stem preparation. Opioid preferentially inhibits the lung rhythm, suggesting an uncoupling of the lung from the buccal oscillator. Reduction of the superfusate chloride concentration alters both the buccal and the lung rhythms. Joint application of opioid and reduced-chloride superfusate leads to an increase in the variability of the buccal burst-to-lung burst intervals. This increase in variability suggests that chloride-mediated mechanisms are involved in coupling the buccal oscillator to the lung oscillator. Given the results from these interventions, we propose a simple schematic model of the frog respiratory rhythm generator, outlining the coupling of the lung and buccal oscillators.     

Marine Ecosystems as Complex Adaptive Systems: Emergent Patterns,Critical Transitions,and Public Goods

Complex adaptive systems provide a unified framework for explaining ecosystem phenomena. In the past 20 years, complex adaptive systems have been sharpened from an abstract concept into a series of tools that can be used to solve concrete problems. These advances have been led by the development of new techniques for coupling ecological and evolutionary dynamics, for integrating dynamics across multiple scales of organization, and for using data to infer the complex interactions among different components of ecological systems. Focusing on the development and usage of these new methods, we discuss how they have led to an improved understanding of three universal features of complex adaptive systems, emergent patterns; tipping points and critical phenomena; and cooperative behavior. We restrict our attention primarily to marine ecosystems, which provide numerous successful examples of the application of complex adaptive systems. Many of these are currently undergoing dramatic changes due to anthropogenic perturbations, and we take the opportunity to discuss how complex adaptive systems can be used to improve the management of public goods and to better preserve critical ecosystem services.     

Environmental entrainment of tidally rhythmic behaviour in marine animals

Field and laboratory experiments show that endogenous circatidal rhythms in coastal animals are entrained by exposure to real or simulated tidal cycles of hydrostatic pressure, temperature, salinity, wave agitation, immersion and light. Short pulses (2–3 h) of simulated high tide induce slight phase advances or delays in the free-running circatidal rhythm of groups of experimental animals, depending upon the time of application. Phase-response curves derived in this way are less clear-cut than for typical circadian rhythms, but their pattern suggests that tidally rhythmic behaviour is controlled by truly circatidal (and not circadian) oscillators. The underlying circatidal oscillators appear, in general, to be fairly stable, suggesting that populations of coastal animals are relatively unsusceptible to irregularly timed environmental stimuli associated, say, with severe storms.     

Co‐adaptation impacts the robustness of predator–prey dynamics against perturbations

Global change threatens the maintenance of ecosystem functions that are shaped by the persistence and dynamics of populations. It has been shown that the persistence of species increases if they possess larger trait adaptability. Here, we investigate whether trait adaptability also affects the robustness of population dynamics of interacting species and thereby shapes the reliability of ecosystem functions that are driven by these dynamics. We model co‐adaptation in a predator–prey system as changes to predator offense and prey defense due to evolution or phenotypic plasticity. We investigate how trait adaptation affects the robustness of population dynamics against press perturbations to environmental parameters and against pulse perturbations targeting species abundances and their trait values. Robustness of population dynamics is characterized by resilience, elasticity, and resistance. In addition to employing established measures for resilience and elasticity against pulse perturbations (extinction probability and return time), we propose the warping distance as a new measure for resistance against press perturbations, which compares the shapes and amplitudes of pre‐ and post‐perturbation population dynamics. As expected, we find that the robustness of population dynamics depends on the speed of adaptation, but in nontrivial ways. Elasticity increases with speed of adaptation as the system returns more rapidly to the pre‐perturbation state. Resilience, in turn, is enhanced by intermediate speeds of adaptation, as here trait adaptation dampens biomass oscillations. The resistance of population dynamics strongly depends on the target of the press perturbation, preventing a simple relationship with the adaptation speed. In general, we find that low robustness often coincides with high amplitudes of population dynamics. Hence, amplitudes may indicate the robustness against perturbations also in other natural systems with similar dynamics. Our findings show that besides counteracting extinctions, trait adaptation indeed strongly affects the robustness of population dynamics against press and pulse perturbations.     

Perturbations both trigger and delay seizures due to generic properties of slow-fast relaxation oscillators

The mechanisms underlying the emergence of seizures are one of the most important unresolved issues in epilepsy research. In this paper, we study how perturbations, exogenous or endogenous, may promote or delay seizure emergence. To this aim, due to the increasingly adopted view of epileptic dynamics in terms of slow-fast systems, we perform a theoretical analysis of the phase response of a generic relaxation oscillator. As relaxation oscillators are effectively bistable systems at the fast time scale, it is intuitive that perturbations of the non-seizing state with a suitable direction and amplitude may cause an immediate transition to seizure. By contrast, and perhaps less intuitively, smaller amplitude perturbations have been found to delay the spontaneous seizure initiation. By studying the isochrons of relaxation oscillators, we show that this is a generic phenomenon, with the size of such delay depending on the slow flow component. Therefore, depending on perturbation amplitudes, frequency and timing, a train of perturbations causes an occurrence increase, decrease or complete suppression of seizures. This dependence lends itself to analysis and mechanistic understanding through methods outlined in this paper. We illustrate this methodology by computing the isochrons, phase response curves and the response to perturbations in several epileptic models possessing different slow vector fields. While our theoretical results are applicable to any planar relaxation oscillator, in the motivating context of epilepsy they elucidate mechanisms of triggering and abating seizures, thus suggesting stimulation strategies with effects ranging from mere delaying to full suppression of seizures.     

Dispersal promotes compensatory dynamics and stability in forced metacommunities

Understanding the factors that govern the stability of populations and communities has gained increasing importance as habitat fragmentation and environmental perturbations continue to escalate due to human activities. Dispersal is commonly viewed as essential to the maintenance of diversity in spatially subdivided communities, but few experiments have explored how dispersal interacts with the spatiotemporal components of environmental perturbations to determine community-level stability. We examined these processes using an experimental planktonic system composed of three competing species of zooplankton. We subjected zooplankton metacommunities to varying levels of dispersal and pH perturbations that varied in their degree of spatial synchrony. We show that dispersal can reverse the destabilizing effects of environmental forcing when perturbations are spatially asynchronous. Asynchrony in pH perturbations generated spatially and temporally varying species refugia that promoted source-sink dynamics and allowed prolonged persistence of zooplankton species that were otherwise extirpated in synchronously varying metacommunities. This, in turn, increased local species diversity, promoted compensatory population dynamics, and enhanced local community-level stability. Our results indicate that patterns of spatial covariation in environmental variability are critical to predicting the effects of dispersal on the dynamics and persistence of communities.     

Period and phase control in a multioscillatory circadian system (Iguana iguana)

The circadian system of the lizard Iguana iguana is composed of several independent pacemakers that work in concert: the pineal gland, retinae of the lateral eyes, and a fourth oscillator presumed to be located in the hypothalamus. These pacemakers govern the circadian expression of multiple behaviors and physiological processes, including rhythms in locomotor activity, endogenous body temperature, electroretinogram, and melatonin synthesis. The numerous, easily measurable rhythmic outputs make the iguana an ideal organism for examining the contributions of individual oscillators and their interactions in governing the expression of overt circadian rhythms. The authors have examined the effects of pinealectomy and enucleation on the endogenous body temperature rhythm (BTR) and locomotor activity rhythm (LAR) of juvenile iguanas at constant temperature both in LD cycles and in constant darkness (DD). They measured the periods (tau) of the circadian rhythms of LAR and BTR, the phase relationships between them in DD (psiAT), and the phase relationship between each rhythm and the light cycle (psiRL). Pinealectomy lengthened tau of locomotor activity in all animals tested and abolished the BTR in two-thirds of the animals. In those animals in which the BTR did persist following pinealectomy, tau lengthened to the same extent as that of locomotor activity. Pinealectomy also delayed the onset of activity with respect to its normal phase relationship with body temperature in DD. Enucleation alone had no significant effect on tau of LAR or BTR; however, after enucleation, BTR became 180 degrees out of phase from LAR in DD. After both pinealectomy and enucleation, 4 of 16 animals became arrhythmic in both activity and body temperature. Their data suggest that rhythmicity, period, and phase of overt circadian behaviors are regulated through the combined output of multiple endogenous circadian oscillators.