Effects of stochasticity on the length and behaviour of ecological transients

There is a growing recognition that ecological systems can spend extended periods of time far away from an asymptotic state, and that ecological understanding will therefore require a deeper appreciation for how long ecological transients arise. Recent work has defined classes of deterministic mechanisms that can lead to long transients. Given the ubiquity of stochasticity in ecological systems, a similar systematic treatment of transients that includes the influence of stochasticity is important. Stochasticity can of course promote the appearance of transient dynamics by preventing systems from settling permanently near their asymptotic state, but stochasticity also interacts with deterministic features to create qualitatively new dynamics. As such, stochasticity may shorten, extend or fundamentally change a system's transient dynamics. Here, we describe a general framework that is developing for understanding the range of possible outcomes when random processes impact the dynamics of ecological systems over realistic time scales. We emphasize that we can understand the ways in which stochasticity can either extend or reduce the lifetime of transients by studying the interactions between the stochastic and deterministic processes present, and we summarize both the current state of knowledge and avenues for future advances.

Management implications of long transients in ecological systems

The underlying biological processes that govern many ecological systems can create very long periods of transient dynamics. It is often difficult or impossible to distinguish this transient behaviour from similar dynamics that would persist indefinitely. In some cases, a shift from the transient to the long-term, stable dynamics may occur in the absence of any exogenous forces. Recognizing the possibility that the state of an ecosystem may be less stable than it appears is crucial to the long-term success of management strategies in systems with long transient periods. Here we demonstrate the importance of considering the potential of transient system behaviour for management actions across a range of ecosystem organizational scales and natural system types. Developing mechanistic models that capture essential system dynamics will be crucial for promoting system resilience and avoiding system collapses.

Long transients in ecology: Theory and applications

This paper discusses the recent progress in understanding the properties of transient dynamics in complex ecological systems. Predicting long-term trends as well as sudden changes and regime shifts in ecosystems dynamics is a major issue for ecology as such changes often result in population collapse and extinctions. Analysis of population dynamics has traditionally been focused on their long-term, asymptotic behavior whilst largely disregarding the effect of transients. However, there is a growing understanding that in ecosystems the asymptotic behavior is rarely seen. A big new challenge for theoretical and empirical ecology is to understand the implications of long transients. It is believed that the identification of the corresponding mechanisms along with the knowledge of scaling laws of the transient's lifetime should substantially improve the quality of long-term forecasting and crisis anticipation. Although transient dynamics have received considerable attention in physical literature, research into ecological transients is in its infancy and systematic studies are lacking. This text aims to partially bridge this gap and facilitate further progress in quantitative analysis of long transients in ecology. By revisiting and critically examining a broad variety of mathematical models used in ecological applications as well as empirical facts, we reveal several main mechanisms leading to the emergence of long transients and hence lays the basis for a unifying theory.

Transient phenomena in ecology

The importance of transient dynamics in ecological systems and in the models that describe them has become increasingly recognized. However, previous work has typically treated each instance of these dynamics separately. We review both empirical examples and model systems, and outline a classification of transient dynamics based on ideas and concepts from dynamical systems theory. This classification provides ways to understand the likelihood of transients for particular systems, and to guide investigations to determine the timing of sudden switches in dynamics and other characteristics of transients. Implications for both management and underlying ecological theories emerge.

Animal-to-animal variability in the phasing of the crustacean cardiac motor pattern: an experimental and computational analysis

The cardiac ganglion (CG) of Homarus americanus is a central pattern generator that consists of two oscillatory groups of neurons: "small cells" (SCs) and "large cells" (LCs). We have shown that SCs and LCs begin their bursts nearly simultaneously but end their bursts at variable phases. This variability contrasts with many other central pattern generator systems in which phase is well maintained. To determine both the consequences of this variability and how CG phasing is controlled, we modeled the CG as a pair of Morris-Lecar oscillators coupled by electrical and excitatory synapses and constructed a database of 15,000 simulated networks using random parameter sets. These simulations, like our experimental results, displayed variable phase relationships, with the bursts beginning together but ending at variable phases. The model suggests that the variable phasing of the pattern has important implications for the functional role of the excitatory synapses. In networks in which the two oscillators had similar duty cycles, the excitatory coupling functioned to increase cycle frequency. In networks with disparate duty cycles, it functioned to decrease network frequency. Overall, we suggest that the phasing of the CG may vary without compromising appropriate motor output and that this variability may critically determine how the network behaves in response to manipulations.

Pituitary network connectivity as a mechanism for the luteinising hormone surge

Ovulation in vertebrates is caused by a surge of luteinising hormone (LH) from the pituitary. The LH surge is initiated by rising oestradiol concentration, although the precise mechanism of oestradiol action in humans and primates is not yet understood. Recent advances in labelling and three-dimensional imaging have revealed a rich pituitary structure of interwoven networks of different cell types. In the present study, we develop a mathematical model to test the hypothesis that oestradiol modulation of connectivity between pituitary cells can underlie the LH surge. In the model, gonadotrophin-releasing hormone (GnRH) pulses stimulate LH secretion by two independent mechanisms. The first mechanism corresponds to the well known direct action of GnRH on gonadotrophs, which is inhibited by the rising oestradiol concentration. The second mechanism of GnRH action is to stimulate a recurrent network of pituitary cells; in this case, the folliculostellate cells, which in turn stimulate LH secretion from the gonadotrophs. The network activity is modelled by a one-dimensional ordinary differential equation. The key to the LH surge in the model lies in the assumption that oestradiol modulates network connectivity. When the circulating oestradiol concentration is low, the network is barely connected, and cannot maintain a recurrent signal. When the oestradiol concentration is high, the network is highly connected, and maintains a high level of activity even after GnRH stimulation, thereby leading to a surge of LH secretion.

Social stress alters expression of large conductance calcium-activated potassium channel subunits in mouse adrenal medulla and pituitary glands

Large conductance calcium-activated potassium (BK) channels are very prominently expressed in adrenal chromaffin and many anterior pituitary cells, where they shape intrinsic excitability complexly. Stress- and sex-steroids regulate alternative splicing of Slo-alpha, the pore-forming subunit of BK channels, and chronic behavioural stress has been shown to alter Slo splicing in tree shrew adrenals. In the present study, we focus on mice, measuring the effects of chronic behavioural stress on total mRNA expression of the Slo-alpha gene, two key BK channel beta subunit genes (beta2 and beta4), and the 'STREX' splice variant of Slo-alpha. As a chronic stressor, males of the relatively aggressive SJL strain were housed with a different unfamiliar SJL male every 24 h for 19 days. This 'social-instability' paradigm stressed all individuals, as demonstrated by reduced weight gain and elevated corticosterone levels. Five quantitative reverse transcriptase-polymerase chain assays were performed in parallel, including beta-actin, each calibrated against a dilution series of its corresponding cDNA template. Stress-related changes in BK expression were larger in mice tested at 6 weeks than 9 weeks. In younger animals, Slo-alpha mRNA levels were elevated 44% and 116% in the adrenal medulla and pituitary, respectively, compared to individually-housed controls. beta2 and beta4 mRNAs were elevated 162% and 194% in the pituitary, but slightly reduced in the adrenals of stressed animals. In the pituitary, dominance scores of stressed animals correlated negatively with alpha and beta subunit expression, with more subordinate individuals exhibiting levels that were three- to four-fold higher than controls or dominant individuals. STREX variant representation was lower in the subordinate subset. Thus, the combination of subunits responding to stress differs markedly between adrenal and pituitary glands. These data suggest that early stress will differentially affect neuroendocrine cell excitability, and call for detailed analysis of functional consequences.

Beta2 and beta4 subunits of BK channels confer differential sensitivity to acute modulation by steroid hormones

Membrane-associated receptors for rapid, steroidal neuromodulation remain elusive. Estradiol has been reported to facilitate activation of voltage- and Ca(2+)-dependent BK potassium channels encoded by Slo, if associated with beta1 subunits. We show here that 1) multiple members of the beta family confer sensitivity to multiple steroids on BK channels, 2) that beta subunits differentiate between steroids, and 3) that different betas have distinct relative preferences for particular steroids. Expressed in HEK 293 cells, inside-out patches with channels composed of Slo-alpha alone showed no steroid sensitivity. Cells expressing alphabeta4 exhibited potent, rapid, reversible, and dose-dependent potentiation by corticosterone (CORT; a glucocorticoid), and were potentiated to a lesser degree by other sex and stress steroids. In contrast, alphabeta2 channels were potentiated more strongly by dehydroepiandrosterone (DHEA; an enigmatic, stress-related adrenal androgen), and to a lesser extent by CORT, estradiol, testosterone, and DHEA-S. Cholesterol had no effect on any BK channel compositions tested. Conductance-voltage plots of channels composed of alpha plus beta2 or beta4 subunits were shifted in the negative direction by steroids, indicating greater activation at negative voltages. Thus our results argue that the variety of Slo-beta subunit coexpression patterns occurring in vivo expands the repertoire of Slo channel gating in yet another dimension not fully appreciated, rendering BK gating responsive to dynamic fluctuations in a multiple of steroid hormones.

A potential role of modulating inositol 1,4,5-trisphosphate receptor desensitization and recovery rates in regulating ovulation

Regulation of the human menstrual cycle is a frequency dependent process controlled in part by the pulsatile release of gonadotropin releasing hormone (GnRH) from the hypothalamus. The binding of GnRH to gonadotroph cells in the pituitary stimulates inositol 1,4,5-trisphosphate (IP3) mediated release of calcium from the endoplasmic reticulum, resulting in calcium oscillations and the secretion of luteinizing hormone (LH). A sudden increase in serum LH concentrations known as the LH surge triggers ovulation. Here we model the intracellular calcium dynamics of gonadotroph cells by adapting the model of Li and Rinzel (J. Theor. Biol. 166 (1994) 461) to include the desensitization of IP3 receptors to IP3. Allowing the resensitization rate of these receptors to vary over the course of the cycle suffices to explain the LH surge in both the normal menstrual cycle, and in the treatment of Kallmann's syndrome (a condition where endogenous production of GnRH is absent).

Resonance in the menstrual cycle: a new model of the LH surge

In vertebrates, ovulation is triggered by a surge of LH from the pituitary. The precise mechanism by which rising oestradiol concentrations initiate the LH surge in the human menstrual cycle remains a fundamental open question of reproductive biology. It is well known that sampling of serum LH on a time scale of minutes reveals pulsatile release from the pituitary in response to pulses of gonadotrophin releasing hormone from the hypothalamus. The LH pulse frequency and amplitude vary considerably over the cycle, with the highest frequency and amplitude at the midcycle surge. Here a new mathematical model is presented of the pituitary as a damped oscillator (pulse generator) driven by the hypothalamus. The model LH surge is consistent with LH data on the time scales of both minutes and days. The model is used to explain the surprising pulse frequency characteristics required to treat human infertility disorders such as Kallmann's syndrome, and new experimental predictions are made.