Chaos and Ultradian Rhythms

Systems in a chaotic state have apparently random outputs despite a simple underlying kinetic mechanism. For instance, the interaction of two coupled oscillators (the mitotic oscillator and the ultradian clock) can produce chaotic behaviour over a limited range of parameter values. Mathematical modelling shows that physiologically realistic characteristics are thereby exhibited. Cell division cycles of lower eukaryotes (protozoa and yeasts) show both deterministic and stochastic properties. Both dispersion of cell cycle times and quantized values can be generated, as a deterministic chaotic consequence of oscillator interaction rather than from noisy limit cycles. Advantages may stem from chaotic operation; a controlled chaotic attractor could provide multifrequency outputs that determine rhythmic behaviour on different time scales (e.g. ultradian and circadian) with the facility for rapid state changes from one periodicity to another.     

The cell division cycle: a physiologically plausible dynamic model can exhibit chaotic solutions.

A mitotic oscillator with one slowly increasing variable (tau L of the order of hours) and one rapidly increasing variable (tau R of the order of minutes) modulated by a timer (ultradian clock) gives an auto-oscillating solution: cells divide when this relaxation oscillator reaches a critical threshold to initiate a rapid phase of the limit cycle. Increasing values of the velocity constant in the slow equation give quasi-periodic, chaotic and periodic solutions. Thus dispersed and quantized cell cycle times are consequences of a chaotic trajectory and have a purely deterministic basis. This model of the dispersion of cell cycle times contrasts with many previous ones in which cell cycle variability is a consequence of stochastic properties inherent in a sequence of many thousands of reactions or the random nature of a key transition step.     

Only noise can induce chaos in discrete populations

Motivated by the papers from Ellner and Turchin 2005 and Dennis et al. 2003 we investigate the possibility to detect chaos in noisy ecological systems. One message of our paper is that if a dynamic model is available and if this model predicts chaotic behaviour, one should consider its discrete-state, noisy version when fitting numerical predictions to observations. We emphasize that deterministic discrete-state models behave periodically, thus only the interaction of these deterministic skeletons with random noise can produce non-regular dynamics. We detect and describe a relatively sharply defined range of the noise (the grey zone) where the gradual transition from periodic to chaotic behaviour happens. This zone, the upper border of which can be predicted analytically, is identified in experimental data as well as in numerical simulations. In the grey zone the global, statistical behaviour will approach the statistics produced by the chaotic, continuous model, and in this sense we claim that noise can produce chaos.     

The origin of glucocorticoid hormone oscillations

Oscillating levels of adrenal glucocorticoid hormones are essential for optimal gene expression, and for maintaining physiological and behavioural responsiveness to stress. The biological basis for these oscillations is not known, but a neuronal "pulse generator" within the hypothalamus has remained a popular hypothesis. We demonstrate that pulsatile hypothalamic activity is not required for generating ultradian glucocorticoid oscillations. We show that a constant level of corticotrophin-releasing hormone (CRH) can activate a dynamic pituitary-adrenal peripheral network to produce ultradian adrenocorticotrophic hormone and glucocorticoid oscillations with a physiological frequency. This oscillatory response to CRH is dose dependent and becomes disrupted for higher levels of CRH. These data suggest that glucocorticoid oscillations result from a sub-hypothalamic pituitary-adrenal system, which functions as a deterministic peripheral hormone oscillator with a characteristic ultradian frequency. This constitutes a novel mechanism by which the level, rather than the pattern, of CRH determines the dynamics of glucocorticoid hormone secretion.     

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.     

Structural perturbations to population skeletons: transient dynamics, coexistence of attractors and the rarity of chaos

BACKGROUND: Simple models of insect populations with non-overlapping generations have been instrumental in understanding the mechanisms behind population cycles, including wild (chaotic) fluctuations. The presence of deterministic chaos in natural populations, however, has never been unequivocally accepted. Recently, it has been proposed that the application of chaos control theory can be useful in unravelling the complexity observed in real population data. This approach is based on structural perturbations to simple population models (population skeletons). The mechanism behind such perturbations to control chaotic dynamics thus far is model dependent and constant (in size and direction) through time. In addition, the outcome of such structurally perturbed models is [almost] always equilibrium type, which fails to commensurate with the patterns observed in population data. METHODOLOGY/PRINCIPAL FINDINGS: We present a proportional feedback mechanism that is independent of model formulation and capable of perturbing population skeletons in an evolutionary way, as opposed to requiring constant feedbacks. We observe the same repertoire of patterns, from equilibrium states to non-chaotic aperiodic oscillations to chaotic behaviour, across different population models, in agreement with observations in real population data. Model outputs also indicate the existence of multiple attractors in some parameter regimes and this coexistence is found to depend on initial population densities or the duration of transient dynamics. Our results suggest that such a feedback mechanism may enable a better understanding of the regulatory processes in natural populations.     

Autogenous Formation of Spiral Waves by Coupled Chaotic Attractors

When large arrays of strange attractors are coupled diffusively through one of the variables, chaotic systems become periodic and form large archimedean spirals or concentric bands. This observation may have importance for many applications in the field of deterministic chaos and seems particularly relevant to the question of the formal temporal structure of the biological clock in metazoan organisms. In particular, although individual cellular oscillators, as manifested in the cell cycle, may have deep basins of attraction and appear to be more or less periodic, we suggest that cells oscillate with chaotic dynamics in the ultradian domain. Only when large aggregates of these cells are tightly coupled can a precise circadian clock emerge. For changing coupling strength or parameter values, period increase occurs through quantal or integral multiple increments of the fundamental. All calculations were implemented on a 386AT, using a Mercury MC6400 floating point processor.     

Autogenous Formation of Spiral Waves by Coupled Chaotic Attractors

When large arrays of strange attractors are coupled diffusively through one of the variables, chaotic systems become periodic and form large archimedean spirals or concentric bands. This observation may have importance for many applications in the field of deterministic chaos and seems particularly relevant to the question of the formal temporal structure of the biological clock in metazoan organisms. In particular, although individual cellular oscillators, as manifested in the cell cycle, may have deep basins of attraction and appear to be more or less periodic, we suggest that cells oscillate with chaotic dynamics in the ultradian domain. Only when large aggregates of these cells are tightly coupled can a precise circadian clock emerge. For changing coupling strength or parameter values, period increase occurs through quantal or integral multiple increments of the fundamental. All calculations were implemented on a 386AT, using a Mercury MC6400 floating point processor.     

Ultradian Rhythms in Arousal-The Problem of Masking

Studies utilizing widely different experimental techniques provided evidence that there are spontaneous ultradian cycles in arousal during the waking state. These comprised of cyclic fluctuations between increased and decreased sleep propensity with a periodicity of about 1.5 hr. Being of relatively low amplitude, these cycles are vulnerable to masking effects by a variety of experimental conditions. Masking can be exerted by varying the tonic level of arousal, by coexisting slow ultradian components which are particularly prominent during the second half of the day, or by some specific experimental conditions. Furthermore, increased sleepiness was shown to enhance the slow ultradian components and suppress the 1.5-hr cycles in EEG indices of arousal on the one hand, and to emphasize the 1.5-hr cycles in motor activity and reaction time performance on the other hand. Much more attention should be paid to the problem of masking of ultradian cycles in arousal. Recognizing the sources and reasons for masking will ad vance our knowledge of the characteristics of these cycles and their function.     

On the existence and the role of chaotic processes in the nervous system

Chaos theory is a rapidly growing field. As a technical term, “chaos” refers to deterministic but unpredictable processes being sensitively dependent upon initial conditions. Neurobiological models and experimental results are very complicated and some research groups have tried to pursue the “neuronal chaos”. Babloyantz's group has studied the fractal dimension (d) of electroencephalograms (EEG) in various physiological and pathological states. From deep sleep (d=4) to full awakening (d>8), a hierarchy of “strange” attractors paralles the hierarchy of states of consciousness. In epilepsy (petit mal), despite the turbulent aspect of a seizure, the attractor dimension was near to 2. In Creutzfeld-Jacob disease, the regular EEG activity corresponded to an attractor dimension less than the one measured in deep sleep. Is it healthy to be chaotic? An “active desynchronisation” could be favourable to a physiological system. Rapp's group reported variations of fractal dimension according to particular tasks. During a mental arithmetic task, this dimension increased. In another task, a P300 fractal index decreased when a target was identified. It is clear that the EEG is not representing noise. Its underlying dynamics depends on only a few degrees of freedom despite yet it is difficult to compute accurately the relevant parameters. What is the cognitive role of such a chaotic dynamics? Freeman has studied the olfactory bulb in rabbits and rats for 15 years. Multi-electrode recordings of a few mm² showed a chaotic hierarchy from deep anaesthesia to alert state. When an animal identified a previously learned odour, the fractal dimension of the dynamics dropped off (near limit cycles). The chaotic activity corresponding to an alert-and-waiting state seems to be a field of all possibilities and a focused activity corresponds to a reduction of the attractor in state space. For a couple of years, Freeman has developed a model of the olfactory bulb-cortex system. The behaviour of the simple model “without learning” was quite similar to the real behaviour and a model “with learning” is developed. Recently, more and more authors insisted on the importance of the dynamic aspect of nervous functioning in cognitive modelling. Most of the models in the neural-network field are designed to converge to a stable state (fixed point) because such behaviour is easy to understand and to control. However, some theoretical studies in physics try to understand how a chaotic behaviour can emerge from neural networks. Sompolinsky's group showed that a sharp transition from a stable state to a chaotic state occurred in totally interconnected networks depending on the value of one control parameter. Learning in such systems is an open field. In conclusion, chaos does exist in neurophysiological processes. It is neither a kind of noise nor a pathological sign. Its main role could be to provide diversity and flexibility to physiological processes. Could “strange” attractors in nervous system embody mental forms? This is a difficult but fascinating question.     

Ultradian rhythm unmasked in the Pdf clock mutant of Drosophila

A diverse range of organisms shows physiological and behavioural rhythms with various periods. Extensive studies have been performed to elucidate the molecular mechanisms of circadian rhythms with an approximately 24 h period in both Drosophila and mammals, while less attention has been paid to ultradian rhythms with shorter periods. We used a video-tracking method to monitor the movement of single flies, and clear ultradian rhythms were detected in the locomotor behaviour of wild type and clock mutant flies kept under constant dark conditions. In particular, the Pigment-dispersing factor mutant (Pdf ⁰¹ ) demonstrated a precise and robust ultradian rhythmicity, which was not temperature compensated. Our results suggest that Drosophila has an endogenous ultradian oscillator that is masked by circadian rhythmic behaviours.     

Stochastic models for circadian rhythms: effect of molecular noise on periodic and chaotic behaviour

Circadian rhythms are endogenous oscillations that occur with a period close to 24 h in nearly all living organisms. These rhythms originate from the negative autoregulation of gene expression. Deterministic models based on such genetic regulatory processes account for the occurrence of circadian rhythms in constant environmental conditions (e.g., constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. When the numbers of protein and mRNA molecules involved in the oscillations are small, as may occur in cellular conditions, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering the stochastic version of a deterministic model previously proposed for circadian oscillations of the PER and TIM proteins and their mRNAs in Drosophila. The model is based on repression of the per and tim genes by a complex between the PER and TIM proteins. Numerical simulations of the stochastic version of the model are performed by means of the Gillespie method. The predictions of the stochastic approach compare well with those of the deterministic model with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity from a bifurcation point beyond which the system evolves to stable steady state. Stochastic simulations indicate that robust circadian oscillations can emerge at the cellular level even when the maximum numbers of mRNA and protein molecules involved in the oscillations are of the order of only a few tens or hundreds. The stochastic model also reproduces the evolution to a strange attractor in conditions where the deterministic PER-TIM model admits chaotic behaviour. The difference between periodic oscillations of the limit cycle type and aperiodic oscillations (i.e. chaos) persists in the presence of molecular noise, as shown by means of Poincaré sections. The progressive obliteration of periodicity observed as the number of molecules decreases can thus be distinguished from the aperiodicity originating from chaotic dynamics. As long as the numbers of molecules involved in the oscillations remain sufficiently large (of the order of a few tens or hundreds, or more), stochastic models therefore provide good agreement with the predictions of the deterministic model for circadian rhythms.     

TEMPORAL PATTERN OF LH SECRETION: REGULATION BY MULTIPLE ULTRADIAN OSCILLATORS VERSUS A SINGLE CIRCADIAN OSCILLATOR

The possibility that the 24h rhythm output is the composite expression of ultradian oscillators of varying periodicities was examined by assessing the effect of external continuously or pulsed (20-minute) Gonadotropinreleasing hormone (GnRH) infusions on in vitro luteinizing hormone (LH) release patterns from female mouse pituitaries during 38h study spans. Applying stepwise analyses (spectral, cosine fit, best-fit curve, and peak detection analyses) revealed the waveform shape of LH release output patterns over time is composed of several ultradian oscillations of different periods. The results further substantiated previous observations indicating the pituitary functions as an autonomous clock. The GnRH oscillator functions as a pulse generator and amplitude regulator, but it is not the oscillator that drives the ultradian LH release rhythms. At different stages of the estrus cycle, the effect of GnRH on the expression of ultradian periodicities varies, resulting in the modification of their amplitudes but not their periods. The functional output from the system of ultradian oscillators may superimpose a “circadian or infradian phenotype” on the observed secretion pattern. An “amplitude control” hypothesis is proposed: The temporal pattern of LH release is governed by several oscillators that function in conjunction with one another and are regulated by an amplitude-controlled mechanism. Simulated models show that such a mechanism results in better adaptive response to environmental requirements than does a single circadian oscillator. (Chronobiology International, 18(3), 399–412, 2001)     

Temporal pattern of LH secretion: regulation by multiple ultradian oscillators versus a single circadian oscillator

The possibility that the 24h rhythm output is the composite expression of ultradian oscillators of varying periodicities was examined by assessing the effect of external continuously or pulsed (20-minute) Gonadotropinreleasing hormone (GnRH) infusions on in vitro luteinizing hormone (LH) release patterns from female mouse pituitaries during 38h study spans. Applying stepwise analyses (spectral, cosine fit, best-fit curve, and peak detection analyses) revealed the waveform shape of LH release output patterns over time is composed of several ultradian oscillations of different periods. The results further substantiated previous observations indicating the pituitary functions as an autonomous clock. The GnRH oscillator functions as a pulse generator and amplitude regulator, but it is not the oscillator that drives the ultradian LH release rhythms. At different stages of the estrus cycle, the effect of GnRH on the expression of ultradian periodicities varies, resulting in the modification of their amplitudes but not their periods. The functional output from the system of ultradian oscillators may superimpose a “circadian or infradian phenotype” on the observed secretion pattern. An “amplitude control” hypothesis is proposed: The temporal pattern of LH release is governed by several oscillators that function in conjunction with one another and are regulated by an amplitude-controlled mechanism. Simulated models show that such a mechanism results in better adaptive response to environmental requirements than does a single circadian oscillator. (Chronobiology International, 18(3), 399-412, 2001)     

Chaotic dynamics and fractal space in biochemistry: simplicity underlies complexity

In this work we attempt to analyze the coupling between the dynamics of biochemical reactions (especially chaotic dynamics), and the geometry of cytoarchitecture (especially fractal ultrastructure), because of its importance and consequences for the ultradian dynamic behaviour of cells. Fractal geometry in intracellular macromolecular assemblies suggests that chaotic dynamics occur during their organization. Non-linear interactions in and between spatial and temporal domains and over wide ranges of scales underlie the emergent properties of complex biological systems.     

From Ultradian to Infradian Rhythms: LH Releases Patterns In Vitro

In the present study, we examined in vitro luteinizing hormone (LH) release patterns from pituitaries and from pituitary cell cultures (3 and 7 days in culture) to elucidate the endogenous period generated by the gonadotroph cell population and to evaluate the relationship between the basic period generated at the cellular level and the output pattern observed at the organ level. In addition, we examined the effect of photic environmental signals perceived by the animals on LH release patterns from pituitaries in vitro. When the animals were exposed to circadian photoperiodic signals, the in vitro LH release pattern from the pituitaries exhibited ultradian, circadian, and infra-dian frequencies. When the animals were exposed to continuous illumination, the in vitro patterns exhibited only ultradian and infradian frequencies. Furthermore, free running is a process, not a state. This process is driven by a change in the relative dominance of different frequencies that construct the pattern without changing the basic period length. Evaluation of the relative dominance of the different frequencies that construct the pattern indicates that, although infradian oscillators may take part in shaping the output pattern, the basic rhythm generated by the pituitary cells is in the ultradian domain. The results obtained from the examined system suggest that an endogenous oscillator is a cellular entity with ultradian periodicity, and that the rhythmic output of many biological variables is structured by various ultradian components that construct the circadian and infradian output rhythms.     

From Ultradian to Infradian Rhythms: LH Releases Patterns In Vitro

In the present study, we examined in vitro luteinizing hormone (LH) release patterns from pituitaries and from pituitary cell cultures (3 and 7 days in culture) to elucidate the endogenous period generated by the gonadotroph cell population and to evaluate the relationship between the basic period generated at the cellular level and the output pattern observed at the organ level. In addition, we examined the effect of photic environmental signals perceived by the animals on LH release patterns from pituitaries in vitro. When the animals were exposed to circadian photoperiodic signals, the in vitro LH release pattern from the pituitaries exhibited ultradian, circadian, and infra-dian frequencies. When the animals were exposed to continuous illumination, the in vitro patterns exhibited only ultradian and infradian frequencies. Furthermore, free running is a process, not a state. This process is driven by a change in the relative dominance of different frequencies that construct the pattern without changing the basic period length. Evaluation of the relative dominance of the different frequencies that construct the pattern indicates that, although infradian oscillators may take part in shaping the output pattern, the basic rhythm generated by the pituitary cells is in the ultradian domain. The results obtained from the examined system suggest that an endogenous oscillator is a cellular entity with ultradian periodicity, and that the rhythmic output of many biological variables is structured by various ultradian components that construct the circadian and infradian output rhythms.     

Study of a tri-trophic prey-dependent food chain model of interacting populations

The current paper accounts for the influence of intra-specific competition among predators in a prey dependent tri-trophic food chain model of interacting populations. We offer a detailed mathematical analysis of the proposed food chain model to illustrate some of the significant results that has arisen from the interplay of deterministic ecological phenomena and processes. Biologically feasible equilibria of the system are observed and the behaviours of the system around each of them are described. In particular, persistence, stability (local and global) and bifurcation (saddle-node, transcritical, Hopf–Andronov) analysis of this model are obtained. Relevant results from previous well known food chain models are compared with the current findings. Global stability analysis is also carried out by constructing appropriate Lyapunov functions. Numerical simulations show that the present system is capable enough to produce chaotic dynamics when the rate of self-interaction is very low. On the other hand such chaotic behaviour disappears for a certain value of the rate of self interaction. In addition, numerical simulations with experimented parameters values confirm the analytical results and shows that intra-specific competitions bears a potential role in controlling the chaotic dynamics of the system; and thus the role of self interactions in food chain model is illustrated first time. Finally, a discussion of the ecological applications of the analytical and numerical findings concludes the paper.     

Probing the circadian oscillator of a mammal by two-pulse perturbations

When organisms are maintained under constant conditions of light and temperature, their endogenous circadian rhythms free run, manifesting their intrinsic period. The phases of these free-running rhythms can be shifted by stimuli of light, temperature, and drugs. The change from one free-running steady state to another following a perturbation often involves several transient cycles (cycles of free-running rhythm drifting slowly to catch up with the postperturbation steady state). Although the investigation of oscillator kinetics in circadian rhythms of both insects and mammals has revealed that the circadian pacemaker phase shifts instantaneously, the phenomenon of transient cycles has remained an enigma. We probed the phases of the transient cycles in the locomotor activity rhythm of the field mouse Mus booduga, evoked by a single light pulse (LP), using LPs at critically timed phases. The results of our experiments indicate that the transient cycles generated during transition from one steady state to another steady state do not represent the state of the circadian pacemaker (basic oscillator) controlling the locomotor activity rhythm in Mus booduga. (Chronobiology International, 17(2), 129-136, 2000)