Finding complex oscillatory phenomena in biochemical systems. An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.     

Properties of strange attractors in yeast glycolysis

The properties of periodic and aperiodic glycolytic oscillations observed in yeast extracts under sinusoidal glucose input were analyzed by the following methods. (1) Spectral analysis, rendering sharp peaks for periodic responses and enhanced broad-band noise for aperiodic oscillations. (2) Phase plane analysis, leading to closed and to open trajectories for periodic and aperiodic oscillations, respectively. (3) Rotation of a phase plane proportionally to time, revealing strange attractors associated with the aperiodic oscillations. (4) Stroboscopic plot on the phase plane, showing that the strange attractors follow a stretch-fold-press process, if the stroboscoping phase is varied. (5) Stroboscopic transfer plot, admitting a period of three transfer processes and thus implying chaos according to the Li-Yorke theorem. (6) Determination of the rate of information production by differentiation of the transfer plot, yielding approx. 0.21 bits per min for the chaotically glycolyzing yeast extract.     

Phase resetting and bifurcation in the ventricular myocardium.

With the dynamic differential equations of Beeler, G. W., and H. Reuter (1977, J. Physiol. [Lond.]. 268:177-210), we have studied the oscillatory behavior of the ventricular muscle fiber stimulated by a depolarizing applied current I app. The dynamic solutions of BR equations revealed that as I app increases, a periodic repetitive spiking mode appears above the subthreshold I app, which transforms to a periodic spiking-bursting mode of oscillations, and finally to chaos near the suprathreshold I app (i.e., near the termination of the periodic state). Phase resetting and annihilation of repetitive firing in the ventricular myocardium were demonstrated by a brief current pulse of the proper magnitude applied at the proper phase. These phenomena were further examined by a bifurcation analysis. A bifurcation diagram constructed as a function of I app revealed the existence of a stable periodic solution for a certain range of current values. Two Hopf bifurcation points exist in the solution, one just above the lower periodic limit point and the other substantially below the upper periodic limit point. Between each periodic limit point and the Hopf bifurcation, the cell exhibited the coexistence of two different stable modes of operation; the oscillatory repetitive firing state and the time-independent steady state. As in the Hodgkin-Huxley case, there was a low amplitude unstable periodic state, which separates the domain of the stable periodic state from the stable steady state. Thus, in support of the dynamic perturbation methods, the bifurcation diagram of the BR equation predicts the region where instantaneous perturbations, such as brief current pulses, can send the stable repetitive rhythmic state into the stable steady state.     

Periodicity and chaos in the response of Halobacterium to temporal light gradients

Halobacteria spontaneously reverse their swimming direction about every 10 s. This periodicity can be altered by light stimuli. We found that temporal exponential changes in light intensity, depending on wavelength and sign, lengthened or shortened the intervals between reversals. Within a limited range of steepness, light gradients enforced a new stable periodicity upon the system. Outside this range, they caused period doubling or induced a sequence of reversal events without any obvious regularity. An analysis of a functional relationship between apparently irregular periods by plotting each period as a function on the preceding one yielded a clearly discernible non-random structure, which shows some similarities to the one obtained by a model calculation for a periodically perturbed limit cycle oscillator. These results indicate that external forcing of the system may generate chaos. When the decay of intracellular sensory signals is delayed by inhibition of protein methylation the transition from periodic to aperiodic behavior occurs at a lower steepness of the gradient. We therefore assume that the generation of either periodic or deterministic chaotic behavior is determined by the relation between the signal lifetime and the frequency of stimulus inputs. The strong indications for transitions from periodic to chaotic behavior can be regarded as a further support of our hypothesis that the behavioral pattern of Halobacterium is controlled by an endogeneous oscillator.     

Periodicity, Mixed-Mode Oscillations, and Quasiperiodicityin a Rhythm-Generating Neural Network

We studied patterns of oscillatory neural activity in the network that generates respiratory rhythm in mammals. When isolated in vitro, this network spontaneously generates an inspiratory-related motor rhythm, with stable amplitude from cycle to cycle. We show that progressively elevating neuronal excitability in vitro causes periodic modulation of this inspiratory rhythm, evoking (in order): mixed-mode oscillations, quasiperiodicity, and ultimately disorganized aperiodic activity. Thus, the respiratory network oscillator follows a well defined sequence of behavioral states characterized by dynamical systems theory, which includes discrete stages of periodic and quasiperiodic amplitude modulation and progresses (according to theory) to aperiodic chaos-like behavior. We also observed periodic, mixed-mode periodic, and quasiperiodic breathing patterns in neonatal rodents and human infants in vivo, suggesting that breathing patterns generated by the intact nervous system reflect deterministic neural activity patterns in the underlying rhythm-generating network.     

Birhythmicity, trirhythmicity and chaos in bursting calcium oscillations

We have analyzed various types of complex calcium oscillations. The oscillations are explained with a model based on calcium-induced calcium release (CICR). In addition to the endoplasmic reticulum as the main intracellular Ca2+ store, mitochondrial and cytosolic Ca2+ binding proteins are also taken into account. This model was previously proposed for the study of the physiological role of mitochondria and the cytosolic proteins in gene rating complex Ca2+ oscillations [1]. Here, we investigated the occurrence of different types of Ca2+ oscillations obtained by the model, i.e. simple oscillations, bursting, and chaos. In a bifurcation diagram, we have shown that all these various modes of oscillatory behavior are obtained by a change of only one model parameter, which corresponds to the physiological variability of an agonist. Bursting oscillations were studied in more detail because they express birhythmicity, trirhythmicity and chaotic behavior. Two different routes to chaos are observed in the model: in addition to the usual period doubling cascade, we also show intermittency. For the characterization of the chaotic behavior, we made use of return maps and Lyapunov exponents. The potential biological role of chaos in intracellular signaling is discussed.     

Bursting, chaos and birhythmicity originating from self-modulation of the inositol 1,4,5-trisphosphate signal in a model for intracellular Ca2+ oscillations

We investigate the various types of complex Ca²⁺ oscillations which can arise in a model based on the mechanism of Ca²⁺-induced Ca²⁺ release (CICR), that takes into account the Ca²⁺-stimulated degradation of inositol 1,4,5-trisphosphate (InsP₃) by a 3-kinase. This model was previously proposed in the course of an investigation of plausible mechanisms capable of generating complex Ca²⁺ oscillations (Borghans et al., 1997). Besides simple periodic behavior, this model for cytosolic Ca²⁺ oscillations in nonexcitable cells shows complex oscillatory phenomena like bursting or chaos. We show that the model also admits a coexistence between two stable regimes of sustained oscillations (birhythmicity). The occurrence of these various modes of oscillatory behavior is analysed by means of bifurcation diagrams. Complex oscillations are characterized by means of Poincaré sections, power spectra and Lyapounov exponents. The results point to the role of self-modulation of the InsP₃ signal by 3-kinase as a possible source for complex temporal patterns in Ca²⁺ signaling.     

Diversity of temporal self-organized behaviors in a biochemical system.

The numerical study of a glycolytic model formed by a system of three delay-differential equations revealed a notable richness of temporal structures which included the three main routes to chaos, as well as a multiplicity of stable coexisting states. The Feigenbaum, intermitency and quasiperiodicity routes to chaos can emerge in the biochemical oscillator. Moreover, different types of birhythmicity, trirhythmicity and hard excitation emerge in the phase space. For a single range of the control parameter it can be observed the coexistence of two quasiperiodicity routes to chaos, the coexistence of a stable steady state with a stable torus, and the coexistence of a strange attractor with different stable regimes such as chaos with different periodic regimes, chaos with bursting behavior, and chaos with torus. In most of the numerical studies, the biochemical oscillator has been considered under periodic input flux being the mean input flux rate 6 mM/h. On the other hand, several investigators have observed quasiperiodic time patterns and chaotic oscillations by monitoring the fluorescence of NADH in glycolyzing yeast under sinusoidal glucose input flux. Our numerical results match well with these experimental studies.     

Dynamic properties of cardiovascular systems.

A recently derived mathematical model of an isolated heart is extended here to a closed-loop cardiovascular system. Taking the end-diastolic volume as state variable, the authors show that the closed-loop cardiovascular system can be described by a one-dimensional nonlinear discrete dynamical system that depends on parameters describing the systolic and diastolic properties of the heart, heart rate, total peripheral resistance, and arterial capacitance. Studies of this model show that the system possesses a rich spectrum of dynamical behavior, from stable points through stable cycles to a "chaotic" behavior. It is shown that such an analysis of dynamic behavior yields those domains in the parameter space that correspond to a normal and abnormal beating heart, when the heart ejects time-invariant and time-variant (periodic or aperiodic) stable stroke volumes, respectively. Determination of such domains may lead to better understanding of the specific pathologic mechanism involved in the evolution of an abnormal beating heart.     

Infinite subharmonic bifurcation in an SEIR epidemic model

The existence of both periodic and aperiodic behavior in recurrent epidemics is now well-documented. In this paper, it is proven that for epidemic models that incur permanent immunity with seasonal variations in the contact rate, there exists an infinite number of stable subharmonic solutions. Random effects in the environment could perturb the state of the dynamics from the domain of attraction from one subharmonic to another, thus producing aperiodic levels of incidence.     

Analysis of chaotic oscillations induced in two coupled Wilson–Cowan models

Although it is known that two coupled Wilson–Cowan models with reciprocal connections induce aperiodic oscillations, little attention has been paid to the dynamical mechanism for such oscillations so far. In this study, we aim to elucidate the fundamental mechanism to induce the aperiodic oscillations in the coupled model. First, aperiodic oscillations observed are investigated for the case when the connections are unidirectional and when the input signal is a periodic oscillation. By the phase portrait analysis, we determine that the aperiodic oscillations are caused by periodically forced state transitions between a stable equilibrium and a stable limit cycle attractors around the saddle-node and saddle separatrix loop bifurcation points. It is revealed that the dynamical mechanism where the state crosses over the saddle-node and saddle separatrix loop bifurcations significantly contributes to the occurrence of chaotic oscillations forced by a periodic input. In addition, this mechanism can also give rise to chaotic oscillations in reciprocally connected Wilson–Cowan models. These results suggest that the dynamic attractor transition underlies chaotic behaviors in two coupled Wilson–Cowan oscillators.     

Modeling the segmentation clock as a network of coupled oscillations in the Notch, Wnt and FGF signaling pathways

The formation of somites in the course of vertebrate segmentation is governed by an oscillator known as the segmentation clock, which is characterized by a period ranging from 30 min to a few hours depending on the organism. This oscillator permits the synchronized activation of segmentation genes in successive cohorts of cells in the presomitic mesoderm in response to a periodic signal emitted by the segmentation clock, thereby defining the future segments. Recent microarray experiments [Dequeant, M.L., Glynn, E., Gaudenz, K., Wahl, M., Chen, J., Mushegian, A., Pourquie, O., 2006. A complex oscillating network of signaling genes underlies the mouse segmentation clock. Science 314, 1595-1598] indicate that the Notch, Wnt and Fibroblast Growth Factor (FGF) signaling pathways are involved in the mechanism of the segmentation clock. By means of computational modeling, we investigate the conditions in which sustained oscillations occur in these three signaling pathways. First we show that negative feedback mediated by the Lunatic Fringe protein on intracellular Notch activation can give rise to periodic behavior in the Notch pathway. We then show that negative feedback exerted by Axin2 on the degradation of β-catenin through formation of the Axin2 destruction complex can produce oscillations in the Wnt pathway. Likewise, negative feedback on FGF signaling mediated by the phosphatase product of the gene MKP3/Dusp6 can produce oscillatory gene expression in the FGF pathway. Coupling the Wnt, Notch and FGF oscillators through common intermediates can lead to synchronized oscillations in the three signaling pathways or to complex periodic behavior, depending on the relative periods of oscillations in the three pathways. The phase relationships between cycling genes in the three pathways depend on the nature of the coupling between the pathways and on their relative autonomous periods. The model provides a framework for analyzing the dynamics of the segmentation clock in terms of a network of oscillating modules involving the Wnt, Notch and FGF signaling pathways.     

Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns

Period-doubling bifurcation to chaos were discovered in spontaneous firings of Onchidium pacemaker neurons. In this paper, we provide three cases of bifurcation processes related to period-doubling bifurcation cascades to chaos observed in the spontaneous firing patterns recorded from an injured site of rat sciatic nerve as a pacemaker. Period-doubling bifurcation cascades to period-4 (π(2,2)) firstly, and then to chaos, at last to a periodicity, which can be period-5, period-4 (π(4)) and period-3, respectively, in different pacemakers. The three bifurcation processes are labeled as case I, II and III, respectively, manifesting procedures different to those of period-adding bifurcation. Higher-dimensional unstable periodic orbits (UPOs) can be detected in the chaos, built close relationships to the periodic firing patterns. Case III bifurcation process is similar to that discovered in the Onchidium pacemaker neurons and simulated in theoretical model-Chay model. The extra-large Feigenbaum constant manifesting in the period-doubling bifurcation process, induced by quasi-discontinuous characteristics exhibited in the first return maps of both ISI series and slow variable of Chay model, shows that higher-dimensional periodic behaviors appeared difficult within the period-doubling bifurcation cascades. The results not only provide examples of period-doubling bifurcation to chaos and chaos with higher-dimensional UPOs, but also reveal the dynamical features of the period-doubling bifurcation cascades to chaos.     

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.     

Computer simulations of synchrony and oscillations evoked by two coherent inputs

Coherent oscillations have been reported in multiple cortical areas. This study examines the characteristics of output spikes through computer simulations when the neural network model receives periodic/aperiodic spatiotemporal spikes with modulated/constant populational activity from two pathways. Synchronous oscillations which have the same period as the input are observed in response to periodic input patterns regardless of populational activity. The results confirm that the output frequency of synchrony is essentially determined by the period of the repeated input patterns. On the other hand, weak periodic outputs are observed when aperiodic spikes are input with modulated populational activity. In this case, higher firing rates are necessary to input for higher frequency oscillations. The spike-timing-dependent plasticity suppresses the spikes which do not contribute to the synchrony for periodic inputs. This effect corresponds to the experimental reports that learning sharpens the synchrony in the motor cortex. These results suggest that spatiotemporal spike patterns should be entrained on modulated populational activity to transmit oscillatory information effectively in the convergent pathway.     

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.     

Sustained and transient oscillations and chaos induced by delayed antiviral immune response in an immunosuppressive infection model

Sustained and transient oscillations are frequently observed in clinical data for immune responses in viral infections such as human immunodeficiency virus, hepatitis B virus, and hepatitis C virus. To account for these oscillations, we incorporate the time lag needed for the expansion of immune cells into an immunosuppressive infection model. It is shown that the delayed antiviral immune response can induce sustained periodic oscillations, transient oscillations and even sustained aperiodic oscillations (chaos). Both local and global Hopf bifurcation theorems are applied to show the existence of periodic solutions, which are illustrated by bifurcation diagrams and numerical simulations. Two types of bistability are shown to be possible: (i) a stable equilibrium can coexist with another stable equilibrium, and (ii) a stable equilibrium can coexist with a stable periodic solution.