We illustrate the appearance of oscillating solutions in delay differential equationsmodeling hematopoietic stem cell dynamics. We focus on autonomous oscillations, arising asconsequences of a destabilization of the system, for instance through a Hopf bifurcation.Models of hematopoietic stem cell dynamics are considered for their abilities to describeperiodic hematological diseases, such as chronic myelogenous leukemia and cyclicalneutropenia. After a review of delay models exhibiting oscillations, we focus on threeexamples, describing different delays: a discrete delay, a continuous distributed delay,and a state-dependent delay. In each case, we show how the system can have oscillatingsolutions, and we characterize these solutions in terms of periods and amplitudes.

Recently, Wang and Xiao studied a four-dimensional competitive Lotka-Volterra systemwithin a deterministic environment in [11]. Withthe help of numerical example they showed the existence of a chaotic attractor through theperiod doubling route. In this paper, we are interested to study the dynamics of the samemodel in presence of environmental driving forces. To incorporate the environmentaldriving force into the deterministic system, we perturb the growth rates of each speciesby white noise terms. Then we prove that the unique positive global solution exists forthe noise added system and the general p-th order moment of it is boundedfor p ≥ 1, which ensures that the solution is stochasticallybounded. It is also shown that the solution of the stochastic system is stochasticallypermanent under some simple conditions. Finally , we demonstrate the noise inducedoscillation for the concerned model with the help of numerical example..

The dynamic model of plant growth GreenLab describes plant architecture and functionalgrowth at the level of individual organs. Structural development is controlled by formalgrammars and empirical equations compute the amount of biomass produced by the plant, andits partitioning among the growing organs, such as leaves, stems and fruits. The number oforgans initiated at each time step depends on the trophic state of the plant, which isevaluated by the ratio of biomass available in plant to the demand of all the organs. Thecontrol of the plant organogenesis by this variable induces oscillations in the simulatedplant behaviour. The mathematical framework of the GreenLab model allows to compute theconditions for the generation of oscillations and the value of the period according to theset of parameters. Two case-studies are presented, corresponding to emergence ofoscillations associated to fructification and branching.

Similar alternating patterns arecommonly reported by botanists. In this article, two examples were selected: alternatepatterns of fruits in cucumber plants and alternate appearances of branches inCecropia trees. The model was calibrated from experimental datacollected on these plants. It shows that a simple feedback hypothesis of trophic controlon plant structure allows the emergence of cyclic patterns corresponding to the observedones.

In this work we discuss and analyze spiking patterns in a generic mathematical model oftwo coupled non-identical nonlinear oscillators supplied with a spike-timing dependentplasticity (STDP) mechanism. Spiking patterns in the system are shown to converge to aphase-locked state in a broad range of parameters. Precision of the phase locking, i.e.the amplitude of relative phase deviations from a given reference, depends on the naturalfrequencies of oscillators and, additionally, on parameters of the STDP law. Thesedeviations can be optimized by appropriate tuning of gains (i.e. sensitivity tospike-timing mismatches) of the STDP mechanisms. The deviations, however, can not be madearbitrarily small neither by mere tuning of STDP gains nor by adjusting synaptic weights.Thus if accurate phase-locking in the system is required then an additional tuningmechanism is generally needed. We found that adding a very simple adaptation dynamics inthe form of slow fluctuations of the base line in the STDP mechanism enables accuratephase tuning in the system with arbitrary high precision. The scheme applies to systems inwhich individual oscillators operate in the oscillatory mode. If the dynamics ofoscillators becomes bistable then relative phase may fail to converge to a given valuegiving rise to the emergence of complex spiking sequences.

Unveiling the mechanisms through which the somitogenesis regulatory network exertsspatiotemporal control of the somitic patterning has required a combination ofexperimental and mathematical modeling strategies. Significant progress has been made forthe zebrafish clockwork. However, due to its complexity, the clockwork of the amniotesegmentation regulatory network has not been fully elucidated. Here, we address thequestion of how oscillations are arrested in the amniote segmentation clock. We do this byconstructing a minimal model of the regulatory network, which privileges architecturalinformation over molecular details. With a suitable choice of parameters, our model isable to reproduce the oscillatory behavior of the Wnt, Notch and FGF signaling pathways inpresomitic mesoderm (PSM) cells. By introducing positional information via a single Wnt3agradient, we show that oscillations are arrested following an infinite-period bifurcation.Notably: the oscillations increase their amplitude as cells approach the anterior PSM andremain in an upregulated state when arrested; the transition from the oscillatory regimeto the upregulated state exhibits hysteresis; and opposing Fgf8 and RA gradients along thePSM naturally arise in our simulations. We hypothesize that the interaction between alimit cycle (originated by the Notch delayed-negative feedback loop) and a bistable switch(originated by the Wnt-Notch positive cross-regulation) is responsible for the observedsegmentation patterning. Our results agree with previously unexplained experimentalobservations and suggest a simple plausible mechanism for spatiotemporal control ofsomitogenesis in amniotes.

Biological rhythms occur at different levels in the organism. In single cells, the celldivision cycle shows rhythmicity in the way its molecular regulators, the cyclin dependantkinases (CDKs), modulate their activity periodically to ensure a healthy progression. Intissues, cell proliferation is driven by the circadian clock, which modulates theprogression through the cell cycle along the day. The circadian clock shows endogenousrhythmicity through a robust network of transcription-translation feedback loops thatcreate sustained oscillations. Rhythmicity is preserved in cell populations by thecoordination of the clocks among cells, through rhythmic synchronization signals. Here wediscuss mechanisms for generating rhythmic activities in cell populations by reviewingsome of the mathematical models that deal with them. We discuss the implication ofbiological rhythms for tissue growth and the possible application to chronomodulatedcancer treatments.

Progression along the successive phases of the mammalian cell cycle is driven by anetwork of cyclin-dependent kinases (Cdks). This network is regulated by a variety ofnegative and positive feedback loops. We previously proposed a detailed, 39-variable modelfor the Cdk network and showed that it is capable of temporal self-organization in theform of sustained oscillations, which correspond to the repetitive, transient, sequentialactivation of the cyclin- Cdk complexes that govern the successive phases of the cellcycle [Gérard and Goldbeter (2009) Proc Natl Acad Sci 106, 21643-8]. Here we compare thedynamical behavior of three models of different complexity for the Cdk network driving themammalian cell cycle. The first is the detailed model that counts 39 variables and isbased on Michaelis-Menten kinetics for the enzymatic steps. From this detailed model, webuild a version based only on mass-action kinetics, which counts 80 variables. In thisversion we do not need to assume that enzymes are present in much smaller amounts thattheir substrates, which is not necessarily the case in the cell cycle. We show that thesetwo versions of the model for the Cdk network yield similar results. In particular theypredict sustained oscillations of the limit cycle type. We show that the model for the Cdknetwork can be reduced to a version containing only 5 variables, which is more amenable tostochastic simulations. This skeleton version retains the dynamic properties of the morecomplex versions of the model for the Cdk network in regard to Cdk oscillations. Theregulatory wiring of the Cdk network therefore governs its dynamic behavior, regardless ofthe degree of molecular detail. We discuss the relative advantages of each version of themodel, all of which support the view that the mammalian cell cycle behaves as a limitcycle oscillator.

Mitochondria are one of the most important organelles determining Ca2+regulatory pathway in the cell. Recent experiments suggested the existence of cytosolicmicrodomains with locally elevated calcium concentration (CMDs) in the nearest vicinity ofthe outer mitochondrial membrane (OMM). These intermediate physical connections betweenendoplasmic reticulum (ER) and mitochodria are called MAM (mitochondria-associated ERmembrane) complexes.

The aim of this paper is to take into account the direct calcium flowfrom ER to mitochondria implied by the existence of MAMs and perform detailed numericalanalysis of the influence of this flow on the type and shape of calcium oscillations.Depending on the permeability of MAMs interface and ER channels, different patterns ofoscillations appear (simple, bursting and chaotic). For some parameters the oscillatorypattern disappear and the system tends to a steady state with extremely high calcium levelin mitochondria, which can be interpreted as a crucial point at the beginning of anapoptotic pathway.