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.

We introduce a model, similar to diffusion limited aggregation (DLA), which serves as adiscrete analog of the continuous dynamics of evaporation of thin liquid films. Withinmean field approximation the dynamics of this model, averaged over many realizations ofthe growing cluster, reduces to that of the idealized evaporation model in which surfacetension is neglected. However fluctuations beyond the mean field level play an importantrole, and we study their effect on the conserved quantities of the idealized evaporationmodel. Assuming the cluster to be a fractal, a heuristic approach is developed in order toexplain the distinctive increase of the fractal dimension with the cluster size.

About twenty five years ago the first discrete mathematical model of the immune systemwas proposed. It was very simple and stylized. Later, many other computational models havebeen proposed each one adding a certain level of sophistication and detail to thedescription of the system. One of these, the Celada-Seiden model published back in 1992,was already mature at its birth, setting apart from the topic-specific nature of the othermodels. This one was not just a model but rather a framework with which one couldimplement his own immunological theories.

Here we describe this computational framework, developed to perform simulations ofdifferent pathologies that are directly or indirectly connected to the immune system. Webriefly describe the system first, then we report on few applications so to give thereader a clear idea of its practical utility in clinical research problems.

A computational framework for testing the effects of cytotoxic molecules, specific to agiven phase of the cell cycle, and vascular disrupting agents (VDAs) is presented. Themodel is based on a cellular automaton to describe tumour cell states transitions fromproliferation to death. It is coupled with a model describing the tumour vasculature andits adaptation to the blood rheological constraints when alterations are induced by VDAstreatment. Several therapeutic protocols in two structurally different vascular networkswere tested by varying the duration of cytotoxic drug perfusion and the periodicity oftreatment cycles. The impact of VDAs were also tested both experimentally from intravitalmicroscopy through a dorsal skinfold chamber on a mouse and numerically. Simulationresults show that combining cytotoxic treatment with a post treatment of VDA through ajudicious timing could favour the rapid eradication of the tumour. The computationalframework thus gives some insights into the outcome of cytotoxic and VDAs treatments on aqualitative basis. Future validation from our experimental setup could open up newperspectives towards Computer-Assisted Therapeutic Strategies.

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.

Influenza has been responsible for human suffering and economic burden worldwide. Isolation is one of the most effective means to control the disease spread. In this work, we incorporate isolation into a two-strain model of influenza. We find that whether strains of influenza die out or coexist, or only one of them persists, it depends on the basic reproductive number of each influenza strain, cross-immunity between strains, and isolation rate. We propose criteria that may be useful for controlling influenza. Furthermore, we investigate how effective isolation is by considering the host’s mean age at infection and the invasion rate of a novel strain. Our results suggest that isolation may help to extend the host’s mean age at infection and reduce the invasion rate of a new strain. When there is a delay in isolation, we show that it may lead to more serious outbreaks as compared to no delay.

The measurement of CFSE dilution by flow cytometry is a powerful experimental tool tomeasure lymphocyte proliferation. CFSE fluorescence precisely halves after each celldivision in a highly predictable manner and is thus highly amenable to mathematicalmodelling. However, there are several biological and experimental conditions that canaffect the quality of the proliferation data generated, which may be important to considerwhen modelling dye dilution data sets. Here we overview several of these variablesincluding the type of fluorescent dye used to monitor cell division, dye labellingmethodology, lymphocyte subset differences, in vitro versus in vivo experimental assays,cell autofluorescence, and dye transfer between cells.

The invasive capability is fundamental in determining the malignancy of a solid tumor.Revealing biomedical strategies that are able to partially decrease cancer invasiveness istherefore an important approach in the treatment of the disease and has given rise tomultiple in vitro and in silico models. We here developa hybrid computational framework, whose aim is to characterize the effects of thedifferent cellular and subcellular mechanisms involved in the invasion of a malignantmass. In particular, a discrete Cellular Potts Model is used to represent the populationof cancer cells at the mesoscopic scale, while a continuous approach of reaction-diffusionequations is employed to describe the evolution of microscopic variables, as the nutrientsand the proteins present in the microenvironment and the matrix degrading enzymes secretedby the tumor. The behavior of each cell is then determined by a balance of forces, such ashomotypic (cell-cell) and heterotypic (cell-matrix) adhesions and haptotaxis, and ismediated by the internal state of the individual, i.e. its motility. The resultingcomposite model quantifies the influence of changes in the mechanisms involved in tumorinvasion and, more interestingly, puts in evidence possible therapeutic approaches, thatare potentially effective in decreasing the malignancy of the disease, such as thealteration in the adhesive properties of the cells, the inhibition in their ability toremodel and the disruption of the haptotactic movement. We also extend the simulationframework by including cell proliferation which, following experimental evidence, isregulated by the intracellular level of growth factors. Interestingly, in spite of theincrement in cellular density, the depth of invasion is not significantly increased, asone could have expected.

Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie’s group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.

A small vicinity of a contact line, with well-defined (micro)scales (henceforth the“microstructure”), is studied theoretically for a system of a perfectly wetting liquid,its pure vapor and a superheated flat substrate. At one end, the microstructure terminatesin a non-evaporating microfilm owing to the disjoining-pressure-induced Kelvin effect. Atthe other end, for motionless contact lines, it terminates in a constant film slope(apparent contact angle as seen on a larger scale), the angle being non-vanishing despitethe perfect wetting due to an overall dynamic situation engendered by evaporation. Here wego one step beyond the standard one-sided model by incorporating the effect of vaporpressure non-uniformity as caused by a locally intense evaporation flow, treated in theStokes approximation. Thereby, the film dynamics is primarily affected throughthermodynamics (shift of the local saturation temperature and evaporation rate), thedirect mechanical impact being rather negligible. The resulting integro-differentiallubrication film equation is solved by handling the newly introduced effect (giving riseto the “integro” part) as a perturbation. In the ammonia (at 300   K) example dealt withhere, it proves to be rather weak indeed: the contact angle decreases while the integralevaporation flux increases just by a few percent for a superheat of  ~1   K.However, the numbers grow (roughly linearly) with the superheat.

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.

We analyze the problem of shear-induced electrokinetic lift on a particle freely suspended near a solid wall, subject to a homogeneous (simple) shear. To this end, we apply the large-Péclet-number generic scheme recently developed by Yariv et al. (J. Fluid Mech., Vol. 685, 2011, p. 306). For a force- and torque-free particle, the driving flow comprises three components, respectively describing (i) a particle translating parallel to the wall; (ii) a particle rotating with an angular velocity vector normal to the plane of shear; and (iiii) a stationary particle in a shear flow. Symmetry arguments reveal that the electro-viscous lift, normal to the wall, is contributed by Maxwell stresses accompanying the induced electric field, while electro-viscous drag and torque corrections, parallel to the wall, are contributed by the Newtonian stresses accompanying the induced flow. We focus upon the near-contact limit, where all electro-viscous contributions are dominated by the intense electric field in the narrow gap between the particle and the wall. This field is determined by the gap-region pressure distributions associated with the translational and rotational components of the driving Stokes flow, with the shear-component contribution directly affecting only higher-order terms. Owing to the similarity of the corresponding pressure distributions, the induced electric field for equal particle–wall zeta potentials is proportional to the sum of translation and rotation speeds. The electro-viscous loads result in induced particle velocities, normal and tangential to the wall, inversely proportional to the second power of particle–wall separation.

In this paper we give a survey on modeling efforts concerning the CVRS. The material wediscuss is organized in accordance with modeling goals and stresses control and transportissues. We also address basic modeling approaches and discuss some of the challenges formathematical modeling methodologies in the context of parameter estimation and modelvalidation.

A decision analytical model is presented and analysed to assess the effectiveness andcost-effectiveness of routine vaccination against varicella and herpes-zoster, orshingles. These diseases have as common aetiological agent the varicella-zoster virus(VZV). Zoster can more likely occur in aged people with declining cell-mediated immunity.The general concern is that universal varicella vaccination might lead to more cases ofzoster: with more vaccinated children exposure of the general population to varicellainfectives become smaller and thus a larger proportion of older people will have weakerimmunity to VZV, leading to more cases of reactivation of zoster. Our compartment modelshows that only two possible equilibria exist, one without varicella and the other onewhere varicella and zoster both thrive. Threshold quantities to distinguish these casesare derived. Cost estimates on a possible herd vaccination program are discussedindicating a possible tradeoff choice.

Gliomas are highly invasive brain tumors that exhibit high and spatially heterogeneouscell proliferation and motility rates. The interplay of proliferation and migrationdynamics plays an important role in the invasion of these malignant tumors. We analyze theregulation of proliferation and migration processes with a lattice-gas cellular automaton(LGCA). We study and characterize the influence of the migration/proliferation dichotomy(also known as the “GO-or-Grow" mechanism) on avascular glioma invasion, in terms ofinvasion speed and width of the infiltration zone. We show that the invasive behavior ofthe (macroscopic) tumor colony is a highly complex phenomenon that cannot be extrapolatedby the sole knowledge of the (microscopic) individual cell phenotype.

In this paper, we show finite time blow-up of solutions of the p−waveequation in ℝN, with critical Sobolev exponent. Our workextends a result by Galaktionov and Pohozaev [4]

HIV infection is multi-faceted and a multi-step process. The virus-induced pathogenicmechanisms are manifold and mediated through a range of positive and negative feedbackregulations of immune and physiological processes engaged in virus-host interactions. Thefundamental questions towards understanding the pathogenesis of HIV infection are nowshifting to ‘dynamic’ categories: (i) why is the HIV-immune response equilibrium finallydisrupted? (ii) can one modify the dynamic equilibrium for host benefit? (iii) can onepredict the outcome of a system perturbation via antiviral drugs or drugs modulating thehost immune response dynamics? Answering these questions requires a majorinterdisciplinary effort, and in particular, the development of novel mathematicalapproaches for a coherent quantitative description and prediction of intra-patient HIVevolution, the immunological responses to HIV infection, and the systems level homeostaticregulation of specific effector and regulatory lymphocyte populations in correlation withdisease status. Here we summarized fundamental biological features of HIV infection andcurrent mathematical modelling attempts to understand HIV pathogenesis.

Two-dimensional inviscid channel flow of an incompressible fluid is considered. It isshown that if the flow is steady and features no horizontal stagnation, then the flow mustnecessarily be a parallel shear flow.

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.

There is evidence that cancer develops when cells acquire a sequence of mutations thatalter normal cell characteristics. This sequence determines a hierarchy among the cells,based on how many more mutations they need to accumulate in order to become cancerous.When cells divide, they exhibit telomere loss and differentiate, which defines anothercell hierarchy, on top of which is the stem cell. We propose a mutation-generation model,which combines the mutation-accumulation hierarchy with the differentiation hierarchy ofthe cells, allowing us to take a step further in examining cancer development and growth.The results of the model support the hypothesis of the cancer stem cell’s role in cancerpathogenesis: a very small fraction of the cancer cell population is responsible for thecancer growth and development. Also, according to the model, the nature of mutationaccumulation is sufficient to explain the faster growth of the cancer cell population.However, numerical results show that in order for a cancer to develop within a reasonabletime frame, cancer cells need to exhibit a higher proliferation rate than normalcells.