The theory of dynamical systems and of nonlinear partial differential evolution equations and their applications in biology, physiology and medicine are fostered with the following papers comprising the present issue. They form a focused issue, which has its roots in an international congress on the same topic held by the Société Francophone de Biologie Théorique (SFBT) from 8 to 11 June 2008 in Saint-Flour dedicated to the memory of Gilbert Chauvet. It was partly sponsored by the Agence National de la Recherche via the ANAR and the SAPHIR projects.

The SFBT was created in the early 1980s under the impetus of Pierre Delattre, Michel Thellier and René Thom with the objective to promote the development of methods and theoretical formalisms useful for fundamental biological research as well as for practical applications. The strength and originality of the SFBT are to bring together scientists from various fields and with heterogeneous backgrounds to debate hot topics in theoretical biology. These cross-disciplinary connections often raise interesting and fruitful discussions and sometimes give rise to new collaborations. For most of the participants, the SFBT meeting, which takes place once a year in Saint-Flour (France) and once every 3 years in a French-speaking country (e.g. Canada and Morocco), is the place where they can exchange new ideas and discover new theoretical work in mathematical biology. Another important vocation for the SFBT is to provide young researchers an overview of the latest research in their area and to give them a different reading of their research topics through meetings with scientists whose training is often quite different from their usual working environment. In the past, various meetings were regularly organized, essentially at the Solignac Abbey, the Saint-Flour Grand Séminaire and the Centre International de Rencontres Mathématiques (CIRM) at Luminy, and then published by various publishing houses (see Delattre *et al*. 1979; Legay *et al*. 1980; Della Dora *et al.* 1981; Le Guyader *et al*. 1981; Chauvet *et al*. 1982; Cosnard *et al*. 1983; Gallis 1984; Benchetrit & Demongeot 1985; Demongeot *et al*. 1985).

Recent years have seen the emergence of an integrated approach called systems biology (Auffray *et al*. 2003). In parallel, model-driven medical and biological data acquisition (see Demongeot 2002) has been sponsored by the French Academy of Sciences and by the programme ‘Technologies pour la Santé’ (Technologies for Health) of the French ministry of research and technology. At the European level, the EC’s Seventh Framework Programme has identified the Virtual Physiological Human as a core target (see Fenner *et al*. 2008). Previous issues of *Philosophical Transactions A* (see Clapworthy *et al.* 2008; Gavaghan *et al*. 2009; Kohl *et al*. 2009) have emphasized the development of tools and ontologies for increasingly predictive modelling of organ systems.

The theme we propose for this issue also discusses the complex cross-disciplinary interactions among the various domains involved in the study of a living system (biology, mathematics and computer sciences). The usual way to formalize in a rational form the structure of a biological system is to propose a mathematical formulation of the key processes and of interactions among them which have been identified as fundamental for the studied system. This approach allows one to study, from a mathematical point of view, the properties arising from the mathematical model. It is then possible to return to reality with proposals for new experiments in order to validate (or to invalidate) the emergent properties predicted by the mathematical model. However, the complexity of living systems precludes a complete model of their behaviour, and models of the subsystems of interest are sometimes mathematically intractable. Moreover, compared with a biological system, the simplified hypotheses required to construct a mathematical model may be too unrealistic. In these cases, the model may be too simple to reproduce the interesting behaviour or, at the opposite extreme, too complex to be well understood. In the latter case, the only possible use of such mathematical objects is to mimic the behaviour of the biological system while accepting a limited knowledge of the dynamic behaviour of the mathematical model. All these questions raise the need for the development of mathematical methods capable either of reducing the complexity of a mathematical model while keeping the richness of its dynamics or of constructing simple comprehensible models that can generate complex dynamics. The contributions in this issue present original approaches in many fields of mathematical biology, mainly in the field of dynamical systems and evolution PDEs. There are articles on continuous differential systems of finite dimension and their bifurcations, discrete dynamical systems, impulsional systems, hybrid systems, stochastic dynamical systems, infinite-dimensional dynamical systems with PDEs of reaction–diffusion type, calcium waves, delay equations and Boolean and topological dynamics on metric spaces.

## Presentation of the articles

The continuous dynamical systems approach provides an efficient and powerful method to investigate the geometric and topological properties of the solutions of ordinary differential equations. As the parameters of the equation vary, the characteristics of the solutions (stability of periodic orbits and of stationary points, basins of attraction, attractors) may display discontinuous changes. Bifurcation theory provides the mathematical methods for their classification. The first two articles of this issue use both bifurcation theory and finite-dimensional continuous dynamical systems.

The article ‘Dependence of the period on the rate of protein degradation in minimal models for circadian oscillations’ (Gérard *et al.* 2009) discusses the periodic profiles in models of circadian oscillations. It focuses on the role of clock protein degradation on the circadian period by means of mathematical modelling. Precise control of the period of circadian rhythms is crucial for a good phase relationship between the clock and periodic external signals such as the light–dark cycle. The dynamics generated by three different models are compared by means of numerical simulations and bifurcation diagrams. The period profile of a family of ordinary differential equations and the existence of critical points for the period is a current subject of investigation for mathematician specialists in dynamical systems.

The second article is ‘Study of a virus–bacteria interaction model in a chemostat: application of geometrical singular perturbation theory’ by Poggiale *et al.* (2009). The authors show that geometric singular perturbation theory can be used to investigate a simplified case of a model introduced by Middelboe. The model is represented as a four-dimensional differential system. The phase portrait is as usual restricted to positive values of the variable. The positive orbits remain inside a compact set and hence there is global existence of the solutions. The authors associate a five-dimensional fast–slow system and use the Fenichel theorem on the existence of normally stable invariant manifolds. They also use the extension of the Poincaré–Bendixson theorem to asymptotically autonomous systems found by Markus and developed by Thieme. As a consequence, they prove the existence of a Hopf bifurcation. They interpret the consequences and the biological pertinence of the model.

The third article is ‘Exploration of beneficial and deleterious effects of inflammation in stroke: dynamics of inflammation cells’ (Lelekov-Boissard *et al.* 2009). The authors provide here an example of the genesis of a differential system emerging from clinical experience.

Another such example is provided by ‘A model of mechanical interactions between heart and lungs’ by Fontecave *et al.* (2009). This article introduces time dependence in the systems model of Guyton *et al.* (1972). In the spirit of the theory of systems (Bertalanffy, Ashby, Rosen), Guyton *et al.* realized a *tour de force* by modelling overall fluid, electrolyte and circulatory regulation (see related article by Randall Thomas *et al*. 2008). The addition of dynamical inputs to the cardiovascular system is proposed here. Further analyses would be fruitful such as sensitivity analysis, parameter identification and identification of time scales.

Three articles follow on the theme of discrete finite dimensional systems, impulsional systems and hybrid systems.

Article 5, ‘Application of interval iterations to the entrainment problem in respiratory physiology’ by Demongeot & Waku (2009), yields novel use of one-dimensional discrete dynamical systems to investigate frequency locking phenomena in respiratory physiology. It relies on a rich collection of clinical data accumulated for years in Grenoble hospitals.

Article 6 is ‘Endogenous circannual rhythm in LH secretion: insight from signal analysis coupled with mathematical modelling’ by Vidal *et al.* (2009). The experimental study relies here on animals and it was carried out with five black-belly ewes exposed for 3 years to a constant short day length of 8 h light/16 h dark regime. Signal processing analysis displays an interface between the experimental data and the mathematical model which is a hybrid impulsional dynamical system. The time–frequency analysis is performed using the Scilab-Scicos environment.

Article 7 is ‘A note on semi-discrete modelling in the life sciences’ by Mailleret & Lemesle (2009). An overview of the literature on hybrid systems is proposed there. The reviewed results are classified with respect to their fields of application: epidemiology (vaccination strategies and plant diseases), medicine (drug therapy) and population dynamics (harvesting and pest management programmes, seasonal phenomena and chemostat modelling). When applicable, the authors highlight the added value of hybrid models with respect to their continuous (or discrete) counterparts.

Article 8, ‘Approximating the distribution of population size in stochastic multi-regional matrix models with fast migrations’ by Sanz & Alonso (2009), develops aggregation methods for stochastic dynamical systems.

With the three next articles, dynamical systems methods for infinite-dimensional systems are used to investigate models of partial differential evolution equations of the reaction–diffusion type.

Article 9 is ‘Synchrony in reaction–diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves’ by Abbas *et al.* (2009). This article relates to probabilistic dynamics (Kolmogoroff–Sinai entropy), which are used to describe cell dynamics incorporating the fundamental cell cycle. The principal novelty of this discrete approach is provided by the synchronization index *H*. This is then applied to reaction–diffusion PDE systems with a morphogenetic gradient. The authors define a constraint by favouring cell proliferation as a function of the concavity of the level lines. One of the most striking features of this new approach is the special concentration phenomenon which arises along the null mean Gaussian curvature lines.

Article 10, ‘Propagation of bursting oscillations’ by Ambrosio & Françoise (2009), concerns periodically slowly driven excitable dynamics and their bifurcations. The mathematical model can be seen either as a very large number of linearly coupled dynamical systems of this type or as the reaction–diffusion PDE system obtained by adding a Laplacian (dimension 2 is considered here). Existence of an attractor is shown using functional analysis techniques developed by Temam (1988) and Haraux (1991). The authors focus next on two propagation phenomena called ‘DeathSpot’ and ‘stimulus–response ratio’. This relates to physiological experiments in cardio-dynamics (Chialvo & Jalife 1987) where a central pacemaker nucleus is vagally driven. It also relates to the mechanism for signal transmission in pancreatic islets of Langerhans (Aslanidi *et al*. 2001).

Article 11 is ‘Mathematical modelling of atherosclerosis as an inflammatory disease’ by El Khatib *et al.* (2009). The authors investigate the inflammatory process resulting in the development of atherosclerosis. They develop one- and two-dimensional models based on reaction–diffusion systems to describe the onset of a chronic inflammatory response in the intima of an artery vessel wall.

Article 12 is ‘An integrated formulation of anisotropic force–calcium relations driving spatio-temporal contractions of cardiac myocytes’ by Tracqui & Ohayon (2009). This article focuses on the dynamics of calcium waves and applies a PDE model to experiments on cardiomyocytes.

Article 13 is ‘A model of a fishery with fish stock involving delay equations’ by Auger & Ducrot (2009). In this article, part of the harvested resource is stored. One can think of a fish resource which can enter into a frozen fish stock for some time before being re-injected into the market. Therefore, a third equation with a distributed delay is added to a classical fishery model governing the time evolution of a fish stock variable. The authors discuss the stability of stationary solutions and the existence of a Hopf bifurcation.

Article 14 is ‘A multiformalism and multiresolution modelling environment: application to the cardiovascular system and its regulation’ by Hernández *et al.* (2009). This article makes the bridge with previous issues of this journal (e.g. Randall Thomas *et al*. 2008, SAPHIR) related to expanding the classic multi-organ Guyton model of blood pressure regulation. It comes at the interface of signal processing, informatics and dynamical systems. As in Fontecave *et al.* (2009), a time dependence is added to the Guyton model. The library is capable of solving coupled models developed under different dynamical systems types (discrete versus continuous or impulsional) with different time scales. Here as well, important new issues appear such as sensitivity analysis, parameter identification and multi-scale approaches.

Article 15, ‘Micro-RNAs: viral genome and robustness of the genes expression in host’ by Demongeot *et al.* (2009), displays use of metric topology and dynamical systems to genetic regulation networks.

Finally, ‘Distance-driven adaptive trees in biological metric spaces: uninformed accretion does not prevent convergence’ by Kergosien (2009) ends this issue with the most abstract setting of topological dynamical systems on metric spaces. The author shows the relevance to interventional medical imaging and artificial vision and discusses the interpretations within the Darwinian paradigm.

## Footnotes

One contribution of 17 to a Theme Issue ‘From biological and clinical experiments to mathematical models’.

- © 2009 The Royal Society