arXiv subjects

Etienne Farcot

Publications and source records attributed to Etienne Farcot.

7 recordsLinked to original sources

arXiv2026

Conservative deterministic Markov models in mathematical biology: uniqueness of steady states, reversibility and computational methods

Ordinary differential equations are commonly used throughout the sciences to build mechanistic models of time-dependent processes. Often, such models are Markov models describing the time-evolution of different interconnected "states". When these models have no "sources" or "sinks", they naturally conserve the total population of the system. When each state is reachable (directly or indirectly) from any other state, these models are called irreducible. For many applications, a deterministic system of ordinary differential equations (ODE) is the most suitable modelling approach. We summarise important mathematical results which show that this irreducibility property guarantees the existence and uniqueness of global stable equilibria, and discuss the computationally-efficient implementation of such ODE-based models. We also discuss the condition of microscopic reversibility and show how it guarantees non-oscillatory behaviour, which enables additional efficiencies in computations. These properties and methods are demonstrated through example models of biological phenomena, where we demonstrate their importance for efficient model fitting and simulation.

q-bio.OT↗

arXiv2025

A Phase-field Model for Apoptotic Cell Death

The process of programmed cell death, namely apoptosis, is a natural mechanism that regulates healthy tissue, multicellular structures, and homeostasis. An improved understanding of apoptosis can significantly enhance our knowledge of biological processes and systems. For instance, pathogens can manipulate the apoptotic process to either evade immune detection or to facilitate their spread. Furthermore, of particular clinical interest is the ability of cancer cells to evade apoptosis, hence allowing them to survive and proliferate uncontrollably. Thus, in this work, we propose a phase-field framework for simulating intrinsic or extrinsic apoptosis induced by an activation field, including deriving the configurational mechanics underlying such phenomena. Along with exploring varying conditions needed to initiate or reduce apoptosis, this can serve as a starting point for computational therapeutic testing. To showcase model capabilities, we present simulations exhibiting different types of cellular dynamics produced when varying the mechanisms underlying apoptosis. The model is subsequently applied to probe different morphological transitions, such as cell shrinkage, membrane blebbing, cavity formation and fragmentation. Lastly, we compare the characteristics observed in our simulations to electron microscopy images, providing additional support for the model.

q-bio.CB↗

arXiv2024

Computer-aided analysis of high-dimensional Glass networks: periodicity, chaos, and bifurcations in a ring circuit

Glass networks model systems of variables that interact via sharp switching. A body of theory has been developed over several decades that, in principle, allows rigorous proof of dynamical properties in high dimensions that is not normally feasible in nonlinear dynamical systems. Previous work has, however, used examples of dimension no higher than 6 to illustrate the methods. Here we show that the same tools can be applied in dimensions at least as high as 20. An important application of Glass networks is to a recently-proposed design of a True Random Number Generator that is based on an intrinsically chaotic electronic circuit. In order for analysis to be meaningful for the application, the dimension must be at least 20. Bifurcation diagrams show what appear to be periodic and chaotic bands. Here we demonstrate that the analytic tools for Glass networks can be used to rigorously show where periodic orbits are lost, and the types of bifurcations that occur there. The main tools are linear algebra and the stability theory of Poincaré maps. All main steps can be automated, and we provide computer code. The methods reviewed here have the potential for many other applications involving sharply switching interactions, such as artificial neural networks.

nlin.CD↗

arXiv2018

Local negative circuits and cyclic attractors in Boolean networks with at most five components

We consider the following question on the relationship between the asymptotic behaviours of asynchronous dynamics of Boolean networks and their regulatory structures: does the presence of a cyclic attractor imply the existence of a local negative circuit in the regulatory graph? When the number of model components $n$ verifies $n \geq 6$, the answer is known to be negative. We show that the question can be translated into a Boolean satisfiability problem on $n \cdot 2^n$ variables. A Boolean formula expressing the absence of local negative circuits and a necessary condition for the existence of cyclic attractors is found unsatisfiable for $n \leq 5$. In other words, for Boolean networks with up to $5$ components, the presence of a cyclic attractor requires the existence of a local negative circuit.

cs.DM↗

arXiv2009

Qualitative control of periodic solutions in piecewise affine systems; application to genetic networks

Hybrid systems, and especially piecewise affine (PWA) systems, are often used to model gene regulatory networks. In this paper we elaborate on previous work about control problems for this class of models, using also some recent results guaranteeing the existence and uniqueness of limit cycles, based solely on a discrete abstraction of the system and its interaction structure. Our aim is to control the transition graph of the PWA system to obtain an oscillatory behaviour, which is indeed of primary functional importance in numerous biological networks; we show how it is possible to control the appearance or disappearance of a unique stable limit cycle by hybrid qualitative action on the degradation rates of the PWA system, both by static and dynamic feedback, i.e. the adequate coupling of a controlling subnetwork. This is illustrated on two classical gene network modules, having the structure of mixed feedback loops.

math.DS↗

arXiv2009

Limit cycles in piecewise-affine gene network models with multiple interaction loops

In this paper we consider piecewise affine differential equations modeling gene networks. We work with arbitrary decay rates, and under a local hypothesis expressed as an alignment condition of successive focal points. The interaction graph of the system may be rather complex (multiple intricate loops of any sign, multiple thresholds...). Our main result is an alternative theorem showing that, if a sequence of region is periodically visited by trajectories, then under our hypotheses, there exists either a unique stable periodic solution, or the origin attracts all trajectories in this sequence of regions. This result extends greatly our previous work on a single negative feedback loop. We give several examples and simulations illustrating different cases.

math.DS↗

arXiv2006

Periodic solutions of piecewise affine gene network models: the case of a negative feedback loop

In this paper the existence and unicity of a stable periodic orbit is proven, for a class of piecewise affine differential equations in dimension 3 or more, provided their interaction structure is a negative feedback loop. It is also shown that the same systems converge toward a unique stable equilibrium point in dimension 2. This extends a theorem of Snoussi, which showed the existence of these orbits only. The considered class of equations is usually studied as a model of gene regulatory networks. It is not assumed that all decay rates are identical, which is biologically irrelevant, but has been done in the vast majority of previous studies. Our work relies on classical results about fixed points of monotone, concave operators acting on positive variables. Moreover, the used techniques are very likely to apply in more general contexts, opening directions for future work.

q-bio.QM↗