## Section: New Results

### Modeling, observation and control in biosciences - Reproductive system

#### Numerical simulation of the selection process of the ovarian follicles

Participants : Benjamin Aymard, Frédérique Clément.

Collaboration with Frédéric Coquel and Marie Postel.

*Implementation of a parallelized numerical scheme based on finite volumes.* We have designed and implemented a numerical method to simulate a multiscale model describing the selection process in ovarian follicles [11] , [10] . The PDE model consists in a quasi-linear hyperbolic system of large size, namely ${N}_{f}\times {N}_{f}$, ruling the time evolution of the cell density functions of ${N}_{f}$ follicles (in practice ${N}_{f}$ is of the order of a few to twenty). These equations are weakly coupled through the sum of the first order moments of the density functions. The time-dependent equations make use of two structuring variables, age and maturity, which play the roles of space variables. The problem is naturally set over a compact domain of ${\mathbf{R}}^{2}$. The formulation of the time-dependent controlled transport coefficients accounts for available biological knowledge on follicular cell kinetics. We introduce a dedicated numerical scheme that is amenable to parallelization, by taking advantage of the weak coupling. Numerical illustrations assess the relevance of the proposed method both in term of accuracy and HPC achievements [32] .

*A numerical method for cell dynamics: kinetic equations with discontinuous coefficients.* The motivation of this work is the numerical treatment of the mitosis in biological models involving cell dynamics. More generally we study hyperbolic PDEs with flux transmission conditions at interfaces between subdomains where coefficients are discontinuous. A dedicated finite volume scheme with a limited high order enhancement is adapted to treat the discontinuities arising at interfaces. The validation of the method is done on 1D and 2D toy problems for which exact solutions are available, allowing us to do a thorough convergence study. A simulation on the original biological model illustrates the full potentialities of the scheme [72] .

#### Optimal control of cell mass and maturity in a model of follicular ovulation

Participants : Frédérique Clément, Peipei Shang.

Collaboration with Jean-Michel Coron

We have studied some optimal control problems associated with a scalar hyperbolic conservation law modeling the development of ovarian follicles. Changes in the age and maturity of follicular cells are described by a 2D conservation law, where the control terms act on the velocities. The control problem consists in optimizing the follicular cell resources so that the follicular maturity reaches a maximal value in fixed time. Formulating the optimal control problem within a hybrid framework, we have proved necessary optimality conditions in the form of Hybrid Maximum Principle [36] . We have then derived the optimal strategy and shown that there exists at least one optimal bang-bang control with one single switching time.

#### Multiscale analysis of mixed-mode oscillations in a phantom bursting model

Participants : Frédérique Clément, Mathieu Desroches, Maciej Krupa, Alexandre Vidal.

We have studied mixed mode oscillations in a model of secretion of GnRH (gonadotropin releasing hormone). The model is a phantom burster consisting of two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The forcing system (Regulator) evolves on the slowest scale and acts by moving the slow null-cline of the forced system (Secretor). There are three modes of dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady state) and small oscillations related to the passage of the slow null-cline through a fold point of the fast null-cline. We have derived a variety of reductions, taking advantage of the mentioned features of the system. We have obtained two results; one on the local dynamics near the fold in the parameter regime corresponding to the presence of small oscillations and the other on the global dynamics, more specifically on the existence of an attracting limit cycle. Our local result is a rigorous characterization of small canards and sectors of rotation in the case of folded node with an additional time scale, a feature allowing for a clear geometric argument. The global result gives the existence of an attracting unique limit cycle, which, in some parameter regimes, remains attracting and unique even during passages through a canard explosion [43] .

#### A network model of the periodic synchronization process in the dynamics of calcium concentration in GnRH neurons

Participants : Frédérique Clément, Maciej Krupa, Alexandre Vidal.

Mathematical neuroendocrinology is a branch of mathematical neurosciences that is specifically interested in endocrine neurons, which have the uncommon ability of secreting neurohormones into the blood. One of the most striking features of neuroendocrine networks is their ability to exhibit very slow rhythms of neurosecretion, on the order of one or several hours. A prototypical instance is that of the pulsatile secretion pattern of GnRH (gonadotropin releasing hormone), the master hormone controlling the reproductive function, whose origin remains a puzzle issue since its discovery in the seventies. We have investigated the question of GnRH neuron synchronization on a mesoscopic scale and study how synchronized events in calcium dynamics can arise from the average electric activity of individual neurons. We have used as reference seminal experiments performed on embryonic GnRH neurons from rhesus monkeys, where calcium imaging series were recorded simultaneously in tens of neurons, and which have clearly shown the occurrence of synchronized calcium peaks associated with GnRH pulses, superposed on asynchronous, yet oscillatory individual background dynamics [100] . We have designed a network model by coupling 3D individual dynamics of FitzHugh-Nagumo type. Using phase-plane analysis, we have constrained the model behavior so that it meets qualitative and quantitative specifications derived from the experiments, including the precise control of the frequency of the synchronization episodes. In particular, we have shown how the time scales of the model can be tuned to fit the individual and synchronized time scales of the experiments. Finally, we have illustrated the ability of the model to reproduce additional experimental observations, such as partial recruitment of cells within the synchronization process or the occurrence of doublets of synchronization [76] .