## 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.

We have designed and implemented a numerical method to simulate a multiscale model describing the selection process in ovarian follicles [9] , [8] . 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 [50] , [51] .

#### Multiscale modeling of follicular ovulation as a mass and maturity dynamical system

Participants : Frédérique Clément, Philippe Michel, Danielle Monniaux.

We have analyzed the dynamics of the solutions using bifurcation tools on a reduced, ODE model [73] . In a first stage, the 2D PDE model is reduced to a 1D PDE model, where the only remaining variable is the age. This reduction is based on a result of exponential convergence in maturity ; we have proved that the granulosa cell density of each follicle converges to a “dirac mass in maturity”, which can be understood as: “the follicle becomes uniform in maturity”. The proof is based on the crucial decay property of the maturity speed rate with respect to the maturity variable, so that the support of the cell density of each follicle concentrates its mass around a curve given by a characteristic equation. In a second stage, the mitosis rate is averaged in age, reducing the 1D PDE to a simpler system of two coupled nonlinear ODE, where each follicle is characterized by its cell number (the follicle mass) and global maturity. These variables correspond respectively to the zero-order moment and first-order moment in maturity of the cell density in the original model. The dynamics of one given follicle can then be studied with respect to the pressure exerted collectively by all other growing follicles, in the framework of dynamical games. In some sense, the pressure can be considered as an exogenous parameter, so that we can detect dynamical bifurcations according to the pressure value. Each follicle plays against the others for its survival. In the course of its terminal development, a follicle first remains in the proliferative zone of the mass-maturity plane and then enters the differentiated zone. At the transition from proliferation to differentiation, the follicle is highly sensitive to the pressure. In the worst (doomed) case, the follicle becomes atretic, due to prolongated cell loss. In the best (saved) case, it manages to go through the vulnerability zone and becomes insensitive to the pressure of other follicles.

#### Optimal control for a conservation law modeling the development of ovulation

Participants : Frédérique Clément, Peipei Shang.

Collaboration with Jean-Michel Coron

We are now investigating control problems associated to the multiscale model of follicle selection. The conditions for the triggering of the ovulatory surge, coupled with the sorting of the ovulatory follicles, define a complex, nested reachability problem. We have considered a more tractable version of that problem, which is centered on defining the optimal local control corresponding to a single ovulatory trajectory. Under some simplifying assumptions (no loss term and constant aging velocity), we have obtained analytical and numerical results in the case when the density is idealized by one or several Dirac mass. We are extending our results to the PDE original model.

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

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

We have carried on the study of our fast-slow model of the GnRH (gonadotropin-releasing hormone) pulse and surge generator [5] , [4] . If we relax a little the constraints imposed by the biological specifications on the parameters, very rich and complex behaviours can be further exhibited by the model. More precisely, both a delay to the surge and a post-surge pause (before pulsatility resumption) may occur. A detailed examination of the pause has revealed that it is shaped by mixed-mode oscillations (MMO). We are currently investigating how the precise sequence of MMO is determined by the global return map from the surge to the pulse regime.

#### Transient synchronization of calcium oscillations in cultures of GnRH neurons.

Participants : Frédérique Clément, Maciej Krupa, Alexandre Vidal.

We have started to study the individual dynamics of GnRH neurons and the conditions under which they may synchronize. We are more specifically tacking the issue of synchronization of calcium oscillations in cultures obtained from the olfactory placodes of rhesus monkey embryos [81] . We have introduced a class of models explaining the synchronization events; their main idea was to introduce a global variable controlling the onset of synchronization that was subsequently reset by the subsequent high firing rate caused by the activation of an adaptation current.