Alexandre Vidal | Evry University (original) (raw)
Papers by Alexandre Vidal
arXiv: Dynamical Systems, Dec 4, 2007
Mathematical Modelling of Natural Phenomena
In [16], the authors analyzed the synchronization features between two identical 3D slow-fast osc... more In [16], the authors analyzed the synchronization features between two identical 3D slow-fast oscillators, symmetrically coupled, built as an extension of the FitzHugh–Nagumo dynamics generating Mixed-Mode Oscillations. The third variable, which is slow, represents the intracellular calcium concentration in neurons. Here, we consider an extension of this model in two directions. First, we consider heterogeneity among cells and analyze the coupling of two oscillators with different values for one parameter tuning the intrinsic frequency. We identify new patterns of antiphasic synchronization, with non-trivial signatures and that exhibit a Devil’s Staircase phenomenon in transitions. Second, we introduce a network of N cells divided into two clusters: the coupling between neurons in each cluster is excitatory, while between the two clusters is inhibitory. Such system models the interactions between neurons tending to synchronization in two subpopulations inhibiting each other, like ip...
In this work we study mixed mode oscillations in a model of secretion of GnRH (Gonadotropin Relea... more In this work we study 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 nullcline 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 nullcline through a fold point of the fast nullcline. We derive a variety of reductions, taking advantage of the mentioned features of the system. We obtain 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...
Abstract. The biological models- particularly the ecological ones- must be understood through the... more Abstract. The biological models- particularly the ecological ones- must be understood through the bifurcations they undergo as the parameters vary. However, the transition between two dynamical behaviours of a same system for diverse values of parameters may be sometimes quite involved. For instance, the analysis of the non generic motions near the transition states is the first step to understand fully the bifurcations occurring in complex dynamics. In this article, we address the question to describe and explain a double bursting behaviour occuring for a tritrophic slow–fast system. We focus therefore on the appearance of a double homoclinic bifurcation of the fast subsystem as the predator death rate parameter evolves. The first part of this article introduces the slow–fast system which extends Lotka–Volterra dynamics by adding a superpredator. The second part displays the analysis of singular points and bifurcations undergone by fast dynamics. The third part is devoted to the fl...
Bulletin of the Belgian Mathematical Society - Simon Stevin, 2008
SIAM Journal on Applied Dynamical Systems, 2018
Discrete and Continuous Dynamical Systems - Series B, 2017
Journal of mathematical neuroscience, 2016
Recent experimental evidence on the clustering of glutamate and GABA transporters on astrocytic p... more Recent experimental evidence on the clustering of glutamate and GABA transporters on astrocytic processes surrounding synaptic terminals pose the question of the functional relevance of the astrocytes in the regulation of neural activity. In this perspective, we introduce a new computational model that embeds recent findings on neuron-astrocyte coupling at the mesoscopic scale intra- and inter-layer local neural circuits. The model consists of a mass model for the neural compartment and an astrocyte compartment which controls dynamics of extracellular glutamate and GABA concentrations. By arguments based on bifurcation theory, we use the model to study the impact of deficiency of astrocytic glutamate and GABA uptakes on neural activity. While deficient astrocytic GABA uptake naturally results in increased neuronal inhibition, which in turn results in a decreased neuronal firing, deficient glutamate uptake by astrocytes may either decrease or increase neuronal firing either transient...
We investigate a model of the GnRH pulse and surge generator, with the definite aim of constraini... more We investigate a model of the GnRH pulse and surge generator, with the definite aim of constraining the model GnRH output with respect to a physiologically relevant list of specifications. The alternating pulse and surge pattern of secretion results from the interaction between a GnRH secreting system and a regulating system exhibiting fast-slow dynamics. The mechanisms underlying the behavior of the model are reminded from the study of the Boundary-Layer System according to the "dissection method" principle. Using singular perturbation theory, we describe the sequence of bifurcations undergone by the regulating (FitzHugh-Nagumo) system, encompassing the rarely investigated case of homoclinic connexion. Basing on pure dynamical considerations, we restrict the space of parameter search for the regulating system and describe a foliation of this restricted space, whose leaves define constant duration ratios between the surge and the pulsatility phase in the whole system. We p...
Fast-slow systems are studied usually by “geometrical dissection” [4]. The fast dynamics exhibit ... more Fast-slow systems are studied usually by “geometrical dissection” [4]. The fast dynamics exhibit attractors which may bifurcate under the influence of the slow dynamics which is seen as a parameter of the fast dynamics. A generic solution comes close to a connected component of the stable invariant sets of the fast dynamics. As the slow dynamics evolves, this attractor may lose its stability and the solution eventually reaches quickly another connected component of attractors of the fast dynamics and the process may repeat. This scenario explains quite well relaxation oscillations and more complicated oscillations like bursting. More recently, in relation both with theory of dynamical systems [11] and with applications to physiology [10, 26], a new interest has emerged in canard cycles. These orbits share the property that they remain for a while close to an unstable invariant set (either singular set or periodic orbits of the fast dynamics). Although canards were first discovered w...
INRA-CNRS-Université de Tours-Haras Nationaux, UMR Physiologie de la Reproduction et des Comporte... more INRA-CNRS-Université de Tours-Haras Nationaux, UMR Physiologie de la Reproduction et des Comportements. Abstract. In sheep as in many vertebrates, the seasonal pattern of reproduction is timed by the annual photoperiodic cycle, characterized by seasonal changes in the daylength. The photoperiodic information is translated into a circadian profile of melatonin secretion. After multiple neuronal relays (within the hypothalamus), melatonin impacts GnRH (gonadotrophin releasing hormone) secretion that in turn controls ovarian cyclicity. The pattern of GnRH secretion is mirrored into that of LH (luteinizing hormone) secretion, whose plasmatic level can be easily measured. We addressed the question of whether there exists an endogenous circannual rhythm in a tropical sheep (Black-belly) population that exhibits clear seasonal ovarian activity when ewes are subjected to temperate latitudes. We based our analysis on LH time series collected in the course of 3 years from ewes subjected to a ...
arXiv: Dynamical Systems, 2015
We exhibit and investigate a new type of mechanism for generating complex oscillations featuring ... more We exhibit and investigate a new type of mechanism for generating complex oscillations featuring an alternation of small oscillations with spikes (MMOs) or bursts (MMBOs) in a class of hybrid dynamical systems modeling neuronal activity. These dynamical systems, called nonlinear adaptive integrate-and-fire neurons, combine nonlinear dynamics modeling input integration in a nerve cell with discrete resets modeling the emission of an action potential and the subsequent return to reversal potential. We show that presence of complex oscillations in these models relies on a fundamentally hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts. The decomposition into these two mechanisms leads us to propose a purely geometrical interpretation of these complex trajectories, and this relative simplicity allows to finely characterize the MMO patterns through the study of iterates of ...
ABSTRACT Neural mass modeling is a field of computational neuroscience that aims at studying the ... more ABSTRACT Neural mass modeling is a field of computational neuroscience that aims at studying the activity of neuronal populations without explicit representation of single neurons. This type of mesoscopic model is able to generate output signals that can be compared with experimental data such as stereo-electroencephalograms. Classically, neural mass models consider two interconnected populations: excitatory pyramidal cells and inhibitory interneurons. Regarding the excitatory feedbacks on the pyramidal cell population, two distinct approaches have been proposed. A “direct feedback” on the main pyramidal cell population or an “indirect feedback” via a secondary pyramidal cell population. In this article, we propose a new neural mass model that couples both these approaches. We analyze the model bifurcations in two specific cases and describe the corresponding time series. We then explain the typical features of experimental records in epileptic mice. Finally, we show that the model is able to reproduce two different regimes identified in experimental data. Our study also reveals the similarity in the proper 4Hz frequency of epileptic discharges in experimental data and generated time series.
Les oscillations en salves apparaissent dans de nombreux systemes biologiques, physiologiques et ... more Les oscillations en salves apparaissent dans de nombreux systemes biologiques, physiologiques et ecologiques. Elles se caracterisent par l’alternance de phases dites silencieuses ou quiescentes separees par des phases dites actives ou pulsatiles. Cette these est consacree a l'analyse mathematique des systemes dynamiques lents-rapides proposes pour la modelisation des oscillations en salves. Grâce a la theorie des bifurcations, la theorie des perturbations singulieres et l’utilisation d’eclatements a parametres, nous caracterisons les differents comportements de tels systemes. En particulier, nous utilisons des developpements asymptotiques pour les applications de transition entre phases d’evolution lentes et rapides. Nous montrons des resultats d’existence d’orbites periodiques non triviales, de convergence d’un continuum d’orbites vers un ensemble limite-periodique et d’apparition de « canards ».
Abstract. We investigate a model of the GnRH pulse and surge generator, with the definite aim of ... more Abstract. We investigate a model of the GnRH pulse and surge generator, with the definite aim of constraining the model GnRH output with respect to a physiologically relevant list of specifications. The alternating pulse and surge pattern of secretion results from the interaction between a GnRH secreting system and a regulating system exhibiting fast-slow dynamics. The mechanisms underlying the behavior of the model are reminded from the study of the Boundary-Layer System according to the ”dissection method ” principle. Using singular perturbation theory, we describe the sequence of bifurcations undergone by the regulating (FitzHugh-Nagumo) system, encompassing the rarely investigated case of homoclinic connexion. Basing on pure dynamical considerations, we restrict the space of parameter search for the regulating system and describe a foliation of this restricted space, whose leaves define constant duration ratios between the surge and the pulsatility phase in the whole system. We ...
Physica D: Nonlinear Phenomena
Leng/Computational Neuroendocrinology, 2016
In mammals, ovulation is a key limiting step of the reproductive success. Ovulation is controlled... more In mammals, ovulation is a key limiting step of the reproductive success. Ovulation is controlled centrally, on the hypothalamus level, and triggered by the so-called “GnRH surge”, which corresponds to a dramatic and quite sudden increase in GnRH (gonadotropin-releasing hormone) secretion occurring in response to elevated levels of estradiol secreted by the ovaries. The GnRH neurosecretory system, often referred to as “the GnRH pulse generator”, involves both the GnRH-secreting neurons and associated brain cells responsible for detecting and relaying the steroid signals to GnRH neurons. The dynamical pattern of GnRH secretion is not easily accessible to experimental procedures, but the advent of a surgical technique enabling direct withdrawal of hypothalamo- pituitary portal blood to assess GnRH levels in different steroid environments has fostered our knowledge of GnRH secretion pattern, especially at the transition from the usual pulsatile regime to the surge regime at the end of the follicular phase of the ovarian cycle. Here, we present a mathematical model (Vidal and Clément 2010) that proposes a single dynamical framework for both the surge and pulse regime of GnRH secretion, and that accounts for the qualitative (i.e. the right sequence of secretory events) and quantitative (i.e. the frequency, duration, amplitude of secretory events) specifications drawn from experimental studies, which amounts to embedding time- and dose-dependent steroid control within the model.
arXiv: Dynamical Systems, Dec 4, 2007
Mathematical Modelling of Natural Phenomena
In [16], the authors analyzed the synchronization features between two identical 3D slow-fast osc... more In [16], the authors analyzed the synchronization features between two identical 3D slow-fast oscillators, symmetrically coupled, built as an extension of the FitzHugh–Nagumo dynamics generating Mixed-Mode Oscillations. The third variable, which is slow, represents the intracellular calcium concentration in neurons. Here, we consider an extension of this model in two directions. First, we consider heterogeneity among cells and analyze the coupling of two oscillators with different values for one parameter tuning the intrinsic frequency. We identify new patterns of antiphasic synchronization, with non-trivial signatures and that exhibit a Devil’s Staircase phenomenon in transitions. Second, we introduce a network of N cells divided into two clusters: the coupling between neurons in each cluster is excitatory, while between the two clusters is inhibitory. Such system models the interactions between neurons tending to synchronization in two subpopulations inhibiting each other, like ip...
In this work we study mixed mode oscillations in a model of secretion of GnRH (Gonadotropin Relea... more In this work we study 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 nullcline 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 nullcline through a fold point of the fast nullcline. We derive a variety of reductions, taking advantage of the mentioned features of the system. We obtain 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...
Abstract. The biological models- particularly the ecological ones- must be understood through the... more Abstract. The biological models- particularly the ecological ones- must be understood through the bifurcations they undergo as the parameters vary. However, the transition between two dynamical behaviours of a same system for diverse values of parameters may be sometimes quite involved. For instance, the analysis of the non generic motions near the transition states is the first step to understand fully the bifurcations occurring in complex dynamics. In this article, we address the question to describe and explain a double bursting behaviour occuring for a tritrophic slow–fast system. We focus therefore on the appearance of a double homoclinic bifurcation of the fast subsystem as the predator death rate parameter evolves. The first part of this article introduces the slow–fast system which extends Lotka–Volterra dynamics by adding a superpredator. The second part displays the analysis of singular points and bifurcations undergone by fast dynamics. The third part is devoted to the fl...
Bulletin of the Belgian Mathematical Society - Simon Stevin, 2008
SIAM Journal on Applied Dynamical Systems, 2018
Discrete and Continuous Dynamical Systems - Series B, 2017
Journal of mathematical neuroscience, 2016
Recent experimental evidence on the clustering of glutamate and GABA transporters on astrocytic p... more Recent experimental evidence on the clustering of glutamate and GABA transporters on astrocytic processes surrounding synaptic terminals pose the question of the functional relevance of the astrocytes in the regulation of neural activity. In this perspective, we introduce a new computational model that embeds recent findings on neuron-astrocyte coupling at the mesoscopic scale intra- and inter-layer local neural circuits. The model consists of a mass model for the neural compartment and an astrocyte compartment which controls dynamics of extracellular glutamate and GABA concentrations. By arguments based on bifurcation theory, we use the model to study the impact of deficiency of astrocytic glutamate and GABA uptakes on neural activity. While deficient astrocytic GABA uptake naturally results in increased neuronal inhibition, which in turn results in a decreased neuronal firing, deficient glutamate uptake by astrocytes may either decrease or increase neuronal firing either transient...
We investigate a model of the GnRH pulse and surge generator, with the definite aim of constraini... more We investigate a model of the GnRH pulse and surge generator, with the definite aim of constraining the model GnRH output with respect to a physiologically relevant list of specifications. The alternating pulse and surge pattern of secretion results from the interaction between a GnRH secreting system and a regulating system exhibiting fast-slow dynamics. The mechanisms underlying the behavior of the model are reminded from the study of the Boundary-Layer System according to the "dissection method" principle. Using singular perturbation theory, we describe the sequence of bifurcations undergone by the regulating (FitzHugh-Nagumo) system, encompassing the rarely investigated case of homoclinic connexion. Basing on pure dynamical considerations, we restrict the space of parameter search for the regulating system and describe a foliation of this restricted space, whose leaves define constant duration ratios between the surge and the pulsatility phase in the whole system. We p...
Fast-slow systems are studied usually by “geometrical dissection” [4]. The fast dynamics exhibit ... more Fast-slow systems are studied usually by “geometrical dissection” [4]. The fast dynamics exhibit attractors which may bifurcate under the influence of the slow dynamics which is seen as a parameter of the fast dynamics. A generic solution comes close to a connected component of the stable invariant sets of the fast dynamics. As the slow dynamics evolves, this attractor may lose its stability and the solution eventually reaches quickly another connected component of attractors of the fast dynamics and the process may repeat. This scenario explains quite well relaxation oscillations and more complicated oscillations like bursting. More recently, in relation both with theory of dynamical systems [11] and with applications to physiology [10, 26], a new interest has emerged in canard cycles. These orbits share the property that they remain for a while close to an unstable invariant set (either singular set or periodic orbits of the fast dynamics). Although canards were first discovered w...
INRA-CNRS-Université de Tours-Haras Nationaux, UMR Physiologie de la Reproduction et des Comporte... more INRA-CNRS-Université de Tours-Haras Nationaux, UMR Physiologie de la Reproduction et des Comportements. Abstract. In sheep as in many vertebrates, the seasonal pattern of reproduction is timed by the annual photoperiodic cycle, characterized by seasonal changes in the daylength. The photoperiodic information is translated into a circadian profile of melatonin secretion. After multiple neuronal relays (within the hypothalamus), melatonin impacts GnRH (gonadotrophin releasing hormone) secretion that in turn controls ovarian cyclicity. The pattern of GnRH secretion is mirrored into that of LH (luteinizing hormone) secretion, whose plasmatic level can be easily measured. We addressed the question of whether there exists an endogenous circannual rhythm in a tropical sheep (Black-belly) population that exhibits clear seasonal ovarian activity when ewes are subjected to temperate latitudes. We based our analysis on LH time series collected in the course of 3 years from ewes subjected to a ...
arXiv: Dynamical Systems, 2015
We exhibit and investigate a new type of mechanism for generating complex oscillations featuring ... more We exhibit and investigate a new type of mechanism for generating complex oscillations featuring an alternation of small oscillations with spikes (MMOs) or bursts (MMBOs) in a class of hybrid dynamical systems modeling neuronal activity. These dynamical systems, called nonlinear adaptive integrate-and-fire neurons, combine nonlinear dynamics modeling input integration in a nerve cell with discrete resets modeling the emission of an action potential and the subsequent return to reversal potential. We show that presence of complex oscillations in these models relies on a fundamentally hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts. The decomposition into these two mechanisms leads us to propose a purely geometrical interpretation of these complex trajectories, and this relative simplicity allows to finely characterize the MMO patterns through the study of iterates of ...
ABSTRACT Neural mass modeling is a field of computational neuroscience that aims at studying the ... more ABSTRACT Neural mass modeling is a field of computational neuroscience that aims at studying the activity of neuronal populations without explicit representation of single neurons. This type of mesoscopic model is able to generate output signals that can be compared with experimental data such as stereo-electroencephalograms. Classically, neural mass models consider two interconnected populations: excitatory pyramidal cells and inhibitory interneurons. Regarding the excitatory feedbacks on the pyramidal cell population, two distinct approaches have been proposed. A “direct feedback” on the main pyramidal cell population or an “indirect feedback” via a secondary pyramidal cell population. In this article, we propose a new neural mass model that couples both these approaches. We analyze the model bifurcations in two specific cases and describe the corresponding time series. We then explain the typical features of experimental records in epileptic mice. Finally, we show that the model is able to reproduce two different regimes identified in experimental data. Our study also reveals the similarity in the proper 4Hz frequency of epileptic discharges in experimental data and generated time series.
Les oscillations en salves apparaissent dans de nombreux systemes biologiques, physiologiques et ... more Les oscillations en salves apparaissent dans de nombreux systemes biologiques, physiologiques et ecologiques. Elles se caracterisent par l’alternance de phases dites silencieuses ou quiescentes separees par des phases dites actives ou pulsatiles. Cette these est consacree a l'analyse mathematique des systemes dynamiques lents-rapides proposes pour la modelisation des oscillations en salves. Grâce a la theorie des bifurcations, la theorie des perturbations singulieres et l’utilisation d’eclatements a parametres, nous caracterisons les differents comportements de tels systemes. En particulier, nous utilisons des developpements asymptotiques pour les applications de transition entre phases d’evolution lentes et rapides. Nous montrons des resultats d’existence d’orbites periodiques non triviales, de convergence d’un continuum d’orbites vers un ensemble limite-periodique et d’apparition de « canards ».
Abstract. We investigate a model of the GnRH pulse and surge generator, with the definite aim of ... more Abstract. We investigate a model of the GnRH pulse and surge generator, with the definite aim of constraining the model GnRH output with respect to a physiologically relevant list of specifications. The alternating pulse and surge pattern of secretion results from the interaction between a GnRH secreting system and a regulating system exhibiting fast-slow dynamics. The mechanisms underlying the behavior of the model are reminded from the study of the Boundary-Layer System according to the ”dissection method ” principle. Using singular perturbation theory, we describe the sequence of bifurcations undergone by the regulating (FitzHugh-Nagumo) system, encompassing the rarely investigated case of homoclinic connexion. Basing on pure dynamical considerations, we restrict the space of parameter search for the regulating system and describe a foliation of this restricted space, whose leaves define constant duration ratios between the surge and the pulsatility phase in the whole system. We ...
Physica D: Nonlinear Phenomena
Leng/Computational Neuroendocrinology, 2016
In mammals, ovulation is a key limiting step of the reproductive success. Ovulation is controlled... more In mammals, ovulation is a key limiting step of the reproductive success. Ovulation is controlled centrally, on the hypothalamus level, and triggered by the so-called “GnRH surge”, which corresponds to a dramatic and quite sudden increase in GnRH (gonadotropin-releasing hormone) secretion occurring in response to elevated levels of estradiol secreted by the ovaries. The GnRH neurosecretory system, often referred to as “the GnRH pulse generator”, involves both the GnRH-secreting neurons and associated brain cells responsible for detecting and relaying the steroid signals to GnRH neurons. The dynamical pattern of GnRH secretion is not easily accessible to experimental procedures, but the advent of a surgical technique enabling direct withdrawal of hypothalamo- pituitary portal blood to assess GnRH levels in different steroid environments has fostered our knowledge of GnRH secretion pattern, especially at the transition from the usual pulsatile regime to the surge regime at the end of the follicular phase of the ovarian cycle. Here, we present a mathematical model (Vidal and Clément 2010) that proposes a single dynamical framework for both the surge and pulse regime of GnRH secretion, and that accounts for the qualitative (i.e. the right sequence of secretory events) and quantitative (i.e. the frequency, duration, amplitude of secretory events) specifications drawn from experimental studies, which amounts to embedding time- and dose-dependent steroid control within the model.
Neural mass modeling is a field of computational neu-roscience that aims at studying the activity... more Neural mass modeling is a field of computational neu-roscience that aims at studying the activity of neuronal populations without explicit representation of single neurons. This type of meso-scopic model is able to generate output signals that can be compared with experimental data such as stereo-electroencephalograms. Classically, neural mass models consider two interconnected populations: excitatory pyramidal cells and inhibitory interneurons. Regarding the excitatory feedbacks on the pyramidal cell population, two distinct approaches have been proposed. A " direct feedback " on the main pyramidal cell population or an " indirect feedback " via a secondary pyramidal cell population. In this article, we propose a new neural mass model that couples both these approaches. We analyze the model bifurcations in two specific cases and describe the corresponding time series. We then explain the typical features of experimental records in epileptic mice. Finally, we show that the model is able to reproduce two di↵erent regimes identified in experimental data. Our study also reveals the similarity in the proper 4Hz frequency of epileptic discharges in experimental data and generated time series.
In mammals, ovulation is a key limiting step of the reproductive success. Ovulation is controlled... more In mammals, ovulation is a key limiting step of the reproductive success. Ovulation is controlled centrally, on the hypothalamus level, and triggered by the so-called “GnRH surge”, which corresponds to a dramatic and quite sudden increase in GnRH (gonadotropin-releasing hormone) secretion occurring in response to elevated levels of estradiol secreted by the ovaries. The GnRH neurosecretory system, often referred to as “the GnRH pulse generator”, involves both the GnRH-secreting neurons and associated brain cells responsible for detecting and relaying the steroid signals to GnRH neurons. The dynamical pattern of GnRH secretion is not easily accessible to experimental procedures, but the advent of a surgical technique enabling direct withdrawal of hypothalamo- pituitary portal blood to assess GnRH levels in different steroid environments has fostered our knowledge of GnRH secretion pattern, especially at the transition from the usual pulsatile regime to the surge regime at the end of the follicular phase of the ovarian cycle. Here, we present a mathematical model (Vidal and Clément 2010) that proposes a single dynamical framework for both the surge and pulse regime of GnRH secretion, and that accounts for the qualitative (i.e. the right sequence of secretory events) and quantitative (i.e. the frequency, duration, amplitude of secretory events) specifications drawn from experimental studies, which amounts to embedding time- and dose-dependent steroid control within the model.
HDR Thesis (French habilitation to lead researches) in Applied Mathematics, 2016
HDR thesis (French habilitation to lead researches) in Applied Mathematics
Ph.D. Thesis (Mathematics), 2007