Border collision bifurcations of stroboscopic maps in periodically driven spiking models

In this work we consider a general non-autonomous hybrid system based on the integrate-and-fire model, widely used as simplified version of neuronal models and other types of excitable systems. Our unique assumption is that the system is monotonic, possesses an attracting subthreshold equilibrium point and is forced by means of periodic pulsatile (square wave) function.\\ In contrast to classical methods, in our approach we use the stroboscopic map (time-$T$ return map) instead of the so-called firing-map. It becomes a discontinuous map potentially defined in an infinite number of partitions. By applying theory for piecewise-smooth systems, we avoid relying on particular computations and we develop a novel approach that can be easily extended to systems with other topologies (expansive dynamics) and higher dimensions.\\ More precisely, we rigorously study the bifurcation structure in the two-dimensional parameter space formed by the amplitude and the duty cycle of the pulse. We show that it is covered by regions of existence of periodic orbits given by period adding structures. They do not only completely describe all the possible spiking asymptotic dynamics but also the behavior of the firing rate, which is a devil's staircase as a function of the parameters

Optimal prefix codes for pairs of geometrically-distributed random variables

Optimal prefix codes are studied for pairs of independent, integer-valued symbols emitted by a source with a geometric probability distribution of parameter $q$, $0{<}q{<}1$. By encoding pairs of symbols, it is possible to reduce the redundancy penalty of symbol-by-symbol encoding, while preserving the simplicity of the encoding and decoding procedures typical of Golomb codes and their variants. It is shown that optimal codes for these so-called two-dimensional geometric distributions are \emph{singular}, in the sense that a prefix code that is optimal for one value of the parameter $q$ cannot be optimal for any other value of $q$. This is in sharp contrast to the one-dimensional case, where codes are optimal for positive-length intervals of the parameter $q$. Thus, in the two-dimensional case, it is infeasible to give a compact characterization of optimal codes for all values of the parameter $q$, as was done in the one-dimensional case. Instead, optimal codes are characterized for a discrete sequence of values of $q$ that provide good coverage of the unit interval. Specifically, optimal prefix codes are described for $q=2^{-1/k}$ ($k\ge 1$), covering the range $q\ge 1/2$, and $q=2^{-k}$ ($k>1$), covering the range $q<1/2$. The described codes produce the expected reduction in redundancy with respect to the one-dimensional case, while maintaining low complexity coding operations.Comment: To appear in IEEE Transactions on Information Theor

Bifurcation-based parameter tuning in a model of the GnRH pulse and surge generator

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 propose an algorithm to fix the parameter values to also meet the other prescribed ratios dealing with amplitude and frequency features of the secretion signal. We finally apply these results to illustrate the dynamics of GnRH secretion in the ovine species and the rhesus monkey

Modélisation biomédicale : des neurosciences à la neuroendocrinologie

National audienceLes neurones sont connus pour leur rôle dans le traitement de l'information via des signaux électriques. On sait moins que certains d'entre eux sécrètent des signaux hormonaux et agissent à distance sur leur cellules cibles : un chapitre des neurosciences encore peu exploré

The Standard Factorization of Lyndon Words: an Average Point of View

International audienceA non-empty word w is a Lyndon word if and only if it is strictly smaller for the lexicographical order than any of its proper suffixes. Such a word w is either a letter or admits a standard factorization uv where v is its smallest proper suffix. For any Lyndon word v, we show that the set of Lyndon words having v as right factor of the standard factorization is regular and compute explicitly the associated generating function. Next, considering the Lyndon words of length n over a twoletter alphabet, we establish that, for the uniform distribution, the average length of the right factor v of the standard factorization is asymptotically 3n/4

Multiscale mathematical modeling of the hypothalamo-pituitary-gonadal axis

International audienceAlthough the fields of systems and integrative biology are in full expansion, few teams are involved worldwide into the study of reproductive function from the mathematical modeling viewpoint. This may be due to the fact that the reproductive function is not compulsory for individual organism survival, even if it is for species survival. Alternatively, the complexity of reproductive physiology may be discouraging. Indeed, the hypothalamo-pituitary-gonadal (HPG) axis involves not only several organs and tissues, but also intricate time (from the neuronal millisecond timescale to circannual rhythmicity) and space (from molecules to organs) scales. Yet, mathematical modeling, and especially multiscale modeling, can renew our approaches of the molecular, cellular and physiological processes underlying the control of reproductive functions. In turn, the remarkable dynamic features exhibited by the HPG axis raise intriguing and challenging questions to modelers and applied mathematicians. In this article, we draw a panoramic review of some mathematical models designed in the framework of the female HPG, with a special focus on the gonadal and central control of follicular development. On the gonadal side, the modeling of follicular development calls to the generic formalism of structured cell populations, that allows one to make mechanistic links between the control of cell fate (proliferation, differentiation or apoptosis) and that of the follicle fate (ovulation or degeneration) or to investigate how the functional interactions between the oocyte and its surrounding cells shape the follicle morphogenesis. On the central, mainly hypothalamic side, models based on dynamical systems with multiple timescales allow one to represent within a single framework both the pulsatile and surge patterns of the neurohormone GnRH (gonadotropin-releasing hormone). Beyond their interest in basic research investigations, mathematical models can also be at the source of useful tools to study the encoding and decoding of the (neuro-)hormonal signals at play within the HPG axis and detect complex, possibly hidden rhythms, in experimental time series

Constructions for Clumps Statistics.

International audienceWe consider a component of the word statistics known as clump; starting from a finite set of words, clumps are maximal overlapping sets of these occurrences. This object has first been studied by Schbath with the aim of counting the number of occurrences of words in random texts. Later work with similar probabilistic approach used the Chen-Stein approximation for a compound Poisson distribution, where the number of clumps follows a law close to Poisson. Presently there is no combinatorial counterpart to this approach, and we fill the gap here. We also provide a construction for the yet unsolved problem of clumps of an arbitrary finite set of words. In contrast with the probabilistic approach which only provides asymptotic results, the combinatorial method provides exact results that are useful when considering short sequences

Endogenous circannual rhythm in LH secretion: insight from signal analysis coupled with mathematical modelling

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 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 constant photoperiodic regime. Due to intra- and inter- animal variability and unequal sampling times, the existence of an endogenous rhythm is not straightforward. We have used time-frequency signal processing methods to extract hidden rhythms from the data. To further investigate the LF (low frequency) and HF (high frequency) components of the signals, we have designed a mathematical model of LH plasmatic level accounting for the effect of experimental sampling times. The model enables us to confirm the existence of an endogenous circannual rhythm, to investigate the action mechanism of photoperiod on the pulsatile pattern of LH secretion (control of the interpulse interval) and to conclude that the HF component is mainly due to the experimental sampling protocol