The success of filamentous fungi in colonizing most natural environments can be largely attributed to their ability to form an expanding interconnected network, the mycelium, or thallus, constituted by a collection of hyphal apexes in motion producing hyphae and subject to branching and fusion. In this work, we characterize the hyphal network expansion and the structure of the fungus Podospora anserina under controlled culture conditions. To this end, temporal series of pictures of the network dynamics are produced, starting from germinating ascospores and ending when the network reaches a few centimeters width, with a typical image resolution of several micrometers. The completely automated image reconstruction steps allow an easy post-processing and a quantitative analysis of the dynamics. The main features of the evolution of the hyphal network, such as the total length L of the mycelium, the number of "nodes" (or crossing points) N and the number of apexes A, can then be precisely quantified. Beyond these main features, the determination of the distribution of the intrathallus surfaces (S i ) and the statistical analysis of some local measures of N, A and L give new insights on the dynamics of expanding fungal networks. Based on these results, we now aim at developing robust and versatile discrete/continuous mathematical models to further understand the key mechanisms driving the development of the fungus thallus.Due to their paramount role in most ecosystems and in different sectors of human economy, filamentous fungi have attracted a lot of attention in the last decades. Indeed, these organisms are able to grow in complex environments, leading to their ubiquitous occurrence 1,2 . The success of filamentous fungi in colonizing most natural environments can be largely attributed to their ability to form hyphae, whose width typically ranges from 2 to 20 µm, and which are usually composed of multiple cells separated by septa 3 . At the edge of the fungal colony, apical cells explore new territories in search for food and are mainly involved in nutrient acquisition through the production of extracellular enzymes for the decomposition of organic matter, as well as through the sensing of their local environment 4 . Sub-apical cells generate new hyphae by lateral branching, thus increasing the density and surface area of the colony. In ascomycetes and basidiomycetes, the branches also mediate the hyphal fusion events, or anastomosis, that occur in the thallus and appear to be crucial for the exchange of nutrients and signals

Abstract. We study the nonparametric estimation of the branching rate B(x) of a supercritical Bellman-Harris population: a particle with age x has a random lifetime governed by B(x); at its death time, it gives rise to k ≥ 2 children with lifetimes governed by the same division rate and so on. We observe in continuous time the process over [0, T ]. Asymptotics are taken as T → ∞; the data are stochastically dependent and one has to face simultaneously censoring, bias selection and non-ancillarity of the number of observations. In this setting, under appropriate ergodicity properties, we construct a kernel-based estimator of B(x) that achieves the rate of convergence exp(−λ B β 2β+1T ), where λ B is the Malthus parameter and β > 0 is the smoothness of the function B(x) in a vicinity of x. We prove that this rate is optimal in a minimax sense and we relate it explicitly to classical nonparametric models such as density estimation observed on an appropriate (parameter dependent) scale. We also shed some light on the fact that estimation with kernel estimators based on data alive at time T only is not sufficient to obtain optimal rates of convergence, a phenomenon which is specific to nonparametric estimation and that has been observed in other related growth-fragmentation models.Mathematics Subject Classification (2010): 35A05, 35B40, 45C05, 45K05, 82D60, 92D25, 62G05, 62G20.