Felix Cheysson

This paper presents a parametric estimation method for ill-observed linear stationary Hawkes processes. When the exact locations of points are not observed, but only counts over time intervals of fixed size, methods based on the likelihood are not feasible. We show that spectral estimation based on Whittle's method is adapted to this case and provides consistent and asymptotically normal estimators, provided a mild moment condition on the reproduction function. Simulated datasets and a case-study illustrate the performances of the estimation, notably of the reproduction function even when time intervals are relatively large.

Neural stem cell (NSC) populations persist in the adult vertebrate brain over a life time, and their homeostasis is controlled at the population level. The nature and properties of these coordination mechanisms remain unknown. Here we combine dynamic imaging of entire NSC populations in their in vivo niche over weeks, pharmacological manipulations, mathematical modeling and spatial statistics, and demonstrate that NSCs use spatiotemporally resolved local feedbacks to coordinate their decision to divide. These involve a Notch-mediated inhibition from transient neural progenitors, and a dispersion effect from dividing NSCs themselves, exerted with a delay of 9-12 days. Simulations from a stochastic NSC lattice model capturing these interactions demonstrate that they are linked by lineage progression and control the spatiotemporal distribution of output neurons. These results highlight how local and temporally delayed interactions occurring between brain germinal cells generate self-propagating dynamics that maintain NSC population homeostasis with specific spatiotemporal correlations.

This work is motivated by the modelling of contagious diseases and the estimation of the impact of explanatory variables on their diffusion. Models of choice are the Hawkes processes, a family of stochastic processes for which the occurrence of any event increases the probability of further events occurring, allowing for elegant and straightforward modelling. When count data are only observed in discrete time, we propose a spectral approach for the estimation of stationary Hawkes processes, based on the Bartlett spectrum (i.e. the Fourier transform of the autocovariance of the process) and the Whittle likelihood. This approach yields consistent and asymptotically normal estimates for the parameters of the Hawkes process. Simulated datasets and an application to the incidence of measles in France illustrate the performances of the estimation, notably of the excitation function, even when the time between observations is large.