Multifractional Poisson process, multistable subordinator and related limit theorems

We introduce a multistable subordinator, which generalizes the stable subordinator to the case of time-varying stability index. This enables us to define a multifractional Poisson process. We study properties of these processes and establish the convergence of a continuous-time random walk to the multifractional Poisson process.

Phase relationships between orbital forcing and the composition of air trapped in Antarctic ice cores

Orbital tuning is central for ice core chronologies beyond annual layer counting, available back to 60 ka (i.e. thousands of years before 1950) for Greenland ice cores. While several complementary orbital tuning tools have recently been developed using δ¹⁸Oatm, δO₂⁄N₂ and air content with different orbital targets, quantifying their uncertainties remains a challenge. Indeed, the exact processes linking variations of these parameters, measured in the air trapped in ice, to their orbital targets are not yet fully understood. Here, we provide new series of δO₂∕N₂ and δ¹⁸Oatm data encompassing Marine Isotopic Stage (MIS) 5 (between 100 and 160 ka) and the oldest part (340–800 ka) of the East Antarctic EPICA Dome C (EDC) ice core. For the first time, the measurements over MIS 5 allow an inter-comparison of δO₂∕N₂ and δ¹⁸Oatm records from three East Antarctic ice core sites (EDC, Vostok and Dome F). This comparison highlights some site-specific δO₂∕N₂ variations. Such an observation, the evidence of a 100 ka periodicity in the δO₂∕N₂ signal and the difficulty to identify extrema and mid-slopes in δO2∕N2 increase the uncertainty associated with the use of δO₂∕N₂ as an orbital tuning tool, now calculated to be 3–4 ka. When combining records of δ¹⁸Oatm and δO₂∕N₂ from Vostok and EDC, we find a loss of orbital signature for these two parameters during periods of minimum eccentricity (∼ 400 ka, ∼ 720–800 ka). Our data set reveals a time-varying offset between δO₂∕N₂ and δ¹⁸Oatm records over the last 800 ka that we interpret as variations in the lagged response of δ¹⁸Oatm to precession. The largest offsets are identified during Terminations II, MIS 8 and MIS 16, corresponding to periods of destabilization of the Northern polar ice sheets. We therefore suggest that the occurrence of Heinrich–like events influences the response of δ¹⁸Oatm to precession.

Parameter-free extraction of the functional network topology from intracranial EEG recordings: a time-resolved study of graph properties in focal onset seizures

Rationale: Focal onset epileptic seizures are due to abnormal interactions between distributed brain areas. By estimating the cross-correlation matrix of multi-site intra-cerebral EEG recordings (iEEG), one can quantify these interactions. To assess the topology of the underlying functional network, the binary connectivity matrix has to be derived from the cross-correlation matrix by use of a threshold. Classically, a unique threshold is used that constrains the topology [1]. Our method aims to set the threshold in a data-driven way by separating genuine from random cross-correlation. We compare our approach to the fixed threshold method and study the dynamics of the functional topology. Methods: We investigate the iEEG of patients suffering from focal onset seizures who underwent evaluation for the possibility of surgery. The equal-time cross-correlation matrices are evaluated using a sliding time window. We then compare 3 approaches assessing the corresponding binary networks. For each time window: * Our parameter-free method derives from the cross-correlation strength matrix (CCS)[2]. It aims at disentangling genuine from random correlations (due to finite length and varying frequency content of the signals). In practice, a threshold is evaluated for each pair of channels independently, in a data-driven way. * The fixed mean degree (FMD) uses a unique threshold on the whole connectivity matrix so as to ensure a user defined mean degree. * The varying mean degree (VMD) uses the mean degree of the CCS network to set a unique threshold for the entire connectivity matrix. * Finally, the connectivity (c), connectedness (given by k, the number of disconnected sub-networks), mean global and local efficiencies (Eg, El, resp.) are computed from FMD, CCS, VMD, and their corresponding random and lattice networks. Results: Compared to FMD and VMD, CCS networks present: *topologies that are different in terms of c, k, Eg and El. *from the pre-ictal to the ictal and then post-ictal period, topological features time courses that are more stable within a period, and more contrasted from one period to the next. For CCS, pre-ictal connectivity is low, increases to a high level during the seizure, then decreases at offset. k shows a ‘‘U-curve’’ underlining the synchronization of all electrodes during the seizure. Eg and El time courses fluctuate between the corresponding random and lattice networks values in a reproducible manner. Conclusions: The definition of a data-driven threshold provides new insights into the topology of the epileptic functional networks.