The stochastic block model and its variants have been a popular tool in analyzing large network data with community structures. In this paper we develop an efficient network cross-validation (NCV) approach to determine the number of communities, as well as to choose between the regular stochastic block model and the degree corrected block model. The proposed NCV method is based on a block-wise node-pair splitting technique, combined with an integrated step of community recovery using sub-blocks of the adjacency matrix. We prove that the probability of under selection vanishes as the number of node increases, under mild conditions satisfied by a wide range of popular community recovery algorithms. The solid performance of our method is also demonstrated in extensive simulations and a data example.

Parvalbumin-positive (PV) neurons control the timing of pyramidal cell output in cortical neuron networks. In the prefrontal cortex (PFC), PV neuron activity is involved in cognitive function, suggesting that PV neuron maturation is critical for cognitive development. The two major PV neuron subtypes found in the PFC, chandelier cells (ChCs) and basket cells (BCs), are thought to play different roles in cortical circuits, but the trajectories of their physiological maturation have not been compared. Using two separate mouse lines, we found that in the mature PFC, both ChCs and BCs are abundant in superficial layer 2, but only BCs are present in deeper laminar locations. This distinctive laminar distribution was observed by postnatal day 12 (P12), when we first identified ChCs by the presence of axon cartridges. Electrophysiology analysis of excitatory synapse development, starting at P12, showed that excitatory drive remains low throughout development in ChCs, but increases rapidly before puberty in BCs, with an earlier time course in deeper-layer BCs. Consistent with a role of excitatory synaptic drive in the maturation of PV neuron firing properties, the fast-spiking phenotype showed different maturation trajectories between ChCs and BCs, and between superficial versus deep-layer BCs. ChC and BC maturation was nearly completed, via different trajectories, before the onset of puberty. These findings suggest that ChC and BC maturation may contribute differentially to the emergence of cognitive function, primarily during prepubertal development. Parvalbumin-positive (PV) neurons tightly control pyramidal cell output. Thus PV neuron maturation in the prefrontal cortex (PFC) is crucial for cognitive development. However, the relative physiological maturation of the two major subtypes of PV neurons, chandelier cells (ChCs) and basket cells (BCs), has not been determined. We assessed the maturation of ChCs and BCs in different layers of the mouse PFC, and found that, from early postnatal age, ChCs and BCs differ in laminar location. Excitatory synapses and fast-spiking properties matured before the onset of puberty in both cell types, but following cell type-specific developmental trajectories. Hence, the physiological maturation of ChCs and BCs may contribute to the emergence of cognitive function differentially, and predominantly during prepubertal development.

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We introduce a new methodological framework for repeatedly observed and thus dependent functional data, aiming at situations where curves are recorded repeatedly for each subject in a sample. Our methodology covers the case where the recordings of the curves are scheduled on a regular and dense grid and also situations more typical for longitudinal studies, where the timing of recordings is often sparse and random. The proposed models lead to an interpretable and straightforward decomposition of the inherent variation in repeatedly observed functional data and are implemented through a straightforward two-step Functional Principal Component Analysis. We provide consistency results and asymptotic convergence rates for the estimated model components. We compare the proposed model with an alternative approach via a two-dimensional Karhunen-Loève expansion and illustrate it through the analysis of longitudinal mortality data from period lifetables that are repeatedly observed for a sample of countries over many years, and also through simulation studies.