Summary Circuits distributed across cortico-limbic brain regions compose the networks that mediate emotional behavior. The prefrontal cortex (PFC) regulates ultraslow (<1Hz) dynamics across these networks, and PFC dysfunction is implicated in stress-related illnesses including major depressive disorder (MDD). To uncover the mechanism whereby stress-induced changes in PFC circuitry alter emotional networks to yield pathology, we used a multi-disciplinary approach including in vivo recordings in mice and chronic social-defeat stress. Our network model, inferred using machine learning, linked stress-induced behavioral pathology to the capacity of PFC to synchronize amygdala and VTA activity. Direct stimulation of PFC-amygdala circuitry with DREADDs normalized PFC-dependent limbic synchrony in stress-susceptible animals and restored normal behavior. In addition to providing insights into MDD mechanisms, our findings demonstrate an interdisciplinary approach that can be used to identify the large-scale network changes that underlie complex emotional pathologies and the specific network nodes that can be used to develop targeted interventions.

Shape constrained regression analysis has applications in dose-response modeling, environmental risk assessment, disease screening and many other areas. Incorporating the shape constraints can improve estimation efficiency and avoid implausible results. We propose two novel methods focusing on Bayesian monotone curve and surface estimation using Gaussian process projections. The first projects samples from an unconstrained prior, while the second projects samples from the Gaussian process posterior. Theory is developed on continuity of the projection, posterior consistency and rates of contraction. The second approach is shown to have an empirical Bayes justification and to lead to simple computation with good performance in finite samples. Our projection approach can be applied in other constrained function estimation problems including in multivariate settings.

Two central limit theorems for sample Fréchet means are derived, both significant for nonparametric inference on non-Euclidean spaces. The first one, Theorem 2.2, encompasses and improves upon most earlier CLTs on Fréchet means and broadens the scope of the methodology beyond manifolds to diverse new non-Euclidean data including those on certain stratified spaces which are important in the study of phylogenetic trees. It does not require that the underlying distribution Q have a density, and applies to both intrinsic and extrinsic analysis. The second theorem, Theorem 3.3, focuses on intrinsic means on Riemannian manifolds of dimensions d > 2 and breaks new ground by providing a broad CLT without any of the earlier restrictive support assumptions. It makes the statistically reasonable assumption of a somewhat smooth density of Q. The excluded case of dimension d = 2 proves to be an enigma, although the first theorem does provide a CLT in this case as well under a support restriction. Theorem 3.3 immediately applies to spheres S d , d > 2, which are also of considerable importance in applications to axial spaces and to landmarks based image analysis, as these spaces are quotients of spheres under a Lie group G of isometries of S d . RABI BHATTACHARYA AND LIZHEN LINif the distance ρ is the Euclidean distance inherited by the embedding J of a ddimensional manifold M in an Euclidean space E N , such that J(M ) is closed. Indeed, under the relabeling of M by J(M ), the Fréchet mean set in this case is given byThus the minimizer is unique if and only if the projection of the Euclidean mean on the image J(M ) of M is unique, in which case it is called an extrinsic mean. On the other hand, if ρ g is the geodesic distance on a Riemannian manifold M with metric tensor g having positive sectional curvature (in some region of M ), then conditions for uniqueness are known only for Q with support in a relatively small geodesic ball [1,30,31], which is too restrictive an assumption from the point of view of statistical applications. If the Fréchet mean exists under ρ g it is called the intrinsic mean. A complete characterization of uniqueness of (1.1) for ρ = ρ g on the circle S 1 for probabilities Q with a continuous density ([12], [10]) indicates that the intrinsic mean exists broadly, without any support restrictions, if Q has a smooth density. An important question that arises in the use of Fréchet means in nonparametric statistics is the choice of the distance ρ on M . There are in general uncountably many embeddings J and metric tensors g on a manifold M . For intrinsic analysis there are often natural choices for the metric tensor g. A good choice for extrinsic analysis is to find an embedding J: M → E N with J(M ) closed, which is equivariant under a large Lie group G of actions on M . This means that there is a homomorphism g → Φ g on G into the general linear group GL(N, R) such that J • g = Φ g • J ∀g ∈ G. Such embeddings and extrinsic means under them have been derived for Kendall type shape spaces in [14], [15], [4...

When patients transition from PD-NCI to PD-MCI, there appears to be an increase in functional connectivity in the PCC, suggesting an expansion of the cortical network. Another new network (a compensatory prefrontal cortical-cerebellar loop) later develops during the transition from PD-MCI to PDD.