It has been noticed in recent years that the unfavorable effects of the gut microbiota could exhaust host vigor and life, yet knowledge and theory are just beginning to be established. Increasing documentation suggests that the microbiota–gut–brain axis not only impacts brain cognition and psychiatric symptoms but also precipitates neurodegenerative diseases, such as Alzheimer’s disease (AD), Parkinson’s disease (PD), and multiple sclerosis (MS). How the blood–brain barrier (BBB), a machinery protecting the central nervous system (CNS) from the systemic circulation, allows the risky factors derived from the gut to be translocated into the brain seems paradoxical. For the unique anatomical, histological, and immunological properties underpinning its permeable dynamics, the BBB has been regarded as a biomarker associated with neural pathogenesis. The BBB permeability of mice and rats caused by GM dysbiosis raises the question of how the GM and its metabolites change BBB permeability and causes the brain pathophysiology of neuroinflammation and neurodegeneration (NF&ND) and brain aging, a pivotal multidisciplinary field tightly associated with immune and chronic systemic inflammation. If not all, gut microbiota-induced systemic chronic inflammation (GM-SCI) mainly refers to excessive gut inflammation caused by gut mucosal immunity dysregulation, which is often influenced by dietary components and age, is produced at the interface of the intestinal barrier (IB) or exacerbated after IB disruption, initiates various common chronic diseases along its dispersal routes, and eventually impairs BBB integrity to cause NF&ND and brain aging. To illustrate the immune roles of the BBB in pathophysiology affected by inflammatory or “leaky” IB resulting from GM and their metabolites, we reviewed the selected publications, including the role of the BBB as the immune barrier, systemic chronic inflammation and inflammation influences on BBB permeability, NF&ND, and brain aging. To add depth to the bridging role of systemic chronic inflammation, a plausible mechanism indispensable for BBB corruption was highlighted; namely, BBB maintenance cues are affected by inflammatory cytokines, which may help to understand how GM and its metabolites play a major role in NF&ND and aging.

Bivariate splines with various degrees are considered in this paper. A matrix form of the extended smoothness conditions for these splines is presented. Upon this form, the multivariate spline method for numerical solution of partial differential equations (PDEs) proposed by Awanou, Lai, and Wenston in [The multivariate spline method for scattered data fitting and numerical solutions of partial differential equations, in Wavelets and Splines, G. Chen and M. J. Lai, eds., Nashboro Press, Brentwood, TN, 2006, pp. 24-76] is generalized to obtain a new spline method. It is observed that, combined with prelocal refinement of triangulation and automatic degree raising over triangles of interest, the new spline method of bivariate splines of various degrees is able to solve linear PDEs very effectively and efficiently.

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We experimentally demonstrate two types of programmable, low-threshold, optically controlled nonlinear activation functions, which are challenging to realize in photonic neural networks (PNNs). These devices rely on on-chip integrated Ge–Si photoelectric detectors and silicon electro-optical switches, and they generate rectified linear unit (ReLU) or sigmoid functions with arbitrary slopes without additional electrical processing. Both devices function at an extremely low threshold of 0.2 mW. The embedding of these nonlinear activation functions into convolutional neural networks facilitates the attainment of high inference accuracies of up to 95% when applied to Modified National Institute of Standards and Technology (MNIST) handwritten digit-classification tasks. The devices are suitable for low-power PNNs with an arbitrary number of propagation layers in photonic-computing chips.