In this paper we provide stability results for algebraic neural networks (AlgNNs) based on non commutative algebras. AlgNNs are stacked layered structures with each layer associated to an algebraic signal model (ASM) determined by an algebra, a vector space, and a homomorphism. Signals are modeled as elements of the vector space, filters are elements in the algebra, while the homomorphism provides a realization of the filters as concrete operators. We study the stability of the algebraic filters in non commutative algebras to perturbations on the homomorphisms, and we provide conditions under which stability is guaranteed. We show that the commutativity between shift operators and between shifts and perturbations does not affect the property of an architecture of being stable. This provides an answer to the question of whether shift invariance was a necessary attribute of convolutional architectures to guarantee stability. Additionally, we show that although the frequency responses of filters in non commutative algebras exhibit substantial differences with respect to filters in commutative algebras, their derivatives for stable filters have a similar behavior.

Glia cells provide supportive functions to the central nervous system and can be compromised by environmental contaminants. The primary objective of this study was to characterize the effects of in vitro exposure to perfluorooctanoic acid, a persistent environmental contaminant and/or monocrotophos (MCP), a neurotoxic organophosphate that is rapidly metabolized, to astroglia SVG p12 cells. The endpoints evaluated include cell viability, intracellular glutamate levels as a marker of astrocyte homeostasis function, differential gene expression for selected proteins, which include inflammatory markers (tachykinin), astrocytosis (nestin), S100B, and metabolism enzymes (CYP1A1). The results from cell viability revealed significant differences from the controls at some of the concentrations tested. Also, intracellular glutamate levels were elevated at the 10‐μM concentration for perfluorooctanoic acid (PFOA) as well as the 10‐μM PFOA/5‐μM MCP concentration. Gene expression results at 80‐μM PFOA concentration revealed a significant increase in the expression of S100B, tachykinin and CYP1A1. A combination of 10‐μM PFOA/20‐μM MCP caused a significant decrease in the expression of tachykinin. Gene expression for MCP exposures produced a decrease at the 20‐μM MCP concentration. Immunofluorescence results indicated an increase in nestin protein expression for the 20‐μM concentration of MCP, which contradicted the gene expression at the same concentration tested. The results indicate that toxicity to glia cells can compromise critical glia functions and could be implicated in neurodegenerative diseases.

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