Coupled cell networks: Boolean perspective

During the 1980s and early 1990s, Martin Golubitsky and Ian Stewart formulated and developed a theory of "coupled cell networks" (CCNs). Their research was primarily focused onquadrupeds' gaits and they applied the framework of differential equations. Golubitsky and Stewart were particularly interested in change of synchrony between $4$ legs of an animal. For example what happens when the animal speeds up from walk to gallop. The most important concept of their theory is a {\it cell}. The cell captures the dynamics of one unit and a dynamical system consists of many identical (governed by the same principles) cells influencing (coupling to) each other. Models based on identical cooperating units are fairly common in many areas, especially in biology, ecology and sociology. The goal of investigation in Coupled Cell Networks theory is understanding the dependencies and interplay between dynamics of an individual cell, graph of connections between cells, and the nature of couplings. \vspace*{0.2em}In this paper, I redefine Coupled Cell Networks using framework of Boolean functions. This moves the entire theory to a new setting. Some phenomena proved to be very similar as for continuous networks and some are completely different. Also, for discrete networks we ask questions differently and study different phenomena. The paper presents two examples: networks that bring 2-cell bidirectional ring as a quotient and networks that bring 3-cell bidirectional ring as a quotient.

Coupled cell networks of astrocytes and chondrocytes are target cells of inflammation

AimsSystemic low-grade inflammation can be initiated in vivo after traumatic injury or in chronic diseases as neurodegenerative, metabolic and autoimmune diseases. Coupled cell networks are target cells leading to the spread of inflammation and changes in biochemical cellular parameters. Do astrocytes and chondrocytes behave in a similar way in an inflammatory reactive state with respect to Ca2+ signaling, actin filaments rearrangement, receptor properties, pro-inflammatory cytokine release etc?MethodsPrimary cultures of astrocytes and chondrocytes, respectively, were incubated with lipopolysaccharide (LPS) (10 ng/ml, 24h) or interleukin-1β (IL-1β) (5ng/ml, 24 h) to induce inflammatory reactivity. Ca2+ signaling, Na+/K+-ATPase-, connexin 43 (Cx43)-, and Toll-like receptor 4 (TLR4)- expressions, actin filament organization, and IL-1β release were analyzed.ResultsStimulation with IL-1β or LPS altered the Ca2+ signaling from single peaks to oscillating waves and increased the expression of Cx43 and TLR4, and decreased expression of Na+/K+-ATPase. A disruption of the actin filaments with more pronounced ring-formed structures was found in inflammatory induced astrocytes and chondrocytes which in turn affects Ca2+ oscillations. Additionally a release of active matrix metallopeptidase-13 was found in media from IL-1β stimulated chondrocytes.ConclusionsOur data show that cellular mechanisms of healthy chondrocytes as well as inflamed, resemble the coupled cell networks of astrocytes. Chronic, low-grade inflammation can influence coupled cell networks in one or several organs, leading to co-morbidity. It is crucial that inflammatory affected cells in various organs are restored back to a physiological non-inflammatory homeostasis in order to prevent tissue degradation and pain.