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We investigate the Sandpile Model and Chip Firing Game and an extension of these models on cycle graphs. The extended model consists of allowing a negative number of chips at each vertex. We give the characterization of reachable configurations and of fixed points of each model. At the end, we give explicit formula for the number of their fixed points. Mathematics Subject Classification. 05C57, 91A43, 68R15, 68Rxx. Wiesenfied  to study a popular phenomena in physics called Self-Organizeddoi:10.1051/ita/2012023 fatcat:7prnflccgre2rnqv6buuaycoty