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The biologist Ren\'e Thomas conjectured, twenty years ago, that the presenceof a negative feedback circuit in the interaction graph of a dynamical systemis a necessary condition for this system to produce sustained oscillations. Inthis paper, we state and prove this conjecture for asynchronous automatanetworks, a class of discrete dynamical systems extensively used to model thebehaviors of gene networks. As a corollary, we obtain the following fixed pointtheorem: given a product $X$ of $n$ finite intervals of integers, and a map $F$from $X$ to itself, if the interaction graph associated with $F$ has nonegative circuit, then $F$ has at least one fixed point.

Negative circuits and sustained oscillations in asynchronous automata networks