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Journal Articles SIAM Journal on Applied Dynamical Systems Year : 2016

Periodic Oscillations for Non Monotonic Smooth Negative Feedback Circuits

Abstract

Negative feedback circuits are a recurrent motif in regulatory biological networks, strongly linked to the emergence of oscillatory behavior. The theoretical analysis of the existence of oscillations is a difficult problem and typically involves many constraints on the monotonicity of the activity functions. Here, we study the occurrence of periodic solutions in an n-dimensional class of nega- tive feedback systems defined by smooth vector fields with a window of not necessarily monotonic activity. Our method consists in circumscribing the smooth system by two piecewise linear ones, each admitting a periodic solution. It can then be shown that the smooth negative feedback system also has a periodic orbit, inscribed in the topological solid torus constructed from the two piecewise linear orbits. The interest of our approach lies in first, adopting a general class of functions, with a nonmonotonicity window, which permits a better fitting between theoretical models and experimen- tal data, and second, establishing a more accurate location for the periodic solution, which is useful for computational purposes in high dimensions. As an illustration, a model for the “repressilator” synthetic system is analyzed and compared to real data, and shown to admit a periodic orbit, for a range of activity functions.
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Dates and versions

hal-01242157 , version 1 (24-03-2016)

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Camille Poignard, Madalena Chaves, Jean-Luc Gouzé. Periodic Oscillations for Non Monotonic Smooth Negative Feedback Circuits. SIAM Journal on Applied Dynamical Systems, 2016, 15 (1), pp.257-286. ⟨10.1137/15M1033368⟩. ⟨hal-01242157⟩
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