Regulatory dynamics has mathematical descriptions in terms of rate equations for continuous variables and, after discretization of state space and time , as Boolean maps. Here we study the effects of discretization for the notion of stability. In particular, we define the stability of a Boolean state sequence in consistency with the stability  of the original continuous trajectory that has been discretized. For a class of randomly connected systems with randomly drawn Boolean functions, so-called Kauffman networks, we find that the dynamics is stable for almost all choices of parameter values. The so-called chaotic regime in Kauffman networks appears only as a damage spreading effect after flip perturbations . We conclude that regulatory systems amenable to state discretization do not exhibit chaotic behaviour. Both resilience to noise and sensitivity to input signals come "for free" as typical properties of sufficiently dense regulatory networks.
Regulatory dynamics in systems biology is often described by continuous rate equations for continuously varying chemical concentrations. Binary discretization of state space and time leads to Boolean dynamics. In the latter, the distinction between stable and unstable dynamics is usually made by checking damage after flip perturbations. Here we point out that this notion of stability is incompatible with the stability properties of the original continuous dynamics probed by small perturbations. In particular, random networks of nodes with large sensitivity yield stable dynamics, in contrast to the prediction of chaos obtained by flip perturbations in Boolean networks.
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has mathematical descriptions in terms of
in systems biology is often described by continuous
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continuous variables and, after
continuously varying chemical concentrations. Binary
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, as Boolean maps. Here we study the effects of discretization for the notion of stability. In particular, we define the stability of a Boolean state sequence in consistency
leads to Boolean dynamics. In the latter, the distinction between stable and unstable dynamics is usually made by checking damage after flip perturbations. Here we point out that this notion of stability is incompatible
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properties
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trajectory that has been discretized. For a class of randomly connected systems with randomly drawn Boolean functions, so-called Kauffman networks, we find that the dynamics is stable for almost all choices of parameter values. The so-called chaotic regime in Kauffman networks appears only as a damage spreading effect after flip perturbations . We conclude that regulatory systems amenable to state discretization do not exhibit chaotic behaviour. Both resilience to noise and sensitivity to input signals come "for free" as typical properties of sufficiently dense regulatory
dynamics probed by small perturbations. In particular, random networks of nodes with large sensitivity yield stable dynamics, in contrast to the prediction of chaos obtained by flip perturbations in Boolean