Comments and Corrections on 'Stability of Genetic Regulatory Networks with Time Delay'

This paper shows that the necessary and sufficient condition for the local stability of gene regulatory networks with time delay (Theorem 2) presented in the highly cited paper [L. Chen and K. Aihara, IEEE TCAS-I, vol. 49, No. 5, pp. 602-608, 2002] is erroneous. An illustrative counterexample is presented, and it is pointed out that there are essentially two errors in their derivation. A corrected theorem is then derived using the approach shown in the aforementioned paper. It turns out that the corrected theorem provides only a sufficient condition for the local stability. Finally, a necessary and sufficient condition that was recently presented by the authors' of this paper is presented. Both the sufficient and the necessary and sufficient conditions are demonstrated using the Repressilator, the oscillatory gene regulatory network, which was also used for the illustration of the Chen and Aihara's original result.