On excitable dynamic systems with delays

The purpose of this paper is to discuss the properties of transparency and excitability of positive linear time‐invariant systems under internal point delays.

The problem is solved by combining the algebraic conditions for positivity, excitability, and transparency for the case of linear and time‐invariant dynamic systems in the presence of discrete lags.

It is shown that the excitability independent of delay is guaranteed if an auxiliary delay‐free system is excitable. Necessary and sufficient conditions for excitability and transparency are formulated in terms of the parameterization of the dynamics and control matrices, and equivalently, in terms of strict positivity of a matrix of an associate system obtained from the influence graph of the original system. Such conditions are testable through simple algebraic tests involving moderate computational effort.

The practical implications mainly rely on some biological and medical problems where delays are present by nature, excitability means the activation of all the state components under positive controls after a short time in the sense that it cannot remain identically zero. In the same way, it relays on the activation of all the output components under zero controls and non‐negative initial conditions.

The paper extends the concepts of excitability and transparency to dynamic systems with point time‐lags which are very common in nature and some practical problems.