*2.1. Boolean Differential Equations*

Let us consider a system with state variables {*v*1, *<sup>v</sup>*2, ... , *vn*}, *vi* <sup>∈</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 1, ... , *<sup>n</sup>*. If a Boolean variable *xi* is related to each state *vi*, depending on a set of thresholds *<sup>σ</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup>. Then, the set of Boolean variables *<sup>x</sup>* = {*x*1, *<sup>x</sup>*2, ... , *xn*} gives a simple qualitative description of the system with 2*<sup>n</sup>* possible states. By adding the time dependence through a set of delays {*τij*}, *i* = 1, ... , *n*, *j* = 1, ... , *n*, *τij* > 0, where *τij* is the time it takes for *xj* to affect *xi*, there is an associated time delay for each pair of state variables not necessarily obeying *τij* = *τji*. In this manner, the feedbacks among the Boolean variables can be described by a system of Boolean differential equations as follows [33,34]:

$$\begin{aligned} \mathbf{x}\_1(t) &= f\_1(\mathbf{x}\_1(t-\tau\_{11}), \mathbf{x}\_2(t-\tau\_{12}), \dots, \mathbf{x}\_n(t-\tau\_{1n})),\\ \mathbf{x}\_2(t) &= f\_2(\mathbf{x}\_1(t-\tau\_{21}), \mathbf{x}\_2(t-\tau\_{22}), \dots, \mathbf{x}\_n(t-\tau\_{2n})),\\ &\vdots\\ \mathbf{x}\_n(t) &= f\_n(\mathbf{x}\_1(t-\tau\_{n1}), \mathbf{x}\_2(t-\tau\_{n2}), \dots, \mathbf{x}\_n(t-\tau\_{nn})),\end{aligned} \tag{1}$$

with *fi* : <sup>B</sup>*<sup>n</sup>* <sup>→</sup> <sup>B</sup>, *<sup>i</sup>* <sup>=</sup> 1, ... , *<sup>n</sup>*, being a set of Boolean functions where <sup>B</sup> <sup>=</sup> {0, 1}. The system (1) determines the dynamics of a Boolean network considering time delays, thereby defining an *Autonomous Boolean Network* (ABN) [33,34]. The dynamics of the ABN given by Equation (1) is numerically solved once the Boolean functions are defined with initial conditions on an interval *xi*(*t*) = *xi*0(*t*) for *t*<sup>0</sup> − *τ* ≤ *t* ≤ *t*0, *i* = 1, ... , *n*, where *τ* = max{*τij*} is the memory length of the system.
