On one side there are those systems whose regime trajectory in the space state is a deterministic cycle. For such systems the observation provides an effective information of log2 k bits1(where k is the number of the nodes on the cycle, i.e. its length). Since a deterministic closed trajectory of length k in the state space corresponds to a suitable subset of k rows of the state transition matrix each containing exactly one value 1, and since before measuring the system the uncertainty is maximum - given that the system can be in any of these k states - while after measuring the systems it is univocally known the predecessor of the current state, the information acquired through observation is maximum and equal, according to the standard way of measuring information, to log k bits.

On the other side there are those systems whose behavior in the state space is uniformly random, that is those systems where each state can be, with equal probability, the predecessor of the current state.

Measuring the actual current state in these systems provides an effective information of 0 bits since no reduction in uncertainty is provided through the observation (complete uncertainty both before and after the measurement). Also for completely static systems, that is systems whose state is constant while time runs there is no reduction in uncertainty provided through the observation (no uncertainty either before or after the measurement).

## 3.2 Formal Definition

We define the effective information obtained by observing that system X is in state xi at instant t as

$$\flat_{K L}(B_{i}(t)\mid\mid\!\!\!\mid X_{t-1})$$

ei(**t, x**i) ⊜ DKL(Bi(t) ||Xt−1) (2)
where DKL is the Kullback-Leibler divergence2. Then

$$\begin{array}{r c l}{{}}&{{}}&{{}}\\ {{e i(t,x_{i})}}&{{=}}&{{\sum_{j}b_{i j}(t)\log\frac{b_{i j}(t)}{p(X_{t-1}=x_{j})}}}\\ {{}}&{{}}&{{}}\\ {{}}&{{}}&{{=}}&{{-H(B_{i}(t))-\sum_{j}b_{i j}(t)\log p(X_{t-1}=x_{j})}}\end{array}$$

Our definition is a generalization of the one provided by Tononi and coauthors (cfr. equations 1A and 1B of [1]). Ours in fact allows to study system behavior for each time instant and for each probability distribution X0, while in [1] the time instant under investigation is always t = 1 and it is always assumed probability distribution X0 is the uniform one. Our formulation hence allows to model both the transient and the stationary regime of a system.

For the case when the state probability distribution Xt−1 is uniform the formula above becomes:

$$\begin{array}{r c l}{{e i(t,x_{i})}}&{{=}}&{{-H(B_{i}(t))-\sum_{j}b_{i j}(t)\log{\frac{1}{2^{n}}}}}\\ {{}}&{{}}&{{}}\\ {{}}&{{}}&{{=}}&{{-H(B_{i}(t))+n\sum_{j}b_{i j}(t)}}\\ {{}}&{{}}&{{}}\\ {{}}&{{}}&{{=}}&{{n-H(B_{i}(t))}}\end{array}$$
