*Memristor - An Emerging Device for Post-Moore's Computing and Applications*

eventually stable structure since the combination of passive circuit elements through a resistive coupling cannot give rise to instability. Thus, to be able to observe the emergence of complexity in a diffusively coupled homogenous network, it is vital to accommodate locally active cells, which cannot be represented by positive real complexity functions in the small-signal regime. This fact at the same time implies that local activity can be simply judged as a violation of local passivity [9]. Furthermore, a locally active cell is defined to be on the edge-of-chaos, if it is biased at an asymptotically stable operating point. Most importantly, an uncoupled cell, which is poised on the edge-of-chaos and therefore considered as a "dead" or "silent" cell due to its stability, can be potentially destabilized from its quite state *via* resistive coupling, which leads to the generation of complex patterns across the homogenous medium. Correspondingly, the edge-of-chaos domain involves a subset called the sharp-edge-of-chaos domain, which defines the set of parameters that destabilize the cell after coupling is introduced. Strictly speaking, a one-port cell is said to be locally active if and only if its small-signal transfer function (e.g., *Z s*ð Þ given by Eq. (3) in our case) satisfies any of the conditions below [9]:


real number or a complex number.

4. *Re Z j* ½ � ð Þ *ω* <0 for some *ω* ∈ð Þ �∞, ∞ .

To confirm the locally active dynamics of the complexity function *Z s*ð Þ of Eq. (3), which is associated with the uncoupled cell of **Figure 3(a)**, a direct approach would be to check whether these four rigorously defined criteria are satisfied or not, an approach we had applied in a previous work [28]. However, in this work, we apply a quick inspection method of local activity criteria, and rather check if *Z s*ð Þ clearly violates local passivity. In consistent with this approach, it can be seen that *Z s*ð Þ possesses a right half plane (RHP) zero (i.e., *z* ¼ �*R*1*R*2*=L R*ð Þ <sup>1</sup> þ *R*<sup>2</sup> > 0, for *R*<sup>1</sup> <0), a feature that cannot be realized with any locally passive transfer function. Thus, without a need for a further examination, we can infer that once it is biased in the NDR region, the uncoupled cell operates in the locally active regime. Furthermore, since the uncoupled cell is designed to be asymptotically stable (*via* tuning *Rb* and *C* accordingly), we can directly conclude that it is both locally active and on the edge-of-chaos across the entire NDR region. Finally, we would like to note that the RHP zero is a strong indicator of a possible destabilization scenario after the coupling is established.
