## 4 A Communication Scheme Based on Self-Organizing Patterns  
As mentioned previously, SOP can be successfully employed to transmit secret visual information between the communicating parties. A typical communication scenario is presented in a scheme that consists of the following steps. Initially, Sender and Receiver must use an asymmetric (arbitrary) protocol in order to determine the initial value \(a_{0}\) and the number of time-forward iterations \(n\) (parameters \(\
u\), \(\tau\), \(\delta\), \(L_{x}\), \(L_{y}\) must be predetermined beforehand). 1. Sender generates pseudo-random matrices of the initial cell excitation states and the connection map by using the Logistic map with the initial value \(a_{0}\); the size of the matrix is set to \(L_{x}\times L_{y}\); the parameters of the AF model are \(\
u\) and \(\delta\). Initial random matrix is dichotomous (cells contain binary values 0 or 1); connection between cells are random. 2. Sender modifies the pseudo-random matrix of initial cell excitation states by inverting pixels corresponding to the dot-skeleton representation of the secret image. 3. Sender executes n time-forward iterations starting from the modified matrix of initial conditions and sends the image of the self-organized pattern to the Receiver. 4. Receiver generates the identical copy of pseudo-random matrices of initial cell excitation states and the connection map by using the chaotic Logistic map with the initial value \(a_{0}\) and parameters of the AF model (\(\
u\) and \(\delta\)); the size of the matrix is \(L_{x}\times L_{y}\) (this is an identical copy of the matrix generated by the Sender in Step 1). 5. Receiver executes \(n\) time-forward iterations starting from the non-modified initial conditions. 6. Finally, the difference between the digital images of the patterns produced by the non-modified and modified initial conditions reveals the secret. Unfortunately, straightforward inversion of pixel values in the areas occupied by the secret image (in the digital image of initial conditions) does not work well in general (Figure 7). The \"secret\" image is shown in Figure 7a; the digital image representing initial conditions (with inverted pixels in the zones occupied by the secret information)--in Figure 7b. The self-organized pattern is illustrated in Figure 7c. It is clear that such pattern is not safe--a naked eye can detect the contours of the secret original image. The difference image is shown in Figure 7d; the binarized difference image--in Figure 7e. Thus, the image hiding scheme proposed in [4] cannot be used with the AF model. However, a single dot (\(1\times 1\) pixel) inversions in the initial conditions are not interpretable in self-organized patterns (Figure 7)--thus the further image communication scheme is based on single pixel inversion strategy. A huge difference in the shape of the \"secret\" images can be seen between Figure 7a,e. It is clear that not all of the parameters considered in the SOP model (\(a_{0}\), \(\
u\), \(\delta\)) are equally applicable for the proposed communication scheme (even if an adequate formation of the self-organizing pattern is ensured). Firstly, the initial random background used for initial excit  
Figure 7: Big clusters of initially excited cells result in interpretable self-organized patterns (\(\
u=0.5\), \(n=\tau=20\), \(\delta=0.01\)). (**a**) The secret image (**b**) Initially excited cells (shown in white) (**c**) The self-organized pattern (**d**) The difference image (**e**) The binarised initially excited cells. uniform (or homogeneous) in order to avoid empty spaces and larger initial cell clusters. Secondly, \(\delta\) directly affects clearness of the resulting image (Figure 7c) as well as an easier interpretation of the difference image (Figure 7d). Meanwhile, probability \(\
u\) does not have a big effect except stretching--however it is recommended to keep it in the range \((0,1)\) and not use the trivial cases (0 or 1). Let us consider four groups of a regular array dots (Figure 8a).