A generalized neural network was adapted for the simulation of processes strongly dependent upon the history, imposed by the inner own history of an individual neuronal activation. This involves the dependence of the neural network parameters upon the cumulated values of the corresponding neuron activations. When in the neural network weakly coupled blocks with strong inner couplings can be identified, the activation wave on the entire network (associated with the overall epidemic) can be decomposed into quasi-independent intra-block local activation waves, with characteristic delays between them (corresponding to the simultaneous and successive local epidemics). Special simulations on strongly connex neural network determine the typical local activation waves for various block parameters and the mentioned delays between two such successive activation waves in two coupled blocks. Another type of neural network is used to achieve the empirical decomposition of the overall epidemic into simultaneous (corresponding to a layer) and successive local epidemics (corresponding to the various epidemic waves, associated with different layers). A simpler approximative algorithm for the estimation of the number of the mentioned simultaneous local typical epidemics is also presented.