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The study of nonlinear long-term correlations in neuronal signals is a central topic for advanced neural signal processing. In particular, the existence of long-term correlations in neural signals recorded via multielectrode array (MEA) could provide interesting information about changes in interneuron communications. In this study we propose a new method for long-term correlation analysis of neuronal burst activity based on the periodogram alpha slope estimation of the MEA signal. We applied our method to recordings taken from cultured networks of dissociated rat cortical neurons. We show the effectiveness of the method in analyzing the activity changes as well as the temporal dynamics that take place during the development of such cultures. Results demonstrate that the alpha parameter is able to divide the network development in three well-defined stages, showing pronounced variations in the long-term correlation among bursts.