The proportion of older people will soon include nearly a quarter of the world population. This leads to an increased prevalence of non-communicable diseases such as Alzheimer's disease (AD), a progressive neurodegenerative disorder and the most common dementia. mild cognitive impairment (MCI) can be considered its prodromal stage. The early diagnosis of AD is a huge issue. We face it by solving these classification tasks: MCI-AD and cognitively normal (CN)-MCI-AD. An intelligent computing system has been developed and implemented to face both challenges. A non-neural preprocessing module was followed by a processing one based on a hybrid and ontogenetic neural architecture, the modular hybrid growing neural gas (MyGNG). The MyGNG is hierarchically organized, with a growing neural gas (GNG) for clustering followed by a perceptron for labeling. For each task, 495 and 819 patients from the Alzheimer's disease neuroimaging initiative (ADNI) database were used, respectively, each with 211 characteristics. Encouraging results have been obtained in the MCI-AD classification task, reaching values of area under the curve (AUC) of 0.96 and sensitivity of 0.91, whereas 0.86 and 0.9 in CN-MCI-AD. Furthermore, a comparative study with popular machine learning (ML) models was also performed for each of these tasks. The MyGNG proved to be a better computational solution than the other ML methods analyzed. Also, it had a similar performance to other deep learning schemes with neuroimaging. Our findings suggest that our proposal may be an interesting computing solution for the early diagnosis of AD.