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The present work is dedicated to a very active branch of Natural Computing (which tries to discover the way nature computes, especially at a biological level), namely Membrane Computing, more precisely, to those models of membrane systems mainly inspired from the functioning of the neural cell. Membrane computing area was initiated by Gh. Paun at the end of 1998 and it deals with seeking of computing models inspired by the structure and functioning of the living cell. The models obtained in literature they are called P systems process multisets of symbols (which are usually called objects) in a distributed parallel manner, inside a membrane structure of a cellular type or of a tissue-like type. The present book introduces a new way of defining the result of a computation by means of following the traces of a specified object within a cell structure or a neural structure. Then, we get closer to the biology of the brain by investigating in a great detail a class of P systems inspired from the way neurons cooperate by means of spikes, electrical pulses of identical shapes.