The purpose of this article is to demonstrate that coordinated communication spontaneously emerges in a population composed of agents that are capable of specific cognitive activities. Internal states of agents are characterized by meaning vectors. Simple neural networks composed of one layer of hidden neurons perform cognitive activities of agents. An elementary communication act consists of the following: (a) two agents are selected, where one of them is declared the speaker and the other the listener; (b) the speaker codes a selected meaning vector onto a sequence of symbols and sends it to the listener as a message; and finally, (c) the listener decodes this message into a meaning vector and adapts his or her neural network such that the differences between speaker and listener meaning vectors are decreased. A Darwinian evolution enlarged by ideas from the Baldwin effect and Dawkins' memes is simulated by a simple version of an evolutionary algorithm without crossover. The agent fitness is determined by success of the mutual pairwise communications. It is demonstrated that agents in the course of evolution gradually do a better job of decoding received messages (they are closer to meaning vectors of speakers) and all agents gradually start to use the same vocabulary for the common communication. Moreover, if agent meaning vectors contain regularities, then these regularities are manifested also in messages created by agent speakers, that is, similar parts of meaning vectors are coded by similar symbol substrings. This observation is considered a manifestation of the emergence of a grammar system in the common coordinated communication.