We present a simple model based on microscopic automata to describe the clonal expansion process. The model is based on a repertoire of antigens and T lymphocytes interacting via antigen-presenting cells which present the antigens peptides. Each cell is represented by an automaton moving randomly on a two-dimensional lattice.We use this simplified model in order to introduce local and spatial considerations in the mathematical models of clonal expansion based on differential equations, and at the same time to attempt an analytical interpretation of the results of computer simulations. For this reason we also derive a mean field theory, whose results are in good agreement with the solutions of the microscopic model, at least for situations that are not too far from equilibrium. This model may be used as the basis of a more realistic one that could follow the clonal expansion process on a simplified version of the lymphatic network.