In this paper, we mathematically study a particular process for assimilation in the brain. The research aims to establish a theoretical model at computational level of the mechanism in a cognitive process operated by the mirror-neuron system, to generate a multi-dimensional system from this model, and to analyze the fundamentals of the related cognitive process in terms of dynamical systems. Finally, to understand the interactions between two individual mirror-neuron systems, we formulate and examine coupled systems that are composed of two distinct systems. We also carry out various numerical simulations to illustrate the assimilation process.