The paper is devoted to the adaptive control problem of synchronization in the network of Hindmarsh-Rose neurons. The case of identical neurons was predominantly considered in the previous studies, which are concerned with the synchronization of the biological neuron models. However, the natural networks of biological neurons are heterogeneous, since living neurons can have different shapes and characteristics. In order to take into account the effect of the natural network heterogeneity, the neuronal models are considered with different parameters in our paper. It was shown that the adaptive synchronization of the heterogeneous Hindmarsh-Rose network is possible, and it is achieved by using the developed controller. The obtained results were mathematically proved and confirmed by the simulations.