The attractor-based complexity of a Boolean neural network is a measure which refers to the ability of the network to perform more or less complicated classification tasks of its inputs via the manifestation of meaningful or spurious attractor dynamics. Here, we study the attractor-based complexity of a Boolean model of the basal ganglia-thalamocortical network. We show that the regulation of the interactive feedback is significantly involved in the maintenance of an optimal level of complexity. We also show that the complexity of the network depends sensitively on the values of its synaptic connections. These considerations support the general rationale that the synaptic plasticity and the interactive architecture play a crucial role in the computational and dynamical capabilities of biological neural networks.