A neural-network model is proposed as a test bed for the characterization of the chaotic dynamics emerging in a context where the coupling is, on the average, neither excitatory nor inhibitory. The proposed discrete-time model generalizes within a single framework two different setups previously studied in the literature. With the help of theoretical mean field arguments and numerical simulations on GPUs, we characterize the transition and show that the chaotic dynamics is extensive (i.e., that the number of active degrees of freedom is proportional to the network size) from the very beginning. Besides the coupling strength, two parameters play a crucial role: (1) one controls the local dissipation and determines the shape of the initial part of the Lyapunov spectrum as well as the shape of the correlation function; (2) the other, which corresponds to the amplitude of an effective random field, determines the nature of the transition.