In this paper, by using the Terman-Wang small-world neuronal network with electrical synapse coupling, we investigate the synchronous dynamics of neuronal network system subjected to spatially correlated white noise. First, the dynamical mean-field approximation theory is extended to the small-world network system under spatially correlated white noise, through which the original 2N-dimensional stochastic differential equations of the network system are transformed to 11-dimensional deterministic moment differential equations. Then, based on this set of moment differential equations, the key effects of spatially correlated noise and network structure on the synchronous firing property are discussed in the Terman-Wang neuronal network system. The results show that the synchronization ratio of this considered neuronal network system becomes higher not only as the noise correlation coefficient is increased but also as the coupling strength and the average vertex degree are added. Those results imply that the noise spatial correlation coefficient, the coupling strength, and the average vertex degree can play a positive role in inducing synchronous neuronal behaviors. Furthermore, the synchronous dynamics of the original neuronal network system, obtained by direct numerical simulations, is compared with those obtained by the dynamical mean-field approximation theory, and good consistence between them is revealed.