Many real-life systems can be modeled as a complex network made up of nodes whose dynamics are governed by nonlinear differential equations. Observability on such networks is one fundamental problem consisting in the capability of inferring the states of all/some target nodes from the knowledge of some node states. In this paper, we present a new approach for studying the full and the target observability in networks of nonlinear systems with linear and/or nonlinear couplings. The proposed method is based on a mathematical result ensuring the existence of specific local relations obtained from equations of each node. Two consequences in terms of observability are deduced from this theoretical result and are used to elaborate an algorithm. This algorithm, named TargetObservability, determines sets of nodes ensuring the observability of a given target set. We exemplify our approach on a biological neural network of C. elegans, made up of Hodgkin-Huxley type models coupled through linear and nonlinear terms. This provides a testable hypothesis that is likely to accelerate the discovery and analysis of the biological circuitry in C. elegans.