A topologically oriented neural network is very efficient in real-time path planning of a mobile robot in dynamic environments. Using a dynamic recurrent neural network to solve the partial differential equation of a potential field in a discrete manner, the problem of obstacle avoidance and path planning of a moving robot can be efficiently solved. A dimensional network used to represent the topology of the robot's workspace, where each network node represents a state associated with a local workspace point. In this paper, two approaches associated with different boundary conditions are proposed, namely, Dirichlet and Neumann conditions. The first approach relies on a field of attraction distributed around the moving target, acting as a unique local extreme in the local network space. The steepest gradients of the network state variables will aim towards the source of the potential field. The second approach considers two attractive and repulsive potential sources associated with the start and destination points. A dynamic neural mesh is used to model the robot workspace. A simulation package has been built and extensive computer experiments were conducted to demonstrate and validate the reliability of the presented approach.