The self-organising neural network with weight normalisation (SONN-WN) for solving combinatorial optimisation problems (COPs) is investigated in terms of its performance and dynamical characteristics. A simplified computational model of the weight normalisation process is constructed, which reveals symmetry-breaking bifurcations in a typical node outside the winning neighbourhood. Experimental results with the N-queen problem show that bifurcations can enhance solution qualities in a consistent manner. A mechanism based on the weights’ transient trajectories is proposed to account for the neural network’s capacity to escape local minima.