We propose a novel way to regularise deep learning models by reducing high correlations between neurons. For this, we present two regularisation terms computed from the weights of a minimum spanning tree of the clique whose vertices are the neurons of a given network (or a sample of those), where weights on edges are correlation dissimilarities. We explore their efficacy by performing a set of proof-of-concept experiments, for which our new regularisation terms outperform some popular ones. We demonstrate that, in these experiments, naive minimisation of all correlations between neurons obtains lower accuracies than our regularisation terms. This suggests that redundancies play a significant role in artificial neural networks, as evidenced by some studies in neuroscience for real networks. We include a proof of differentiability of our regularisers, thus developing the first effective topological persistence-based regularisation terms that consider the whole set of neurons and that can be applied to a feedforward architecture in any deep learning task such as classification, data generation, or regression.