Several recent studies have shown that neural activity in vivo tends to be constrained to a low-dimensional manifold. Such activity does not arise in simulated neural networks with homogeneous connectivity and it has been suggested that it is indicative of some other connectivity pattern in neuronal networks. In particular, this connectivity pattern appears to be constraining learning so that only neural activity patterns falling within the intrinsic manifold can be learned and elicited. Here, we use three different models of spiking neural networks (echo-state networks, the Neural Engineering Framework and Efficient Coding) to demonstrate how the intrinsic manifold can be made a direct consequence of the circuit connectivity. Using this relationship between the circuit connectivity and the intrinsic manifold, we show that learning of patterns outside the intrinsic manifold corresponds to much larger changes in synaptic weights than learning of patterns within the intrinsic manifold. Assuming larger changes to synaptic weights requires extensive learning, this observation provides an explanation of why learning is easier when it does not require the neural activity to leave its intrinsic manifold.