Two perspectives can be distinguished in computational neuroscience towards understanding the link between neuronal activity and behaviour.$^1$ On the one hand, focusing on single-neuron responses highlighted that a few functional properties, such as neuronal tuning$^2$ or selectivity,$^3$ often characterise individual responses. These properties determine each neuron's “position” in an abstract space, which we call the “(functional) similarity space”: for example, each cell's preferred head direction (HD) in the mouse HD system determines its position on an abstract ring.$^4$ It is commonly hypothesised that these functional properties structure network connectivity.$^{4-7}$ In contrast, the recent advent of large-scale recordings has allowed for a population-level approach, revealing that the activity of large networks is often restricted to low-dimensional subsets of the state-space of the joint activity of all neurons, called the neural manifold.$^{1,8,9}$ The neuronal activity can thus be parameterised by a set of collective variables, whose dynamical flow is thought to reflect neural computations.$^{10,11}$ However, the link between the neural manifold and the structure of the underlying recurrent connectivity remains elusive. We aim at connecting the two perspectives in models of Recurrent Neural Networks (RNN). First, we propose a theoretical framework that explicitly links the circuit structure and the flow of the collective dynamics in the neural manifold. Relying on the theory of neural fields $^{12}$ and large low-rank networks,$^{13}$ it connects functional properties of single neurons to emergent low-dimensional dynamics in a model of RNN. Maybe surprisingly, the theory predicts no one-to-one relationship between the functional similarity space and the neural manifold. In particular, their respective dimensionality can be different; consequently, the same collective dynamics can result from different circuit structures. Yet, we show that they are linked through the symmetries of the connectivity. Second, we propose an approach for extracting the circuit structure from the neuronal activity, by exploiting the similarities between the activity of single neurons. We apply it to artificial data to retrieve the circuit structure in different networks performing the same simple task. Unlike the characterisation of neuronal tuning, this approach is task-agnostic, allowing us to blindly detect even a task-irrelevant circuit structure.