  Reservoir computing is a popular approach to design recurrent neural
networks, due to its training simplicity and approximation performance. The
recurrent part of these networks is not trained (e.g., via gradient descent),
making them appealing for analytical studies by a large community of
researchers with backgrounds spanning from dynamical systems to neuroscience.
However, even in the simple linear case, the working principle of these
networks is not fully understood and their design is usually driven by
heuristics. A novel analysis of the dynamics of such networks is proposed,
which allows the investigator to express the state evolution using the
controllability matrix. Such a matrix encodes salient characteristics of the
network dynamics; in particular, its rank represents an input-indepedent
measure of the memory capacity of the network. Using the proposed approach, it
is possible to compare different reservoir architectures and explain why a
cyclic topology achieves favourable results as verified by practitioners.
