Analysis of Structure and Dynamics in Three-Neuron Motifs

Recurrent neural networks can produce ongoing state-to-state transitions without any driving inputs, and the dynamical properties of these transitions are determined by the neuronal connection strengths. Due to non-linearity, it is not clear how strongly the system dynamics is affected by discrete local changes in the connection structure, such as the removal, addition, or sign-switching of individual connections. Moreover, there are no suitable metrics to quantify structural and dynamical differences between two given networks with arbitrarily indexed neurons. In this work, we present such permutation-invariant metrics and apply them to motifs of three binary neurons with discrete ternary connection strengths, an important class of building blocks in biological networks. Using multidimensional scaling, we then study the similarity relations between all 3,411 topologically distinct motifs with regard to structure and dynamics, revealing a strong clustering and various symmetries. As expected, the structural and dynamical distance between pairs of motifs show a significant positive correlation. Strikingly, however, the key parameter controlling motif dynamics turns out to be the ratio of excitatory to inhibitory connections.