Emergence of Slow-Switching Assemblies in Structured Neuronal Networks
Fig 5
Unraveling the association between the leading Schur vectors of the weight matrix and the observed SSA activity.
A Illustration of a weight matrix of the N = 2000 neuron LIF network with c = 20 groups of excitatory neurons (REE = 3.4) shown together with the heatmap of the real parts of its first 25 Schur vectors. The first 19 Schur vectors exhibit block-uniform patterns relatively constant within each of the 20 groups of neurons, whereas no such pattern is observed for the other Schur vectors. Note that there is no pattern discernible over the inhibitory neurons. The leading Schur vectors correspond to the cloud of ‘slow’ eigenvalues above the gap, as indicated in C, and span a patterned dominant subspace that induces grouped dynamics in the network. B A long simulation (80s) of the LIF network dynamics (only the first 2s shown) was analyzed using PCA and the first 25 principal components (PCs) are shown. Reflecting the banded structure of the simulated dynamics, the leading PCs also show a block-patterned structure consistent with the neuronal groups. C On the spectrum of W, we indicate the group of leading eigenvalues above the gap associated with the dominant subspace. D The alignment between the dominant Schur subspace of the W matrix and the subspace of the strongest principal components is measured by the first principal angle θ Eq (21). Above a threshold of the clustering strength REE, both subspaces become highly aligned in line with the observations in Fig 2 (dots: raw data from simulations; line: mean; shading: standard deviation). E The same effect is observed when the clustering is introduced in the weights by varying WEE, as in Fig 4.