Figure 3. Properties of PINned Connectivity Matrices.

A. Synaptic connectivity matrix of a 500-neuron network with p = 8% producing an idealized sequence with pVar = 92% (highlighted by a red circle in Figure S2B), denoted by J_{PINned, 8%}, and that of the randomly initialized network with p = 0, denoted by J_Rand, are shown here. Colorbars indicate, in pseudocolor, the magnitudes of synaptic strengths after PINning.

B. Influence of neurons away from the sequentially active neurons is estimated by computing the mean (circles) and the standard deviation (lines) of the elements of J_Rand (in blue) and J_{PINned, 8%} (in orange) in successive off-diagonal “stripes” away from the principal diagonal. These quantities are plotted as a function of the “inter-neuron distance”, i − j. In units of i − j, 0 corresponds to the principal diagonal or self-interactions, and the positive and the negative terms are the successive interaction magnitudes a distance i − j away from the primary sequential neurons.

C. Dynamics from the band-averages of J_{PINned, 8%} are shown here. Left panel is a synthetic matrix generated by replacing the elements of J_{PINned, 8%} by their means (orange circles in (B)). The normalized activity from a network with this synthetic connectivity is shown on the right. Although there is a localized “bump” around i − j = 0 and long range inhibition, these features lead to fixed point activity.

D. Same as panel (A), except for the connectivity matrix from a fully PINned (p = 100%) network, denoted by J_{PINned, 100%}.

E. Same as (B), except comparing J_{PINned, 100%} and J_Rand.. Band-averages (orange circles) are bigger and more asymmetric compared to those for J_{PINned, 8%}. Notably, these band-averages are also negative for i − j = 0 and in the neighborhood of 0.

F. Log of the probability density of the elements of J_Rand (gray squares), J_{PINned, 8%} (yellow squares) and for comparison, J_{PINned, 100%} (red squares) are shown here.

G. Same as panel (C), except showing the firing rates from the mean J_{PINned, 100%}. In contrast with J_{PINned, 8%}, the band averages of J_{PINned, 100%} can be sufficient to evoke a “Gaussian bump” qualitatively similar to the moving bump evoked by a ring attractor model, since its movement is driven by the asymmetry in the mean connectivity.