Systematically map eigenspectrum configurations beyond unit-rank mean connectivity

Systematically map the configurations of outlying eigenvalues and associated eigenmodes that arise when the mean connectivity has rank greater than one and the number of neuronal populations exceeds two in recurrent networks with chain motifs, thereby characterizing how relaxing the unit-rank assumption and increasing population complexity alter the eigenspectrum.

Background

The analysis in this work focuses on two-population networks with unit-rank mean connectivity, yielding a single mean-induced eigenmode and one additional chain-motif-induced mode. Relaxing these constraints—by increasing the number of populations or the rank of the mean connectivity—introduces additional non-trivial eigenmodes and potentially multiple outliers. A comprehensive taxonomy of these eigenspectrum configurations is needed to generalize the theory to richer cortical circuits with multiple inhibitory subtypes and more complex population structures.

References

Relaxing the unit-rank constraint, and increasing the number of populations leads to additional non-trivial eigenmodes of the mean connectivity J0, and additional outlying eigenvalues induced by chain motifs. Systematically mapping the resulting eigenspectrum configurations is left for future work.

— Identifying the impact of local connectivity patterns on dynamics in excitatory-inhibitory networks  (2411.06802 - Shao et al., 2024) in Discussion

Another open question concerns population-specific wandering on intermediate timescales.

— Stability and Wandering of Bumps in Neural Fields with Interneuron Subtypes  (2609.13074 - Ahmed et al., 11 Sep 2026) in Section 7, Discussion, paragraph beginning “Another open question concerns population-specific wandering on intermediate timescales.”