Emma Reyner-Fuentes, Sebastian Rudolph, Carmen Peláez-Moreno, Francisco J. Valverde-Albacete
Brains do not simply add up every signal they receive: they select which routes dominate. Yet connectomes are usually analyzed with tools that sum over all paths and place neural states in Euclidean spaces, blurring the feedback circuits that may sustain internal states. Here we treat neural signaling as routing instead of diffusion and describe it with max-plus algebra, in which signals accumulate along a path and the strongest route prevails. We propose a model in which the functional connectome of Caenorhabditis elegans emerges from the time-varying combination of electrical, synaptic, monoaminergic, neuropeptidergic and hormonal layers, and analyze five networks within this framework. Each collapses onto a handful of dominant feedback loops, which recover well-characterized circuits and point to new candidates for experimental study. Unlike linear spectra, these structures keep the biological meaning of the data, and their order-based description suggests that brain states live in lattices rather than Euclidean spaces.
Priyanka Grover, Hemant K. Mishra, Sarishti Singh, Temjensangba
The symplectic interlacing theorem states that the symplectic spectrum of a $2n \times 2n$ real symmetric positive definite matrix satisfies a $2k$-step interlacing relation with the symplectic spectrum of every $2(n-k) \times 2(n-k)$ $s$-principal submatrix. We prove the converse in this paper. More precisely, given a positive global symplectic spectrum of size $n$ and a positive local symplectic spectrum of size $n-k$ satisfying the $2k$-step interlacing inequalities, we construct a $2n \times 2n$ real symmetric positive definite matrix having the prescribed global symplectic spectrum and a $2(n-k) \times 2(n-k)$ $s$-principal submatrix of it having the prescribed local symplectic spectrum. Our main tool is a reduction of $s$-principal compression to a rank-$k$ singular value perturbation. In the context of the Gaussian quantum marginal problem, our result implies that the symplectic interlacing inequalities are not merely necessary conditions on Gaussian marginals, but are also the complete spectral compatibility conditions for the global-versus-local-mode marginal problem.