Every route counts, but some count more: a max-plus view of the C. elegans brain

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.