States and sequences of paired subspace ideals and their relationship to patterned brain function
It is found here that the state of a network of coupled ordinary differential equations is partially localizable through a pair of contractive ideal subspaces, chosen from dual complete lattices related to the synchrony and synchronization of cells within the network. The first lattice is comprised of polydiagonal subspaces, corresponding to synchronous activity patterns that arise from functional equivalences of cell receptive fields. This lattice is dual to a transdiagonal subspace lattice ordering subspaces transverse to these network-compatible synchronies. Combinatorial consideration of contracting polydiagonal and transdiagonal subspace pairs yields a rich array of dynamical possibilities for structured networks. After proving that contraction commutes with the lattice ordering, it is shown that subpopulations of cells are left at fixed potentials when pairs of contracting subspaces span the cells' local coordinates - a phenomenon named glyph formation here. Treatment of mappings between paired states then leads to a theory of network-compatible sequence generation. The theory's utility is illustrated with examples ranging from the construction of a minimal circuit for encoding a simple phoneme to a model of the primary visual cortex including high-dimensional environmental inputs, laminar speficicity, spiking discontinuities, and time delays. In this model, glyph formation and dissolution provide one account for an unexplained anomaly in electroencephalographic recordings under periodic flicker, where stimulus frequencies differing by as little as 1 Hz generate responses varying by an order of magnitude in alpha-band spectral power. Further links between coupled-cell systems and neural dynamics are drawn through a review of synchronization in the brain and its relationship to aggregate observables, focusing again on electroencephalography. Given previous theoretical work relating the geometry of visual hallucinations to symmetries in visual cortex, periodic perturbation of the visual system along a putative symmetry axis is hypothesized to lead to a greater concentration of harmonic spectral energy than asymmetric perturbations; preliminary experimental evidence affirms this hypothesis. To conclude, connections drawn between dynamics, sensation, and behavior are distilled to seven hypotheses, and the potential medical uses of the theory are illustrated with a lattice depiction of ketamine xylazine anaesthesia and a reinterpretation of hemifield neglect.