This talk covers some efforts to extend the formalism of informational flow from the first-order and mono-layers to the second-order and double-layers, as well as symmetry and skew-symmetry in manifolds focusing on our synaptic excitability, neuronal circuitry, and connectivity.
 
The first part is about the immersed boundary method with advection-electrodiffusion retaining the non-electroneutral space charge layers along the fluctuating membrane. This is realized with regularized singular integrals as chemical potential barriers. In local mesh refinement, the second-order interpolation is prescribed at the coarse-fine interface. When the concentration gets diluted, the ionic transport is represented by a stochastic transition in the second order on lattice spaces. In the microenvironment of compression, the Eulerian-Lagrangian interaction is turned into a port-Hamiltonian formalism. Foldable rigidity (origami) and wrinkling dynamics will be introduced shortly.
 
The next is about the neuronal circuitry of the hypothalamus for characterizing energy balance. In pubertal maturation, the ventral premammillary nucleus (PMv), the upstream for energy sensing in the reproductive axis, shows a developmental plasticity in the connectome. The role of the dopamine transporter is characterized by the integration of single RNA-seq data and conductance electrophysiology models and recapitulates the awakening of ovulation and fertility. A working model on pregnancy will be introduced briefly.
 
Finally, the up-scaling of synaptic transmission and dendritic integration is represented by covariance matrices of brain connectivity. In the presence of the uncertainty principle, the divergence between two brain states or subjects is defined from dynamical systems points of view of optimal transport, applicable in the classes of two-sample tests, classification, and regression with kernel methods. Specifically, sliced Wasserstein flows on covariance matrices are reformulated from Liouville PDE with normalizing flows on manifolds. Some signatures of competence are shown in the computation of the Wasserstein barycenter with openness to high-dimensional sparsity.