Biology-inspired joint distribution neurons based on Hierarchical Correlation Reconstruction allowing for multidirectional neural networks

Biological neural networks seem qualitatively superior (e.g. in learning, flexibility, robustness) to current artificial like Multi-Layer Perceptron (MLP) or Kolmogorov-Arnold Network (KAN). Simultaneously, in contrast to them: biological have fundamentally multidirectional signal propagation \cite{axon}, also of probability distributions e.g. for uncertainty estimation, and are believed not being able to use standard backpropagation training \cite{backprop}. There are proposed novel artificial neurons based on HCR (Hierarchical Correlation Reconstruction) allowing to remove the above low level differences: with neurons containing local joint distribution model (of its connections), representing joint density on normalized variables as just linear combination of $(f_\mathbf{j})$ orthonormal polynomials: $\rho(\mathbf{x})=\sum_{\mathbf{j}\in B} a_\mathbf{j} f_\mathbf{j}(\mathbf{x})$ for $\mathbf{x} \in [0,1]^d$ and $B\subset \mathbb{N}^d$ some chosen basis. By various index summations of such $(a_\mathbf{j})_{\mathbf{j}\in B}$ tensor as neuron parameters, we get simple formulas for e.g. conditional expected values for propagation in any direction, like $E[x|y,z]$, $E[y|x]$, which degenerate to KAN-like parametrization if restricting to pairwise dependencies. Such HCR network can also propagate probability distributions (also joint) like $\rho(y,z|x)$. It also allows for additional training approaches, like direct $(a_\mathbf{j})$ estimation, through tensor decomposition, or more biologically plausible information bottleneck training: layers directly influencing only neighbors, optimizing content to maximize information about the next layer, and minimizing about the previous to remove noise, extract crucial information.