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

Popular artificial neural networks (ANN) optimize parameters for unidirectional value propagation, assuming some guessed parametrization type like Multi-Layer Perceptron (MLP) or Kolmogorov-Arnold Network (KAN). In contrast, for biological neurons e.g. "it is not uncommon for axonal propagation of action potentials to happen in both directions" \cite{axon} - suggesting they are optimized to continuously operate in multidirectional way. Additionally, statistical dependencies a single neuron could model is not just (expected) value dependence, but entire joint distributions including also higher moments. Such agnostic joint distribution neuron would allow for multidirectional propagation (of distributions or values) e.g. $\rho(x|y,z)$ or $\rho(y,z|x)$ by substituting to $\rho(x,y,z)$ and normalizing. There will be discussed Hierarchical Correlation Reconstruction (HCR) for such neuron model: assuming $\rho(x,y,z)=\sum_{ijk} a_{ijk} f_i(x) f_j(y) f_k(z)$ type parametrization of joint distribution with polynomial basis $f_i$, which allows for flexible, inexpensive processing including nonlinearities, direct model estimation and update, trained through standard backpropagation or novel ways for such structure up to tensor decomposition. Using only pairwise (input-output) dependencies, its expected value prediction becomes KAN-like with trained activation functions as polynomials, can be extended by adding higher order dependencies through included products - in conscious interpretable way, allowing for multidirectional propagation of both values and probability densities.