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.

人工神经网络对参数进行优化，而生物神经元是多向传播的，可以建模整体联合分布，本文讨论了一种基于多项式基函数的分布神经元的层次相关重构，并通过标准反向传播或新方法进行训练，实现多向传播的数值和概率密度。