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.

翻译：当前流行的人工神经网络通过假设某种预定义参数化形式（如多层感知器或Kolmogorov-Arnold网络），优化参数以实现单向数值传播。而生物神经元则不同，例如"轴突动作电位的双向传导并不罕见" \cite{axon}——这表明生物神经元经过优化可连续在多方条件下工作。此外，单个神经元所能建模的统计依赖性不仅限于（期望）数值依赖，更包括包含高阶矩的完整联合分布。这种无偏的联合分布神经元可通过代入$\rho(x,y,z)$并进行归一化，实现多方传播（分布或数值），例如$\rho(x|y,z)$或$\rho(y,z|x)$。本文将讨论用于此类神经元模型的层级相关性重建方法：假设$\rho(x,y,z)=\sum_{ijk} a_{ijk} f_i(x) f_j(y) f_k(z)$这种基于多项式基底$f_i$的联合分布参数化形式，使其能够灵活、低开销地进行包含非线性的处理，实现模型的直接估计与更新，并通过标准反向传播或适用于此类结构的张量分解新型训练方法进行优化。仅利用成对（输入-输出）依赖性时，其期望值预测将呈现类似KAN的特性，其中训练激活函数为多项式形式；通过引入乘积项加入高阶依赖性后，该模型能以可解释的方式拓展，最终实现数值与概率密度的多方传播。