  Biological neural networks seem qualitatively superior (e.g. in learning,
flexibility, robustness) from current artificial like Multi-Layer Perceptron
(MLP) or Kolmogorov-Arnold Network (KAN). Simultaneously, in contrast to them:
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) removing 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 among
$(f_\mathbf{j})$ 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 $B$
some chosen basis, with basis growth approaching complete description of joint
distribution. By various index summations of such $(a_\mathbf{j})$ 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 minimize the noise.
