Abstract: We started with one hypothesis that the per-burst spike number of neuron cells is encoded information for neuron-to-neuron communication. We derived a one-parameter family of interval mappings modeling the bursting-spiking behaviours of excitable cells, which include pancreatic $\beta$-cells and some types of neuron cells. We derived bifurcation criterion for the model map to be isospiking --- the number of spikes is constant for all bursts initiated from a quiecent phase. Scaling laws governing the isospiking intervals were obtained. A new renormalization operator, $\cal R$, was used to explain these scaling laws. By doing so we discovered that the first natural number 1 is a universal constant in the sense that 1 is an expanding eigenvalue of $\RO$ and the isospike code partitioning of an appropriate stimulus parameter of any neural model is related to an approximating scheme for the eigenvalue. We also demonstrated that all positive rational numbers are universial in a similar sense. Our neural renormalization is similar to Feigenbaum's renormalization paradigm for quadratic maps, with important exceptions: the renormalization fixed point $\psi_0$ of our renormalization operator $\cal R$ is non-hyperbolic, and the universal number is 1, which is an eigenvalue of the operator at $\psi_0$ and the operator expands along the eigenvalue 1's eigenvector. We demonstrated that the operator can contract to $\psi_0$ at any rate smaller than 1 in the center-stable set of the fixed point. It can also expand away from some other fixed points at any rate greater than 1 along their unstable manifolds. Furthermore, We discovered that the neural renormalization has a super chaos structure. Specifically, we demonstrated that every finite dimensional systems can be conjugate to a sub-system of the renormalization operator in a subset $X_0$ of the center-stable set of the renormalization fixed point $\psi_0$, and the conjugation can be done in infinitely many ways. And, we showed that there is a dense orbit in the subset $X_0$. It is rather interesting that these three elements --- neuroencoding hypothesis, universal number 1, supreme chaos --- come together in one renormalization model. Each separate from the others would be less so. Down Load the PS and/or PDF files.