Collatz Stopping Time as a Mathematical Model for Neuronal Refractory Period Duration: A Discrete Dynamical Systems Approach

Abstract

The Collatz conjecture — one of the most celebrated unsolved problems in mathematics —asserts that iterative application of a simple branching rule on any positive integer eventually converges to 1. In this paper, we propose a novel theoretical framework that maps the structural dynamics of the Collatz sequence onto the phases of neuronal signal transmission, with particular focus on the convergence to resting membrane potential (?70mV). We demonstrate that the odd-step rule (3n+1) is mathematically analogous to Na?-mediated depolarization, the even-step rule (n/2) mirrors K?-driven re polarization, and the Collatz stopping time provides a computable upper bound estimate for neuronal refractory period duration. This cross-disciplinary framework connects number theory, discrete dynamical systems, and computational neuroscience, opening a new avenue for modeling neuronal convergence behavior using integer sequence theory.

Introduction

This paper proposes a novel Collatz–Neuron Model, drawing an analogy between the mathematical behavior of the Collatz conjecture and the electrical activity of biological neurons. The Collatz sequence, which alternates between the operations n/2n/2n/2 for even numbers and 3n+13n+13n+1 for odd numbers, exhibits complex oscillatory behavior before converging to 1. The authors observe similarities between this process and neuronal dynamics described by the Hodgkin–Huxley model, where neurons undergo depolarization, repolarization, hyperpolarization, refractory recovery, and return to a resting state.

The framework establishes a mapping between Collatz sequence properties and neuronal behavior. Odd steps (3n+13n+13n+1) are associated with neuronal excitation and depolarization, while even steps (n/2n/2n/2) correspond to repolarization and recovery processes. Sequence maxima represent peak action potentials, convergence to 1 represents a return to the resting membrane potential, and stopping time corresponds to the neuron’s refractory period. A logarithmic voltage-mapping function is introduced to convert Collatz sequence values into biologically inspired membrane voltage levels between −70 mV and +40 mV.

The authors hypothesize that the Collatz stopping time is proportional to a neuron's refractory period, suggesting that stronger stimuli require longer recovery times. Computational simulations were conducted for starting values ranging from small integers to large numbers, generating voltage trajectories and phase classifications. Results showed that larger starting values produced longer stopping times and more complex voltage patterns. A key observation was that even steps consistently outnumber odd steps by approximately log?(3) ≈ 1.585, reflecting the biological principle that neurons spend more time in recovery than excitation.

The study suggests that integer-sequence-based neuron models could provide a computationally efficient alternative to traditional differential-equation models such as Hodgkin–Huxley and FitzHugh–Nagumo. Potential applications include large-scale neural simulations, refractory period estimation, and interdisciplinary links between number theory and neuroscience.

However, the authors acknowledge several limitations. The proposed relationship is currently a mathematical analogy rather than a validated biological mechanism. Real neuronal dynamics are continuous, whereas the Collatz process is discrete, and stopping times lack direct physical units. Future work includes calibrating stopping times with experimental neuronal data, extending the model to neural networks using Collatz trees, exploring stochastic variants, and applying machine learning techniques to study the relationship between stopping times and neuronal behavior.

Conclusion

We have proposed a novel theoretical analogy between the Collatz conjecture and neuronalsignal transmission. The branching rules of the Collatz map — 3n+1 for odd inputs(depolarization) and n/2 for even inputs (repolarization) — structurally mirror the ionicmechanisms of action potential generation and recovery. The stopping time T(n) emergesas a natural discrete estimator of refractory period duration. While this framework iscurrently a mathematical model rather than a biophysical theory, it opens a newinterdisciplinary direction connecting number theory and computational neuroscience.

References

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Copyright

Copyright © 2026 Vinay Kumar Havinal , Dr. Shilpa . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.