East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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INPUT-DRIVEN MEMORY DYNAMICS VIA CONVOLUTION OPERATORS

  • YongDae Jeong (Department of Mathematics and Institute of Mathematical Sciences, Pusan National University) ;
  • Jong Hyuk Byun (Department of Mathematics and Institute of Mathematical Sciences, Pusan National University, Republic of Korea Institute for Future Earth, Pusan National University)
  • Received : 2026.01.07
  • Accepted : 2026.02.03
  • Published : 2026.05.31
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Abstract

We study memory effects generated by a general time-dependent input through a convolution operator with an integrable kernel. Focusing on the role of the input rather than the specific kernel structure, we analyze how qualitative properties of g(t) determine the long-term behavior of the resulting memory response. We establish general conditions under which the memory output converges to a constant equilibrium when the input approaches a steady value, showing that the equilibrium depends only on the total mass of the kernel. For time-dependent inputs, including periodic functions, we prove the existence of steady responses that attract all transient dynamics. Numerical simulations illustrate representative phenomena such as transient overshoot, phase lag, and memory-induced smoothing. This work provides a kernel-agnostic framework for understanding input-driven memory dynamics with broad relevance to biological and medical applications.

Keywords

Acknowledgement

This work was supported by a 2-Year Research Grant of Pusan National University (Jong Hyuk Byun).

References

  1. H. T. Banks, S. Hu, and W. C. Thompson, Modeling and inverse problems in the presence of uncertainty, CRC Press, 2014.
  2. J. J. Batzel and F. Kappel, Time delay in physiological systems: Analyzing and modeling its impact, Math. Biosci. (2011).
  3. S. Biswas, S. Manicka, E. Hoel, and M. Levin, Gene regulatory networks exhibit several kinds of memory: Quantification of memory in biological and random transcriptional networks, iScience 24 (2021), no. 3, 102131.
  4. J. H. Byun and I. H. Jung, Phase-specific cancer-immune model considering acquired resistance to therapeutic agents, Appl. Math. Comput. 391 (2021), 125555. https://doi.org/10.1016/j.amc.2020.125555
  5. J. H. Byun and I. H. Jung, Stochastic target-mediated drug disposition model based on birth-death process and its parameter inference using approximate bayesian computation-mcmc, Appl. Math. Modelling 105 (2022), 81-94. https://doi.org/10.1016/j.apm.2021.12.032
  6. J. H. Byun, A. Park, and I. H. Jung, Receptor-mediated endocytosis modeling of antibody-drug conjugates to the released payload within the intracellular space considering target antigen expression levels, J. Appl. Anal. Comput. 10 (2020), no. 5, 1848-1868.
  7. J. H. Byun, I. S. Yoon, S. Y. Lee, H. J Cho, and I. H. Jung, Extended transit compartment model to describe tumor delay using coxian distribution, Sci. Rep. 12 (2022), no. 1, 10086.
  8. C. M. Cheng, Z. K. Peng, W. M. Zhang, and G. Meng, Volterra-series-based nonlinear system modeling and its engineering applications: A state-of-the-art review, Mech. Syst. Signal Process. 87 (2017), 340-364. https://doi.org/10.1016/j.ymssp.2016.10.029
  9. I. Goychuk, Fractional kinetics and anomalous transport, Phys. Rep. 505 (2012), no. 4-5, 109-179.
  10. G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Cambridge University Press, 1990.
  11. M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal. 31 (1968), no. 2, 113-126. https://doi.org/10.1007/BF00281373
  12. K. S. Kim, I. H. Jung, and J. H. Byun, Exploring immune responses through dynamic modeling of cell-immune interactions in viral infection and vaccination, Complexity 2024 (2024), no. 1, 3357763.
  13. Y. Kuang and Y. Yang, Delay differential equations with applications in population dynamics, Academic Press, 2006.
  14. M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science 197 (1977), no. 4300, 287-289. https://doi.org/10.1126/science.267326
  15. D. E. Mager and W. J. Jusko, General pharmacodynamic model for drugs administered by multiple dosing regimens, J. Pharmacokinet. Pharmacodyn. 28 (2001), no. 6, 507-532. https://doi.org/10.1023/A:1014414520282
  16. F. Mainardi, Fractional calculus and waves in linear viscoelasticity, Imperial College Press, 2010.
  17. B. Perthame, Transport equations in biology, Birkhäuser, 2007.
  18. I. Podlubny, Fractional differential equations, Academic Press, 1999.
  19. J. Prüss, Evolutionary integral equations and applications, Birkhäuser, 1993.
  20. A. F. Wang and V. S. Ayyar, Pharmacodynamic models of indirect effects and irreversible inactioation with turnover: Applicability to mechanism-based modeling of gene silencing and targeted protein degradation, J. Pharm. Sci. 113 (2024), no. 1, 191-201. https://doi.org/10.1016/j.xphs.2023.10.027