Modulation Analysis for Stochastic FitzHugh–Nagumo Type Equation with Time Delay

• Yu Liu ^[1] ; Guanggan Chen ^[1] ; Shuyong Li ^[2]
  1. [1] Sichuan Normal University

     Sichuan Normal University

     China

     
  2. [2] Mianyang Teachers’ College
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 2, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This work is dedicated to investigating the splitting expression of solutions for a stochastic FitzHugh–Nagumo type equation with time delay. At first, through the construction of upper and lower solutions for second-order delay differential equations, it is examined that the deterministic FitzHugh–Nagumo type equation with time delay admits the existence of traveling wave front solutions. Then, by means of an appropriate weight function and evaluating low-order fluctuations within the weight function space, the solution of the stochastic FitzHugh–Nagumo type equation with time delay can be decomposed into the sum of a randomly modulated travelling wave and a small remainder term.

• Referencias bibliográficas
  • 1. Adimy, M., Chekroun, A., Kazmierczak, B.: Traveling waves in a coupled reaction–diffusion and difference model of hematopoiesis. J. Differ....
  • 2. Bátkai, A., Piazzera, S.: Semigroups and linear partial differential equations with delay. J. Math. Anal. Appl. 264(1), 1–20 (2001)
  • 3. Bátkai, A., Piazzera, S.: Semigroups for Delay Equations. Research Notes in Mathematics, vol. 10. A K Peters, Wellesley (2005)
  • 4. Becker, S., Richter, L.: Model order reduction for (stochastic-) delay equations with error bounds. J. Comput. Dyn. 9(4), 575–600 (2022)
  • 5. Caraballo, T., Liu, K., Truman, A.: Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property....
  • 6. Chen, C.-N., Choi, Y.S.: Traveling pulse solutions to FitzHugh–Nagumo equations. Calc. Var. Part. Differ. 5(4), 1–45 (2015)
  • 7. Cornwell, P., Jones, C.: On the existence and stability of fast traveling waves in a doubly diffusive FitzHugh–Nagumo system. SIAM J. Appl....
  • 8. De Angelis, M.: A priori estimates for solutions of FitzHugh–Rinzel system. Meccanica 5(7), 1035– 1045 (2022)
  • 9. De Angelis, M.: Hopf bifurcations in dynamics of excitable systems. Ricerche Mat. 7(3), 2591–2604 (2024)
  • 10. De Angelis, M.: Transport phenomena in excitable systems: existence of bounded solutions and absorbing sets. Mathematics 10(12), 2041...
  • 11. Eichinger, K., Gnann, M., Kuehn, C.: Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations. Ann....
  • 12. FitzHugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biol. 17(4), 257–278 (1955)
  • 13. Huang, J.H., Zou, X.F.: Traveling wavefronts in diffusive and cooperative Lotka–Volterra system with delays. J. Math. Anal. Appl. 271,...
  • 14. Krüger, J., Stannat, W.: A multiscale-analysis of stochastic bistable reaction–diffusion equations. Nonlinear Anal. 162, 197–223 (2017)
  • 15. Krüger, J., Stannat, W.: Front propagation in stochastic neural fields: a rigorous mathematical framework. SIAM J. Appl. Dyn. Syst. 13(3),...
  • 16. Kudryashov, N.A.: On Integrability of the FitzHugh–Rinzel model. Russ. J. Nonlinear Dyn. 15(1), 13–19 (2019)
  • 17. Lang, E.: A multiscale analysis of traveling waves in stochastic neural fields. SIAM J. Appl. Dyn. Syst. 15(3), 1581–1614 (2016)
  • 18. Liu, W.S., Van Vleck, E.: Turning points and traveling waves in FitzHugh–Nagumo type equations. J. Differ. Equ. 225, 381–410 (2006)
  • 19. Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50(10), 2061–2070 (1962)
  • 20. Njitacke, Z.T., Parthasarathy, S., Takembo, C.N., Rajagopal, K., Awrejcewicz, J.: Dynamics of a memristive FitzHugh–Rinzel neuron model:...
  • 21. Schaaf, K.W.: Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations. Trans. Am. Math. Soc....
  • 22. Upadhyay, R.K., Sharma, S.K., Mondal, A.: Emergence of hidden dynamics in different neuronal network architecture. Appl. Math. Model....
  • 23. Van Vuuren, J.H.: The existence of traveling plane waves in a general class of competition–diffusion systems. IMA J. Appl. Math. 5(5),...
  • 24. Wang, K., Du, Z.J., Liu, J.: Traveling pulses of coupled FitzHugh–Nagumo equations with doublydiffusive effect. J. Differ. Equ. 374, 316–338...
  • 25. Wu, J.H., Zou, X.F.: Traveling wave fronts of reaction–diffusion systems with delay. J. Dyn. Differ. Equ. 13(3), 651–687 (2001)
  • 26. Zemskov, E.P., Epstein, I.R.: Wave propagation in a FitzHugh–Nagumo-type model with modified excitability. Phys. Rev. E 82, 026207 (2010)