Have a personal or library account? Click to login
Spike Patterns and Chaos in a Map–Based Neuron Model Cover

References

  1. Afraimovich, V. and Hsu, S. (2002). Lectures on Chaotic Dynamical Systems, American Mathematical Society, Providence.
  2. Alsedà, L., Llibre, J., Misiurewicz, M. and Tresser, C. (1989). Periods and entropy for Lorenz-like maps, Annales de l’Institut Fourier (Grenoble) 39(4): 929–952, DOI: 10.5802/aif.1195.
  3. Cholewa, Ł. and Oprocha, P. (2021a). On α-limit sets in Lorenz maps, Entropy 23(9): 1153. DOI: 10.3390/e23091153.
  4. Cholewa, Ł. and Oprocha, P. (2021b). Renormalization in Lorenz maps—Completely invariant sets and periodic orbits. arXiv: 2104.00110[math.DS].
  5. Courbage, M., Maslennikov, O.V. and Nekorkin, V.I. (2012). Synchronization in time-discrete model of two electrically coupled spike-bursting neurons, Chaos, Solitons, Fractals 45(05): 645–659, DOI: 10.1016/j.chaos.2011.12.018.
  6. Courbage, M. and Nekorkin, V.I. (2010). Map based models in neurodynamics, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 20(06): 1631–1651, DOI: 10.1142/S0218127410026733.
  7. Courbage, M., Nekorkin, V.I. and Vdovin, L.V. (2007). Chaotic oscillations in a map-based model of neural activity, Chaos 17(4): 043109, DOI: 10.1063/1.2795435.
  8. Derks, G., Glendinning, P.A. and Skeldon, A.C. (2021). Creation of discontinuities in circle maps, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477(2251): 20200872, DOI: 10.1098/rspa.2020.0872.
  9. Ding, Y.M., Fan, A.H. and Yu, J.H. (2010). Absolutely continuous invariant measures of piecewise linear Lorenz maps. arXiv: 1001.3014 [math.DS].
  10. FitzHugh, R. (1955). Mathematical models of the threshold phenomena in the nerve membrane, The Bulletin of Mathematical Biophysics 17: 257–278, DOI: 10.1007/BF02477753.
  11. Geller, W. and Misiurewicz, M. (2018). Farey–Lorenz permutations for interval maps, International Journal of Bifurcation and Chaos 28(02): 1850021, DOI: 10.1142/S0218127418500219.
  12. Hess, A., Yu, L., Klein, I., Mazancourt, M.D., Jebrak, G. and Mal, H. (2013). Neural mechanisms underlying breathing complexity, PLoS ONE 8(10): e75740, DOI: 10.1371/journal.pone.0075740.
  13. Hofbauer, F. (1979). Maximal measures for piecewise monotonically increasing transformations on [0,1], in M. Denker and K. Jacobs (Eds), Ergodic Theory, Springer, Berlin/Heidelberg, pp. 66–77.
  14. Hofbauer, F. (1981). The maximal measure for linear mod. one transformations, Journal of the London Mathematical Society s2–23(1): 92–112, DOI: 10.1112/jlms/s2-23.1.92.
  15. Ibarz, B., Casado, J.M. and Sanjuán, M.A.F. (2011). Map-based models in neuronal dynamics, Physics Reports 501(1–2): 1–74, DOI: 10.1016/j.physrep.2010.12.003.
  16. Kameyama, A. (2002). Topological transitivity and strong transitivity, Acta Mathematica Universitatis Comenianae 71(2): 139–145.
  17. Korbicz, J., Patan, K. and Obuchowicz, A. (1999). Dynamic neural networks for process modelling in fault detection and isolation systems, International Journal of Applied Mathematics and Computer Science 9(3): 519–546.
  18. Llovera-Trujillo, F., Signerska-Rynkowska, J. and Bartłomiejczyk, P. (2023). Periodic and chaotic dynamics in a map-based neuron model, Mathematical Methods in the Applied Sciences 46(11): 11906–11931.
  19. Maslennikov, O.V. and Nekorkin, V.I. (2012). Discrete model of the olivo-cerebellar system: Structure and dynamics, Radiophysics and Quantum Electronics 55(3): 198–214, DOI: 10.1007/s11141-012-9360-6.
  20. Maslennikov, O.V. and Nekorkin, V.I. (2013). Dynamic boundary crisis in the Lorenz-type map, Chaos 23(2): 023129, DOI: 10.1063/1.4811545.
  21. Maslennikov, O.V., Nekorkin, V.I. and Kurths, J. (2018). Transient chaos in the Lorenz-type map with periodic forcing, Chaos 28(3): 033107, DOI:10.1063/1.5018265.
  22. Oprocha, P., Potorski, P. and Raith, P. (2019). Mixing properties in expanding Lorenz maps, Advances in Mathematics. 343: 712–755, DOI: 10.1016/j.aim.2018.11.015.
  23. Palmer, R. (1979). On the Classification of Measure Preserving Transformations of Lebesgue Spaces, PhD thesis, University of Warwick, Warwick, https://wrap.warwick.ac.uk/88796/1/WRAP_Theses_Palmer_2016.pdf.
  24. Parry, W. (1979). The Lorenz attractor and a related population model, in M. Denker and K. Jacobs (eds), Ergodic Theory, Springer, Berlin/Heidelberg, pp. 169–187, DOI: 10.1007/BFb0063293.
  25. Patan, K., Witczak, M. and Korbicz, J. (2008). Towards robustness in neural network based fault diagnosis, International Journal of Applied Mathematics and Computer Science 18(4): 443–454, DOI: 10.2478/v10006-008-0039-2.
  26. Rubin, J.E., Touboul, J.D., Signerska-Rynkowska, J. and Vidal, A. (2017). Wild oscillations in a nonlinear neuron model with resets. II: Mixed-mode oscillations, Discrete and Continuous Dynamical Systems B 22(10): 4003–4039, DOI: 10.3934/dcdsb.2017205.
  27. Yu, L., Mazancourt, M.D. and Hess, A. (2016). Functional connectivity and information flow of the respiratory neural network in chronic obstructive pulmonary disease, Human Brain Mapping 37(8): 2736–2754, DOI: 10.1002/hbm.23205.
  28. Yue, Y., Liu, Y.J., Song, Y.L., Chen, Y. and Yu, L. (2017). Information capacity and transmission in a Courbage–Nekorkin–Vdovin map-based neuron model, Chinese Physics Letters 34(4): 048701, DOI: 10.1088/0256-307x/34/4/048701.
DOI: https://doi.org/10.34768/amcs-2023-0028 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X
Language: English
Page range: 395 - 408
Submitted on: Oct 11, 2022
Accepted on: Apr 13, 2023
Published on: Sep 21, 2023
Published by: University of Zielona Góra
In partnership with: Paradigm Publishing Services
Publication frequency: 4 issues per year

© 2023 Piotr Bartłomiejczyk, Frank Llovera Trujillo, Justyna Signerska-Rynkowska, published by University of Zielona Góra
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.