Geodesic
Spike patterns and chaos in a map-based neuron model
International Journal of Applied Mathematics and Computer Science, Tome 33 (2023) no. 3, pp. 395-408.

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The work studies the well-known map-based model of neuronal dynamics introduced in 2007 by Courbage, Nekorkin and Vdovin, important due to various medical applications. We also review and extend some of the existing results concerning β-transformations and (expanding) Lorenz mappings. Then we apply them for deducing important properties of spike-trains generated by the CNV model and explain their implications for neuron behaviour. In particular, using recent theorems of rotation theory for Lorenz-like maps, we provide a classification of periodic spiking patterns in this model.
Keywords: neuronal dynamics, β-transformations, Farey–Lorenz permutations, periodic spiking, chaos
Mots-clés : dynamika neuronalna, permutacje Fareya-Lorenza, chaos 
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Bartłomiejczyk, Piotr; Llovera Trujillo; Signerska-Rynkowska, Justyna. Spike patterns and chaos in a map-based neuron model. International Journal of Applied Mathematics and Computer Science, Tome 33 (2023) no. 3, pp. 395-408. http://geodesic.mathdoc.fr/item/IJAMCS_2023_33_3_a3/

[1] [1] Afraimovich, V. and Hsu, S. (2002). Lectures on Chaotic Dynamical Systems, American Mathematical Society, Providence.

[2] [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] [3] Cholewa, Ł. and Oprocha, P. (2021a). On α-limit sets in Lorenz maps, Entropy 23(9): 1153. DOI: 10.3390/e23091153.

[4] [4] Cholewa, Ł. and Oprocha, P. (2021b). Renormalization in Lorenz maps-Completely invariant sets and periodic orbits. arXiv: 2104.00110[math.DS].

[5] [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] [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] [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] [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] [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] [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] [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] [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] [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] [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] [15] Ibarz, B., Casado, J.M. and Sanju´an, 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] [16] Kameyama, A. (2002). Topological transitivity and strong transitivity, Acta Mathematica Universitatis Comenianae 71(2): 139-145.

[17] [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] [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] [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] [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] [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] [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] [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] [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] [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] [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] [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] [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.