Analisis Dinamik Kelangsungan Hidup Neuron Berdasarkan Aktivasi Mikroglia Menggunakan Model Stroke-Microglia-Damage

Sofiyah Fuaidah, Usman Pagalay, Erna Herawati, Mohammad Jamhuri

Abstract


Penelitian ini mengkaji dinamika model matematika neuroinflamasi yang dinyatakan dalam sistem persamaan diferensial nonlinier. Model melibatkan interaksi antara mikroglia istirahat, mikroglia pro-inflamasi, mikroglia anti-inflamasi, tingkat kerusakan jaringan, dan neuron hidup. Analisis dilakukan melalui penentuan titik kesetimbangan, linearisasi menggunakan matriks Jacobian, perhitungan nilai eigen, serta analisis kestabilan untuk memahami perilaku sistem di sekitar titik kesetimbangan. Selain itu, dianalisis pengaruh beberapa parameter model terhadap kestabilan neuron hidup. Hasil penelitian menunjukkan bahwa model memiliki tiga titik kesetimbangan, yaitu $E_1$, $E_2$, dan $E_3$. Berdasarkan analisis Jacobian dan nilai eigen, diperoleh bahwa sebagian titik kesetimbangan bersifat stabil, sedangkan yang lain tidak stabil, bergantung pada nilai parameter yang digunakan. Kondisi ini menunjukkan bahwa sistem neuroinflamasi dapat mencapai keadaan seimbang apabila faktor peradangan dapat dikendalikan. Analisis variabel memperlihatkan bahwa peningkatan mikroglia pro-inflamasi dan kerusakan jaringan menyebabkan penurunan jumlah neuron hidup, sedangkan peningkatan mikroglia anti-inflamasi berperan dalam mempertahankan kestabilan neuron. Simulasi numerik menggunakan MATLAB menunjukkan bahwa solusi sistem cenderung menuju titik kesetimbangan stabil yang diperoleh secara teoritis. Variasi parameter $b$, $c$, $\delta$, dan $q$ juga memberikan pengaruh signifikan terhadap kestabilan neuron hidup. Hasil simulasi mendukung analisis teoritis dan memberikan gambaran matematis mengenai dinamika neuroinflamasi pasca-stroke.

Keywords


Analisis Dinamik; Kestabilan; Model Matematika; Neuroinflamasi; Stroke-Microglia-Damage

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References


[1] S. Amato and A. Arnold, “Modeling microglia activation and inflammation-based neuroprotectant strategies during ischemic stroke,” Bulletin of Mathematical Biology, vol. 83, no. 6, pp. 1–27, 2021. doi: 10.1007/s11538-021-00905-4.

[2] S. Amato and A. Arnold, “A data-informed mathematical model of microglial cell dynamics during ischemic stroke in the middle cerebral artery,” Bulletin of Mathematical Biology, vol. 87, no. 2, pp. 1–25, 2025. doi: 10.1007/s11538-025-01412-6.

[3] Q. Zhang et al., “M2 microglia-derived small extracellular vesicles modulate nsc fate after ischemic stroke via mir-25-3p/mir-93-5p-tgfbr/pten/foxo3 axis,” Journal of Nanobiotechnology, vol. 23, p. 311, 2025. doi: 10.1186/s12951-025-03390-2.

[4] A. J. Alqarni, A. S. Rambely, and I. Hashim, “Dynamic modelling of interactions between microglia and endogenous neural stem cells in the brain during a stroke,” Mathematics, vol. 8, no. 1, pp. 1–21, 2020. doi: 10.3390/math8010132.

[5] R. K. Leak and J. Chen, “Microglial and macrophage polarization: New prospects for brain repair,” Nature Reviews Neurology, vol. 11, no. 1, pp. 56–64, 2015.

[6] L. E. Vaughan, P. R. Ranganathan, R. G. Kumar, A. K. Wagner, and J. E. Rubin, “A mathematical model of neuroinflammation in severe clinical traumatic brain injury,” Journal of Neuroinflammation, vol. 15, no. 1, pp. 1–19, 2018. doi: 10.1186/s12974-018-1384-1.

[7] U. Dirnagl, C. Iadecola, and M. A. Moskowitz, “Pathobiology of ischaemic stroke: An integrated view,” Trends in Neurosciences, vol. 22, no. 9, pp. 391–397, 1999. doi: 10.1016/S0166-2236(99)01401-0.

[8] W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems. Wiley, 2009.

[9] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd ed. CRC Press, 2018.

[10] C. Iadecola and J. Anrather, “The immunology of stroke: From mechanisms to translation,” Nature Medicine, vol. 17, no. 7, pp. 796–808, 2011. doi: 10.1038/nm.2399.

[11] L. Perko, Differential Equations and Dynamical Systems, 3rd ed. Springer, 2010.

[12] D. G. Zill, A First Course in Differential Equations with Modeling Applications, 11th ed. Cengage Learning, 2017.

[13] E. R. Scheinerman, Invitation to Dynamical Systems. Prentice Hall, 1996.

[14] Widowati, S. P. Putro, and Silfiana, “Stability analysis of the phytoplankton effect model on changes in nitrogen concentration on integrated multi-trophic aquaculture systems,” Journal of Physics: Conference Series, vol. 1025, no. 1, 2018. doi: 10.1088/1742-6596/1025/1/012088.

[15] M. Prinz and J. Priller, “Microglia and brain macrophages in the molecular age: From origin to neuropsychiatric disease,” Nature Reviews Neuroscience, vol. 15, no. 5, pp. 300–312, 2014. doi: 10.1038/nrn3722.

[16] S. Saleem, A. Raza, M. Lampart, M. Rafiq, N. Ahmed, and M. S. Arif, “Numerical solutions for norovirus epidemic spread: Implications for public health control,” Scientific Reports, vol. 15, no. 1, pp. 1–19, 2025. doi: 10.1038/s41598-025-14688-4.




DOI: https://doi.org/10.18860/jrmm.v5i5.43698

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