Özel Başlangıç Koşulları Altında Lineer LIF Nöron Modelinin Analizi ve Çözüm Metodu

Leaky Integrate and Fire (LIF) modeli, nöronların matematiksel olarak modellenmesi ve çalışma prensiplerinin anlaşılması için yaygın olarak kullanılmaktadır. Birçok metot ve yöntem sayesinde modelin simülasyonu ve analizi yapılsa da mühendislik çalışmalarına uygun çözümlerin azlığından söz etmek mümkündür. Birinci dereceden adi diferansiyel denklemler içeren LIF modelinin çözümüne ideal başlangıç koşulları altında kolayca ulaşılırken, karmaşık şartlar sunulduğunda sonucu bulmak zorlaşmaktadır. Bu çalışmada nöronun, birim adım akımı, darbe akımı ve rastgele seçilen akım girişleri için çözümleri yapılmıştır. Böylece literatürde yer alan metotların özel durumlar ortaya çıktığında nasıl uygulanması gerektiği gösterilmiştir.

Anahtar Kelimeler:

LIF Nöron Modeli, Pasif Membran, Diferansiyel Denklemler

Analysis of Linear LIF Neuron Model under Particular Initial Value Conditions and Solution Method

The Leaky Integrate and Fire (LIF) model is widely used for mathematical modelling of neurons and understanding their working principles. Even though model is simulated and analyzed thanks to many methods and procedures, it is possible to mention about rarity of appropriate solutions for engineering studies. While solution of LIF model involving first order ordinary differential equations is easily obtained under ideal initial conditions, finding result is getting difficult when represented with complicated circumstances. In this study solutions for step current input, pulse current and arbitrary current input of neuron are elucidated. Therefore, it is demonstrated how to apply methods in literature when particular conditions emerged.

Keywords:

LIF Neuron Model, Passive Membrane, Differential Equations,

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Cavarretta F., Naldi G. Mathematical study of a nonlinear neuron model with active dendrites. AIMS Mathematics, 2019; 4(3): 831-846.
Coşkun Ö., Kahriman M., Çömlekçi S., Özkorucuklu S. Sinir hücresinin pasif kablo modellemesi ve simülasyonu. Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi 2012; 27(1): 1-9.
Daliri M., Ferreira PM., Klisnick G., Delai AB. A comparative study between E-neurons mathematical model and circuit model. IET Circuits, Devices and Systems 2021; 15(2): 175-182.
Edwards CH., Penney DE., Calvis DT. Differential equations and boundary value problems: computing and modelling. 5th ed. BOS: Pearson; 2014.
Fatoorehchi H., Abolghasemi H. Series solution of nonlinear differential equations by a novel extension of the laplace transform method. International Journal of Computer Mathematics 2016; 93(8): 1299-1319.
Gerstner W., Kistler WM., Naud R., Paninski L. Neuronal Dynamics: from single neurons to networks and models of cognition. CB: Cambridge University Press; 2014.
Hasan MM., Holleman J. Hardware model based simulation of spiking neuron using phase plane. IEEE International Symposium on Circuits and Systems (ISCAS), 22-28 Mayıs 2021, sayfa no:1-5, Daegu.
He JH. Variational iteration method- a kind of non-linear analytical technique: some examples. International Journal of Non-Linear Mechanics 1999; 34(4): 699-708.
Hodgkın AL., Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology 1952; 117(4):500-44.
Howell KB. Ordinary differential equations an introduction to the fundamentals. 2nd ed. FL: CRC Press;2020
https://academy.neuromatch.io/
https://briansimulator.org/
Li S., McLaughlin DW., Zhou D. Mathematical modeling and analysis of spatial neuron dynamics: dendritic integration and beyond. Communications on Pure and Applied Mathematics, 2021; https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.22020
Mishra HK., Tripathi R. Homotopy perturbation method of delay differential equation using he’s polynomial with laplace transform. The Proceedings of the National Academy of Sciences, India, Section A: Physical Sciences 2020; 90(2): 289–298.
Stöckel A., Eliasmith C. Passive nonlinear dendritic interactions as a computational resource in spiking neural networks 2021; 33(1): 96-128.
Thunibat RG., Jaradat EK., Khalifeh JM. Solution of non-linear rlc circuit equation using the homotopy perturbation transform method. Jordan Journal of Physics 2021; 14(1): 89-100.
Trench WF. Elementary differential equations with boundary value problems. CA: Brooks Cole; 2001.