Fraksiyonel dereceli FitzHugh-Nagumo nöron modelinin devre sentezi için alternatif bir yaklaşım

Year 2022, Volume: 28 Issue: 2, 248 - 254, 30.04.2022

Nimet Korkmaz İbrahim Ethem Saçu

Abstract

Bu çalışmada FitzHugh-Nagumo (FHN) nöron modelinin fraksiyonel versiyonu üzerinde durulmuştur. Öncelikle fraksiyonel dereceli FHN nöron modelinin kararlılık analizleri yapılarak, sistemin dinamik davranış sergileyebileceği minimum fraksiyonel derece belirlenmiştir. Ardından fraksiyonel derece ile temsil edilen sistemlerin nümerik analizlerinde kullanılan yöntemlerden biri olan Grünwald-Letnikov (G-L) fraksiyonel türev yöntemi ile fraksiyonel dereceli FHN nöron modelinin yanıtları elde edilmiştir.Nöron modellerinin donanımsal çözümleri sayesinde matematiksel olarak tanımlanan sistemlerin yanıtları gerçek zamanlı işaretler şeklinde elde edilebilir; nöronların hücre zarı özellikleri elektromekanik olarak tanımlanabilir ve nöronların dinamik davranışlarını etkileyen parametreler, donanım çözümlerinde kullanılan elektronik elemanların karakteristikleri ile ilişkilendirilebilir. Biyolojiden esinlenilerek geliştirilen sistemlerde fraksiyonel dereceli hesaplamaların kullanılabilirliğinin görülmesi amacıyla, bu çalışmada fraksiyonel dereceli FHN nöron modelinin devre gerçekleştirimi üzerinde durulmuştur. Bu kapsamda, diferansiyel denklemlerin donanım çözümlerinde op-amp, direnç ve kapasitör elemanları kullanılarak tasarlanan integratör devrelerinde; fraksiyonel derecenin karşılanması için klasik kapasitör elemanları yerine R-C taklit devreleri kullanılmıştır. R-C taklit devrelerinin tasarımının ilk aşamasında Matsuda yaklaşıklık metodu ile üçüncü dereceden bir transfer fonksiyonu elde edilmiştir. Elde edilen bu transfer fonksiyonu, FOSTER-I R-C ağına dönüştürülerek tamsayı dereceli FHN nöron modelinin devre gerçekleştirim çözümü için tarafımızca tasarlanan devredeki integratör bloklarında, klasik kapasitör elemanı yerine kullanılmıştır. Böylece fraksiyonel dereceli FHN nöron modelinin devre çözümü için alternatif bir yaklaşım ortaya konmuştur ve bu yapının doğrulaması SPICE devre simülasyonu ile yapılmıştır.

Keywords

FitzHugh-Nagumo Nöron modeli, Fraksiyonel kapasitör, Devre sentezi, Matsuda yaklaşıklık yöntemi, FOSTER-I ağı

An alternative approach for the circuit synthesis of the fractional-order FitzHugh-Nagumo neuron model

Abstract

This study focuses on the fractional version of the FitzHugh-Nagumo (FHN) neuron model. Firstly, the stability analysis of the fractionalorder FHN neuron model has been performed and the minimum fractional degree, at which the system could exhibit dynamic behavior, has been determined. Then, the responses of the fractional-order FHN neuron model have been obtained using the Grünwald-Letnikov (G-L) fractional derivative method. This method is one of the methods used in the numerical analysis of the systems that are represented by fractional order. Thanks to the hardware solutions of neuron models; the responses of mathematically defined systems can be obtained in the form of real-time signals, the cell membrane properties of the neurons can be described electromechanically, and the parameters that affect the dynamic behavior of neurons can be associated with the characteristics of the electronic components used in hardware solutions. In this study, the circuit implementation of the fractionalorder FHN neuron model is emphasized in order to see the usability of fractional-order calculations in systems that are inspired by biology. In this context, the R-C mimetic circuits have been used instead of classical capacitor elements to compensate for the fractional order in the integrator circuits that are designed by using op-amp, resistor and capacitor elements for the hardware solutions of the differential equations. In the first stage of the design of these R-C imitation circuits, a third-order transfer function has been obtained by the Matsuda approximation method. This obtained transfer function has been transformed into FOSTER-I R-C network and it has been used instead of the classical capacitor element in the integrator blocks of the circuit that is designed by us for the circuit implementation solution of the integer-order FHN neuron model. Thus, an alternative approach for circuit solution of the fractional-order FHN neuron model has been introduced and the verification of this structure has been made by the SPICE circuit simulation.

Keywords

FitzHugh-Nagumo neuron model, Fractional capacitor, Circuit synthesis, Matsuda approximation method, FOSTER-Inetwork

References

• [1] Hodgkin AL, Huxley A F. “A quantitative description of membrane current and its application to conduction and excitation in nerve”. The Journal of Physiology, 117(4), 500-544, 1952.
• [2] Özer M. “İyonik kanal aktivasyon ve inaktivasyon kapılarının dinamik davranışı için alternatif denklemler”. Pamukkale Üniversitesi Mühendislik Bilimleri Dergisi, 9(3), 349-356, 2003.
• [3] FitzHugh R. Mathematical Models for Excitation and Propagation in Nerve. Editor: Schawn HP. Biological Engineering, 1-85, New York, USA, McGraw-Hill, 1969.
• [4] Azar AT, Radwan AG, Vaidyanathan S. Fractional Order Systems, 1st ed. San Diego, USA, Elsevier, Academic Press, 2018.
• [5] Podlubny I. "Fractional-order systems and PIλDαcontrollers". IEEE Transactions on Automatic Control, 44(1), 208-214, 1999.
• [6] Freeborn TJ. “A survey of fractional-order circuit models for biology and biomedicine”. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 3(3), 416-424, 2013.
• [7] Sacu IE, Alci M. “Low-power OTA-C based tuneable fractional order filters”. Journal of Microelectronics, Electronic Components and Materials, 48(3), 135-144, 2018.
• [8] Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. 1st ed. California, USA, Elsevier, Academic Press, 1998.
• [9] Atangana A, Alkahtani BST. “Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel”. Advances in Mechanical Engineering, 7(6), 1-6, 2015.
• [10] Hindmarsh JL, Rose RM. “A model of neural bursting using three couple first order differential equations”. Proceedings of the Royal society of London. Series B. Biological Sciences, 221(1222), 87-102, 1984.
• [11] Izhikevich EM. “Simple model of spiking neurons”. IEEE Transactions on Neural Networks, 14(6), 1569-1572, 2003.
• [12] Korkmaz N, Öztürk İ, Kılıç R. "Multiple perspectives on the hardware implementations of biological neuron models and programmable design aspects". Turkish Journal of Electrical Engineering & Computer Sciences, 24(3), 1729-1746, 2016.
• [13] Lazaridis E, Drakakis EM. “Full analogue electronic realisation of the Hodgkin-Huxley neuronal dynamics in weak-inversion CMOS”. 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Lyon, France, 22-26 August 2007.
• [14] Linares-Barranco B, Sanchez-Sinencio E, RodriguezVazquez A, Huertas JL. “A CMOS implementation of FitzHugh-Nagumo neuron model”. IEEE Journal of Solid-State Circuits, 26(7), 956-965, 1991.
• [15] Weinstein RK, Lee RH. “Architectures for highperformance FPGA implementations of neural models”. Journal of Neural Engineering, 3, 21-34, 2006.
• [16] Malik S A, Mir A H. “FPGA realization of fractional order neuron”. Applied Mathematical Modelling, 81, 372-385, 2020.
• [17] Tolba MF, Elsafty AH, Armanyos M, Said LA, Madian AH, Radwan AG. “Synchronization and FPGA realization of fractional-order Izhikevich neuron model”. Microelectronics Journal, 89, 56-69, 2019.
• [18] Khanday FA, Kant NA, Dar MR, Zulkifli TZA, Psychalinos C. “Low-voltage low-power integrable CMOS circuit implementation of integer-and fractional-order FitzHughNagumo neuron model”. IEEE Transactions on Neural Networks and Learning Systems, 30(7), 2108-2122, 2018.
• [19] Matsuda K, Fujii H. “H (infinity) optimized wave-absorbing control-analytical and experimental results”. Journal of Guidance, Control, and Dynamics, 16(6), 1146-1153, 1993.
• [20] Elwy O, Rashad SH, Said LA, Radwan AG. “Comparison between three approximation methods on oscillator circuits”. Microelectronics Journal, 81, 162-178, 2018.
• [21] Tavazoei MS, Haeri M. “A necessary condition for double scroll attractor existence in fractional-order systems”. Physics Letters A, 367(1-2), 102-113, 2007.
• [22] Tavazoei MS, Haeri M. “A note on the stability of fractional order systems”. Mathematics and Computers in Simulation, 79(5), 1566-1576, 2009.