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A Conformal Fractional Derivative-based Leaky Integrate-and-Fire Neuron Model

Yıl 2022, Cilt: 26 Sayı: 3, 568 - 578, 30.06.2022

İsmail Devecioğlu Reşat Mutlu

https://doi.org/10.16984/saufenbilder.1041088
Cited By: 1

Öz

Neuron model have been extensively studied and different models have been proposed. Nobel laureate Hodgkin-Huxley model is physiologically relevant and can demonstrate different neural behaviors, but it is mathematically complex. For this reason, simplified neuron models such as integrate-and-fire model and its derivatives are more popular in the literature to study neural populations. Lapicque’s integrate-and-fire model is proposed in 1907 and its leaky integrate-and-fire version is very popular due to its simplicity. In order to improve this simple model and capture different aspects of neurons, a variety of it have been proposed. Fractional order derivative-based neuron models are one of those varieties, which can show adaptation without necessitating additional differential equations. However, fractional-order derivatives could be computationally costly. Recently, a conformal fractional derivative (CFD) is suggested in literature. It is easy to understand and implement compared to the other methods. In this study, a CFD-based leaky integrate-and-fire neuron model is proposed. The model captures the adaptation in firing rate under sustained current injection. Results suggest that it could be used to easily and efficiently implement network models as well as to model different sensory afferents.

Anahtar Kelimeler

fractional order derivatives conformal fractional derivative leaky integrate-and-fire neuron model

Teşekkür

We thank to editors and reviewers for their time and effort to evaluate our manuscript and for their valuable feedbacks.

Kaynakça

  • [1] L. Lapicque, “Recherches quantitatives sur l'excitation electrique des nerfs traitee comme une polarization,” Journal de Physiologie et de Pathologie Generalej, vol. 9, pp. 620-635, 1907.
  • [2] N. Brunel and M.C. Van Rossum, “Lapicque’s 1907 paper: from frogs to integrate-and-fire,” Biological Cybernetics, vol. 97, no. 5-6, pp. 337-339, 2007.
  • [3] A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” The Journal of Physiology, vol. 117, no. 4, pp. 500-544, 1952.
  • [4] R. FitzHugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophysical Journal, vol. 1, no. 6, pp. 445-466, 1961.
  • [5] J. Nagumo, S. Arimoto and S. Yoshizawa, “An active pulse transmission line simulating nerve axon,” Proceedings of the IRE, vol. 50, no. 10, pp. 2061-2070, 1962.
  • [6] R. FitzHugh, “Mathematical models of excitation and propagation in nerve,” Biological Engineering, vol. 1, no. 9, pp. 1-85, 1969.
  • [7] E. M. Izhikevich, “Simple model of spiking neurons,” IEEE Transactions on Neural Networks, vol. 14, no. 6, pp. 1569-1572, 2003.
  • [8] E. M. Izhikevich, “Which model to use for cortical spiking neurons?” IEEE Transactions on Neural Networks, vol. 15, no. 5, pp. 1063-1070, 2004.
  • [9] E. M. Izhikevich and F. Hoppensteadt, “Classification of bursting mappings,”. International Journal of Bifurcation and Chaos, vol. 14, no. 11, pp. 3847-3854, 2004.
  • [10] E. M. Izhikevich, J. A. Gally and G. M. Edelman, “Spike-timing dynamics of neuronal groups,” Cerebral Cortex, vol. 14, no. 8, pp. 933-944, 2004.
  • [11] B. Ross, “The development of fractional calculus 1695–1900,” Historia Mathematica, vol. 4, no. 1, pp. 75-89, 1977.
  • [12] I. Podlubny “Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications,” Elsevier, 1998.
  • [13] A. A. Kilbas, H.M Srivastava and J.J. Trujillo, “Theory and applications of fractional differential equations,” Elsevier, 2006.
  • [14] A. Babiarz, A. Czornik, J. Klamka and M. Niezabitowski, “Theory and applications of non-integer order systems,” Lecture Notes Electrical Engineering, vol. 407, 2017
  • [15] X.J. Yang, “General fractional derivatives: theory, methods, and applications,” Chapman and Hall/CRC, 2019.
  • [16] T.J. Freeborn, “A survey of fractional-order circuit models for biology and biomedicine,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 3, no. 3, pp. 416-424, 2013.
  • [17] G.W. Leibnitz, “Leibnitzen's Mathematische Schriften,” Hildesheim, vol.2, pp. 301-302, 1962.
  • [18] A. Karcı, “Fractional order derivative and relationship between derivative and complex functions,” Mathematical Sciences and Applications E-Notes, vol. 2, no. 1, pp. 44-54, 2004.
  • [19] R. Khalil, M. al Horani, A. Yousef, M. Sababheh, “A new definition of fractional derivative,” Journal of Computational and Applied Mathematics, vol. 264, pp. 65–70, 2014.
  • [20] T. Abdeljawad, “On conformable fractional calculus,” Journal of Computational and Applied Mathematics, vol. 279, pp. 57-66, 2015.
  • [21] D. Zhao and M. Luo, “General conformable fractional derivative and its physical interpretation,” Calcolo, vol. 54, no. 3, pp. 903-917, 2017.
  • [22] R. Sikora, “Fractional derivatives in electrical circuit theory–critical remarks,” Archives of Electrical Engineering, vol. 66, no. 1, pp. 155-163, 2017.
  • [23] M. Lewandowski and M. Orzyłowski, “Fractional-order models: The case study of the supercapacitor capacitance measurement,” Bulletin of the Polish Academy of Sciences Technical Sciences, vol. 65, no. 4, pp. 449-457, 2017.
  • [24] R. Kopka “Estimation of supercapacitor energy storage based on fractional differential equations,” Nanoscale Research Letters, vol. 12, no. 1, pp. 636, 2017.
  • [25] T. J. Freeborn, A. S. Elwakil and A. Allagui, “Supercapacitor fractional-order model discharging from polynomial time-varying currents,” in 2018 IEEE International Symposium on Circuits and Systems (ISCAS), pp. 1-5, 2018.
  • [26] T. J. Freeborn, B. Maundy and A.S. Elwakil, “Measurement of supercapacitor fractional-order model parameters from voltage-excited step response,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 3, no. 3, pp. 367-376, 2013.
  • [27] A. Kartci, A. Agambayev, N. Herencsar and K. N. Salama, “Series-, parallel-, and inter-connection of solid-state arbitrary fractional-order capacitors: theoretical study and experimental verification,” IEEE Access, vol. 6, pp. 10933-10943, 2018.
  • [28] E. Piotrowska, “Analysis the conformable fractional derivative and Caputo definitions in the action of an electric circuit containing a supercapacitor,” in “Photonics applications in astronomy, communications, industry, and high-energy physics experiments 2018” vol. 10808, pp. 108081T, International Society for Optics and Photonics, 2018.
  • [29] U. Palaz and R. Mutlu, “Analysis of a capacitor modelled with conformable fractional derivative under dc and sinusoidal signals,” Celal Bayar University Journal of Science, vol. 17, no. 2, pp. 193-198, 2021.
  • [30] A. A. H. A. Mohammed, K. Kandemir and R. Mutlu, “Analysis of parallel resonance circuit consisting of a capacitor modelled using conformal fractional order derivative using Simulink,” European Journal of Engineering and Applied Sciences, vol. 3, no. 1, pp.13-18, 2020.
  • [31] U. Palaz and R. Mutlu, “Two capacitor problem with a lti capacitor and a capacitor modelled using conformal fractional order derivative,” European Journal of Engineering and Applied Sciences, vol. 4, no. 1, pp. 8-13, 2021.
  • [32] E. Piotrowska, “Analysis of fractional capacitor and coil by the use of the Conformable Fractional Derivative and Caputo definitions,” In 2018 IEEE International Interdisciplinary PhD Workshop (IIPhDW) (pp. 103-107), 2018.
  • [33] E. Piotrowska, “Analysis of fractional electrical circuit with sinusoidal input signal using Caputo and conformable derivative definitions,” Poznan University of Technology Academic Journals. Electrical Engineering, vol. 97, pp.155-167, 2019.
  • [34] E. Piotrowska and L. Sajewski, “Analysis of an electrical circuit using two-parameter conformable operator in the Caputo sense,” Symmetry, vol. 13, no. 5, pp. 771, 2021.
  • [35] B. N. Lundstrom, M. H. Higgs, W. J. Spain and A. L. Fairhall, “Fractional differentiation by neocortical pyramidal neurons,” Nature Neuroscience, vol. 11, no. 11, pp. 1335-1342, 2008.
  • [36] W. Teka, T.M. Marinov and F. Santamaria, “Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model,” PLoS Computational Biology, vol. 10, no. 3, pp. e1003526, 2014.
  • [37] O. Taşbozan and A. Kurt, “New exact solutions of fractional Fitzhugh-Nagumo equation,” Journal of the Institute of Science and Technology, vol. 9, no. 3, pp. 1633-1645, 2019.
  • [38] W. Gerstner, W. M. Kistler, R. Naud and L. Paninski, “Neuronal dynamics: From single neurons to networks and models of cognition,” Cambridge University Press, 2014.
  • [39] F. Amzica and D. A. G. Neckelmann,”Membrane capacitance of cortical neurons and glia during sleep oscillations and spike-wave seizures,” Journal of Neurophysiology, vol. 82, no. 5, pp. 2731-2746, 1999.
  • [40] İ. Devecioğlu and B. Güçlü, “Asymmetric response properties of rapidly adapting mechanoreceptive fibers in the rat glabrous skin,” Somatosensory & Motor Research, vol. 30, no. 1, pp. 16-29, 2013.
  • [41] E. P. Gardner and K. O. Johnson, “Touch” in E. R. Kandel, J. H. Schwartz, T. M. Jessell, S. A. Siegelbaum, and A. J. Hudspeth (Eds.), “Principles of neural science”, 5th ed., pp. 498-529. New York: McGraw-Hill, 2012.
  • [42] I. Devecioglu, S. C. Yener and R. Mutlu, “Demonstration of synaptic connections with unipolar junction transistor based neuron emulators,” International Journal of Engineering Transactions B: Applications, vol. 33, no. 11, pp. 2195-2200, 2020.
  • [43] F. Tulumbaci, M. H. Eryildiz and R. Mutlu, “A simple lapicque neuron emulator,” in 6th International Conference on Electrical Engineering and Electronics (EEE’20) (13.08.2020-15.08.2020).

Yıl 2022, Cilt: 26 Sayı: 3, 568 - 578, 30.06.2022

İsmail Devecioğlu Reşat Mutlu

https://doi.org/10.16984/saufenbilder.1041088
Cited By: 1

Öz

Kaynakça

  • [1] L. Lapicque, “Recherches quantitatives sur l'excitation electrique des nerfs traitee comme une polarization,” Journal de Physiologie et de Pathologie Generalej, vol. 9, pp. 620-635, 1907.
  • [2] N. Brunel and M.C. Van Rossum, “Lapicque’s 1907 paper: from frogs to integrate-and-fire,” Biological Cybernetics, vol. 97, no. 5-6, pp. 337-339, 2007.
  • [3] A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” The Journal of Physiology, vol. 117, no. 4, pp. 500-544, 1952.
  • [4] R. FitzHugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophysical Journal, vol. 1, no. 6, pp. 445-466, 1961.
  • [5] J. Nagumo, S. Arimoto and S. Yoshizawa, “An active pulse transmission line simulating nerve axon,” Proceedings of the IRE, vol. 50, no. 10, pp. 2061-2070, 1962.
  • [6] R. FitzHugh, “Mathematical models of excitation and propagation in nerve,” Biological Engineering, vol. 1, no. 9, pp. 1-85, 1969.
  • [7] E. M. Izhikevich, “Simple model of spiking neurons,” IEEE Transactions on Neural Networks, vol. 14, no. 6, pp. 1569-1572, 2003.
  • [8] E. M. Izhikevich, “Which model to use for cortical spiking neurons?” IEEE Transactions on Neural Networks, vol. 15, no. 5, pp. 1063-1070, 2004.
  • [9] E. M. Izhikevich and F. Hoppensteadt, “Classification of bursting mappings,”. International Journal of Bifurcation and Chaos, vol. 14, no. 11, pp. 3847-3854, 2004.
  • [10] E. M. Izhikevich, J. A. Gally and G. M. Edelman, “Spike-timing dynamics of neuronal groups,” Cerebral Cortex, vol. 14, no. 8, pp. 933-944, 2004.
  • [11] B. Ross, “The development of fractional calculus 1695–1900,” Historia Mathematica, vol. 4, no. 1, pp. 75-89, 1977.
  • [12] I. Podlubny “Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications,” Elsevier, 1998.
  • [13] A. A. Kilbas, H.M Srivastava and J.J. Trujillo, “Theory and applications of fractional differential equations,” Elsevier, 2006.
  • [14] A. Babiarz, A. Czornik, J. Klamka and M. Niezabitowski, “Theory and applications of non-integer order systems,” Lecture Notes Electrical Engineering, vol. 407, 2017
  • [15] X.J. Yang, “General fractional derivatives: theory, methods, and applications,” Chapman and Hall/CRC, 2019.
  • [16] T.J. Freeborn, “A survey of fractional-order circuit models for biology and biomedicine,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 3, no. 3, pp. 416-424, 2013.
  • [17] G.W. Leibnitz, “Leibnitzen's Mathematische Schriften,” Hildesheim, vol.2, pp. 301-302, 1962.
  • [18] A. Karcı, “Fractional order derivative and relationship between derivative and complex functions,” Mathematical Sciences and Applications E-Notes, vol. 2, no. 1, pp. 44-54, 2004.
  • [19] R. Khalil, M. al Horani, A. Yousef, M. Sababheh, “A new definition of fractional derivative,” Journal of Computational and Applied Mathematics, vol. 264, pp. 65–70, 2014.
  • [20] T. Abdeljawad, “On conformable fractional calculus,” Journal of Computational and Applied Mathematics, vol. 279, pp. 57-66, 2015.
  • [21] D. Zhao and M. Luo, “General conformable fractional derivative and its physical interpretation,” Calcolo, vol. 54, no. 3, pp. 903-917, 2017.
  • [22] R. Sikora, “Fractional derivatives in electrical circuit theory–critical remarks,” Archives of Electrical Engineering, vol. 66, no. 1, pp. 155-163, 2017.
  • [23] M. Lewandowski and M. Orzyłowski, “Fractional-order models: The case study of the supercapacitor capacitance measurement,” Bulletin of the Polish Academy of Sciences Technical Sciences, vol. 65, no. 4, pp. 449-457, 2017.
  • [24] R. Kopka “Estimation of supercapacitor energy storage based on fractional differential equations,” Nanoscale Research Letters, vol. 12, no. 1, pp. 636, 2017.
  • [25] T. J. Freeborn, A. S. Elwakil and A. Allagui, “Supercapacitor fractional-order model discharging from polynomial time-varying currents,” in 2018 IEEE International Symposium on Circuits and Systems (ISCAS), pp. 1-5, 2018.
  • [26] T. J. Freeborn, B. Maundy and A.S. Elwakil, “Measurement of supercapacitor fractional-order model parameters from voltage-excited step response,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 3, no. 3, pp. 367-376, 2013.
  • [27] A. Kartci, A. Agambayev, N. Herencsar and K. N. Salama, “Series-, parallel-, and inter-connection of solid-state arbitrary fractional-order capacitors: theoretical study and experimental verification,” IEEE Access, vol. 6, pp. 10933-10943, 2018.
  • [28] E. Piotrowska, “Analysis the conformable fractional derivative and Caputo definitions in the action of an electric circuit containing a supercapacitor,” in “Photonics applications in astronomy, communications, industry, and high-energy physics experiments 2018” vol. 10808, pp. 108081T, International Society for Optics and Photonics, 2018.
  • [29] U. Palaz and R. Mutlu, “Analysis of a capacitor modelled with conformable fractional derivative under dc and sinusoidal signals,” Celal Bayar University Journal of Science, vol. 17, no. 2, pp. 193-198, 2021.
  • [30] A. A. H. A. Mohammed, K. Kandemir and R. Mutlu, “Analysis of parallel resonance circuit consisting of a capacitor modelled using conformal fractional order derivative using Simulink,” European Journal of Engineering and Applied Sciences, vol. 3, no. 1, pp.13-18, 2020.
  • [31] U. Palaz and R. Mutlu, “Two capacitor problem with a lti capacitor and a capacitor modelled using conformal fractional order derivative,” European Journal of Engineering and Applied Sciences, vol. 4, no. 1, pp. 8-13, 2021.
  • [32] E. Piotrowska, “Analysis of fractional capacitor and coil by the use of the Conformable Fractional Derivative and Caputo definitions,” In 2018 IEEE International Interdisciplinary PhD Workshop (IIPhDW) (pp. 103-107), 2018.
  • [33] E. Piotrowska, “Analysis of fractional electrical circuit with sinusoidal input signal using Caputo and conformable derivative definitions,” Poznan University of Technology Academic Journals. Electrical Engineering, vol. 97, pp.155-167, 2019.
  • [34] E. Piotrowska and L. Sajewski, “Analysis of an electrical circuit using two-parameter conformable operator in the Caputo sense,” Symmetry, vol. 13, no. 5, pp. 771, 2021.
  • [35] B. N. Lundstrom, M. H. Higgs, W. J. Spain and A. L. Fairhall, “Fractional differentiation by neocortical pyramidal neurons,” Nature Neuroscience, vol. 11, no. 11, pp. 1335-1342, 2008.
  • [36] W. Teka, T.M. Marinov and F. Santamaria, “Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model,” PLoS Computational Biology, vol. 10, no. 3, pp. e1003526, 2014.
  • [37] O. Taşbozan and A. Kurt, “New exact solutions of fractional Fitzhugh-Nagumo equation,” Journal of the Institute of Science and Technology, vol. 9, no. 3, pp. 1633-1645, 2019.
  • [38] W. Gerstner, W. M. Kistler, R. Naud and L. Paninski, “Neuronal dynamics: From single neurons to networks and models of cognition,” Cambridge University Press, 2014.
  • [39] F. Amzica and D. A. G. Neckelmann,”Membrane capacitance of cortical neurons and glia during sleep oscillations and spike-wave seizures,” Journal of Neurophysiology, vol. 82, no. 5, pp. 2731-2746, 1999.
  • [40] İ. Devecioğlu and B. Güçlü, “Asymmetric response properties of rapidly adapting mechanoreceptive fibers in the rat glabrous skin,” Somatosensory & Motor Research, vol. 30, no. 1, pp. 16-29, 2013.
  • [41] E. P. Gardner and K. O. Johnson, “Touch” in E. R. Kandel, J. H. Schwartz, T. M. Jessell, S. A. Siegelbaum, and A. J. Hudspeth (Eds.), “Principles of neural science”, 5th ed., pp. 498-529. New York: McGraw-Hill, 2012.
  • [42] I. Devecioglu, S. C. Yener and R. Mutlu, “Demonstration of synaptic connections with unipolar junction transistor based neuron emulators,” International Journal of Engineering Transactions B: Applications, vol. 33, no. 11, pp. 2195-2200, 2020.
  • [43] F. Tulumbaci, M. H. Eryildiz and R. Mutlu, “A simple lapicque neuron emulator,” in 6th International Conference on Electrical Engineering and Electronics (EEE’20) (13.08.2020-15.08.2020).
Toplam 43 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Araştırma Makalesi
Yazarlar

İsmail Devecioğlu 0000-0003-4119-617X

Reşat Mutlu 0000-0003-0030-7136

Yayımlanma Tarihi 30 Haziran 2022
Gönderilme Tarihi 24 Aralık 2021
Kabul Tarihi 6 Mayıs 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 26 Sayı: 3

Kaynak Göster

APA Devecioğlu, İ., & Mutlu, R. (2022). A Conformal Fractional Derivative-based Leaky Integrate-and-Fire Neuron Model. Sakarya University Journal of Science, 26(3), 568-578. https://doi.org/10.16984/saufenbilder.1041088
AMA Devecioğlu İ, Mutlu R. A Conformal Fractional Derivative-based Leaky Integrate-and-Fire Neuron Model. SAUJS. Haziran 2022;26(3):568-578. doi:10.16984/saufenbilder.1041088
Chicago Devecioğlu, İsmail, ve Reşat Mutlu. “A Conformal Fractional Derivative-Based Leaky Integrate-and-Fire Neuron Model”. Sakarya University Journal of Science 26, sy. 3 (Haziran 2022): 568-78. https://doi.org/10.16984/saufenbilder.1041088.
EndNote Devecioğlu İ, Mutlu R (01 Haziran 2022) A Conformal Fractional Derivative-based Leaky Integrate-and-Fire Neuron Model. Sakarya University Journal of Science 26 3 568–578.
IEEE İ. Devecioğlu ve R. Mutlu, “A Conformal Fractional Derivative-based Leaky Integrate-and-Fire Neuron Model”, SAUJS, c. 26, sy. 3, ss. 568–578, 2022, doi: 10.16984/saufenbilder.1041088.
ISNAD Devecioğlu, İsmail - Mutlu, Reşat. “A Conformal Fractional Derivative-Based Leaky Integrate-and-Fire Neuron Model”. Sakarya University Journal of Science 26/3 (Haziran 2022), 568-578. https://doi.org/10.16984/saufenbilder.1041088.
JAMA Devecioğlu İ, Mutlu R. A Conformal Fractional Derivative-based Leaky Integrate-and-Fire Neuron Model. SAUJS. 2022;26:568–578.
MLA Devecioğlu, İsmail ve Reşat Mutlu. “A Conformal Fractional Derivative-Based Leaky Integrate-and-Fire Neuron Model”. Sakarya University Journal of Science, c. 26, sy. 3, 2022, ss. 568-7, doi:10.16984/saufenbilder.1041088.
Vancouver Devecioğlu İ, Mutlu R. A Conformal Fractional Derivative-based Leaky Integrate-and-Fire Neuron Model. SAUJS. 2022;26(3):568-7.

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30930  This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.