Due to known information processing capabilities of the brain, neurons are modeled at many different levels. Circuit theory is also often used to describe the function of neurons, especially in complex multi-compartment models, but when used for simple models, there is no subsequent biological justification of used parts. We propose a new single-compartment model of excitatory and inhibitory neuron, the capacitor-switch model of excitatory and inhibitory neuron, as an extension of the existing integrate-and-fire model, preserving the signal properties of more complex multi-compartment models. The correspondence to existing structures in the neuronal cell is then discussed for each part of the model. We demonstrate that a few such inter-connected model units are capable of acting as a chaotic oscillator dependent on fire patterns of the input signal providing a complex deterministic and specific response through the output signal. The well-known necessary conditions for constructing a chaotic oscillator are met for our presented model. The capacitor-switch model provides a biologically-plausible concept of chaotic oscillator based on neuronal cells.

Czech Institute of Informatics Robotics and Cybernetics Czech Technical University Prague Prague Czech Republic

Department of Computing and Control Engineering Faculty of Chemical Engineering University of Chemistry and Technology Prague Prague Czech Republic

Department of Neurology Faculty of Medicine in Hradec Králové Charles University Hradec Králové Czech Republic

Independent researcher Uppsala Sweden

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Herz AVM, Gollisch T, Machens CK, Jaeger D. Modeling single-neuron dynamics and computations: A balance of detail and abstraction. Science. 2006;314:80–85. doi: 10.1126/science.1127240. PubMed DOI

Segev I. Single neurone models: oversimple, complex and reduced. Trends Neurosci. 1992;15:414–421. doi: 10.1016/0166-2236(92)90003-q. PubMed DOI

McCulloch WS, Pitts WH. A Logical Calculus of the Ideas Immanent in Nervous Activity. Bull Math Biophys. 1943;5:115–133. doi: 10.1007/BF02478259. PubMed DOI

Maass W. Networks of spiking neurons: The third generation of neural network models. Neural Networks. 1997;10:1659–1671. doi: 10.1016/S0893-6080(97)00011-7. DOI

Poirazi P, Brannon T, Mel BW. Pyramidal neuron as two-layer neural network. Neuron. 2003;37:989–999. doi: 10.1016/S0896-6273(03)00149-1. PubMed DOI

Ashwin, P., Coombes, S. & Nicks, R. Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience. J Math Neurosci6, UNSP 2, 10.1186/s13408-015-0033-6 (2016). PubMed PMC

Izhikevich EM. Which model to use for cortical spiking neurons? IEEE Trans Neural Netw. 2004;15:1063–1070. doi: 10.1109/TNN.2004.832719. PubMed DOI

Hindmarsh JL, Rose RM. A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B. 1984;221:87–102. doi: 10.1098/rspb.1984.0024. PubMed DOI

Koch, C. & Segev, I. Methods in neuronal modeling: from ions to networks. 2nd edn, (MIT Press, 1998).

Izhikevich EM, Edelman GM. Large-scale model of mammalian thalamocortical systems. Proc Natl Acad Sci USA. 2008;105:3593–3598. doi: 10.1073/pnas.0712231105. PubMed DOI PMC

Abarbanel HDI, et al. Synchronization in neural networks. Physics-Uspekhi. 1996;39:337–362. doi: 10.1070/PU1996v039n04ABEH000141. DOI

Hodgkin AL, Huxley AF. A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve. J Physiol-London. 1952;117:500–544. doi: 10.1113/jphysiol.1952.sp004764. PubMed DOI PMC

Bower, J. M. & Beeman, D. The Book of GENESIS: Exploring Realistic Neural Models with the GEneral NEural SImulation System. 2 edn, 458 (Springer-Verlag, 1998).

Carnevale, N. T. & Hines, M. L. The NEURON book. (Cambridge University Press, 2006).

Ziv I, Baxter DA, Byrne JH. Simulator for Neural Networks and Action-Potentials - Description and Application. Journal of Neurophysiology. 1994;71:294–308. doi: 10.1152/jn.1994.71.1.294. PubMed DOI

Squire, L. R. Fundamental neuroscience. 4th edn, (Elsevier/Academic Press, 2013).

Cavanagh JB. The problems of neurons with long axons. Lancet. 1984;1:1284–1287. doi: 10.1016/s0140-6736(84)92457-7. PubMed DOI

Leterrier C. The Axon Initial Segment: An Updated Viewpoint. Journal of Neuroscience. 2018;38:2135–2145. doi: 10.1523/Jneurosci.1922-17.2018. PubMed DOI PMC

Clark BD, Goldberg EM, Rudy B. Electrogenic Tuning of the Axon Initial Segment. Neuroscientist. 2009;15:651–668. doi: 10.1177/1073858409341973. PubMed DOI PMC

Llinas RR. The Intrinsic Electrophysiological Properties of Mammalian Neurons - Insights into Central Nervous-System Function. Science. 1988;242:1654–1664. doi: 10.1126/science.3059497. PubMed DOI

Hormuzdi SG, Filippov MA, Mitropoulou G, Monyer H, Bruzzone R. Electrical synapses: a dynamic signaling system that shapes the activity of neuronal networks. Bba-Biomembranes. 2004;1662:113–137. doi: 10.1016/j.bbamem.2003.10.023. PubMed DOI

Purves, D. Neuroscience. Sixth edition. (Oxford University Press, 2018).

Kandel, E. R. Principles of neural science. 5th edn, (McGraw-Hill, 2013).

Bennett MVL, Zukin RS. Electrical coupling and neuronal synchronization in the mammalian brain. Neuron. 2004;41:495–511. doi: 10.1016/S0896-6273(04)00043-1. PubMed DOI

Gibson JR, Beierlein M, Connors BW. Functional properties of electrical synapses between inhibitory interneurons of neocortical layer 4. Journal of Neurophysiology. 2005;93:467–480. doi: 10.1152/jn.00520.2004. PubMed DOI

Lytton, W. W. From computer to brain: foundations of computational neuroscience. (Springer, 2002).

Guillery RW. Early electron microscopic observations of synaptic structures in the cerebral cortex: a view of the contributions made by George Gray (1924-1999) Trends in Neurosciences. 2000;23:594–598. doi: 10.1016/S0166-2236(00)01635-0. PubMed DOI

Llinas R, Steinberg IZ, Walton K. Relationship between Presynaptic Calcium Current and Postsynaptic Potential in Squid Giant Synapse. Biophys J. 1981;33:323–351. doi: 10.1016/S0006-3495(81)84899-0. PubMed DOI PMC

Chapman ER. Synaptotagmin: a Ca(2+) sensor that triggers exocytosis? Nat Rev Mol Cell Biol. 2002;3:498–508. doi: 10.1038/nrm855. PubMed DOI

Sherwood, L. Human physiology: from cells to systems. 9th edition. edn, (Cengage Learning, 2016).

Haas JS, Zavala B, Landisman CE. Activity-Dependent Long-Term Depression of Electrical Synapses. Science. 2011;334:389–393. doi: 10.1126/science.1207502. PubMed DOI PMC

Gaiarsa JL, Caillard O, Ben-Ari Y. Long-term plasticity at GABAergic and glycinergic synapses: mechanisms and functional significance. Trends Neurosci. 2002;25:564–570. doi: 10.1016/s0166-2236(02)02269-5. PubMed DOI

Gerrow K, Triller A. Synaptic stability and plasticity in a floating world. Curr Opin Neurobiol. 2010;20:631–639. doi: 10.1016/j.conb.2010.06.010. PubMed DOI

Rall W. Branching Dendritic Trees and Motoneuron Membrane Resistivity. Exp Neurol. 1959;1:491–527. doi: 10.1016/0014-4886(59)90046-9. PubMed DOI

Rall, W. Theoretical significance of dendritic trees for neuronal input-output relations. (Stanford University Press, 1964).

Rall W. Distinguishing Theoretical Synaptic Potentials Computed for Different Soma-Dendritic Distributions of Synaptic Input. Journal of Neurophysiology. 1967;30:1138–&. doi: 10.1152/jn.1967.30.5.1138. PubMed DOI

Rall W, Shepherd GM. Theoretical Reconstruction of Field Potentials and Dendrodendritic Synaptic Interactions in Olfactory Bulb. Journal of Neurophysiology. 1968;31:884. doi: 10.1152/jn.1968.31.6.884. PubMed DOI

London M, Hausser M. Dendritic computation. Annu Rev Neurosci. 2005;28:503–532. doi: 10.1146/annurev.neuro.28.061604.135703. PubMed DOI

Polsky A, Mel BW, Schiller J. Computational subunits in thin dendrites of pyramidal cells. Nat Neurosci. 2004;7:621–627. doi: 10.1038/nn1253. PubMed DOI

Läuger, P. Electrogenic ion pumps. (Sinauer Associates, 1991).

Liang M, et al. Identification of a pool of non-pumping Na/K-ATPase. J Biol Chem. 2007;282:10585–10593. doi: 10.1074/jbc.M609181200. PubMed DOI

el Mernissi G, et al. Characterization and localization of ouabain-insensitive Na-dependent ATPase activities along the rat nephron. Biochim Biophys Acta. 1991;1064:205–211. doi: 10.1016/0005-2736(91)90303-p. PubMed DOI

Kety, S. S. In Metabolism of the nervous system. (ed D. Richter), (Pergamon, 1957), 221–237.

Howarth C, Gleeson P, Attwell D. Updated energy budgets for neural computation in the neocortex and cerebellum. J Cerebr Blood F Met. 2012;32:1222–1232. doi: 10.1038/jcbfm.2012.35. PubMed DOI PMC

Hodgkin AL, Huxley AF. Action potentials recorded from inside a nerve fiber. Nat Commun. 1939;144:710–711. doi: 10.1038/144710a0. DOI

Hodgkin AL, Huxley AF. Currents Carried by Sodium and Potassium Ions through the Membrane of the Giant Axon of Loligo. J Physiol-London. 1952;116:449–472. doi: 10.1113/jphysiol.1952.sp004717. PubMed DOI PMC

Hille, B. In Handbook of Physiology. (ed E.R. Kandel), (Am. Physiol. Soc., Bethesda, MD, U.S.A., 1977), 99–136.

Sengupta, B., Stemmler, M., Laughlin, S. B. & Niven, J. E. Action Potential Energy Efficiency Varies Among Neuron Types in Vertebrates and Invertebrates. PLOS Computational Biology6, ARTN e1000840, 10.1371/journal.pcbi.1000840 (2010). PubMed PMC

Coombs JS, Curtis DR, Eccles JC. The Interpretation of Spike Potentials of Motoneurones. J Physiol-London. 1957;139:198–231. doi: 10.1113/jphysiol.1957.sp005887. PubMed DOI PMC

Shu YS, Duque A, Yu YG, Haider B, McCormick DA. Properties of action-potential initiation in neocortical pyramidal cells: Evidence from whole cell axon recordings. Journal of Neurophysiology. 2007;97:746–760. doi: 10.1152/jn.00922.2006. PubMed DOI

Stuart G, Spruston N, Sakmann B, Hausser M. Action potential initiation and backpropagation in neurons of the mammalian CNS. Trends in Neurosciences. 1997;20:125–131. doi: 10.1016/S0166-2236(96)10075-8. PubMed DOI

Coombs JS, Eccles JC, Fatt P. The Specific Ionic Conductances and the Ionic Movements across the Motoneuronal Membrane That Produce the Inhibitory Post-Synaptic Potential. J Physiol-London. 1955;130:326–373. doi: 10.1113/jphysiol.1955.sp005412. PubMed DOI PMC

Bunge RP. Glial Cells and Central Myelin Sheath. Physiol Rev. 1968;48:197–+. doi: 10.1152/physrev.1968.48.1.197. PubMed DOI

Baumann N, Pham-Dinh D. Biology of oligodendrocyte and myelin in the mammalian central nervous system. Physiol Rev. 2001;81:871–927. doi: 10.1152/physrev.2001.81.2.871. PubMed DOI

Ritchie, J. M. In Myelin. (ed Pierre Morell), (Plenum Press, 1984), 117–146.

Rushton WAH. A Theory of the Effects of Fibre Size in Medullated Nerve. J Physiol-London. 1951;115:101–122. doi: 10.1113/jphysiol.1951.sp004655. PubMed DOI PMC

Ritchie, J. M. In The Axon, Structure, Function and Pathophysiology. (eds S. G. Waxman, J. D. Kocsis, & P. K. Stys), (Oxford Univ. Press, 1995), 68–69.

Kellert, S. H. In the wake of chaos: unpredictable order in dynamical systems. (University of Chicago Press, 1993).

Kantz, H. & Schreiber, T. Nonlinear time series analysis. 2nd edn, (Cambridge University Press, 2004).

Wolf A, Swift JB, Swinney HL, Vastano JA. Determining Lyapunov Exponents from a Time-Series. Physica D. 1985;16:285–317. doi: 10.1016/0167-2789(85)90011-9. DOI

Dlask M, Kukal J. Application of rotational spectrum for correlation dimension estimation. Chaos Soliton Fract. 2017;99:256–262. doi: 10.1016/j.chaos.2017.04.026. DOI

Grassberger P, Procaccia I. Measuring the Strangeness of Strange Attractors. Physica D. 1983;9:189–208. doi: 10.1016/0167-2789(83)90298-1. DOI

Duane GS. Synchronicity from Synchronized Chaos. Entropy-Switz. 2015;17:1701–1733. doi: 10.3390/e17041701. DOI

Shilnikov AL, Rulkov NF. Origin of chaos in a two-dimensional map modeling spiking-bursting neural activity. Int J Bifurcat Chaos. 2003;13:3325–3340. doi: 10.1142/S0218127403008521. DOI

Rall W. Theory of Physiological Properties of Dendrites. Ann N Y Acad Sci. 1962;96:1071–+. doi: 10.1111/j.1749-6632.1962.tb54120.x. PubMed DOI

Rall, W., Segev, I., Rinzel, J. & Shepherd, G. M. The theoretical foundation of dendritic function: selected papers of Wilfrid Rall with commentaries. (MIT Press, 1995).

Destexhe A, Neubig M, Ulrich D, Huguenard J. Dendritic low-threshold calcium currents in thalamic relay cells. Journal of Neuroscience. 1998;18:3574–3588. doi: 10.1523/JNEUROSCI.18-10-03574.1998. PubMed DOI PMC

Jaeger D, DeSchutter E, Bower JM. The role of synaptic and voltage-gated currents in the control of Purkinje cell spiking: A modeling study. Journal of Neuroscience. 1997;17:91–106. doi: 10.1523/JNEUROSCI.17-01-00091.1997. PubMed DOI PMC

Rhodes PA, Llinas R. A model of thalamocortical relay cells. J Physiol-London. 2005;565:765–781. doi: 10.1113/jphysiol.2004.070888. PubMed DOI PMC

Morris C, Lecar H. Voltage Oscillations in the Barnacle Giant Muscle-Fiber. Biophys J. 1981;35:193–213. doi: 10.1016/S0006-3495(81)84782-0. PubMed DOI PMC

Wilson HR, Cowan JD. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik. 1973;13:55–80. doi: 10.1007/bf00288786. PubMed DOI

Abbott LF. Lapicque’s introduction of the integrate-and-fire model neuron (1907) Brain Res Bull. 1999;50:303–304. doi: 10.1016/S0361-9230(99)00161-6. PubMed DOI

Brunel, N. & van Rossum, M. C. W. Quantitative investigations of electrical nerve excitation treated as polarization (Reprinted from Journal de Physiologie et de Pathologie Generale, vol. 9, pg 620–635, 1907). Biological Cybernetics97, 341–349, 10.1007/s00422-007-0189-6 (2007). PubMed

Dayan, P. & Abbott, L. F. Theoretical neuroscience: computational and mathematical modeling of neural systems. (Massachusetts Institute of Technology Press, 2001).

Fitzhugh, R. Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophys J1, 445–&, 10.1016/S0006-3495(61)86902-6 (1961). PubMed PMC

Hoppensteadt F. Heuristics for the Hodgkin-Huxley system. Math Biosci. 2013;245:56–60. doi: 10.1016/j.mbs.2012.11.006. PubMed DOI

Giannoukos G, Min M. Mathematical and Physical Modelling of the Dynamic Electrical Impedance of a Neuron. International Journal of Circuits, Systems and Signal Processing. 2012;6:359–366.

Velichko, A., Belyaev, M. & Boriskov, P. A Model of an Oscillatory Neural Network with Multilevel Neurons for Pattern Recognition and Computing. Electronics-Switz8, ARTN 75, 10.3390/electronics8010075 (2019).

Ghaffari BV, Kouhnavard M, Aihara T, Kitajima T. Mathematical Modeling of Subthreshold Resonant Properties in Pyloric Dilator Neurons. Biomed Res Int. 2015 doi: 10.1155/2015/135787. PubMed DOI PMC

Aissi, C. & Kazakos, D. In Proceedings of the 10th WSEAS International Conference on CIRCUITS. (Athens, Greece, 2006), 125–131.

Elwakil AS, Kennedy MP. Chaotic oscillators derived from sinusoidal oscillators based on the current feedback op amp. Analog Integr Circ S. 2000;24:239–251. doi: 10.1023/A:1008369810214. DOI

Hasler MJ. Electrical Circuits with Chaotic Behavior. P Ieee. 1987;75:1009–1021. doi: 10.1109/Proc.1987.13846. DOI

Minati, L., Frasca, M., Oswiecimka, P., Faes, L. & Drozdz, S. Atypical transistor-based chaotic oscillators: Design, realization, and diversity. Chaos27, Artn 073113, 10.1063/1.4994815 (2017). PubMed

Tamasevicius A, Mykolaitis G, Pyragas V, Pyragas K. A simple chaotic oscillator for educational purposes. Eur J Phys. 2005;26:61–63. doi: 10.1088/0143-0807/26/1/007. DOI

Matsumoto T. A chaotic attractor from Chua’s circuit. IEEE Transactions on Circuits and Systems. 1984;31:1055–1058. doi: 10.1109/TCS.1984.1085459. DOI

Minati, L. Experimental dynamical characterization of five autonomous chaotic oscillators with tunable series resistance. Chaos24, Artn 033110, 10.1063/1.4890530 (2014). PubMed

Pinsky PF, Rinzel J. Intrinsic and Network Rhythmogenesis in a Reduced Traub Model for Ca3 Neurons (Vol. 1, Pg. 59, 1994) J Comput Neurosci. 1995;2:275–275. doi: 10.1007/Bf00961439. PubMed DOI

Mainen ZF, Sejnowski TJ. Influence of dendritic structure on firing pattern in model neocortical neurons. Nature. 1996;382:363–366. doi: 10.1038/382363a0. PubMed DOI

Schmitt, O. H. & Schmitt, F. O. The nature of the nerve impulse. Am. J. Physiol. 97 (1931).

Cejnar P, Vysata O, Valis M, Prochazka A. The Complex Behaviour of a Simple Neural Oscillator Model in the Human Cortex. Ieee T Neur Sys Reh. 2019;27:337–347. doi: 10.1109/Tnsre.2018.2883618. PubMed DOI

Manneville, P. Dissipative structures and weak turbulence. (Academic Press, 1990).

Whitney H. Differentiable Manifolds. Annals of Mathematics. 1936;37:645–680. doi: 10.2307/1968482. DOI

Grassberger P, Procaccia I. Characterization of Strange Attractors. Physical Review Letters. 1983;50:346–349. doi: 10.1103/PhysRevLett.50.346. DOI

Moffett, S. X., O’Malley, S. M., Man, S. S., Hong, D. W. & Martin, J. V. Dynamics of high frequency brain activity. Sci Rep7, ARTN 15758, 10.1038/s41598-017-15966-6 (2017). PubMed PMC

Hamada MS, Kole MHP. Myelin Loss and Axonal Ion Channel Adaptations Associated with Gray Matter Neuronal Hyperexcitability. Journal of Neuroscience. 2015;35:7272–7286. doi: 10.1523/Jneurosci.4747-14.2015. PubMed DOI PMC

R Core Team. R: A Language and Environment for Statistical Computing, https://www.R-project.org/ (2018).