Abstract
We sum up our work about neurodynamics in this chapter. It is widely considered that the nervous system in man and animals is a rather complicated nonlinear dynamical system. Therefore, it is both necessary and important to understand the behavior occurred in the nervous system from the perspective of nonlinear dynamics. Actually, a great many of novel and puzzling phenomena are just observed in a single neuron, but their physiological or dynamical mechanisms remain open so far. In other words, single neurons are not simple. We show many firing patterns in theoretical neuronal models or neurophysiological experiments of single neurons in rats in this chapter. And then we introduce three representative examples of best known mathematical neuron models. The two types of neuronal excitability are illustrated by the Hodgkin-Huxley mode and the Morris-Lecar model. Especially, it is shown that we can change the types of neuronal excitability using the methods of bifurcation control. Besides, we display bursting and its topological classification, and explain bifurcation, chaos and crisis by the existing neuronal models. We give emphasis to sensitive responsiveness of aperiodic firing neurons to external stimuli, and show experimental phenomena and their underlying nonlinear mechanisms. The synchronization between neurons is remarked simply. We stress a constructive role of noise in the nervous system, and depict the phenomena of stochastic resonance and coherence resonance, and give their dynamical mechanisms. The common analysis methods are presented for the time series of the interspike intervals. Finally, we give two application examples about controlling chaos and stochastic resonance, and draw some conclusions.
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References
Abed E.H. and Fu J.H., 1986, Local feedback stabilization and bifurcation control I. Hopf bifurcation, Syst. Control Lett., 7, 11–17.
Abed E.H., Wang H.O. and Chen R.C., 1994, Stabilization of period doubling bifurcations and implications for control of chaos, Physica D, 70, 154–164.
Aihara K., and Matsumoto G., 1986, Chaotic oscillations and bifurcations in squid giant axons, In: Chaos (A.V. Holden, ed), Manchester and Princeton University Press, Princeton, NJ, 257–269.
Aihara K., Matsumoto G. and Ichikawa M., 1985, An alternating periodic-chaotic sequence observed in neural oscillators, Phys. Lett. A, 111, 251–255.
Auerbach D., Cvitanovic P., Eckmann J.P., Gunaratne G. and Procaccia I., 1987, Exploring chaotic motion through periodic orbits, Phys. Rev. Lett., 58, 2387–2389.
Baer S.M., Rinzel J. and Carrillo H., 1995, Analysis of an autonomous phase model for neuronal parabolic bursting, J. Math. Biol., 33, 309–333.
Barbi M., Chillemi S., Garbo A.D. and Reale L., 2003, Stochastic resonance in a sinusoidally forced LIF model WIth noisy threshold, BioSystems, 71, 23–28.
Bertram R., Butte M.J., Klemel T. and Sherman A., 1995, TopologIcal and phenomenological classification of bursting oscillations, Bull. Math. Biol., 57, 413–439.
Buonocore A., Nobile A.G., Ricciardi L.M., 1987, A new integral equation for the evaluation of first-passage-time probability density, Adv. Appl. Prob., 19, 784–800.
Bulsara R., Elson T.C., Doering C.R., Lowen S.B. and Lindenberg K., 1996, Coopeative behavior in periodically driven noisy integrate-and-fire models of neuronal dynamics, Phys. Rev. E, 53, 3958–3969.
Braun H.A., Wissing H., Schafer K. and Hirsch M.C., 1994, Oscillation and noise determine signal transduction in shark multimodal sensory cells, Nature, 367. 270–273.
Chay T.R., Fan Y.S. and Lee Y.S., 1995, Bursting, spiking, chaos, fractals, and universality in biological rhythms, International Journal of Bifurcattion and Chaos; 5, 595–635.
Chen D.S., Wang H.O. and Chen G., 2001, Anti-control of Hopf bifurcation, IEEE Trans. Circ. Sys.-I, 48, 661–672.
Chialvo D.R., Longtin A. and Muller-Gerking J., 1997, Stochastic resonance in models of neuronal ensembles, Phys. Rev. E, 55, 1798–1808.
Cvitanovic P., 1998, Invariant measurement of strange sets in terms of cycles, Phys. Rev. Lett., 61, 2729–2732.
Cymbalyuk G. and Shilnikov A., 2005, Coexistence of tonic spikIng oscillations in a leech neuron model, Journal of Computational Neuroscience, 18, 255–263.
Davidchack R.L. and Lai Y.C., 1999, Efficient algorithm for detecting unstable periodic orbits in chaotic systems, Physical Review E, 60, 6172–6175.
Douglass J.K., Wilkens L., Pantazelou E. and Moss F., 1993, Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance, Nature, 365, 337–340.
Ermentrout B., 2002, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia.
Ermentrout G.B. and Kopell N., 1986, Parabolic bursting in an excitable system coupled with a slow oscillation, SIAM J. Appl. Math., 46, 233–253.
Elson R.C., Selverston A.I., Huerta R., Rulkov N.F., Rabinovich M.I. and Abarbanel H.D.I., 1998, Synchronous behavior of two coupled biological neurons, Phys. Rev. Lett., 81, 5692–5695.
Faisal A.A., Selen L.P.J. and Wolpert D.M., 2008, Noise in the nervous system, Nat. Rev. Neurosci., 9, 292–303.
Feudel D., Neiman A., Pei X., Wojtenek W. and Moss F., 2002, Homoclinic bifurcation in a thermally sensitive neuron, In: Experimental chaos, AIP Conference Proceedings, 622, 139–148.
FitzHugh R., 1961, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1, 445–466.
Fries P., 2001, A mechanism for cognitive dynamics: neuronal communication through neuronal coherence, TICS, 9, 474–480.
Gammaitoni L., Hanggi P., Jung P. and Marchesoni F., 1998, Stochastic resonance, Rev. Mod. Phys., 70, 223–287.
Gong P.L., Xu J.X., Long K.P. and Hu S.J., 2002, Chaotic interspike intervals with multipeaked histogram in neurons, International Journal of Bifurcation ands Chaos, 12, 319–328.
Gong Y.F., Xu J.X., Ren W., Hu S.J., and Wang F.Z., 1998, Determining the degree of chaos from analysis of ISI time series in the nervous system: a comparison between correlation dimension and nonlinear forecasting methods, Biological Cybernetics, 78, 159–165.
Grassberger P., 1986, Do climatic attractors exist? Nature, 323, 609–612.
Grassberger P. and Procaccia I., 1983a, Characterization of strange attractors, Phys Rev. Lett., 50, 346–349.
Grassberger P. and Procaccia I., 1983b, Measuring the strangeness of strange attractors, Physica D, 9, 189–208.
Gray C.M., Konig P., Engel A.K. and Singer W., 1989, Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties, Nature, 338, 334–337.
Grebogi C., Ott E. and Yorke J.A., 1986, Critical exponent of chaotic transients in nonlinear dynamical systems, Phys. Rev. Lett., 57, 1284–1287.
Grebogi C., Ott E., Romeiras F. and Yorke J.A., 1987, Critical exponents for crisisinduced intermittency, Phys. Rev. A, 36, 5365–80.
Gu H.G., Ren W., Lu Q.S., Wu S.G., Yang M.H. and Chen W.J., 2001, Integer multiple spiking in neuronal pacemakers without external periodic stimulation, Physics Letter A, 285, 63–68.
Gu H.G., Li L., Yang M.H., Liu Z.Q., and Ren W., 2003, Integer muliple bursting generated in an experimental neural pacemaker, Acta Biophyica Sinica, 19, 68–72.
Gu H., Yang M.H., Li L., Liu Z. and Ren W., 2002, Experimental observation of the stochastic bursting caused by coherence resonance in a neural pacemaker, Neuroreport, 13, 1657–1660.
Guckenheimer J. and Holmes P., 1997, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 5th edition, Springer, New York, 150–152.
Guckenheimer J. and Oliva R.A., 2002, Chaos in the Hodgkin-Huxley model, SIAM J. Applied Dynamical Systems, 1, 105–114.
Gong Y.F., Xu J.X. and Hu S.J., 1998, Stochastic resonance: When does it not occur in neuronal models? Phys. Lett. A, 243, 351–359.
Gong P.L. and Xu J.X., 2001, Globall dynamics and stochastic resonance of the forced FitaHugh-Nagumo neuron model, Phys. Rev. E, 63, 031906.
Han S., Duan Y.B., Jian Z., Xie Y., Xing J.L. and Hu S.J., 2002, Calculating the degree of complexity of interspike interval with the method of approximate entropy, Acta Biophysica Sinica, 18, 448–451.
Hassard B.D., Kazarinoff N.D. and Wan Y.H., 1981, Theory and application of Hopf bifurcation, in: London Mathematical Society Lecture Note Series, 41, Cambridge Univ. Press, Cambridge, 86–91.
Hassouneh M.A., Lee H.C. and Abed E.H., 2004, Washout filters in feedback control: Benefits, Limitations and Extensions, In: Proceeding of the 2004 American Control Conference, Boston, Massachusetts, June 30–July 2, 3950–3955.
Hindmarsh J.L. and Rose R.M., 1984, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. London, Sere B, 221, 87–102.
Hodgkin A., and Huxley A., 1952, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol, 117, 500–544.
Hoffman R.E., Shi W.X. and Bunney B.S., 1995, Nonlinear sequence-dependent structure of nigral dopamine neuron interspike interval firing patterns, Biophys. J., 69, 128–137.
Hu S.J., Yang H.J., Jian Z., Long K.P., Duan Y.B., Wan Y.H., Xing J.L., Xu H., and Ju G., 2000, Adrenergic sensitivity of neurons with non-periodic firing activity in rat injured dorsal root ganglion, Neuroscience, 101, 689–698.
Izhikevich E.M., 2000, Neural excitability, spiking, and bursting, International Jour nal of Bifurcation and Chaos, 10, 1171–1266.
Izhikevich E.M., 2001, Resonate-and-fire neurons, Neural Networks, 14, 883–894.
Izhikevich E.M., 2007, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT, Cambridge, MA.
Jin W.Y., Xu J.X., Wu Y., Hong L. and Wei Y.B., 2006, Crisis of interspike intervals in HodgkIn-Huxley model, Chaos, Solitons and Fractals, 27, 952–958.
Kang Y.M., Xu J.X. and Xle Y., 2005a, A further insight Into stochastic resonance in an integrate-and-fire neuron with noisy periodic input, Chaos, Solitons and Fractals, 25, 165–170.
Kang Y.M., Xu J.X. and Xie Y., 2005b, Signal-to-noise ratio gain of a noisy neuron that transmits subthreshold periodic spike trains, Phys. Rev. E, 72, 021902.
Kim J.H. and Stringer J.,1992, Appl. Chaos, Wiley, New York, 441–455.
Kretzberg J., Warzecha A.K. and Egelhaaf M., 2001, Neural coding with graded membrane potential changes and spikes, Journal of Computational Neuroscience; 11, 153–164.
Lathrop D.P. and Kostelich E.J., 1989, Characterization of an experimental strange attractor by periodic orbits, Phys. Rev. A, 40, 4028–4031.
Levin J.E. and Miller J.P., 1996, Broadband neural encoding in the cricket cereal sensory system enhanced by stochastic resonance, Nature, 380, 165–168.
Li L., Gu H.G., Yang M.H., Liu Z.Q. and Ren W., 2003, Bifurcation scenario rhythm in the firing pattern transition of a neural pacemaker, Acta Biophysica Sinica, 19, 388–394.
Liu W.M., 1994, Criterion of Hopf bifurcation without using eigenvalues, J. Math. Anal. Appl., 182, 250–255.
Longtin A., 1993a, Nonlinear forecasting of spike trains from sensory neurons, International Journal of Bifurcation and Chaos, 3, 651–661.
Longtin A., 1993b, Stochastic resonance in neuron models, Journal of Statistical Physics, 70, 309–327.
Longtin A., 1995, Mechanisms of stochastic phase-locking, Chaos, 5, 209–215.
Longtin A., 1997, Autonomous stochastic resonance in bursting neurons, Phys. Rev. E, 55, 868–876.
Longtin A. and Bulsara A., 1991, Time-interval sequences in bistable systems and the noise-induced transmission of information by sensory neurons, Phys. Rev. Lett., 67, 656–659.
Longtin A., and Hinzer K., 1996, Encoding with bursting, subthreshold oscillations, and noise In mammalian cold receptors, Neural Computation, 8, 215–255.
Mandelblat Y., Etzion Y., Grossman Y. and Golomb D., 2001, Period doubling of calcium spike firing in a model of a Purkinje cell dendrite, Journal of Computational Neuroscience, 11, 43–62.
Matsumoto G., Aihara K., Hanyu Y., Takahashi N., Yoshizawa S. and Nagumo J., 1987, Chaos and phase lockIng in normal squid axons, Physics Letters A, 123, 162–166.
Michael V.L. and Bennett D.P., 2006, Electrical synapses between neurons synchronize gamma oscillations generated during higher level processing in the nervous system, Electroneurobiologia, 14, 227–250.
Mizrachi A.B., Procaccia I. and Grassberger P., 1984, The characterization of experimental (noisy) strange attractor, Phys. Rev. A, 29, 975–977.
Morris C., and Lecar H., 1981, Voltage oscillations in the barnacle giant muscle fiber, Biophys. J., 35, 193–213.
Moss F., Ward L.M., and Sannita W.G., 2004, Stochastic resonance and sensory information processing: a tutorial and review of application, Clin Neurophysiol, 115, 267–281.
Mpitsos G.J., Burton R.M., Creech Jr. H.C. and Soinila S.O., 1988, Evidence for chaos in spike trains of neurons that generate rhythmic motor patterns, Brain Res. Bull., 21, 529–538.
Neiman A.B. and Russell D.F., 2002, Synchronization of noise-induced bursts in noncoupled sensory neurons, Phys. Rev. Lett., 88, 138103.
Osborne A.R., and Provenzale A., 1989, Finite correlation dimension for stochastic systems with power-law spectra, Physica D, 35, 357–381.
Pincus S.M., 1991, Approximate entropy as a measure of system complexity, Proc. Natl. Acad. Sci. USA, 88, 2297–2304.
Pikovsky A.S. and Kurths J., 1997, Coherence resonance in a noise-driven excitable system, Phys. Rev. Lett., 78, 775–778.
Plesser H.E. and Geisel T., 1999, Markov analysis of stochastic resonance in a periodically driven integrate-and-fire neuron, Phys. Rev. E, 59, 7008–7017.
Ren W., Gu H.G., Jian Z., Lu Q.S., and Yang M.H., 2001, Different classifications of UPOs in the parametrically different chaotic lSI series of neural pacemaker, Neuroreport, 12, 2121–2124.
Ren W., Hu S.J., Zhang B.J., Wang F.Z., Gong Y.F. and Xu J.X., 1997, Periodadding bifurcation with chaos in the interspike intervals generated by an experimental neural pacemaker, International Journal of Bifurcation and Chaos, 7, 1867–1872.
Rinzel J., 1985, Bursting oscillations in an excitable membrane model, In: Ordinary and partial Differential Equations Proceedings of the 8th Dundee Conference (B.D. Sleeman and R.J. Jarvis, eds.), 304–316, Lecture Notes in Mathematics 1151, Springer, Berlin.
Rinzel J., 1987, A formal classification of bursting mechanisms in excitable systerns, In: Mathematical Topics in Population Biology; Morphogenesis; and Neurosciences (E. Teramoto, M. Yamaguti, eds), Vol. 71 of Lecture Notes in Biomathematics, Springer, Berlin.
Rinzel J. and Ermentrout B., 1989, Analysis of neural excitability and oscillations, In: Methods in Neuronal Modeling (Koch C, Segev I, eds): MIT, Cambridge, MA.
Rinzel J. and Lee Y.S., 1987, Dissection of a model for neuronal parabolic bursting, J. Math. Biol., 25, 653–675.
Rozental R., Andrade-Rozental A.F., Zheng X., Urban M., Spray D.C. and Chiu F.C., 2001, Gap Junction-Mediated Bidirectional Signaling between Human Fetal Hippocampal Neurons and Astrocytes, Develpmental Neuroscience, 23, 420–431.
Russell D.F., Wilkens L.A. and Moss F., 1999, Use of behavioural stochastic resonance by paddle fish for feeding, Nature, 402, 291–294.
Schmelcher P., and Diakonos F.K., 1998, General approach to the localization of unstable periodic orbits in chaotic dynamical systems, Physical Review E, 57, 2739–2746.
Schnitlzer A. and Gross J., 2005, Normal and pathological oscillatory communication in the brain, Nat. Rev. Neurosci., 6, 285–296.
Shimokawa T., Pakdaman K., Takahata T., Tanabe S. and Sato S., 2000, A first-passage-time analysis of the periodically forced noisy leaky integrate-and-fire model, Biol. Cybern., 83, 327–340.
Shilnikov A. and Cymbalyuk G., 2005, Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe, Physical Review Letters, 94, 048101.
Shimokawa T., Pakdaman K. and Sato S., 1999, Mean discharge frequency locking in the response of a noisy neuron model to subthreshold periodic stimulation, Phys. Rev. E, 60, R33–R36.
Singer W., 1993, Synchronization of cortical activity and its putative role in information processing and learning, Annu. Rev. Physiol., 55, 349–374.
So P., Francis J.T., Netoff T.I., Gluckman B.J. and Schiff S.J., 1998, Periodic orbits: a new language for Neuronal dynamics, Biophys. J., 74, 2776–2785.
So P., Ott E., Schiff S.J., Kaplan D.T., Sauer T. and Grebogi C., 1996, Detecting unstable periodic orbits in chaotic experimental data, Physical Review Letters, 76, 4705–4708.
So P., Ott E., Sauser T., Gluckman B.J., Grebogi C. and Schiff S.J., 1997, Extracting unstable orbits from chaotic time series data, Physical Review E, 55, 5398–5417.
Suzuki H., Aihara K., Murakami J. and Shimozawa T., 2000, Analysis of neural spike trains with interspike interval reconstruction, Biological Cybernetics, 82, 305–311.
Sugihara G., and May R.M., 1990, Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series, Nature, 344, 734–741.
Szczepaski J., Amigo J.M., Wajnryb E. and Sanchez-Vives M.V., 2003, Application of Lempel-Ziv complexity to the analysis of neural discharges, Network: Computation in Neural Systems, 14, 335–350.
Szczepanski J., Amigo J.M., Wajnryb E. and Sanchez-Vives M.V., 2004, Characterizing spike trains with Lempel-Ziv complexity, Neurocomputing, 58-60, 79–84.
Takahashi N., Hanyu Y., Musha T., Kubo R. and Matsumoto G., 1990, Global bifurcation structure in periodically stimulated giant axons of squid, Physica D, 43, 318–334.
Theiler J., Eubank S., Longtin A. Galdrikian B. and Farmer J.D., 1992, Testing for nonlinearity in time series: the method of surrogate data, Physica D, 58, 77–94.
Traub R.D., Wong R.K.S., Miles R. and Michelson H., 1991, A model of a CA3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances, Journal of Neurophysiology, 66, 635–650.
Tsuji S., Veta T., Kawakami H., Fujii H. and Aihara K., 2007, Bifurcations in two-dimensional Hindmarsh-Rose type model, International Journal of Bifurcation and Chaos, 17, 985–998.
Wellens T., Shatokhin V. and Buchleitner A., 2004, Stochastic resonance, Rep. Prog. Phys., 67, 45–105.
Wiesenfeld K. and Moss F., 1995, Stochastic resonance and the benefits of noise: from ice ages to crayfish and Squids, Nature, 373, 33–36.
Wu Y., Xu J.X. and He M., 2005, Synchronous behaviors of Hindmarsh-Rose neu rons with chemical coupling, In: Lecture in Computer Science, 3610, 508–511.
Xie Y., Aihara K. and Kang Y.M., 2008a, Change in types of neuronal excitability via bifurcation control, Physical Review E, 021917.
Xie Y., Chen L., Kang Y.M. and Aihara K., 2008b, Controlling the onset of Hopf bifurcation in the Hodgkin-Huxley model, Physical Review E, 77, 061921.
Xie Y., Duan Y.B., Xu J.X., Kang Y.M. and Hu S.J., 2003a, Parabolic bursting induced by Veratridine in rat injured sciatic nerves, Acta Biochimica et Biophysica Sinica, 35, 806–810.
Xie Y., Duan Y.B., Xu J.X., Kang Y.M. and Hu S.J., 2003b, The interspike interval increases gradually: why? Acta Biophysica Sinica, 19, 401–408.
Xie Y., Xu J.X. and Hu S.J., 2004a, A novel dynamical mechanism of neural excitability for integer multiple spiking, Chaos, Solitons and Fractals, 21, 177–184.
Xie Y., Xu J.X., Hu S.J., Kang Y.M., Yang H.J. and Duan Y.B., 2004b, Dynamical mechanisms for sensitive response of aperiodic firing cells to external stimulation, Chaos, Solitons and Fractals, 22, 151–160.
Xie Y., Xu J.X., Kang Y.M., Hu S.J. and Duan Y.B., 2005, Critical amplitude curves for different periodic stimuliand different dynamical mechanisms of excitability, Communications in Nonlinear Science and Numerical Simulation, 10, 823–832.
Xing J.L., Hu S.J., Xu H., Han S. and Wan Y.H., 2001, Subthreshold membrane oscillations underlying integer multiples firing from injured sensory neurons, Neuroreport, 12, 1311–1313.
Xu J.X., Gong Y.F., Ren W., Hu S.J., and Wang F.Z., 1997, Propagation of periodic and chaotic action potential trains along nerve fibers, Physica D, 100, 212–224.
Yang H.J., Hu S.J., Jian Z., Wan Y.H., and Long K.P., 2000, Relationship between the sensitivity to tetraethylammonium and firing patters of injured dorsal root ganglion neurons, Acta Physiologica Sinica, 52, 39S–401.
Yang H.J., Hu S.J., Han S., Liu G.P., Xie Y. and Xu J.X., 2002, Relation between responsiveness to neurotransmitters and complexity of epileptiform activity in rat hippocampal CAl neurons, Epilepsia, 43, 1330–1336.
Yang Z.Q., Lu Q.S., Gu H.G. and Ren W., 2002, Integer multiple spiking in the stochastic Chay model and its dynamical generation mechanism, Physics Letter A, 299, 499–506.
Zhang G.J., Xu J.X., Wang J., Yue Z.F., Liu C.B., Yao H. and Wang X.B., 2008, The Mechanism of Bifurcation-Dependent Coherence Resonance of Morris-Lecar Neuron Model, In: Advances in Cognitive Neurodynamics, Proceedings, 83–89.
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Xie, Y., Xu, JX. (2011). The Complexity in Activity of Biological Neurons. In: Luo, A.C.J., Sun, JQ. (eds) Complex Systems. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17593-0_6
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