×

zbMATH — the first resource for mathematics

Dynamics of firing patterns, synchronization and resonances in neuronal electrical activities: experiments and analysis. (English) Zbl 1257.92015
Summary: Recent advances in experimental and theoretical studies of dynamics of the neuronal electrical firing activities are reviewed. Firstly, some experimental phenomena of neuronal irregular firing patterns, especially chaotic and stochastic firing patterns, are presented, and practical nonlinear time analysis methods are introduced to distinguish deterministic and stochastic mechanism in time series. Secondly, the dynamics of electrical firing activities in a single neuron is concerned, namely, fast-slow dynamics analysis for classification and mechanism of various bursting patterns, one- or two-parameter bifurcation analysis for transitions of firing patterns, and stochastic dynamics of firing activities (stochastic and coherence resonances, integer multiple and other firing patterns induced by noise, etc.). Thirdly, different types of synchronization of coupled neurons with electrical and chemical synapses are discussed. As noise and time delays are inevitable in nervous systems, it is found that noise and time delays may induce or enhance synchronization and change firing patterns of coupled neurons. Noise-induced resonance and spatiotemporal patterns in coupled neuronal networks are also demonstrated. Finally, some prospects are presented for future research. In consequence, the ideas and methods of nonlinear dynamics are of great significance in the exploration of dynamic processes and physiological functions of nervous systems.

MSC:
92C20 Neural biology
92C05 Biophysics
92B25 Biological rhythms and synchronization
37N25 Dynamical systems in biology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chialvo D.R.: Critical brain networks. Physica A 340, 756–765 (2004)
[2] Gerstner W., Kistler W.M.: Spiking Neuron Models. Cambridge University Press, Cambridge (2002) · Zbl 1100.92501
[3] Stephan K.E., Hilgetag C.C., Burns G.A.P.C. et al.: Computational analysis of functional connectivity between areas of primate cerebral cortex. Phil. Trans. R. Soc. Lond. B Biol. Sci. 355, 111–126 (2000)
[4] Waldeyer W.: Ueber einige neuere Forschungen im Gebiete der Anatomic des Centralnervensystems. V. Dtsche. Med. Wochenzeitschr. 17, 1352–1356 (1891)
[5] Hayashi H., Ishzuka S., Ohta M., Hirakawa K.: Chaotic behavior in the onchidium giant neuron. Phys. Lett. A 88(8–9), 435–438 (1982)
[6] Aihara K., Matsumoto G., Ikegaya Y.: Periodic and non-periodic response of a periodically forced Hodgkin–Huxley oscillator. J. Theor. Biol. 109, 249–269 (1984)
[7] Thomas E., William J.R., Zbigniew J.K., James E.S., Karl E.G., Niels B.: Chaos and physiology: deterministic chaos in excitable cell assemblies. Physiol. Rev. 74(1), 1–47 (1994)
[8] Ren W., Hu S.J., Zhang B.J., Wang F.Z., Gong Y.F., Xu J.X.: Period-adding bifurcation with chaos in the inter-spike intervals generated by an experimental neural pacemaker. Int. J. Bif. Chaos 7, 1867–1872 (1997)
[9] Ren W., Gu H.G., Jian Z., Lu Q.S., Yang M.H.: Different classification of UPOs in the parametrically different chaotic ISI series. NeuroReport 12, 2121–2124 (2001)
[10] Li L., Gu H.G., Yang M.H., Liu Z.Q., Ren W.: A series of bifurcation scenarios in the firing transitions in an experimental neural pacemaker. Int. J. Bif. Chaos 14(5), 1813–1817 (2004) · Zbl 1064.37071
[11] Gu H.G., Yang M.H., Li L., Ren W., Lu Q.S.: Period adding bifurcation with chaotic and stochastic bursting in an experimental neural pacemaker. Dyn. Continuous Discrete Impulsive Syst. (Ser. B Appl. Algorithms) 14(S5), 6–11 (2007)
[12] Wu X.B., Mo J., Yang M.H., Zheng Q.H., Gu H.G., Ren W.: Two different bifurcation scenarios in neural firing rhythms discovered in biological experiments by adjusting two parameters. Chin. Phys. Lett. 25(8), 2799–2802 (2008)
[13] Yang J., Duan Y.B., Xing J.L., Zhu J.L., Duan J.H., Hu S.J.: Responsiveness of a neural pacemaker near the bifurcation point. Neurosci. Lett. 392, 1050–1109 (2006)
[14] Selverston A.I., Rabinovich M.I., Abarbanel H.DI. et al.: Reliable circuits from irregular neurons: a dynamical approach to understanding central pattern generators. J. Physiol. 94, 357–374 (2000)
[15] Elson R.C., Selverston A.I., Huerta R. et al.: Synchronous behavior of two coupled biological neurons. Phys. Rev. Lett. 81, 5692(4) (1998)
[16] Rabinovich M.I., Abarbanel H.D.I.: The role of chaos in neural system. Neuroscience 87(1), 5–14 (1998)
[17] Varona P., Torres J.J., Abarbanel H.D.I. et al.: Dynamics of two electrically coupled chaotic neurons: experimental observations and model analysis. Biol. Cybern. 84(2), 91–101 (2001)
[18] Attila S.Z., Elson R.C., Rabinovich M.I. et al.: Nonlinear behavior of sinusoidally forced pyloric pacemaker neurons. J. Neurophys. 85(4), 1623–1638 (2001)
[19] Hoffman R.E., Shi W.X., Bunney B.S.: Nonlinear sequence-dependent structure of nigral dopamine neuron interspike interval firing patterns. Biophys. J. 69, 128–137 (1995)
[20] Lovejoy L.P., Shepard P.D, Canavier C.C.: Apamin-induced irregular firing in vitro and irregular single-spike firing observed in vivo in dopamine neurons is chaotic. Neuroscience 104(3), 829–840 (2001)
[21] Mascio M.DI., Giovanni G.DI., Matteo V.DI., Espostto E.: Decreased chaos of midbrain dopaminergic neurons after serotonin denervation. Neuroscience 92(1), 237–243 (1999)
[22] Mascio M.DI., Giovanni G.DI., Matteo V.DI., Esposito E.: Reduced chaos of interspike interval of midbrain dopaminergic neurons in aged rats. Neuroscience 89(4), 1003–1008 (1999)
[23] Quyen M.L.V., Martinerie M.J., Adam C., Varela F.J.: Unstable periodic orbits in human epileptic activity. Phys. Rev. E 56, 3401–3411 (1997)
[24] Braun H.A., Schäfer K., Voigt K. et al.: Low-dimensional dynamics in sensory biology 1: thermally sensitive electroreceptors of the catfish. J. Comp. Neurosci. 4, 335–347 (1997) · Zbl 0894.92013
[25] Braun H.A., Dewald M., Schäfer K. et al.: Low-dimensional dynamics in sensory biology 2: facial cold receptors of the rat. J. Comp. Neurosci. 7, 17–32 (1999) · Zbl 0933.92014
[26] Braun H.A., Dewald M., Voigt K. et al.: Finding unstable periodic orbits in electroreceptors, cold receptors and hypothalamic neurons. Neurocomputing 26–27, 79–86 (1999)
[27] Pei X., Moss F.: Characterization of low-dimensional dynamics in the crayfish caudal photoreceptor. Nature 379, 618–621 (1996)
[28] Kanno T., Miyano T., Tokudac I. et al.: Chaotic electrical activity of living {\(\beta\)}-cells in the mouse pancreatic islet. Physica D 226, 107–116 (2007) · Zbl 1105.92004
[29] Hu S.J., Yang H.J., Jian Z. et al.: Adrenergic sensitivity of neurons with non-periodic firing activity in rat injured dorsal root ganglion. Neuroscience 101(3), 689–698 (2000)
[30] Faure, P., Korn, H.: A nonrandom dynamic component in the synaptic noise of a central neuron. In: Proc. Natl. Acad. Sci. USA 94, 6506–6511 (1997)
[31] So P., Francis J.T., Netoff T.I. et al.: Periodic orbits: a new language for neuronal dynamics. Biophys. J. 74, 2776–2785 (1998)
[32] Schiff S.J., Jerger K., Duong D. et al.: Controlling chaos in the brain. Nature 370, 615–620 (1994)
[33] Gong Y.F., Xu J.X., Ren W. et al.: Determining the degree of chaos from analysis of ISI time series in the nervous system: a comparison between correlation dimension and nonlinear forecasting methods. Biol. Cybern. 78(2), 159–165 (1988) · Zbl 0895.92008
[34] Schweighofer, N., Doya, K., Fukai, H., et al.: Chaos may enhance information transmission in the inferior olive. In: Proc. Natl. Acad. Sci. USA 101(13), 4655–4660 (2004)
[35] Wan Y.H., Jian Z., Wen Z.H. et al.: Synaptic transmission of chaotic spike trains between primary afferent fiber and spinal dorsal horn neuron in the rat. Neuroscience 125(4), 1051–1060 (2004)
[36] Canavier C.C., Perla S.R., Shepard P.D.: Scaling of prediction error does not confirm chaotic dynamics underlying irregular firing using interspike intervals from midbrain dopamine neurons. Neuroscience 129, 491–502 (2004)
[37] Longtin A., Bulsara A., Moss F.: Time interval sequences in bistable system and the noise-induced transmission of information by sensory neurons. Phys. Rev. Lett. 67, 656–659 (1991)
[38] Gammaitoni L., Hänggi P., Jung P., Marchesoni F.: Stochastic resonance. Rev. Mod. Phys. 70, 22–287 (1998)
[39] Hu G., Ditzlinger T., Ning C.Z. et al.: Stochastic resonance without external periodic force. Phys. Rev. Lett. 71(6), 807–810 (1993)
[40] Pikovsky A.S., Kurth J.: Coherence resonance in a noise-driven excitable system. Phys. Rev. Lett. 78, 775–778 (1997) · Zbl 0961.70506
[41] Longtin A.: Autonomous stochastic resonance in bursting neurons. Phys. Rev. E 55, 868–876 (1997)
[42] Hou Z., Xin H.: Noise-induced oscillation and stochastic resonance in an autonomous chemical reaction system. Phys. Rev. E 60, 6329–6332 (1999)
[43] Ditzinger T., Ning C.Z., Hu G.: Resonancelike responses of autonomous nonlinear systems to white noise. Phys. Rev. E 50(5), 3508–3516 (1994)
[44] Douglass J.K., Wilkens L., Pantazelou E., Moss F.: Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance. Nature 365, 337–340 (1993)
[45] Wiesenfeld K., Moss F.: Stochastic resonance and the benifits of noise: from ice ages to crayfish and squids. Nature 373, 33–36 (1995)
[46] Levin J.E., Miller J.P.: Broadband neural encoding in the cricket cereal sensory system enhanced by stochastic resonance. Nature 380, 165–168 (1996)
[47] Stacey W.C., Durand D.M.: Stochastic resonance improves signal detection in hippocampal CA1 neurons. J. Neurophysiol. 83(3), 1394–1402 (2000)
[48] Gluckman B.J., Netoff T.I., Neel E.J. et al.: Stochastic resonance in a neuronal network from mammalian brain. Phys. Rev. Lett. 77, 4098–4101 (1996)
[49] Russell D.F., Wilkens L.A., Moss F.: Use of behavioural stochastic resonance by paddlefish for feeding. Nature 402(6759), 241–294 (1999)
[50] Greenwood P.E., Ward L.M., Russell D.F. et al.: Stochastic resonance enhances the electrosensory information available to paddlefish for prey capture. Phys. Rev. Lett. 84, 4773–4776 (2000)
[51] Freund J.A., Schimansky-Geier L., Beisner B., Neiman A., Russell D.F., Yakusheva T., Moss F.: Behavioral stochastic resonance: how the noise from a Daphnia swarm enhances individual prey capture by juvenile paddlefish. J. Theor. Biol. 214(1), 71–83 (2002)
[52] Wilkens L.A., Hofmann M.H., Wojtenek W.: The electric sense of the paddlefish: a passive system for the detection and capture of zooplankton prey. J. Physiol. 96(5–6), 363–377 (2002)
[53] Neiman A.B., Russell D.F.: Stochastic biperiodic oscillations in the electroreceptors of paddlefish. Phys. Rev. Lett. 86, 3443–3446 (2001)
[54] Neiman A.B., Russell D.F.: Two distinct types of noisy oscillators in electroreceptors of paddlefish. J. Neurophysiol. 92, 492–509 (2004)
[55] Gu H.G., Ren W, Lu Q.S., Wu S.G., Yang M.H., Chen W.J.: Integer multiple spiking in neural pacemakers without external periodic stimulation. Phys. Lett. A 285, 63–68 (2001) · Zbl 01616595
[56] Gu H.G., Yang M.H., Li L., Liu Z.Q., Ren W.: Dynamics of autonomous stochastic resonance in neural period adding bifurcation scenarios. Phys. Lett. A 319(1–2), 89–96 (2003) · Zbl 1038.92007
[57] Yang M.H., Gu H.G., Li L., Liu Z.Q., Ren W.: Characteristics of period adding bifurcation without chaos in firing pattern transitions in an experimental neural pacemaker. NeuroReport 14(17), 2153–2157 (2003)
[58] Moss F., Ward L.M., Sannita W.G.: Stochastic resonance and sensory information processing: a tutorial and review of application. Clin. Neurophysiol. 115(2), 267–281 (2004)
[59] Braun H.A., Wissing H., Schäfer K., Hirsch M.C.: Oscillation and noise determine signal transduction in shark multimodal sensory cells. Nature 367, 270–273 (1994)
[60] Chacron M.J., Longtin A., St-Hilaire M., Maler L.: Suprathreshold stochastic firing dynamics in P-type electroreceptors. Phys. Rev. Lett. 85(7), 1576(4) (2000)
[61] Chacron M.J., Linder B., Longtin A.: Noise shaping by interval correlations increases information transfer. Phys. Rev. Lett. 92(8), 080601 (2004)
[62] Chacron M.J., Longtin A., Maler L.: Negative interspike interval correlations increases the neuronal capacity for encoding time-dependent stimuli. J. Neurosci. 21(14), 5328–5343 (2001)
[63] Li C., Tripathi P.K., Armstrong W.E.: Differences in spike train variability in rat vasopressin and oxytoc in neurons and their relationship to synaptic activity. J. Physiol. 581(1), 221–240 (2007)
[64] Lee J.I., Metman L.V., Ohara S. et al.: Internal pallidal neuronal activity during mild drug-related dyskinesias in Parkinson’s disease: decreased firing rates and altered firing patterns. J. Neurophysiol. 97, 2627–2641 (2007)
[65] Kantz H., Schreiber T.: Nonlinear Time Series Analysis. Cambridge university press, Cambridge (1997) · Zbl 0873.62085
[66] Sauer T.: Reconstruction of dynamical system from interspike intervals. Phys. Rev. Lett. 72, 3811–3814 (1994)
[67] Sugihara G., May R.M.: Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series. Nature 344, 734–741 (1990)
[68] Theiler J., Eubank S., Longtin A. et al.: Testing for nonlinearity in time series: the method of surrogate data. Physica D 58, 77–94 (1992) · Zbl 1194.37144
[69] So P., Ott E., Schiff S.J., Kaplan D.T., Sauer T., Grebogi C.: Detecting unstable periodic orbits in chaotic experimental data. Phys. Rev. Lett. 76, 4705–4708 (1996)
[70] Pierson D., Moss F.: Detecting periodic unstable points in noisy chaotic and limit cycle attractors with applications to biology. Phys. Rev. Lett. 75, 2124–2127 (1995)
[71] Kantz H.: A robust method to estimate the maximal Lyapunov exponent of a time series. Phys. Lett. A 185, 77–87 (1994)
[72] Kasper F., Schuster H.G.: Easily calculable measure for the complexity of spatio-temporal patterns. Phys. Rev. A 36, 836–842 (1987)
[73] Deschenes M., Roy J.P., Steriade M.: Thalamic bursting mechanism: an invariant slow current revealed by membrane hyperpolarization. Brain Res. 239, 289–293 (1982)
[74] Harris-Warrick R.M., Flamm R.E.: Multiple mechanisms of bursting in a conditional bursting neuron. J. Neurosci. 7, 2113–2128 (1987)
[75] Ashcroft F., Rorsman P.: Electrophysiology of the pancreatic {\(\beta\)}-cell. Prog. Biophys. Molec. Biol. 54, 87–143 (1989)
[76] Johnson S.W., Seutin V., North R.A.: Burst firing in dopamine neurons induced by N-Methyl-D-Aspartate: role of electrogenic sodium pump. Science 258, 665–667 (1992)
[77] Rinzel J.: Bursting oscillation in an excitable membrane model. In: Sleeman, B.D., Jarvis, R.J. (eds) Ordinary and Partial Differential Equations, pp. 304–316. Springer, Berlin (1985)
[78] Rinzel J.: A formal classification of Bursting mechanisms in excitable systems. In: Teramoto, E., Yamaguti, M. (eds) Mathematical Topics in Population Biology. Morphogenesis and Neurosciences, pp. 267–281. Springer, Berlin (1987) · Zbl 0665.92003
[79] Sherman A., Rinzel J.: Rhythmogenic effects of weak electrotonic coupling in neuronal model. Proc. Natl. Acad. Sci. USA 89, 2471–2474 (1992)
[80] Rinzel J., Lee Y.S.: Dissection of a model for neuronal parabolic bursting. J. Math. Biol. 25, 653–675 (1987) · Zbl 0628.92016
[81] Av-Ron E., Parnas H., Segel L.: A basic biophysical model for bursting neurons. Biol. Cybern. 69, 87–95 (1993)
[82] Holden L., Erneux T.: Slow passage through a Hopf bifurcation: form oscillatory to steady state solutions. SIAM. J. Appl. Math. 53, 1045–1058 (1993) · Zbl 0781.34030
[83] Holden L., Erneux T.: Understanding bursting oscillations as periodic slow passages through bifurcation and limit points. J. Math. Biol. 31, 351–365 (1993) · Zbl 0769.92013
[84] Smolen P., Terman D., Rinzel J.: Properties of a bursting model with two slow inhibitory variables. SIAM. J. Appl. Math. 53, 832–861 (1993) · Zbl 0785.34030
[85] Pernarowski M.: Fast subsystem bifurcations in a slowly varied Lienard system exhibiting bursting. SIAM. J. Appl. Math. 54, 814–832 (1994) · Zbl 0805.34035
[86] Rush M.E., Rinzel J.: Analysis of bursting in thalamic neuron model. Biol. Cybern. 71, 281–291 (1994) · Zbl 0804.92008
[87] Kepecs A., Wang X.J.: Analysis of complex bursting in cortical pyramidal neuron models. Neurocomputing 32–33, 81–187 (2000)
[88] Soto-Trevino C., Kopell N., Watson D.: Parabolic bursting revisited. J. Mat. Biol. 35, 114–128 (1996) · Zbl 0868.92010
[89] Chay T.R., Fan Y.S., Lee Y.S.: Bursting, spiking, chaos, fractals, and university in biological rhythms. Int. J. Bif. Chaos 5, 595–635 (1995) · Zbl 0885.92019
[90] Booth V., Carr T.W., Erneux T.: Near-threshold bursting is delayed by a slow passage near a limit point. SIAM. J. Appl. Math. 57, 1406–1420 (1997) · Zbl 0891.34042
[91] Izhikevich E.M.: Neural excitability, spiking and bursting. Int. J. Bif. Chaos 10, 1171–1266 (2000) · Zbl 1090.92505
[92] Yang Z.Q., Lu Q.S.: Different types of bursting in Chay neuronal model. Sci. China (Ser. G) Phys. Mech. Astron. 51(6), 1–12 (2008) · Zbl 1136.81348
[93] Chay T.R.: Chaos in a three-variable model of an excitable cell. Physica D 16, 233–242 (1985) · Zbl 0582.92007
[94] Mosekide E., Lading B., Yanchuk S., Maistrenko Y.: Bifurcation structure of a model of bursting pancreatic cells. BioSystems 63, 3–13 (2001)
[95] Doi S., Nabetani S., Kumagai S.: Complex nonlinear dynamics of the Hodgkin–Huxley equations induced by time scale changes. Biol. Cybern. 85, 44–51 (2001) · Zbl 1160.92326
[96] Mandelblat Y., Etzion Y., Grossman Y., Golomb D.: Period doubling of calcium spike firing in a model of a Purkinje cell dendrite. J. Comput. Neurosci. 11, 43–62 (2001) · Zbl 05967450
[97] Belykh V.N., Belykh I.V., Colding-Jrgensen M., Mosekilde E.: Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models. Eur. Phys. J. E 3, 205–219 (2000)
[98] Holden A.V., Fan Y.S.: From simple to simple bursting oscillatory behaviour via chaos in the Rose–Hindmarsh model for neuronal activity. Chaos Solitons Fractals 2, 221–236 (1992) · Zbl 0766.92006
[99] Holden A.V., Fan Y.S.: From simple to complex oscillatory behaviour via intermittent chaos in the Rose–Hindmarsh model for neuronal activity. Chaos Solitons Fractals 2, 349–369 (1992) · Zbl 0753.92009
[100] Holden A.V., Fan Y.S.: Crisis-induced chaos in the Rose–Hindmarsh model for neuronal activity. Chaos Solitons Fractals 2, 583–595 (1992) · Zbl 0766.92007
[101] Terman D.: The transition from bursting to continuous spiking in excitable membrane models. J. Nonlin. Sci. 2, 135–182 (1992) · Zbl 0900.92059
[102] Terman D.: Chaotic spikes arising from a model of bursting in excitable membranes. SIAM. J. Appl. Math. 51, 1418–1450 (1991) · Zbl 0754.58026
[103] Yang Z.Q., Lu Q.S.: The bifurcation structure of firing pattern transitions in the Chay neuronal pacemaker model. J. Biol. Syst. 16(1), 33–49 (2008) · Zbl 1149.92311
[104] He J.H., Wu X.H.: A modified Morris–Lecar model for interacting ion channels. Neurocomputing 64, 543–545 (2005) · Zbl 02223971
[105] He J.H.: Resistance in cell membrane and nerve fiber. Neurosci. Lett. 373, 48–50 (2005)
[106] Terada K., Tanaka H., Yoshizawa S.: Two-parameter bifurcation in the Hodgkin–Huxley equations for muscle fibers. Electron. Comm. J. 83, 86–94 (2000)
[107] Fukai H., Doi S., Nomura T., Sato S.: Hopf bifurcations in multiple-parameter space of the Hodgkin–Huxley equations I: global organization of bistable periodic solutions. Biol. Cybern. 82, 215–222 (2000) · Zbl 0962.92005
[108] Liao X.F.: Hopf and resonant codimension two bifurcation in van der Pol equation with two time delays. Chaos, Solitons Fractals 23, 857–871 (2005) · Zbl 1076.34087
[109] Bertram R., Butte M., Kiemel T., Sherman A.: Topological and phenomenological classification of bursting oscillations. Bull. Math. Biol. 57, 413–439 (1995) · Zbl 0813.92010
[110] Shorten P.R., Wall D.J.: A Hodgkin–Huxley model exhibiting bursting oscillations. Bull. Math. Biol. 62, 695–715 (2000) · Zbl 1323.92060
[111] Tsumoto K., Kitajima H., Yoshinaga T., Aihara K., Kawakami H.: Bifurcations in Morris–Lecar neuron model. Neurocomputing 69, 293–316 (2006) · Zbl 05011799
[112] Duan L.X., Lu Q.S.: Codimension-two bifurcation analysis in Hindmarsh–Rose model with two parameters. Chin. Phys. Lett. 22(6), 1325–1328 (2005)
[113] Duan L.X., Lu Q.S.: Codimension-two bifurcation analysis on firing activities in Chay neuron model. Chaos, Solitons Fractals 30, 1172–1179 (2006) · Zbl 1142.37378
[114] Duan, L.X., Lu, Q.S., Wang Q.Y.: Two-parameter bifurcation analysis of firing activities in the Chay neuronal model. Neurocomputing. doi: 10.1016/j.neucom.2008.01.019 (2008)
[115] Longtin A.: Stochastic resonance in neuron models. J. Stat. Phys. 70, 309–327 (1993) · Zbl 1002.92503
[116] Moss, F., Bulsara, A., Shlesinger, M. (eds.): Proc. NATO Adv. Res. Workshop on Stochastic Resonance in Physics and Biology. J. Stat. Phys. 70, 1–514 (1993) · Zbl 0915.00054
[117] Petracchi D., Lucia S., Cercignani G.: Fundamental sources of noise in chemical transduction. Nuovo Cimento D 17, 888–891 (1995)
[118] Teich M.C., Khanna S.M., Guiney P.C.: Spectral characteristics and synchrony in primary auditory-nerve fibers in response to pure-tone acoustic stimuli. J. Stat. Phys. 70, 257–279 (1993)
[119] Pei X., Wilkens L., Moss F.: Light enhances hydrodynamic signaling in the multimodal caudal photoreceptor interneurons of the crayfish. J. Neurophysiol. 76, 3002–3011 (1996)
[120] Lees S., Neiman A., Kim S. et al.: Coherence resonance in a Hodgkin–Huxley neuron. Phys. Rev. Lett. 57, 3292–3297 (1998)
[121] Wang Y., Chik D., Wang Z. et al.: Coherence resonance and noise-induced synchronization in globally coupled Hodgin-Huxley neurons. Phys. Rev. E 61, 740–746 (2000)
[122] Wiesenfeld K., Pierson D., Pantazelou E., Dames C., Moss F.: Stochastic resonance on a circle. Phys. Rev. Lett. 72, 2125–2129 (1994)
[123] Ginzburg S., Pustovoit M.: Bursting dynamics of a model neuron induced by intrinsic channel noise. Fluctuation Noise Lett. 3, 265–274 (2003)
[124] FitzHugh R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)
[125] Hindmars J.L., Rose R.M.: A model of neuronal bursting using three coupled first order differential equations. Proc. R Soc. Lond. Ser. B 221, 87–102 (1984)
[126] Plant R.E.: Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin–Huxley equations. Biophys. J. 16, 227–244 (1976)
[127] Balans J.P., Casado J.M.: Bursting behaviour of the FitzHugh–Nagumo neuron model subject to quasi-monochromatic noise. Physica D 122, 231–240 (1998)
[128] Lindner B., Garc-Ojalvo J., Neiman A., Schimansky-Geier L.: Effects of noise in excitable systems. Phys. Rep. 392, 321–424 (2004)
[129] Wu S.G., Ren W., He K.F., Huang Z.Q.: Burst and coherence resonance in Rose–Hindmarsh model induced by additive noise. Phys. Lett. A 279, 347–354 (2001) · Zbl 0979.92012
[130] Yang Z.Q., Lu Q.S., Gu H.G., Ren W.: Integer multiple spiking in the stochastic Chay model and its dynamical generation mechanism. Phys. Lett. A 299, 499–506 (2002) · Zbl 0996.92002
[131] Yang Z.Q., Lu Q.S., Gu H.G. et al.: GWN-induced bursting, spiking, and random subthreshold impulsing oscillation before Hopf bifurcations in the Chay model. Int. J. Bif. Chaos 14, 4143–4159 (2004) · Zbl 1070.92008
[132] Yang Z.D., Lu Q.S., Gu H.G. et al.: The generation of stochastic integer multiple spiking in the Chay model. Int. J. Mod. Phys. B 17, 4362–4366 (2003)
[133] Rose J.E., Brugge J.F., Arderson D.D., Hind J.E.: Phase-locked response to low-frequency tones in single auditory nerve fibers of the squirrel monkey. J. Neurophysiol. 30, 769–793 (1967)
[134] Siegel R.M.: Nonlinear dynamical system theory and primary visual cortical processing. Physica D 42, 385–395 (1990)
[135] Schäfer K., Braun H.A., Rempe L.: Classification of a calcium conductance in cold receptors. Prog. Brain Res. 74, 29–36 (1988)
[136] Longtin A.: Mechanisms of stochastic phase locking. Chaos 5(1), 209–215 (1995)
[137] Makarov V.A., Nekorkin V.I., Velarde M.G.: Spiking behavior in a noise-driven system combining oscillatory and excitatory properties. Phys. Rev. Lett. 86, 3431–3434 (2001)
[138] Longkin A.: Stochastic aspects of neural phase locking to periodic signals, stochastic dynamics and pattern formation in biological and complex systems. AIP Conf. Proc. 501, 219–239 (2000)
[139] Longtin A., Hinzer K.: Encoding with bursting, subthreshold oscillation, and noise in mammalian cold receptors. Neural Comput. 8, 215–255 (1996) · Zbl 05478615
[140] Braun H.A., Huber M.T., Anthes N., Voigt K., Neiman A. et al.: Noise-induced impulse pattern modifications at different dynamical period-one situations in a computer model of temperature encoding. BioSystems 62, 99–112 (2001)
[141] Yang Z.Q., Lu Q.S.: Bursting and spiking due to additional direct and stochastic currents in neuron models. Chin. Phys. 15(3), 518–525 (2006)
[142] Shi X., Lu Q.S.: Firing patterns and complete synchronization of coupled Hindmarsh–Rose neurons. Chin. Phys. 14, 77–85 (2005)
[143] Wang Q.Y., Lu Q.S., Chen G.R., Guo D.H.: Chaos synchronization of coupled neurons with gap junctions. Phys. Lett. A 356, 17–25 (2006) · Zbl 1160.81304
[144] Shuai J.W., Durand D.M.: Phase synchronization in two coupled chaotic neurons. Phys. Lett. A 264, 289–297 (1999) · Zbl 0949.37015
[145] Shi X., Lu Q.S.: Rhythm synchronization of coupled neurons with temporal coding scheme. Chin. Phys. Lett. 24, 636–639 (2007)
[146] Hansel D., Sompolinsky H.: Synchronization and computation in a chaotic neural network. Phys. Rev. Lett. 68, 718–721 (1992)
[147] Kurths J., Zhou C.S.: Noise, synchronization and coherence in chaotic oscillators. Int. J. Mod. Phys. B 17, 4023–4044 (2003) · Zbl 1073.37033
[148] He D.H., Shi P.L., Stone L.: Noise-induced synchronization in realistic models. Phys. Rev. E 67, 027201 (2003)
[149] Zhou C.S., Kurths J.: Noise-induced synchronization and coherence resonance of a Hodgkin–Huxley model of thermally sensitive neurons. Chaos 13, 401–409 (2003)
[150] Shi X., Lu Q.S.: Coherence resonance and synchronization of Hindmarsh–Rose neurons with noise. Chin. Phys. 14, 1088–1094 (2005)
[151] Casado J.M.: Synchronization of two Hodgkin–Huxley neurons due to internal noise. Phys. Lett. A 310, 400–406 (2003) · Zbl 1017.92010
[152] Casado J.M., Baltanás J.P.: Phase switching in a system of two noisy Hodgkin–Huxley neurons coupled by a diffusive interaction. Phys. Rev. E 68, 061917 (2003)
[153] Postnov D.E., Sosnovtseva O.V., Han S.K., Yim T.G.: Stochastic synchronization of coupled coherence resonance oscillators. Int. J. Bif. Chaos 10, 2541–2550 (2000) · Zbl 0984.37099
[154] Wu Y., Xu J.X., He D.H., Earn D.: Generalized synchronization induced by noise and parameter mismatching in Hindmarsh–Rose neurons. Chaos, Solitons Fractals 23, 1605–1611 (2005) · Zbl 1066.92015
[155] Dhamala M., Jirsa V.K., Ding M.: Enhancement of neural synchrony by time delay. Phys. Rev. Lett. 92, 074101 (2004)
[156] Burić N., Ranković D.: Bursting neurons with coupling delays. Phys. Lett. A 363, 282–289 (2007)
[157] Rossoni E., Chen Y.H., Ding M.Z., Feng J.F.: Stability of synchronous oscillations in a system of Hodgkin–Huxley neurons with delayed diffusive and pulsed coupling. Phys. Rev. E 71, 061904 (2005)
[158] Wang Q.Y., Lu Q.S.: Time delay-enhanced synchronziation and regulation in two coupled chaotic neurons. Chin. Phys. Lett. 22, 543–546 (2005)
[159] Wang Q.Y., Lu Q.S., Zheng Y.H.: Conduction delay-aided synchronization in two inhibitorily synaptic coupled Chay neurons. Acta Biophys. Sinica 21, 449–455 (2005)
[160] Wang Q.Y., Lu Q.S., Chen G.R.: Synchronization transition induced by synaptic delay in coupled fast-spiking neurons. Int. J. Bifur. Chaos 18, 1189–1198 (2008) · Zbl 1147.34334
[161] Chavas J., Marty A.: Coexistence of excitatory and inhibitory GABA synapses in the cerebellar interneuron network. J. Neurosci. 23, 2019–2031 (2003)
[162] Hasegawa H.: Augmented moment method for stochastic ensembles with delayed couplings. II. FitzHugh-Nagumo model. Phys. Rev. E 70, 021912 (2004)
[163] Hasegawa H.: Augmented moment method for stochastic ensembles with delayed couplings. I. Langevin model. Phys. Rev. E 70, 021911 (2004)
[164] Sainz-Trapága M., Masoller C., Braun H.A., Huber M.T.: Influence of time-delayed feedback in the firing pattern of thermally sensitive neurons. Phys. Rev. E 70, 031904 (2004)
[165] Burić N., Todorović K., Vasović N.: Influence of noise on dynamics of coupled bursters. Phys. Rev. E 75, 067204 (2007)
[166] Burić N., Todorović K., Vasović N.: Global stability of synchronization between delay-differential systems with generalized diffusive coupling. Chaos, Solitons Fractals 31, 336–342 (2007) · Zbl 1336.34102
[167] Zhou C.S., Kurth J., Hu B.: Array-enhanced coherence resonance: nontrivial effects of heterogeneity and spatial independence of noise. Phys. Rev. Lett. 87, 098101 (2001)
[168] Balenzuela P., Garćia-Ojalvo J.: Role of chemical synapses in coupled neurons with noise. Phys. Rev. E 72, 021901 (2005)
[169] Perc M.: Spatial coherence resonance in excitable media. Phys. Rev. E 72, 016207 (2005)
[170] Perc M.: Spatial decoherence induced by small-world connectivity in excitable media. New J. Phys. 7, 252 (2005)
[171] Perc M.: Effects of small-world connectivity on noise-induced temporal and spatial order in neural media. Chaos, Solitons Fractals 31, 280–291 (2007) · Zbl 1133.92007
[172] Perc M.: Stochastic resonance on excitable small-world networks via a pacemaker. Phys. Rev. E 76, 066203 (2007)
[173] Gosak M., Marhl M., Perc M.: Spatial coherence resonance in excitable biochemical media induced by internal noise. Biophys. Chem. 128, 210–214 (2007)
[174] Perc M., Gosak M., Marhl M.: Periodic calcium waves in coupled cells induced by internal noise. Chem. Phys. Lett. 437, 143–147 (2007)
[175] Perc M.: Spatial coherence resonance in neuronal media with discrete local dynamics. Chaos, Solitons Fractals 31, 64–69 (2007)
[176] Wang Q.Y., Lu Q.S., Chen G.R.: Spatio-temporal patterns in a square-lattice Hodgkin–Huxley neural network. Eur. Phys. J. B 54, 255–261 (2006)
[177] Wang Q.Y., Lu Q.S., Chen G.R.: Subthreshold stimulus-aided temporal order and synchronization in a square lattice noisy neuronal network. Europhys. Lett. 77, 10004 (2007)
[178] Zheng Y.H., Lu Q.S.: Spatiotemporal patterns and chaotic burst synchronization in a small-world neuronal network. Physica A 387(14), 3719–3728 (2008)
[179] Sun X.J., Perc M., Lu Q.S., Kurths J.: Spatial coherence resonance on diffusive and small-world networks of Hodgkin–Huxley neurons. Chaos 18, 023102 (2008) · Zbl 06417113
[180] Buzsáki G., Geisler C., Henze D.A., Wang X.J.: Interneuron diversity series: circuit complexity and axon wiring economy of cortical interneurons. Trends Neurosci. 27(4), 186–193 (2004)
[181] Buzsáki G., Chrobak J.J.: Temporal structure in spatially organized neuronal ensembles: a role for interneuronal networks. Curr. Opin. Neurobiol. 5(4), 504–510 (1995)
[182] Whittington M.A., Traub R.D., Kopell N., Ermentrout B., Buhl E.H.: Inhibition-based rhythms: experimental and mathematical observations on network dynamics. Int. J. Psycophysiol. 38(3), 315–336 (2000)
[183] Börgers C., Kopell N.: Synchronization in Networks of Excitatory and Inhibitory Neurons with Sparse, Random Connectivity. Neural Comput. 15, 509–538 (2003) · Zbl 1085.68615
[184] Engel A.K., Fries P., Singer W.: Dynamic predictions: oscillations and synchrony in top-down processing. Nat. Rev. Neurosci. 2, 704–716 (2001)
[185] Gray C.M. et al.: Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338, 334–337 (1989)
[186] Roxin A., Riecke H., Solla S.A.: Self-sustained activity in a small-world network of excitable neurons. Phys. Rev. Lett. 92, 198101 (2004)
[187] Byrne, J.H., Roberts, J.L. (eds.): From Molecules to Networks: An Introduction to Cellular and Molecular Neuroscience. Elsevier Science, New York (2004) · Zbl 1081.92006
[188] Coombes, S., Bressloff, P.C. (eds.): Bursting: The Genesis of Rhythm in the Nervous System. World Scientific, Singapore (2005) · Zbl 1094.92500
[189] Schuster S., Marhl M., Höfer T.: Modelling of simple and complex calcium oscillations. From single-cell to intercellular signaling. Eur. J. Biochem. 269, 1333–1355 (2002)
[190] Falcke M.: Reading the patterns in living cells–the physics of Ca2+signaling. Adv. Phys. 53, 255–440 (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.