Identifying distinct stochastic dynamics from chao: A Study on multimodal neural firing patterns

International Journal of Bifurcation and Chaos

Volumn 19, Issue 2, 2009, Pages 453-485

  Yang, Minghao ^a   Liu, Zhiqiang ^a   Li, Li ^b   Xu, Yulin ^b   Liu, Hongjv ^b   Gu, Huaguang ^a   Ren, Wei ^a  

^a SHAANXI NORMAL UNIVERSITY   (China)

^b INSTITUTE OF SPACE MEDICO ENGINEERING   (China)

Author keywords

Bifurcation multimodal characteristics; Chaos; Neural firing; Nonlinear prediction; Stochastic dynamics

Indexed keywords

BIFURCATION (MATHEMATICS); CHAOS THEORY; DELTA SIGMA MODULATION; DYNAMICAL SYSTEMS; DYNAMICS; HOPF BIFURCATION; STOCHASTIC MODELS; TIME SERIES;

LIMIT CYCLE ATTRACTORS; MULTI-MODAL; NEURAL FIRING; NEURAL FIRING PATTERNS; NON-LINEAR PREDICTIONS; NONLINEAR TIME SERIES; STOCHASTIC DYNAMICS; THEORETICAL SIMULATION;

STOCHASTIC SYSTEMS;

EID: 67449138053     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127409023135     Document Type: Article

References (85)

1. Abarbanel, H. D. I., Brown, R., Sidorowich, J. J. & Tsimring, L. S. [1993] "The analysis of observed chaotic data in physical system," Rev. Mod. Phys. 65, 1331-1393.
2. Aihara, K., Matsumoto, G. & Ikegaya, Y. [1984] "Periodic and nonperiodic response of a periodically forced Hodgkin-Huxley oscillator," J. Theor. Biol. 109, 249-269.
3. Aihara, K., Matsumoto, G. & Ichikawa, M. [1985] "An alternating periodic-chaotic sequence observed in neural oscillators," Phys. Lett. A 111, 435-438.
4. Aihara, K. & Matsumoto, G. [1987] "Forced oscillations and routines to chaos in the Hodgkin-Huxley axons and squid giant axons," Chaos in Biological Systems, eds. Degn, H., Holden, A. V. & Olsen L. F., pp. 121-131.
5. Albano, A. M., Muench, J., Schwartz, C., Mees, A. I. & Rapp, P. E. [1988] "Singular value decomposition and the Grassberger-Procaccia algorithm," Phys. Rev. A 38, 3107-3026.
6. Babloyantz, A., Salazar, J. M. & Nicolis, C. [1985] "Evidence of chaotic dynamics of brain activity during the sleep cycle," Phys. Lett. A 111, 152-156.
7. Bennett, G. T. & Xie, Y. K. [1988] "A peripheral mononeuropathy in rat that produces disorders of pain sensation like those seen in man," Pain 33, 87-107.
8. Benzi, R., Sutera, S. & Vulpiani, A. [1981] "The mechanism of stochastic resonance," J. Phys. A 14, L453-L457.
9. Bezrukov, S. M. & Vodyanoy, I. [1995] "Noise-induced enhancement of signal transduction across voltagedependent ion channels," Nature 378, 362-364.
10. Braun, H. A., Wissing, H. & Schäfer, K. [1994] "Oscillation and noise determine signal transduction in shark multi-modal sensory cells," Nature 367, 270-273.
11. Braun, H. A., Schafer, K., Voigt, K., Peters, R., Bretschneider, F., Pei, X., Wilkens, L. & Moss, F. [1997] "Low-dimensional dynamics in sensory biology. 1: Thermally sensitive electroreceptors of the catfish," Comput. Neurosci. 4, 335-347.
12. Braun, H. A., Dewald, M., Schafer, K., Voigt, K., Pei, X., Dolan, K. & Moss, F. [1999] "Low-dimensional dynamics in sensory biology 2: Facial cold receptors of the rat," J. Comput. Neurosci. 7, 17-32.
13. Chay, T. R. [1985] "Chaos in a three-variable model of an excitable cell," Physica D 16, 233-242.
14. Chay, T. R., Fan, Y. S. & Lee, Y. S. [1995] "Bursting, spiking, chaos, fractals and universality in biological rhythms," Int. J. Bifurcation and Chaos 5, 595-635.
15. Chay, T. R. & Rinzel, J. [1985] "Bursting, beating, and chaos in an excitable membrane model," Biophys. J. 47, 357-366.
16. Collins, J. J., Imhoff, T. T. & Grigg, P. [1997] "Noisemediated enhancements and decrements in human tactile sensation," Phys. Rev. E 56, 923-926.
17. Ditto, W. L., Spano, M. L., In, V., Neff, J. & Meadows B. [2000] "Control of human atrial fibrillation," Int. J. Bifurcation and Chaos 10, 593-601.
18. Douglass, J. K., Wilkens, L. & Pantazelou, E. [1993] "Noise enhancement of information transfer in crayfish mechoreceptors by stochastic resonance," Nature 365, 337-340.
19. Fallon, J. B., Richard, W. C. & Morgan, D. L. [2004] "Stochastic resonance in muscle receptors," J. Neurophysiol. 91, 2429-2436.
20. Fan, Y. S. & Holden, A. V. [1993] "Bifurcations, burstings, chaos and crises in the Rose-Hindmarsh model for neuronal activity," Chaos Solit. Fract. 3, 439-449.
21. Fan, Y. S. & Chay, T. R. [1994] "Generation of periodic and chaotic bursting in an excitable cell model," Biol. Cybern. 71, 417-431.
22. Farmer, J. D. & Sidorowich, J. J. [1987] "Predicating chaotic series," Phys. Rev. Lett. 59, 845-849.
23. Gammaitoni, L., Hanggi, P., Jung, P. & Marchesoni, F. [1998] "Stochastic resonance," Rev. Mod. Phys. 70, 223-287.
24. Garfinkel, A., Spano, M. L., Ditto, W. L. & Weiss, J. N. [1992] "Experimental control of cardiac chaos," Science 257, 1230-1235.
25. Gerstein, G. L. & Mandelbrot, B. [1964] "Random walk models for the spike activity of a single neuron," Biophys. J. 4, 41-68.
26. Gong, P. L., Xu, J. X., Long, K. P. & Hu, S. J. [1999] "The deterministic and stochastic integer multiple firing of neurons," Acta Biophys. Sin. 15, 482-487. (In Chinese).
27. Gong, P. L., Xu, J. X., Hu, S. J. & Long, K. P. [2002] "Chaotic interspike intervals with multipeaked in neurons histogram," Int. J. Bifurcation and Chaos 12, 319-328.
28. Gong, Y. F., Xu, J. X., Ren, W., Hu, S. J. & 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," Biol. Cybern. 78, 159-165.
29. Gu, H. G., Ren, W., Lu, Q. S., Wu, S. G., Yang, M. H. & Chen, W. J. [2001] "Integer multiple spiking in neural pacemakers without external periodic stimulation," Phys. Lett. A 285, 63-68.
30. Gu, H. G., Yang, M. H., Li, L., Liu, Z. Q. & Ren, W. [2002] "Experimental observation of stochastic bursting caused by coherence resonance in experimental neural pacemaker," NeuroReport 13, 1657-1660.
31. Gu, H. G., Yang, M. H., Li, L., Liu, Z. Q. & Ren, W. [2003a] "Noise induced multi-mode firing patterns in neural period adding bifurcation scenario," Int. J. Modern. Phys. B 17, 4195-4200.
32. Gu, H. G., Yang, M. H., Li, L., Liu, Z. Q. & Ren, W. [2003b] "Dynamics of autonomous stochastic resonance in neural period adding bifurcation scenarios," Phys. Lett. A 319, 89-96.
33. Gu, H. G., Yang, M. H., Li, L., Liu, Z. Q. & Ren, W. [2004] "Chaotic and ASR induced firing pattern in experimental neural pacemaker," Dyn. Contin. Discr. Impul. Syst. (Series B: Applications & Algorithms) 11a, 19-24.
34. Gu, H. G., Yang, M. H., Li, L., Ren, W. & Lu, Q. S. [2007] "Period adding bifurcation with chaotic and stochastic bursting in an experimental neural pacemaker," Dyn. Contin. Discr. Impul. Syst. (Series B) 14, 6-11.
35. Guevara, M. R., Glass, L. & Shrier, A. [1981] "Phase locking, period doubling bifurcations and irregular dynamics in periodically stimulated cardiac cells," Science 214, 1350-1353.
36. Hayashi, H., Ishzuka, S., Ohta, M. & Hirakawa, K. [1982] "Chaotic behavior in the onchidium giant neuron," Phys. Lett. A 88, 435-438.
37. Hayashi, H., Ishzuka, S. & Hirakawa, K. [1983] "Transition to chaos via intermittency in the onchidium pacemaker neuron," Phys. Lett. A 98, 474-476.
38. Huber, M. T., Krige, J. C., Braun, H. A., Pei, X., Neiman, A. & Moss, F. [2000] "Noisy precursors of bifurcation in a neurodynamical model for disease states of mood disorder," Neurocomputing 32-33, 823-831.
39. Izhikevich, E. M. [2000] "Neural excitability, spiking and bursting," Int. J. Bifurcation and Chaos 14, 1171-1266.
40. Izhikevich, E. M. [2007] Dynamical System in Neuroscience: The Geometry of Excitability and Bursting(The MIT Press, Cambridge MA).
41. Kantz, H. [1994] "A robust method to estimate the maximal Lyapunov exponent of a time series," Phys. Lett. A 185, 77-87.
42. Kantz, H. & Schreiber, T. [1997] Nonlinear Time Series Analysis(Cambridge university press, Cambridge, England); [2000] (Tsinghua University Press, Beijing, China), 2000.
43. Kaplan, D. T., Clay, J. R., Manning, T., Glass, L., Guevara, M. R. & Shrier, A. [1996] "Subthreshold dynamics in periodically stimulated squid giant axons," Phys. Rev. Lett. 76, 4074-4077.
44. Kasper, F. & Schuster, H. G. [1987] "Easily calculable measure for the complexity of spatio-temporal patterns," Phys. Rev. A 36, 836-842.
45. Kugiumtzis, D. [2001] "On the reliability of the surrogate data test for nonlinearity in analysis of the noisy time series," Int. J. Bifurcation and Chaos 11, 1881-1896.
46. Lehrman, M., Rechester, A. B. & White, R. B. [1997] "Symbolic analysis of chaotic signals and turbulent fluctuations," Phys. Rev. Lett. 78, 54-57.
47. Lemple, A. & Ziv, J. [1976] "On the complexity of finite sequence," IEEE Trans. IT 222, 75-81.
48. Li, L., Gu, H. G., Yang, M. H., Liu, Z. Q. & Ren, W. [2004] "A series of bifurcation scenarios in the firing transitions in an experimental neural pacemaker," Int. J. Bifurcation and Chaos 14, 1813-1817.
49. Lindner, B., García-Ojalvo, J., Neiman, A. & Schimansky-Geier, L. [2004] "Effects of noise in excitable systems," Phys. Rep. 392, 321-424.
50. Long, K. P., Hu, S. J., Jian, Z., Duan, Y. B. & Xu, H. [2000] "Pattern and dynamic changes of integer multiples in spontaneous discharge of injured dorsal root ganglion neurons," Acta Biophys. Sin. 16, 250-258 (in Chinese).
51. Longtin, A., Bulsara, A. & Moss, F. [1991] "Time interval sequences in bistable system and the noise-induced transmission of information by sensory neurons," Phys. Rev. Lett. 67, 656-659.
52. Longtin, A. [1993] "Nonlinear forecasting of spike trains from sensory neurons," Int. J. Bifurcation and Chaos 3, 651-661.
53. Longtin, A., Bulsara, A., Pierson, D. & Moss F. [1994] "Bistability and the dynamics of periodically forced sensory neurons," Biol. Cybern 70, 569-578.
54. Longtin, A. [1997] "Autonomous stochastic resonance in bursting neurons," Phys. Rev. E 55, 868-876.
55. Mannella, R. & Palleschi, V. [1989] "Fast and precise algorithm for compute simulation of the stochastic differential equations," Phys. Rev. A 40, 3381-3386.
56. Pei, X. & Moss, F. [1996] "Characterization of lowdimensional dynamics in the crayfish caudal photoreceptor," Nature 379, 618-621.
57. Pei, X., Wilkens, L. & Moss, F. [1996] "Noise-mediated spike timing precision from aperiodic stimuli in an array of Hodgkin-Huxley-type neurons," Phys. Rev. Lett. 77, 4679-4682.
58. Pikovsky, A. S. & Kurth, J. [1997] "Coherence resonance in a noise-driven excitable system," Phys. Rev. Lett. 78, 775-778.
59. Rapp, P. E. [1986] "Oscillations and chaos in cellular metabolism and physiological systems," in Chaos-Nonlinear Science. Theory and Applications, ed. Holden, A. V. (Manchester Univ. Press, Manchester, UK), pp. 179-208.
60. Ren, W., Hu, S. J., Zhang, B. J., Xu, J. X. & Gong, Y. F. [1997] "Period-adding bifurcation with chaos in the interspike intervals generated by an experimental neural pacemaker," Int. J. Bifurcation and Chaos 7, 1867-1872.
61. Ren, W., Gu, H. G., Jian, Z., Lu, Q. S. & Yang, M. H. [2001] "Different classification of UPOs in the parametrically different chaotic ISI series," NeuroReport 12, 2121-2124.
62. Rose, J. E., Brugge, J. F., Arderson, D. D. & Hind, J. E. [1967] "Phase-locked response to low-frequency tones in single auditory nerve fibers of the squirrel monkey," J. Neurophysiol. 30, 769-793.
63. Sauer, T. [1994] "Reconstruction of dynamical system from interspike intervals," Phys. Rev. Lett. 72, 3811-3814.
64. Schiff, S. J., Jerger, K. & Duong, D. H. [1994]"Controlling chaos in the brain," Nature 370, 615-620.
65. Siegel, R. M. [1990] "Nonlinear dynamical system theory and primary visual cortical processing," Physica D 42, 385-395.
66. So, P., Ott, E., Schiff, S. J., Kaplan, D. T., Sauer, T. & Grebogi, C. [1996] "Detecting unstable periodic orbits in chaotic experimental data," Phys. Rev. Lett. 76, 4705-4708.
67. So, P., Ott, E., Sauer, T., Gluckman, B. J., Grebogi, C. & Schiff, S. J. [1997] "Extraction unstable periodic orbits from chaotic experimental data," Phys. Rev. E 55, 5398-5417.
68. So, P., Francis, J. T., Netoff, T. I., Gluckman, B. J. & Schiff, S. J. [1998] "Periodic orbits: A new language for neuronal dynamics," Biophys. J. 74, 2776-2785.
69. Sugihara, G. & May, R. M. [1990] "Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series," Nature 344, 734-741.
70. Suzuki, H., Aihara, K. & Murakami. [2000] "Analysis of neural spike trains with interspike interval reconstruction," Biol. Cybern. 82, 305-311.
71. Takens, F. [1981] "Detecting strange attractors in turbulence," Dynamical Systems and Turbulence, eds. Rand, D. A. & Yong, L.-S., Lecture Notes in Mathematics, Vol. 898 (Springer-Verlag), pp. 366-381.
72. Theiler, J., Eubank, S., Longtin, A., Galdrinkian, B. & Farmer, J. D. [1992] "Testing for nonlinearity in time series: The method of surrogate data," Physica D 58, 77-94.
73. Thomas, E., William, J. R., Zbigniew, J. K., James, E. S., Karl, E. G. & Niels, B. [1994] "Chaos and physiology: Deterministic chaos in excitable cell assemblies," Physiol. Rev. 74, 1-47.
74. Wan, Y. H., Jian, Z. & Hu, S. J. [2000] "Detection of determinism within time series of irregular burst firing from injured sensory neuron," NeuroReport 11, 3295-3298.
75. Wang, X. J. [1994] "Multiple dynamical modes of thalamic relay neurons: Rhythmic bursting and intermittent phase-locking," Neuroscience 59, 21-31.
76. Whitney, H. [1936] "Differentiable manifolds," Ann. Math. 37, 645-680.
77. Wiesenfeld, K., Pierson, D. & Pantazelo, E. [1994] "Stochastic resonance on a circle," Phys. Rev. Lett. 72, 2125-2129.
78. Wiesenfeld, K. & Moss, F. [1995] "Stochastic resonance and the benefits of noise: From ice ages to crayfish and SQUIDs," Nature 373, 33-36.
79. Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. [1985] "Determining Lyapunov exponents from a time series," Physica D 16, 285-317.
80. Xie, Y., Xu, J. X. & Jiang, J. [2006] "Deterministic integer multiple firing depending on initial state in Wang model," Chaos, Solit. Fract. 30, 1250-1259.
81. Xing, J. L., Hu, S. J., Xu, H., Han, S. & Wan, Y. H. [2001] "Subthreshold membrane oscillation underlying integer multiples firing from injured sensory neurons," NeuroReport 12, 1311-1313.
82. Xu, J. X., Gong, Y. F., Ren, W., Hu, S. J. & Wang, F. Z. [1997] "Propagation of periodic and chaos action potential trains along nerve fibers," Physica D 100, 212-225.
83. Xu, Y. L., Li, L., Yang, M. H., Liu, Z. Q., Liu, H. J., Gu, H. G. & Ren, W. [2007] "Three cases of the bifurcation from period 1 to period 2 bursting in theoretical and experimental neural models," Dyn. Contin. Discr. Impul. Syst. (Series B) 14, 41-46.
84. Yang, M. H., Gu, H. G., Li, L., Liu, Z. Q. & Ren, W. [2003] "Characteristics of period adding bifurcation without chaos in firing pattern transitions in an experimental neural pacemaker," NeuroReport 14, 2153-2157.
85. Yang, M. H., An, S. C., Gu, H. G., Liu, Z. Q. & Ren,W. [2006] "Understanding of physiological neural firing patterns through dynamical bifurcation machineries," NeuroReport 17, 995-999.