Spectrum of Lyapunov exponents of non-smooth dynamical systems of integrate-and-fire type

Journal of Computational Neuroscience

Volumn 28, Issue 2, 2010, Pages 229-245

  Zhou, Douglas ^a   Sun, Yi ^a   Rangan, Aaditya V ^a   Cai, David ^a,b  

^a NEW YORK UNIVERSITY   (United States)

^b SHANGHAI JIAO TONG UNIVERSITY   (China)

Author keywords

Firing reset; Integrate and fire; Lyapunov exponents; Non smooth; Refractory induced degeneracy

Indexed keywords

DIFFERENTIAL EQUATIONS; DYNAMICAL SYSTEMS; LYAPUNOV FUNCTIONS; LYAPUNOV METHODS; NEURAL NETWORKS; NEURONS; NUMERICAL METHODS; REFRACTORY MATERIALS;

FIRING-RESET; INTEGRATE AND FIRES; INTEGRATE-AND-FIRE NEURONS; LONG-TIME DYNAMICS; LYAPUNOV EXPONENT; NEURONAL NETWORKS; NON-SMOOTH; NON-SMOOTH DYNAMICAL SYSTEMS; REFRACTORY-INDUCED DEGENERACY; SPECTRA'S;

DYNAMICS;

ACCURACY; ALGORITHM; ARTICLE; ARTIFICIAL NEURAL NETWORK; DYNAMICS; LYAPUNOV EXPONENT; MATHEMATICAL ANALYSIS; MATHEMATICAL PARAMETERS; NERVE CELL NETWORK; PRIORITY JOURNAL; THEORETICAL MODEL;

EID: 77953324824     PISSN: 09295313     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10827-009-0201-3     Document Type: Article

References (65)

1. Alstrom, P., Christiansen, B., & Levinsen, M. T. (1988). Nonchaotic transition from quasiperiodicity to complete phase locking. Physical Review Letters, 61, 1679-1682.
2. Azouz, R., & Gray, C. M. (1999). Cellular mechanisms contributing to response variability of cortical neurons in vivo. Journal of Neuroscience, 19, 2209-2223.
3. Benettin, G., Galgani, L., Giorgilli, A., & Strelcyn, J. M. (1980a). Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them, Part I: Theory. Meccanica, 15, 9-20. (Pubitemid 11453346)
4. Benettin, G., Galgani, L., Giorgilli, A., & Strelcyn, J. M. (1980b). Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them, Part II: Numerical application. Meccanica, 15, 21-30. (Pubitemid 11453347)
5. Brette, R. (2004). Dynamics of one-dimensional spiking neuron models. Journal of Mathematical Biology, 48, 38-56.
6. Brette, R., & Gerstner, W. (2005). Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. Journal of Mathematical Biology, 94, 3637-3642.
7. Burkitt, A. N. (2006a). A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input. Biological Cybernetics, 95, 1-19.
8. Burkitt, A. N. (2006b). A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties. Biological Cybernetics, 95, 97-112.
9. Cai, D., Rangan, A. V., & McLaughlin, D. W. (2005). Architectural and synaptic mechanisms underlying coherent spontaneous activity in V1. Proceedings of National Academy of Sciences of the United States of America, 102, 5868-5873.
10. Carandini, M., Mechler, F., Leonard, C. S., & Movshon, J. A. (1996). Spike train encoding by regular-spiking cells of the visual cortex. Journal of Neurophysiology, 76, 3425-3441.
11. Chacron, M. J., Lindner, B., & Longtin, A. (2007). Threshold fatigue and information transmission. Journal of Computational Neuroscience, 23, 301-311.
12. Chacron, M. J., Longtin, A., & Maler, L. (2001a). Negative interspike interval correlations increase the neuronal capacity for encoding time dependent stimuli. Journal of Neuroscience, 21, 5328-5343.
13. Chacron, M. J., Longtin, A., & Maler, L. (2001b). Simple models of bursting and non-bursting P-type electroreceptors. Neurocomputing, 38, 129-139.
14. Chacron, M. J., Longtin, A., & Pakdaman, K. (2004). Chaotic firing in the sinusoidally forced leaky integrate-and-fire model with threshold fatigue. Physica D, 192, 138-160.
15. Chacron, M. J., Longtin, A., St-Hilaire, M., & Maler, L. (2000). Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors. Physical Review Letters, 85, 1576-1579.
16. Chacron, M. J., Pakdaman, K., & Longtin, A. (2003). Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue. Neural Computation, 15, 253-276.
17. Dayan, P., & Abbott, L. F. (2001). Theoretical Neuroscience. Massachusetts: MIT.
18. de Souza, S. L. T., & Caldas, I. L. (2001). Basins of attraction and transient chaos in a gear-rattling model. Journal of Vibration and Control, 7, 849-862.
19. de Souza, S. L. T., & Caldas, I. L. (2004). Calculation of Lyapunov exponents in systems with impacts. Chaos, Solitons & Fractals, 19, 569-579.
20. Eckmann, J. P., & Ruelle, D. (1985). Ergodic theory of chaos and strange attractors. Reviews of Modern Physics, 57, 617-656.
21. Farmer, J. D., Ott, E., & Yorke, J. A. (1983). The dimension of chaotic attractors. Physica D, 7, 153-180.
22. Fourcaud-Trocme, N., Hansel, D., van Vreeswijk, C., & Brunel, N. (2003). How spike generation mechanisms determine the neuronal response to fluctuating inputs. Journal of Neuroscience, 23, 11628-11640.
23. Frederickson, P., Kaplan, J. L., Yorke, E. D., & Yorke, J. A. (1983). The Lyapunov dimension of strange attractors. Journal of Differential Equations, 49, 185-207.
24. Fujisaka, H., & Yamada, T. (1983). Stability theory of synchronized motion in coupled-oscillator systems. Progress of Theoretical Physics, 69, 32-47.
25. Geisler, C., Brunel, N., & Wang, X. J. (2005). Contributions of intrinsic membrane dynamics to fast network oscillations with irregular neuronal discharges. Journal of Neurophysiology, 94, 4344-4361.
26. Gerstner, W., & Kistler, W. M. (2002). Spiking neuron models. Cambridge: Cambridge University Press.
27. Guckenheimer, J., & Holmes, P. (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer.
28. Han, R. P. S., Luo, A. C. J., & Deng, W. (1995). Chaotic motion of a horizontal impact pair. Journal of Sound & Vibration, 181, 231-250.
29. Harsch, A., & Robinson, H. P. (2000). Postsynaptic variability of firing in rat cortical neurons: The roles of input synchronization and synaptic NMDA receptor conductance. Journal of Neuroscience, 20, 6181-6192.
30. Hegger, R., Kantz, H., & Schreiber, T. (1999). Practical implementation of nonlinear time series methods: The TISEAN package. Chaos, 9, 413-435.
31. Hinrichs, N., Oestreich, M., & Popp, K. (1997). Dynamics of oscillators with impact and friction. Chaos, Solitons & Fractals, 8, 535-558.
32. Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117, 500-544.
33. Horn, D., & Opher, I. (2000). Complex dynamics of neuronal thresholds. Neurocomputing, 32, 161-166.
34. Horn, D., & Usher, M. (1989). Neural networks with dynamical thresholds. Physical Review A, 40, 1036-1044.
35. Kantz, H. (1994). A robust method to estimate the maximal Lyapunov exponent of a time series. Physics letters A, 185, 77-87.
36. Kaplan, J. L., & Yorke, J. A. (1979). Funtional differential equations and approximations of fixed points. Berlin: Springer.
37. Lichtenberg, A. L., & Lieberman, M. A. (1992). Regular and chaotic dynamics. New York: Springer.
38. Lindner, B., & Longtin, A. (2005). Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron. Journal of Theoretical Biology, 232, 505-521.
39. Liu, Y. H., & Wang, X. J. (2001). Spike-frequency adaptation of a generalized integrate-and fire model neuron. Journal of Computational Neuroscience, 10, 25-45.
40. Mainen, Z. F., & Sejnowski, T. J. (1995). Reliability of spike timing in neocortical neurons. Science, 268, 1503-1506.
41. McLaughlin, D. W., Shapley, R., Shelley, M., & Wielaard, D. J. (2000). A neuronal network model of macaque primary visual cortex (V1): Orientation selectivity and dynamics in the input layer 4C alpha. Proceedings of National Academy of Sciences of the United States of America, 97, 8087-8092.
42. Mueller, P. C. (1995). Calculation of Lyapunov exponents for dynamics systems with discontinuities. Chaos, Solitons & Fractals, 5, 1671-1681.
43. Nowak, L. G., Azouz, R., Sanchez-vives, M. V., Gray, C. M., & McCormick, D. A. (2003). Electrophysiological classes of cat primary visual cortical neurons in vivo as revealed by quantitative analyses. Journal of Neurophysiology, 89, 1541-1566.
44. Oseledec, V. I. (1968). A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Transactions of the Moscow Mathematical Society, 19, 197-221.
45. Ott, E. (1993). Chaos in dynamical systems. New York: Cambridge University Press.
46. Parker, T. S., & Chua, L. O. (1989). Practical numerical algorithms for chaotic systems. New York: Springer.
47. Pecora, L. M., & Carroll, T. L. (1990). Synchronization in chaotic systems. Physical Review Letters, 64, 821-824.
48. Rabinovich, M. I., Varona, P., Selverston, A. I., & Abarbanel, H. D. I. (2006). Dynamical principles in neuroscience. Reviews of Modern Physics, 78, 1213-1265.
49. Ramasubramanian, K., & Seiram, M. S. (2000). A comparative study of computation of Lyapunov spectra with different algorithms. Physica D, 139, 72-86.
50. Rangan, A. V., & Cai, D. (2007). Fast numerical methods for simulating large-scale integrate-and-fire neuronal networks. Journal of Computational Neuroscience, 22, 81-100.
51. Rangan, A. V., Cai, D., & McLaughlin, D. W. (2005). Modeling the spatiotemporal cortical activity associated with the linemotion illusion in primary visual cortex. Proceedings of National Academy of Sciences of the United States of America, 102, 18793-18800.
52. Rauch, A., Camera, G. L., Luscher, H., Senn, W., & Fusi, S. (2003). Neocortical pyramidal cells respond as integrateand-fire neurons to in vivo-like input currents. Journal of Neurophysiology, 90, 15980-1612.
53. Shadlen, M. N., & Newsome, W. T. (1998). The variable discharge of cortical neurons: Implications for connectivity, computation and information coding. Journal of Neuroscience, 18, 3870-3896.
54. Softky, W. R., & Koch, C. (1993). The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. Journal of Neuroscience, 13, 334-350.
55. Somers, D. C., Nelson, S. B., & Sur, M. (1995). An emergent model of orientation selectivity in cat visual cortical simple cells. Journal of Neuroscience, 15, 5448-5465.
56. Stefanski, A. (2000). Estimation of the largest Lyapunov exponent in systems with impacts. Chaos, Solitons & Fractals, 11, 2443-2451.
57. Stefanski, A., & Kapitaniak, T. (2000). Using chaos synchronization to estimate the largest Lyapunov exponent of nonsmooth systems. Discrete Dynamics in Nature and Society, 4, 207-215.
58. Stefanski, A., & Kapitaniak, T. (2003). Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization. Chaos, Solitons & Fractals, 15, 233-244.
59. Tao, L., Shelley, M., McLaughlin, D. W., & Shapley, R. (2004). An egalitarian network model for the emergence of simple and complex cells in visual cortex. Proceedings of National Academy of Sciences of the United States of America, 101, 366-371.
60. Tolhurst, D. J., Movshon, J. A., & Dean, A. F. (1983). The statistical reliability of signals in single neurons in cat and monkey visual cortex. Vision Research, 23, 775-785.
61. Troyer, T. W., Krukowski, A. E., Priebe, N. J., & Miller, K. D. (1998). Contrast-invariant orientation tuning in cat visual cortex: Thalamocortical input tuning and correlation-based intracortical connectivity. Journal of Neuroscience, 18, 5908-5927.
62. Wang, X. J., & Buzsaki, G. (1996). Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. Journal of Neuroscience, 16, 6402-6413.
63. Wark, B., Lundstrom, B. N., & Fairhall, A. (2007). Sensory adaptation. Current opinion in Neurobiology, 17, 423-429.
64. Wolf, A., Swift, J. B., Swinney, H. L., & Vastano, J. A. (1985). Determining lyapunov exponents from a time series. Physica D, 16, 285-317.
65. Zhang, L., & McBain, C. J. (1995). Potassium conductances underlying repolarization and after-hyperpolarization in rat CA1 hippocampal interneurones. Journal of Physiology, 488, 661-672.