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FitzHugh-Nagumo equations with generalized diffusive coupling
On a spike train probability model with interacting neural units
1. | Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, I-84084 Fisciano (SA), Italy, Italy |
2. | Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, I-80126 Napoli, Italy |
References:
[1] |
D. J. Amit and N. Brunel, Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex, Cerebral Cortex, 7 (1997), 237-252.
doi: 10.1093/cercor/7.3.237. |
[2] |
M. Barbi, S. Chillemi, A. Di Garbo and L. Reale, Stochastic resonance in a sinusoidally forced LIF model with noisy threshold, Biosystems, 71 (2003), 23-28.
doi: 10.1016/S0303-2647(03)00106-0. |
[3] |
O. Bernander, C. Koch and M. Usher, The effect of synchronized inputs at the single neuron level, Neural Computation, 6 (1994), 622-641. |
[4] |
A. Buonocore, A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, A Markov chain-based model for actomyosin dynamics, Sci. Math. Japon., 70 (2009), 159-174. |
[5] |
A. Buonocore, A. Di Crescenzo, B. Martinucci and L. M. Ricciardi, A stochastic model for the stepwise motion in actomyosin dynamics, Sci. Math. Japon., 58 (2003), 245-254. |
[6] |
M. J. Berry and M. Meister, Refractoriness and neural precision, J. Neurosci., 18 (1998), 2200-2211. |
[7] |
A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input, Biol. Cybern., 95 (2006), 1-19.
doi: 10.1007/s00422-006-0068-6. |
[8] |
A. N. Burkitt, A review of the integrate-and-fire neuron model. II. Inhomogeneous synaptic input and network properties, Biol. Cybern., 95 (2006), 97-112.
doi: 10.1007/s00422-006-0082-8. |
[9] |
H. P. Chan and W.-L. Loh, Some theoretical results on neural spike train probability models, Ann. Stat., 35 (2007), 2691-2722.
doi: 10.1214/009053607000000280. |
[10] |
M. Crowder, Classical Competing Risks, Chapman & Hall/CRC, Boca Raton, 2001.
doi: 10.1201/9781420035902. |
[11] |
M. Deger, S. Cardanobile, M. Helias and S. Rotter, The Poisson process with dead time captures important statistical features of neural activity, BMC Neuroscience, 10 (2009), P110.
doi: 10.1186/1471-2202-10-S1-P110. |
[12] |
A. Di Crescenzo, E. Di Nardo, A. G. Nobile, E. Pirozzi and L. M. Ricciardi, On some computational results for single neurons' activity modeling, BioSystems, 58 (2000), 19-26. |
[13] |
A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, Stochastic population models with interacting species, J. Math. Biol., 42 (2001), 1-25.
doi: 10.1007/PL00000070. |
[14] |
A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, A note on birth-death processes with catastrophes, Stat. Prob. Lett., 78 (2008), 2248-2257.
doi: 10.1016/j.spl.2008.01.093. |
[15] |
A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, On time non-homogeneous stochastic processes with catastrophes, in Cybernetics and Systems 2010 (ed. R. Trappl), Austrian Society for Cybernetic Studies, Vienna, 2010, 169-174. |
[16] |
A. Di Crescenzo and M. Longobardi, On the NBU ageing notion within the competing risks model, J. Stat. Plann. Infer., 136 (2006), 1638-1654.
doi: 10.1016/j.jspi.2004.08.022. |
[17] |
A. Di Crescenzo and M. Longobardi, Competing risks within shock models, Sci. Math. Japon., 67 (2008), 125-135. |
[18] |
A. Di Crescenzo, B. Martinucci, E. Pirozzi and L. M. Ricciardi, On the interaction between two Stein's neuronal units, in Cybernetics and Systems 2004 (ed. R. Trappl), Austrian Society for Cybernetic Studies, Vienna, 2004, 205-210. |
[19] |
G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophy. J., 4 (1964), 41-68.
doi: 10.1016/S0006-3495(64)86768-0. |
[20] |
D. Hampel and P. Lansky, On the estimation of refractory period, J. Neurosci. Meth., 171 (2008), 288-295.
doi: 10.1016/j.jneumeth.2008.03.003. |
[21] |
D. H. Johnson, Point process models of single-neuron discharges, J. Comput. Neurosci., 3 (1996), 275-299.
doi: 10.1007/BF00161089. |
[22] |
D. H. Johnson and A. Swami, The transmission of signals by auditory-nerve fiber discharge patterns, J. Acoust. Soc. Am., 74 (1983), 493-501.
doi: 10.1121/1.389815. |
[23] |
R. E. Kass and V. Ventura, A spike-train probability model, Neural Comput., 13 (2001), 1713-1720.
doi: 10.1162/08997660152469314. |
[24] |
A. Mazzoni, F. D. Broccard, E. Garcia-Perez, P. Bonifazi, M. E. Ruaro and V. Torre, On the dynamics of the spontaneous activity in neuronal networks, PLoS ONE, 2 (2007), e439.
doi: 10.1371/journal.pone.0000439. |
[25] |
M. I. Miller, Algorithms for removing recovery-related distortion from auditory nerve discharge patterns, J. Acoust. Soc. Am., 77 (1985), 1452-1464.
doi: 10.1121/1.392040. |
[26] |
U. Picchini, S. Ditlevsen, A. De Gaetano and P. Lansky, Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal, Neural Comput., 20 (2008), 2696-2714.
doi: 10.1162/neco.2008.11-07-653. |
[27] |
L. M. Ricciardi, Diffusion Processes and Related Topics in Biology, Notes taken by Charles E. Smith, Lecture Notes in Biomathematics, Vol. 14, Springer-Verlag, Berlin-New York, 1977. |
[28] |
L. M. Ricciardi, Modeling single neuron activity in the presence of refractoriness: New contributions to an old problem, in Imagination and Rigor. Essays on Eduardo R. Caianiello's Scientific Heritage (ed. S. Termini), Springer-Verlag Italia, 2006, 133-145.
doi: 10.1007/88-470-0472-1_11. |
[29] |
L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, On the instantaneous return process for neuronal diffusion models, in Structure: from Physics to General Systems - Festschrift Volume in Honour of E.R. Caianiello on his Seventieth Birthday (eds. M. Marinaro and G. Scarpetta), World Scientific, Singapore, 1992, 78-94. |
[30] |
L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling, Math. Japon., 50 (1999), 247-322. |
[31] |
W. R. Softky and C. Koch, The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs, The Journal of Neuroscience, 13 (1993), 334-350. |
[32] |
R. B. Stein, A theoretical analysis of neuronal variability, Biophys. J., 5 (1965), 173-194.
doi: 10.1016/S0006-3495(65)86709-1. |
[33] |
T. Tateno, S. Doi, S. Sato and L. M. Ricciardi, Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach, J. Stat. Phys., 78 (1995), 917-935.
doi: 10.1007/BF02183694. |
[34] |
K. Yoshino, T. Nomura, K. Pakdaman and S. Sato, Synthetic analysis of periodically stimulated excitable and oscillatory membrane models, Phys. Rev. E (3), 59 (1999), 956-969.
doi: 10.1103/PhysRevE.59.956. |
show all references
References:
[1] |
D. J. Amit and N. Brunel, Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex, Cerebral Cortex, 7 (1997), 237-252.
doi: 10.1093/cercor/7.3.237. |
[2] |
M. Barbi, S. Chillemi, A. Di Garbo and L. Reale, Stochastic resonance in a sinusoidally forced LIF model with noisy threshold, Biosystems, 71 (2003), 23-28.
doi: 10.1016/S0303-2647(03)00106-0. |
[3] |
O. Bernander, C. Koch and M. Usher, The effect of synchronized inputs at the single neuron level, Neural Computation, 6 (1994), 622-641. |
[4] |
A. Buonocore, A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, A Markov chain-based model for actomyosin dynamics, Sci. Math. Japon., 70 (2009), 159-174. |
[5] |
A. Buonocore, A. Di Crescenzo, B. Martinucci and L. M. Ricciardi, A stochastic model for the stepwise motion in actomyosin dynamics, Sci. Math. Japon., 58 (2003), 245-254. |
[6] |
M. J. Berry and M. Meister, Refractoriness and neural precision, J. Neurosci., 18 (1998), 2200-2211. |
[7] |
A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input, Biol. Cybern., 95 (2006), 1-19.
doi: 10.1007/s00422-006-0068-6. |
[8] |
A. N. Burkitt, A review of the integrate-and-fire neuron model. II. Inhomogeneous synaptic input and network properties, Biol. Cybern., 95 (2006), 97-112.
doi: 10.1007/s00422-006-0082-8. |
[9] |
H. P. Chan and W.-L. Loh, Some theoretical results on neural spike train probability models, Ann. Stat., 35 (2007), 2691-2722.
doi: 10.1214/009053607000000280. |
[10] |
M. Crowder, Classical Competing Risks, Chapman & Hall/CRC, Boca Raton, 2001.
doi: 10.1201/9781420035902. |
[11] |
M. Deger, S. Cardanobile, M. Helias and S. Rotter, The Poisson process with dead time captures important statistical features of neural activity, BMC Neuroscience, 10 (2009), P110.
doi: 10.1186/1471-2202-10-S1-P110. |
[12] |
A. Di Crescenzo, E. Di Nardo, A. G. Nobile, E. Pirozzi and L. M. Ricciardi, On some computational results for single neurons' activity modeling, BioSystems, 58 (2000), 19-26. |
[13] |
A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, Stochastic population models with interacting species, J. Math. Biol., 42 (2001), 1-25.
doi: 10.1007/PL00000070. |
[14] |
A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, A note on birth-death processes with catastrophes, Stat. Prob. Lett., 78 (2008), 2248-2257.
doi: 10.1016/j.spl.2008.01.093. |
[15] |
A. Di Crescenzo, V. Giorno, A. G. Nobile and L. M. Ricciardi, On time non-homogeneous stochastic processes with catastrophes, in Cybernetics and Systems 2010 (ed. R. Trappl), Austrian Society for Cybernetic Studies, Vienna, 2010, 169-174. |
[16] |
A. Di Crescenzo and M. Longobardi, On the NBU ageing notion within the competing risks model, J. Stat. Plann. Infer., 136 (2006), 1638-1654.
doi: 10.1016/j.jspi.2004.08.022. |
[17] |
A. Di Crescenzo and M. Longobardi, Competing risks within shock models, Sci. Math. Japon., 67 (2008), 125-135. |
[18] |
A. Di Crescenzo, B. Martinucci, E. Pirozzi and L. M. Ricciardi, On the interaction between two Stein's neuronal units, in Cybernetics and Systems 2004 (ed. R. Trappl), Austrian Society for Cybernetic Studies, Vienna, 2004, 205-210. |
[19] |
G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophy. J., 4 (1964), 41-68.
doi: 10.1016/S0006-3495(64)86768-0. |
[20] |
D. Hampel and P. Lansky, On the estimation of refractory period, J. Neurosci. Meth., 171 (2008), 288-295.
doi: 10.1016/j.jneumeth.2008.03.003. |
[21] |
D. H. Johnson, Point process models of single-neuron discharges, J. Comput. Neurosci., 3 (1996), 275-299.
doi: 10.1007/BF00161089. |
[22] |
D. H. Johnson and A. Swami, The transmission of signals by auditory-nerve fiber discharge patterns, J. Acoust. Soc. Am., 74 (1983), 493-501.
doi: 10.1121/1.389815. |
[23] |
R. E. Kass and V. Ventura, A spike-train probability model, Neural Comput., 13 (2001), 1713-1720.
doi: 10.1162/08997660152469314. |
[24] |
A. Mazzoni, F. D. Broccard, E. Garcia-Perez, P. Bonifazi, M. E. Ruaro and V. Torre, On the dynamics of the spontaneous activity in neuronal networks, PLoS ONE, 2 (2007), e439.
doi: 10.1371/journal.pone.0000439. |
[25] |
M. I. Miller, Algorithms for removing recovery-related distortion from auditory nerve discharge patterns, J. Acoust. Soc. Am., 77 (1985), 1452-1464.
doi: 10.1121/1.392040. |
[26] |
U. Picchini, S. Ditlevsen, A. De Gaetano and P. Lansky, Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal, Neural Comput., 20 (2008), 2696-2714.
doi: 10.1162/neco.2008.11-07-653. |
[27] |
L. M. Ricciardi, Diffusion Processes and Related Topics in Biology, Notes taken by Charles E. Smith, Lecture Notes in Biomathematics, Vol. 14, Springer-Verlag, Berlin-New York, 1977. |
[28] |
L. M. Ricciardi, Modeling single neuron activity in the presence of refractoriness: New contributions to an old problem, in Imagination and Rigor. Essays on Eduardo R. Caianiello's Scientific Heritage (ed. S. Termini), Springer-Verlag Italia, 2006, 133-145.
doi: 10.1007/88-470-0472-1_11. |
[29] |
L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, On the instantaneous return process for neuronal diffusion models, in Structure: from Physics to General Systems - Festschrift Volume in Honour of E.R. Caianiello on his Seventieth Birthday (eds. M. Marinaro and G. Scarpetta), World Scientific, Singapore, 1992, 78-94. |
[30] |
L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling, Math. Japon., 50 (1999), 247-322. |
[31] |
W. R. Softky and C. Koch, The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs, The Journal of Neuroscience, 13 (1993), 334-350. |
[32] |
R. B. Stein, A theoretical analysis of neuronal variability, Biophys. J., 5 (1965), 173-194.
doi: 10.1016/S0006-3495(65)86709-1. |
[33] |
T. Tateno, S. Doi, S. Sato and L. M. Ricciardi, Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: A first-passage-time approach, J. Stat. Phys., 78 (1995), 917-935.
doi: 10.1007/BF02183694. |
[34] |
K. Yoshino, T. Nomura, K. Pakdaman and S. Sato, Synthetic analysis of periodically stimulated excitable and oscillatory membrane models, Phys. Rev. E (3), 59 (1999), 956-969.
doi: 10.1103/PhysRevE.59.956. |
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