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Mode locking in a periodically forced resonate-and-fire neuron model

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Khajeh Alijani, Azadeh. (2009) Mode locking in a periodically forced resonate-and-fire neuron model. Physical Review E, Vol.80 (No.5). ISSN 1539-3755

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Official URL: http://dx.doi.org/10.1103/PhysRevE.80.051922

Abstract

The resonate-and-fire (RF) model is a spiking neuron model which from a dynamical systems perspective is a piecewise smooth system (impact oscillator). We analyze the response of the RF neuron oscillator to periodic stimuli by expressing the firing events in terms of an implicit one-dimensional time map. Based on such a firing map, we describe mode-locked solutions and their stability, leading to the so-called Arnol'd tongues. The boundaries of these tongues correspond to either local bifurcations of the firing time map or grazing bifurcations of the discontinuity of the flow. Despite the fact that the periodically driven RF system shows periodic firing, its behavior may become chaotic when the forcing frequency is near the resonant frequency. We compare these results to numerical simulations of the model undergoing sinusoidal forcing. Furthermore, upon varying a system parameter, the RF system can be reduced to the integrate-and-fire system and in this case we show the consistency of the results on mode-locked solutions.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Neurons -- Mathematical models
Journal or Publication Title:	Physical Review E
Publisher:	American Physical Society
ISSN:	1539-3755
Date:	25 November 2009
Volume:	Vol.80
Number:	No.5
Number of Pages:	12
Identification Number:	10.1103/PhysRevE.80.051922
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	University of Warwick
References:	[1] K. Aihara, G. Matsumoto, and Y. Ikegaya, J. Theor. Biol. 109, 249 (1984). [2] C. Ascoli, M. Barbi, S. Chillemi, and D. Petracchi, Biophys. J. 19, 219 (1977). [3] M. di Bernardo, C. J. Budd, A. R. Champneys, and P. Kowalczyk,Piecewise-smooth Dynamical Systems, Theory and Applications (Appllied Mathematical Sciences, Springer, 2008). [4] M. Bezzi, T. Nieus, O. J. M. D. Coenen, and E. D’Angelo, Neurocomputing 58, 593 (2004). [5] P. L. Boyland, Commun. Math. Phys. 106, 353 (1986). [6] N. Brunel, V. Hakim, and M. J. E. Richardson, Phys. Rev. E 67, 051916 (2003). [7] M. J. Chacron, A. Longtin, and K. Pakdaman, Physica D 192, 138 (2004). [8] K. S. Cole, J. Gen. Physiol. 25, 29 (1941). [9] K. S. Cole, R. Guttman, and F. Bezanilia, Proc. Natl. Acad. Sci. USA 65, 884 (1970). [10] S. Coombes and P. C. Bressloff, Phys. Rev. E 60, 2086 (1999). [11] S. Coombes and P. C. Bressloff, Phys. Rev. E 63, 059901(E) (2001). [12] S. Coombes, Phys. Lett. A 255, 49 (1999). [13] S. Coombes, M. R. Owen, and G. D. Smith, Phys. Rev. E 64, 041914 (2001). [14] A. S. French, A. V. Holden, and R. B. Stein. Kybernetika 11, 15 (1972). [15] Y. Gutfreund, Y. Yarom and I. Segev, J. Physiol. (London) 483, 621 (1995). [16] H. Hayashi, S. Ishizuka, M. Ohta, and K. Hirakawa, Phys. Lett. A 88, 435 (1982). [17] B. Hutcheon, R. M. Miura, and E. Puil, J. Neurophysiol. 76, 683 (1996); 26, 698 (1996). [18] B. Hutcheon and Y. Yarom, Trends Neurosci. 23, 216 (2000). [19] B. Hutcheon, R. M. Miura, Y. Yarom, and E. Puil, J. Neurophysiol. 71, 583 (1994). [20] E. Izhikevich, Neural Networks 14, 883 (2001). [21] E. Izhikevich, BioSystems 67, 95 (2007). [22] H. Jahnsen and S. Karnup, Brain Res. 666, 9 (1994). [23] D. T. Kaplan, J. R. Clay, T. Manning, L. Glass, M. R. Guevara, and A. Shrier, Phys. Rev. Lett. 76, 4074 (1996). [24] J. P. Keener, F. C. Hppensteadt, and J. Rinzel, SIAM. J. Appl. Math. 41, 503 (1981). [25] K. L. Kevin, SIAM, J. Appl. Dyn. 5, 179 (2006). [26] C. Koch, Biol. Cybern. 50, 15 (1984). [27] C. Koch, Biophysics of Computation (Oxford University Press, Oxford, 1999). [28] C. R. Laing and S. Coombes, Int. J. Bifurcat. Chaos 15, 1433 (2005). [29] L. S. Leung and H. W. Yu, J. Neurophysiol. 79, 1592 (1998). [30] R. R. Llinas, A. A. Grace, and Y. Yaromange, Proc. Natl. Acad. Sci. USA 88, 897 (1991). [31] R. S. Mackay and C. Tresser, Physica D 19, 206 (1986). [32] P. C. M¨uller, Chaos Soli. Frac. 5, 1671 (1995). [33] K. Nakada, K. Miura, H. Hayashi, Int. J. Bifurcat. Chaos 18, 1249 (2008). [34] K. Yoshino, T. Nomura, K. Pakdaman, and S. Sato, Phys. Rev. E 59, 956 (1999). [35] F. G. Pike, R. S. Goddard, J. M. Suckling, P. Ganter, N. Kasthuri, and O. Paulsen, J. Physiol. 529, 205 (2000). [36] E. Puil, H. Meiri, and Y. Yarom, J. Physiol. 71, 575 (1994). [37] A. Rescigno, R. B. Stein, R. L. Purple, and R. E. Poppele, Bull. Math. Biophys 32, 337 (1970). [38] M. J. E. Richardson, N. Brunel, and V. Hakim,J. Neurophysiol. 89, 2538 (2003). [39] F. Rieke, D. Warland, R. de Ruyter van steveninck, and W. Bialek, Exploring the Neural Code (MIT Press, Cambridge, MA, 1997). [40] F. Schilder and B. B. Peckam, J. Comput. Phys. 220, 932 (2007). [41] R. B. Stein, A. S. French, and A. V. Holden, Biophys. J. 12, 295 (1972). [42] J. F. Vibert and J. P. Segundo, Biol. Cybern. 33, 81 (1979).
URI:	http://wrap.warwick.ac.uk/id/eprint/16830