Abstract
Background oscillations, reflecting the excitability of neurons, are ubiquitous in the brain. Some studies have conjectured that when spikes sent by one population reach the other population in the peaks of excitability, then information transmission between two oscillating neuronal groups is more effective. In this context, phase locking relationships between oscillating neuronal populations may have implications in neuronal communication as they assure synchronous activity between brain areas. To study this relationship, we consider a population rate model and perturb it with a time-dependent input. We use the stroboscopic map and apply powerful computational methods to compute the invariant objects and their bifurcations as the perturbation parameters (frequency and amplitude) are varied. The analysis performed shows the relationship between the appearance of synchronous and asynchronous regimes and the invariant objects of the stroboscopic map.
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References
Arrowsmith, D.K., Place, C.M.: An Introduction to Dynamical Systems. Cambridge University Press, Cambridge (1990)
Berger, H.: Über das elektrenkephalogramm des menschen. Arch. Psychiat. Nerven. 87(1), 527–570 (1929)
Borisyuk, R.M., Kirillov, A.B.: Bifurcation analysis of a neural network model. Biol. Cybern. 66(4), 319–325 (1992)
Buzsáki, G., Draguhn, A.: Neuronal oscillations in cortical networks. Science 304(5679), 1926–1929 (2004)
Canadell, M., Haro, A.: Parameterization method for computing quasi-periodic reducible normally hyperbolic invariant tori. In: F. Casas, V. Martínez (eds.) Advances in Differential Equations and Applications, vol. 4, pp. 85–94. Springer, Berlin (2014)
Dayan, P., Abbott, L.F.: Theoretical Neuroscience. Computational Modeling of Neural Systems. MIT Press, Cambridge (2001)
Fries, P.: A mechanism for cognitive dynamics: neuronal communication through neuronal coherence. Trends Cogn. Sci. (Regul. Ed.) 9(10), 474–480 (2005)
Fries, P., Reynolds, J.H., Rorie, A.E., Desimone, R.: Modulation of oscillatory neuronal synchronization by selective visual attention. Science 291(5508), 1560–1563 (2001)
Gambaudo, J.M.: Perturbation of a Hopf bifurcation by an external time-periodic forcing. J. Differ. Equ. 57(2), 172–199 (1985)
Guillamon, A., Huguet, G.: A computational and geometric approach to phase resetting curves and surfaces. SIAM J. Appl. Dyn. Syst. 8(3), 1005–1042 (2009)
Haro, À., Canadell, M., Figueras, J.L., Luque, A., Mondelo, J.M.: The Parameterization Method for Invariant Manifolds. Springer, Berlin (2016)
Hoppensteadt, F.C., Izhikevich, E.M.: Weakly connected neural networks. In: Mardsen, J.E., Sinovich, L., John, F. (eds.) Applied Mathematical Sciences, vol. 126. Springer Science & Business Media, New York (1997)
Niebur, E., Hsiao, S.S., Johnson, K.O.: Synchrony: a neuronal mechanism for attentional selection? Curr. Opin. Neurobiol. 12(2), 190–194 (2002)
Pinto, D.J., Brumberg, J.C., Simons, D.J., Ermentrout, G.B., Traub, R.: A quantitative population model of whisker barrels: re-examining the Wilson-Cowan equations. J. Comput. Neurosci. 3(3), 247–264 (1996)
Roberts, M.J., Lowet, E., Brunet, N.M., Ter Wal, M., Tiesinga, P., Fries, P., De Weerd, P.: Robust gamma coherence between macaque V1 and V2 by dynamic frequency matching. Neuron 78(3), 523–536 (2013)
Seara, T.M., Villanueva, J.: On the numerical computation of Diophantine rotation numbers of analytic circle maps. Physica D 217(2), 107–120 (2006)
Simó, C.: On the analytical and numerical approximation of invariant manifolds. In: Les Méthodes Modernes de la Mécanique Céleste. Modern Methods in Celestial Mechanics, vol. 1, pp. 285–329 (1990)
Tiesinga, P., Sejnowski, T.J.: Cortical enlightenment: are attentional gamma oscillations driven by ING or PING? Neuron 63(6), 727–732 (2009)
Veltz, R., Sejnowski, T.J.: Periodic forcing of inhibition-stabilized networks: nonlinear resonances and phase-amplitude coupling. Neural. Comput. 27(12), 2477–2509 (2015)
Wilson, H.R., Cowan, J.D.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12(1), 1–24 (1972)
Acknowledgements
A.P, G.H and T.S acknowledge financial support from the Spanish MINECO-FEDER Grants MTM2012-31714, MTM2015-65715-P and the Catalan Grant 2014SGR504.
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Pérez-Cervera, A., Huguet, G., M-Seara, T. (2018). Computation of Invariant Curves in the Analysis of Periodically Forced Neural Oscillators. In: Archilla, J., Palmero, F., Lemos, M., Sánchez-Rey, B., Casado-Pascual, J. (eds) Nonlinear Systems, Vol. 2. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-72218-4_3
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DOI: https://doi.org/10.1007/978-3-319-72218-4_3
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