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Isostables for Stochastic Oscillators

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2021-12-14
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American Physical Society
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Thomas and Lindner [P. J. Thomas and B. Lindner, Phys. Rev. Lett. 113, 254101 (2014).], defined an asymptotic phase for stochastic oscillators as the angle in the complex plane made by the eigenfunction, having a complex eigenvalue with a least negative real part, of the backward Kolmogorov (or stochastic Koopman) operator. We complete the phase-amplitude description of noisy oscillators by defining the stochastic isostable coordinate as the eigenfunction with the least negative nontrivial real eigenvalue. Our results suggest a framework for stochastic limit cycle dynamics that encompasses noise-induced oscillations.
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[1] H. Bryant, Jr., A. R. Marcos, and J. Segundo, J. Neurophysiol. 36, 205 (1973). [2] J. T. Walter, K. Alvina, M. D. Womack, C. Chevez, and K.Khodakhah, Nat. Neurosci. 9, 389 (2006). [3] P. Martin, D. Bozovic, Y. Choe, and A. J. Hudspeth, J. Neurosci. 23, 4533 (2003). [4] A. Skupin, H. Kettenmann, U. Winkler, M. Wartenberg, H. Sauer, S. C. Tovey, C. W. Taylor, and M. Falcke, Biophys. J. 94, 2404 (2008). [5] A. J. McKane and T. J. Newman, Phys. Rev. Lett. 94, 218102 (2005). [6] R. Feistel and W. Ebeling, Physica (Amsterdam) 93A, 114 (1978). [7] A. Ganopolski and S. Rahmstorf, Phys. Rev. Lett. 88, 038501 (2002). [8] B. McNamara, K. Wiesenfeld, and R. Roy, Phys. Rev. Lett. 60, 2626 (1988). [9] A. Guillamon and G. Huguet, SIAM J. Appl. Dyn. Syst. 8, 1005 (2009). [10] D. Wilson and B. Ermentrout, Phys. Rev. Lett. 123, 164101 (2019). [11] D. Wilson, Phys. Rev. E 101, 022220 (2020). [12] A. P´erez-Cervera, T. M-Seara, and G. Huguet, Chaos 30,083117 (2020). [13] A. Pikovsky, J. Kurths, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, England, (2003), Vol. 12. [14] O. Castejón, A. Guillamon, and G. Huguet, J. Math. Neurosci. 3, 13 (2013). [15] S. Shirasaka, W. Kurebayashi, and H. Nakao, Chaos 27, 023119 (2017). [16] D. Wilson and B. Ermentrout, J. Math. Biol. 76, 37 (2018). [17] B. Monga, D. Wilson, T. Matchen, and J. Moehlis, Biol. Cybern. 113, 11 (2019). [18] J. Guckenheimer, J. Math. Biol. 1, 259 (1975). [19] G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1930). [20] J. Giner-Baldó, P. J. Thomas, and B. Lindner, J. Stat. Phys. 168, 447 (2017). [21] B. Lindner, J. Garcıa-Ojalvo, A. Neiman, and L. Schimansky-Geier, Phys. Rep. 392, 321 (2004). [22] For an example in which k ≠ n, see [23,24]. [23] S. Pu and P. J. Thomas, Biol. Cybern. 115, 267 (2021). [24] S. Pu and P. J. Thomas, Neural Comput. 32, 1775 (2020). [25] We choose the Itô interpretation for its mathematical convenience. For every Stratonovich-interpreted SDE there is an equivalent Itô-interpreted SDE [26]. Thus, choosing between the Itô or the Stratonovich interpretation will not change our framework, which is based on the (uniquely defined) backward Kolmogorov operator. [26] C. W. Gardiner, Handbook of Stochastic Methods (Springerverlag, Berlin, 1985). [27] J. T. C. Schwabedal and A. Pikovsky, Phys. Rev. Lett. 110, 204102 (2013). [28] A. Cao and B. Lindner, and P. J. Thomas, SIAM J. Appl. Math. 80, 422 (2020). [29] P. J. Thomas and B. Lindner, Phys. Rev. Lett. 113, 254101 (2014). [30] For an alternative approach to phase reduction for stochastic oscillators see [27,28]. [31] P. J. Thomas and B. Lindner, Phys. Rev. E 99, 062221 (2019). [32] In the n > 2 case, we expect n − 1 stochastic amplitudes Σi corresponding to the n − 1 deterministic Floquet modes. The effective vector field would be determined by a system of n equations, e.g., ∇Q �ðxÞ · FðxÞ ¼ λ�Q �ðxÞ, ∇Σi ðxÞ · FðxÞ ¼ λi FloqΣi ðxÞ, for 1 ≤ i ≤ n − 1. If jλ1 Floqj ≪jλi Floqj for i > 1, we expect the flow generated by F will have a 2D invariant manifold Σi ¼ 0;i> 1, on which the dynamics will be well approximated by our 2D (phase, amplitude) construction. [33] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevLett.127.254101 for the numerical procedure generating the results in this Letter. [34] T. K. Leen, R. Friel, and D. Nielsen, arXiv:1609.01194. [35] A. S. Powanwe and A. Longtin, Sci. Rep. 9, 18335 (2019). [36] N. Črnjarić-Žic, S. Maćešić, and I. Mezić, J. Nonlinear Sci. 30, 2007 (2019). [37] V. S. Afraimovich, M. I. Rabinovich, and P. Varona, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1195 (2004). [38] D. Armbruster, E. Stone, and V. Kirk, Chaos 13, 71 (2003). [39] R. M. May and W. J. Leonard, SIAM J. Appl. Math. 29, 243 (1975). [40] In [29] Thomas and Lindner numerically solved the eigenvalue problem Eq. (4) for the heteroclinic system Eq. (13) using a Fourier mode decomposition method. Due to a subtle error in our treatment of the boundary conditions, the slowest decaying real eigenvalue plotted in Fig. 2(c) of [29] was incorrect. It has been corrected in Fig. 3(a) of this Letter. This error had no effect on the analysis or conclusions in [29], which concerned only the complex-valued eigenvalue and its eigenfunction. [41] C. A. Lugo and A. J. McKane, Phys. Rev. E 78, 051911 (2008). [42] H. A. Brooks and P. C. Bressloff, Phys. Rev. E 92, 012704 (2015). [43] P. C. Bressloff, Phys. Rev. E 82, 051903 (2010). [44] B. Duchet, G. Weerasinghe, C. Bick, and R. Bogacz, J. Neural Eng. 18, 046023 (2021). [45] L. Arnold, Random Dynamical Systems (Springer, New York, 1995). [46] A. S. Pikovsky and J. Kurths, Phys. Rev. Lett. 78, 775 (1997). [47] O. V. Ushakov, H.-J. Wünsche, F. Henneberger, I. A. Khovanov, L. Schimansky-Geier, and M. A. Zaks, Phys.Rev. Lett. 95, 123903 (2005). [48] A. Mauroy and I. Mezić, Chaos 28, 073108 (2018). [49] S. Shirasaka, W. Kurebayashi, and H. Nakao, in The Koopman Operator in Systems and Control (Springer, New York, 2020), pp. 383–417. [50] Y. Kato and H. Nakao, arXiv:2006.00760. [51] M. Engel and C. Kuehn, Commun. Math. Phys. 386, 1603 (2021). [52] Y. Kato, J. Zhu, W. Kurebayashi, and H. Nakao, Mathematics 9, 2188 (2021). [53] M. Budišić, R. Mohr, and I. Mezić, Chaos 22, 047510 (2012). [54] A. Mauroy, Y. Susuki, and I. Mezić, The Koopman Operator in Systems and Control (Springer, New York, (2020). [55] J. L. Proctor, S. L. Brunton, and J. N. Kutz, SIAM J. Appl. Dyn. Syst. 15, 142 (2016). [56] P. J. Schmid, J. Fluid Mech. 656, 5 (2010).
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