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Phase-amplitude descriptions of neural oscillator models. (English) Zbl 1291.92052
Summary: Phase oscillators are a common starting point for the reduced description of many single neuron models that exhibit a strongly attracting limit cycle. The framework for analysing such models in response to weak perturbations is now particularly well advanced, and has allowed for the development of a theory of weakly connected neural networks. However, the strong-attraction assumption may well not be the natural one for many neural oscillator models. For example, the popular conductance based Morris-Lecar model is known to respond to periodic pulsatile stimulation in a chaotic fashion that cannot be adequately described with a phase reduction. In this paper, we generalise the phase description that allows one to track the evolution of distance from the cycle as well as phase on cycle. We use a classical technique from the theory of ordinary differential equations that makes use of a moving coordinate system to analyse periodic orbits. The subsequent phase-amplitude description is shown to be very well suited to understanding the response of the oscillator to external stimuli (which are not necessarily weak). We consider a number of examples of neural oscillator models, ranging from planar through to high dimensional models, to illustrate the effectiveness of this approach in providing an improvement over the standard phase-reduction technique. As an explicit application of this phase-amplitude framework, we consider in some detail the response of a generic planar model where the strong-attraction assumption does not hold, and examine the response of the system to periodic pulsatile forcing. In addition, we explore how the presence of dynamical shear can lead to a chaotic response.

##### MSC:
 92C20 Neural biology 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 37N25 Dynamical systems in biology
##### Keywords:
phase-amplitude; oscillator; chaos; non-weak coupling
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##### References:
 [1] Winfree A: The Geometry of Biological Time. 2nd edition. Springer, Berlin; 2001. · Zbl 1014.92001 [2] Guckenheimer, J, Isochrons and phaseless sets, J Math Biol, 1, 259-273, (1975) · Zbl 0345.92001 [3] Cohen, AH; Rand, RH; Holmes, PJ, Systems of coupled oscillators as models of central pattern generators, (1988), New York [4] Kopell, N; Ermentrout, GB, Symmetry and phaselocking in chains of weakly coupled oscillators, Commun Pure Appl Math, 39, 623-660, (1986) · Zbl 0596.92011 [5] Ermentrout, GB, $$n$$:$$m$$ phase-locking of weakly coupled oscillators, J Math Biol, 12, 327-342, (1981) · Zbl 0476.92007 [6] Izhikevich EM: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, Cambridge; 2007. [7] Ermentrout GB, Terman DH: Mathematical Foundations of Neuroscience. Springer, Berlin; 2010. · Zbl 1320.92002 [8] Josic, K; Shea-Brown, ET, Isochron, (2006) [9] Guillamon, A; Huguet, G, A computational and geometric approach to phase resetting curves and surfaces, SIAM J Appl Dyn Syst, 8, 1005-1042, (2009) · Zbl 1216.34030 [10] Osinga, HM; Moehlis, J, A continuation method for computing global isochrons, SIAM J Appl Dyn Syst, 9, 1201-1228, (2010) · Zbl 1232.37014 [11] Mauroy, A; Mezic, I:, On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics., (2012) · Zbl 1319.70024 [12] Brown, E; Moehlis, J; Holmes, P, On the phase reduction and response dynamics of neural oscillator populations, Neural Comput, 16, 673-715, (2004) · Zbl 1054.92006 [13] Hoppensteadt FC, Izhikevich EM: Weakly Connected Neural Networks. Springer, Berlin; 1997. [14] Achuthan, S; Canavier, CC, Phase-resetting curves determine synchronization, phase locking, and clustering in networks of neural oscillators, J Neurosci, 29, 5218-5233, (2009) [15] Yoshimura, K, Phase reduction of stochastic limit-cycle oscillators, 59-90, (2010), New York · Zbl 1201.37110 [16] Lin, KK; Wedgwood, KCA; Coombes, S; Young, LS, Limitations of perturbative techniques in the analysis of rhythms and oscillations, J Math Biol, 66, 139-161, (2013) · Zbl 1256.92006 [17] Demir, A; Suvak, O, Quadratic approximations for the isochrons of oscillators: a general theory, advanced numerical methods and accurate phase computations, IEEE Trans Comput-Aided Des Integr Circuits Syst, 29, 1215-1228, (2010) [18] Medvedev, GS, Synchronization of coupled stochastic limit cycle oscillators, Phys Lett A, 374, 1712-1720, (2010) · Zbl 1236.34070 [19] Diliberto, SP, On systems of ordinary differential equations, 1-38, (1950), Princeton [20] Hale JK: Ordinary Differential Equations. Wiley, New York; 1969. · Zbl 0186.40901 [21] Ermentrout, GB; Kopell, N, Oscillator death in systems of coupled neural oscillators, SIAM J Appl Math, 50, 125-146, (1990) · Zbl 0686.34033 [22] Ott, W; Stenlund, M, From limit cycles to strange attractors, Commun Math Phys, 296, 215-249, (2010) · Zbl 1202.37046 [23] Morris, C; Lecar, H, Voltage oscillations in the barnacle giant muscle fiber, Biophys J, 35, 193-213, (1981) [24] Rinzel, J; Ermentrout, GB, Analysis of neural excitability and oscillations, 135-169, (1989), Cambridge [25] Connor, JA; Stevens, CF, Prediction of repetitive firing behaviour from voltage clamp data on an isolated neurone soma, J Physiol, 213, 31-53, (1971) [26] Kepler, TB; Abbott, LF; Marder, E, Reduction of conductance-based neuron models, Biol Cybern, 66, 381-387, (1992) · Zbl 0745.92006 [27] Lin, KK; Young, LS, Shear-induced chaos, Nonlinearity, 21, 899-922, (2008) · Zbl 1153.37355 [28] Wang, Q; Young, LS, Strange attractors with one direction of instability, Commun Math Phys, 218, 1-97, (2001) · Zbl 0996.37040 [29] Wang, Q; Young, LS, From invariant curves to strange attractors, Commun Math Phys, 225, 275-304, (2002) · Zbl 1080.37550 [30] Wang, Q, Strange attractors in periodically-kicked limit cycles and Hopf bifurcations, Commun Math Phys, 240, 509-529, (2003) · Zbl 1078.37027 [31] Catllá, AJ; Schaeffer, DG; Witelski, TP; Monson, EE; Lin, AL, On spiking models for synaptic activity and impulsive differential equations, SIAM Rev, 50, 553-569, (2008) · Zbl 1166.34004 [32] Christiansen, F; Rugh, F, Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization, Nonlinearity, 10, 1063-1072, (1997) · Zbl 0910.34055 [33] Ashwin, P, Weak coupling of strongly nonlinear, weakly dissipative identical oscillators, Dyn Syst, 10, 2471-2474, (1989) [34] Ashwin, P; Dangelmayr, G, Isochronicity-induced bifurcations in systems of weakly dissipative coupled oscillators, Dyn Stab Syst, 15, 263-286, (2000) · Zbl 0976.34033 [35] Ashwin, P; Dangelmayr, G, Reduced dynamics and symmetric solutions for globally coupled weakly dissipative oscillators, Dyn Syst, 20, 333-367, (2005) · Zbl 1086.37041 [36] Han, SK; Kurrer, C; Kuramoto, Y, Dephasing and bursting in coupled neural oscillators, Phys Rev Lett, 75, 3190-3193, (1995) [37] Coombes, S, Neuronal networks with gap junctions: a study of piecewise linear planar neuron models, SIAM J Appl Dyn Syst, 7, 1101-1129, (2008) · Zbl 1159.92008
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