Weakly coupled oscillators in a slowly varying world

• DOI: 10.1007/s10827-016-0596-6
• Publication Date: 2016-03-05T06:48:57Z
• Authors: Youngmin Park; Bard Ermentrout
• Keywords: Neurons; Periodicity; Models, Neurological; Biophysics; Dynamical Systems (math.DS); 03 medical and health sciences; 0302 clinical medicine; Quantitative Biology - Neurons and Cognition; FOS: Biological sciences; FOS: Mathematics; Humans; Neurons and Cognition (q-bio.NC); Mathematics - Dynamical Systems; Nerve Net

28 pages, 11 figures<br/>We extend the theory of weakly coupled oscillators to incorporate slowly varying inputs and parameters. We employ a combination of regular perturbation and an adiabatic approximation to derive equations for the phase-difference between a pair of oscillators. We apply this to the simple Hopf oscillator and then to a biophysical model. The latter represents the behavior of a neuron that is subject to slow modulation of a muscarinic current such as would occur during transient attention through cholinergic activation. Our method extends and simplifies the recent work of Kurebayashi (Physical Review Letters, 111, 214101, 2013) to include coupling. We apply the method to an all-to-all network and show that there is a waxing and waning of synchrony of modulated neurons.<br/>