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Weakly coupled oscillators in a slowly varying world

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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.

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BE and YMP were partially supported by NSF DMS 1219753.

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Correspondence to Youngmin Park.

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Action Editor: Charles Wilson

Appendix: Traub model with adaptation

Appendix: Traub model with adaptation

All other equations for the Traub model are defined as follows

$$\begin{array}{@{}rcl@{}} t_{w}(V)&=&\tau_{w}/(3.3\exp((V-V_{wt})/20)\notag\\&&+\exp(-(V-V_{wt})/20))\\ w_{\infty}(V)&=&1/(1+\exp(-(V-V_{wt})/10))\\ a_{m}(V)&=&0.32(54+V)/(1-\exp(-(V+54)/4))\\ b_{m}(V)&=&0.28(V+27)/(\exp((V+27)/5)-1)\\ a_{h}(V)&=&0.128\exp(-(V-V_{hn})/18)\\ b_{h}(V)&=&4/(1+\exp(-(V+27)/5))\\ a_{n}(V)&=&0.032(V+52)/(1-\exp(-(V+52)/5))\\ b_{n}(V)&=&0.5\exp(-(57+V)/40)\\ \alpha(V)&=&a_{0}/(1+\exp(-(V-V_{t})/V_{s})) \end{array} $$

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Park, Y., Ermentrout, B. Weakly coupled oscillators in a slowly varying world. J Comput Neurosci 40, 269–281 (2016).

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