Non-weak inhibition and phase resetting at negative values of phase in cells with fast-slow dynamics at hyperpolarized potentials
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Phase response is a powerful concept in the analysis of both weakly and non-weakly perturbed oscillators such as regularly spiking neurons, and is applicable if the oscillator returns to its limit cycle trajectory between successive perturbations. When the latter condition is violated, a formal application of the phase return map may yield phase values outside of its definition domain; in particular, strong synaptic inhibition may result in negative values of phase. The effect of a second perturbation arriving close to the first one is undetermined in this case. However, here we show that for a Morris–Lecar model of a spiking cell with strong time scale separation, extending the phase response function definition domain to an additional negative value branch allows to retain the accuracy of the phase response approach in the face of such strong inhibitory coupling. We use the resulting extended phase response function to accurately describe the response of a Morris–Lecar oscillator to consecutive non-weak synaptic inputs. This method is particularly useful when analyzing the dynamics of three or more non-weakly coupled cells, whereby more than one synaptic perturbation arrives per oscillation cycle into each cell. The method of perturbation prediction based on the negative-phase extension of the phase response function may be applicable to other excitable cell models characterized by slow voltage dynamics at hyperpolarized potentials.
KeywordsPhase resetting Phase response curve Spike-time response curve Phase return map Pulse coupled oscillators Non-weak coupling Synaptic inhibition Morris–Lecar model
This research was supported in part by the National Science Foundation grant DMS-0817703 (V.M.).
- Izhikevich, E. M. (2006). Dynamics systems in neuroscience: The geometry of excitability and bursting (Chapter 10). Synchronization. Cambridge: MIT.Google Scholar
- Izhikevich, E. M., & Kuramoto, Y. (2006). Weakly coupled oscillators (Vol. 5, p. 448). Elsevier: Encyclopedia of Mathematical Physics.Google Scholar
- Kuramoto, Y. (1984). Chemical oscillations, waves, and turbulence. Berlin: Springer.Google Scholar
- Oprisan, S. A., & Canavier, C. C. (2001). Stability analysis of rings of pulse-coupled oscillators: The effect of phase resetting in the second cycle after the pulse is important at synchrony and for long pulses. Journal of Differential Equations and Dynamical Systems, 9, 243–258.Google Scholar
- Rinzel, J., & Ermentrout, B. (1998). Analysis of neural excitability and oscillations. In C. Koch & I. Segev (Eds.). Methods in neuronal modeling: From ions to networks (2nd ed.). Cambridge: MIT.Google Scholar
- Winfree, A. T. (2001). The geometry of biological time (2nd edn). New York: Springer.Google Scholar