Loss of phase-locking in non-weakly coupled inhibitory networks of type-I model neurons
- 206 Downloads
Synchronization of excitable cells coupled by reciprocal inhibition is a topic of significant interest due to the important role that inhibitory synaptic interaction plays in the generation and regulation of coherent rhythmic activity in a variety of neural systems. While recent work revealed the synchronizing influence of inhibitory coupling on the dynamics of many networks, it is known that strong coupling can destabilize phase-locked firing. Here we examine the loss of synchrony caused by an increase in inhibitory coupling in networks of type-I Morris–Lecar model oscillators, which is characterized by a period-doubling cascade and leads to mode-locked states with alternation in the firing order of the two cells, as reported recently by Maran and Canavier (J Comput Nerosci, 2008) for a network of Wang-Buzsáki model neurons. Although alternating-order firing has been previously reported as a near-synchronous state, we show that the stable phase difference between the spikes of the two Morris–Lecar cells can constitute as much as 70% of the unperturbed oscillation period. Further, we examine the generality of this phenomenon for a class of type-I oscillators that are close to their excitation thresholds, and provide an intuitive geometric description of such “leap-frog” dynamics. In the Morris–Lecar model network, the alternation in the firing order arises under the condition of fast closing of K + channels at hyperpolarized potentials, which leads to slow dynamics of membrane potential upon synaptic inhibition, allowing the presynaptic cell to advance past the postsynaptic cell in each cycle of the oscillation. Further, we show that non-zero synaptic decay time is crucial for the existence of leap-frog firing in networks of phase oscillators. However, we demonstrate that leap-frog spiking can also be obtained in pulse-coupled inhibitory networks of one-dimensional oscillators with a multi-branched phase domain, for instance in a network of quadratic integrate-and-fire model cells. Finally, for the case of a homogeneous network, we establish quantitative conditions on the phase resetting properties of each cell necessary for stable alternating-order spiking, complementing the analysis of Goel and Ermentrout (Physica D 163:191–216, 2002) of the order-preserving phase transition map.
KeywordsSynchronization Non-weak coupling Non-synchronous dynamics Inhibitory network Type-I excitability Synaptic inhibition Leader switching Spike-time response Phase resetting
This work was supported by the National Science Foundation grant DMS-0417416. We wish to thank Amitabha Bose and Farzan Nadim for helpful comments and discussions.
- Hoppensteadt, F. C., & Izhikevich, E. M. (1997). Weakly connected neural networks. New York: Springer.Google Scholar
- Izhikevich, E. M. (2006). Dynamics systems in neuroscience: The geometry of excitability and bursting. Chapter 10: Synchronization. Cambridge: MIT.Google Scholar
- Kopell, N. (1988). Toward a theory of modeling central pattern generators. In A. H. Cohen, S. Rossignol, & S. Grillner (Eds.), Neural control of rhythms. New York: Wiley.Google Scholar
- Kopell, N., & Ermentrout, G. B. (2002). Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators. In B. Fiedler (Ed.), Handbook on Dynamical Systems: Toward Applications. New York: Elsevier.Google Scholar
- Kuramoto, Y. (1984). Chemical oscillations, waves, and turbulence. Berlin: Springer.Google Scholar
- Maran, S. K., & Canavier, C. C. (2008). Using phase resetting to predict 1:1 and 2:2 locking in two neuron networks in which firing order is not always preserved. Journal of Computational Neroscience, 24, 37–55.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 Difference. Equations and Dynamical Systems, 9, 243–258.Google Scholar
- Peskin, C. S. (1975). Mathematical aspects of heart physiology. New York: New York University Courant Institute of Mathematical Sciences.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 edn). Cambridge: MIT.Google Scholar
- Winfree, A. T. (2001). The geometry of biological time (2nd edn). New York: Springer.Google Scholar