# Loss of phase-locking in non-weakly coupled inhibitory networks of type-I model neurons

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## Abstract

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.

## Keywords

Synchronization Non-weak coupling Non-synchronous dynamics Inhibitory network Type-I excitability Synaptic inhibition Leader switching Spike-time response Phase resetting## Notes

### Acknowledgements

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.

## Supplementary material

## References

- Acker, C. D., Kopell, N., & White, J. A. (2003). Synchronization of strongly coupled excitatory neurons: Relating network behavior to biophysics.
*Journal Comparative Neuroscience, 15*, 71–90.CrossRefGoogle Scholar - Bose, A., Kopell, N., & Terman, D. (2000). Almost synchronous solutions for pairs of neurons coupled by excitation.
*Physica D, 140*, 69–94.CrossRefGoogle Scholar - Bressloff, P. C., & Coombes, S. (1998). Desynchronization, mode locking, and bursting in strongly coupled integrate-and-fire oscillators.
*Physical Review Letters, 81*, 2168–2171.CrossRefGoogle Scholar - Bressloff, P. C., & Coombes, S. (2000). Dynamics of strongly-coupled spiking neurons.
*Neural Computation, 12*, 91–129.PubMedCrossRefGoogle Scholar - Brown, E., Moehlis, J., & Holmes, P. (2004). On the phase reduction and response dynamics of neural oscillator populations.
*Neural Computation, 16*, 673–715.PubMedCrossRefGoogle Scholar - Canavier, C. C., Baxter, D. A., Clark, J. W., & Byrne, J. H. (1999). Control of multistability in ring circuits of oscillators.
*Biological Cybernetics, 80*, 87–102.CrossRefGoogle Scholar - Ermentrout, G. B. (1996). Type I membranes, phase resetting curves, and synchrony.
*Neural Computation, 8*, 979–1001.PubMedCrossRefGoogle Scholar - Ermentrout, G. B., & Kopell, N. (1984). Frequency plateaus in a chain of weakly coupled oscillators.
*SIAM Journal on Mathematical Analysis, 15*, 215–237.CrossRefGoogle Scholar - Ermentrout, G. B., & Kopell, N. (1990). Oscillator death in systems of coupled neural oscillators.
*SIAM Journal on Applied Mathematics, 50*, 125–146.CrossRefGoogle Scholar - Ermentrout, G. B., & Kopell, N. (1991). Multiple pulse interactions and averaging in systems of coupled neural oscillators.
*Journal of Mathematical Biology, 29*, 195–217.CrossRefGoogle Scholar - Glass, L., Guevara, M. R., Belair, J., & Shrier, A. (1984). Global bifurcations of a periodically forced biological oscillator.
*Physical Review, A 29*, 1348–1357.CrossRefGoogle Scholar - Goel, P., & Ermentrout, G. B. (2002). Synchrony, stability, and firing patterns in pulse-coupled oscillators.
*Physica D, 163*, 191–216.CrossRefGoogle Scholar - Golubitsky, M., Stewart, I., Buono, P. L., & Collins, J. J. (1999). Symmetry in locomotor central pattern generators and animal gaits.
*Nature, 401*, 693–695.PubMedCrossRefGoogle Scholar - Golubitsky, M., Josic, K., & Shea-Brown, E. (2006). Winding numbers and average frequencies in phase oscillator networks.
*Journal of Nonlinear Science, 16*, 201–231.CrossRefGoogle Scholar - Hansel, D., Mato, G., & Meunier, C. (1995). Synchrony in excitatory neural networks.
*Neural Computation, 7*, 307–337.PubMedCrossRefGoogle Scholar - Hoppensteadt, F. C., & Izhikevich, E. M. (1997).
*Weakly connected neural networks*. New York: Springer.Google Scholar - Izhikevich, E. M. (2000). Phase equations for relaxation oscillators.
*SIAM Journal on Applied Mathematics, 60*, 1789–1805.CrossRefGoogle Scholar - 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.
*Encyclopedia of Mathematical Physics, Elsevier, 5*, 448.CrossRefGoogle Scholar - Jones, S. R., Pinto, D., Kaper, T., & Kopell, N. (2000). Alpha-frequency rhythms desynchronize over long cortical distances: A modelling study.
*Journal Computational Neuroscience, 9*, 271–291.CrossRefGoogle 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., Whittington, M., & Traub, R. D. (2000). Gamma rhythms and beta rhythms have different synchronization properties.
*Proceedings of the National Academy of Sciences of United States America, 97*, 1867–1872.CrossRefGoogle 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 - Mirollo, R. E., & Strogatz, S. H. (1990). Synchronization of pulse-coupled biological oscillators.
*SIAM Journal of Applied Mathemaics, 50*, 1645–1662.CrossRefGoogle Scholar - Morris, C., & Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber.
*Biophysical Journal, 35*, 193–213.PubMedCrossRefGoogle Scholar - Netoff, T. I., Banks, M. I., Dorval, A. D., Acker, C. D., Haas, J. S., Kopell, N., et al. (2005). Synchronization in hybrid neuronal networks of the hippocampal formation.
*Journal of Neurophysiology, 93*, 1197–1208.PubMedCrossRefGoogle 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 - Oprisan, S. A., & Canavier, C. C. (2002). The influence of limit cycle topology on the phase resetting curve.
*Neural Computation, 14*, 1027–1057.PubMedCrossRefGoogle Scholar - Oprisan, S. A., Prinz, A. A., & Canavier, C. C. (2004). Phase resetting and phase locking in hybrid circuits of one model and one biological neuron.
*Biophysical Journal, 87*, 2283–2298.PubMedCrossRefGoogle 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 - Rubin, J., & Terman, D. (2000). Geometric analysis of population rhythms in synaptically coupled neuronal networks.
*Neural Computation, 12*, 597–645PubMedCrossRefGoogle Scholar - Sato, Y. D., & Shiino, M. (2007). Generalization of coupled spiking models and effects of the width of an action potential on synchronization phenomena.
*Physical Review E, 75*, 011909.CrossRefGoogle Scholar - Somers, D., & Kopell, N. (1993). Rapid synchronization through fast threshold modulation.
*Biological Cybernetics, 68*, 393–407.PubMedCrossRefGoogle Scholar - van Vreeswijk, C., Abbott, L. F., & Ermentrout, B. (1994). When inhibition not excitation synchronizes neural firing.
*Journal of Computational Neuroscience, 1*, 313–321.PubMedCrossRefGoogle Scholar - Wang, X. J., Buzsáki, G. (1996). Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model.
*Journal of Neuroscience, 16*, 6402–6413.PubMedGoogle Scholar - White, J. A., Chow, C. C., Ritt, J., Soto-Trevino, C., & Kopell, N. (1998). Dynamics in heterogeneous, mutually inhibited neurons.
*Journal of Computational Neuroscience, 5*, 5–16.PubMedCrossRefGoogle Scholar - Winfree, A. T. (2001).
*The geometry of biological time*(2nd edn). New York: Springer.Google Scholar