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Robust phase-waves in chains of half-center oscillators

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Abstract

Many neuronal circuits driving coordinated locomotion are composed of chains of half-center oscillators (HCOs) of various lengths. The HCO is a common motif in central pattern generating circuits (CPGs); an HCO consists of two neurons, or two neuronal populations, connected by reciprocal inhibition. To maintain appropriate motor coordination for effective locomotion over a broad range of frequencies, chains of CPGs must produce approximately constant phase-differences in a robust manner. In this article, we study phase-locking in chains of nearest-neighbor coupled HCOs and examine how the circuit architecture can promote phase-constancy, i.e., inter-HCO phase-differences that are frequency-invariant. We use two models with different levels of abstraction: (1) a conductance-based model in which each neuron is modeled by the Morris–Lecar equations (the ML-HCO model); and (2) a coupled phase model in which the state of each HCO is captured by its phase (the phase-HCO model). We show that one of four phase-waves with inter-HCO phase-differences at approximately 0, 25, 50 or 75 % arises robustly as a result of the inter-HCO connection topology, and its robust existence is not affected by the number of HCOs in the chain, the difference in strength between the ascending and descending nearest-neighbor connections, or the number of nearest-neighbor connections. Our results show that the internal anti-phase structure of the HCO and an appropriate inter-HCO connection topology together can provide a mechanism for robust (i.e., frequency-independent) limb coordination in segmented animals, such as the 50 % interlimb phase-differences in the tripod gate of stick insects and cockroaches, and the 25 % interlimb phase-differences in crayfish and other long-tailed crustaceans during forward swimming.

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Acknowledgments

CZ and TJL were partially supported by NSF under grant number CRCNS 0905063. CZ is a Courant Instructor.

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Correspondence to Calvin Zhang.

Appendix: Numerical results of the ML-HCO model: inter-HCO connections with unequal strengths between the ascending and descending directions

Appendix: Numerical results of the ML-HCO model: inter-HCO connections with unequal strengths between the ascending and descending directions

In Sect. 4.1, we used the phase-HCO model to show that an approximate 0, 25, 50, or 75 % phase-waves persists regardless of the difference in strength between the ascending and descending connections. Here, we verify the above result using the conductance-based ML-HCO model. We consider a chain of 10 HCOs under the four fundamentally different inter-HCO connection topologies. Figures 6, 7, 8, 9 show that regardless of the ratio of the ascending to descending connection strengths, the deviation of the phase-differences is always in the order of \(O(\epsilon )\) (see Eq. 24 for definition of \(\epsilon \)). Both the phase-HCO model and the ML-HCO model give the same result that an asymmetry in the ascending and descending inter-HCO connection strength does not affect the existence of the four robust approximate phase-waves.

Fig. 6
figure 6

The effect of the difference in the synaptic strength between the ascending and descending inter-HCO connections on the phase-differences in a chain of 10 HCOs with inter-HCO connection topology (a1). The phase-differences \({\varDelta }\phi _k\) (Eq. 36) is plotted as a function of k, \(k=1,2,\ldots ,9\) for four different values of \(a = \alpha /\beta \), where \(\alpha \) and \(\beta \) are the strength of the ascending and descending connections, respectively. Numerical simulation of this ML-HCO model was performed using the same parameters reported in Sect.  2 (color figure online)

Fig. 7
figure 7

The effect of the difference in the synaptic strength between the ascending and descending inter-HCO connections on the phase-differences in a chain of 10 HCOs with inter-HCO connection topology (a2). See the caption and legend of Fig. 6 for descriptions (color figure online)

Fig. 8
figure 8

The effect of the difference in the synaptic strength between the ascending and descending inter-HCO connections on the phase-differences in a chain of 10 HCOs with inter-HCO connection topology (s1). See the caption and legend of Fig. 6 for descriptions (color figure online)

Fig. 9
figure 9

The effect of the difference in the synaptic strength between the ascending and descending inter-HCO connections on the phase-differences in a chain of 10 HCOs with inter-HCO connection topology (s2). See the caption and legend of Fig. 6 for descriptions (color figure online)

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Zhang, C., Lewis, T.J. Robust phase-waves in chains of half-center oscillators. J. Math. Biol. 74, 1627–1656 (2017). https://doi.org/10.1007/s00285-016-1066-5

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  • DOI: https://doi.org/10.1007/s00285-016-1066-5

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