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Loss of synchrony in an inhibitory network of type-I oscillators

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Understanding the dynamics and synchronization of inhibitory neurons is a question of fundamental importance in neuroscience, since inhibitory networks play a crucial role in rhythmogenesis, both in invertebrate motor pattern generators [1] and in the mammalian hippocampus and neocortex [2]. Invertebrate CPGs in particular often contain simple two-cell inhibitory sub-networks that play a crucial role in the control of rhythmic motor behaviors. Therefore, characterizing the dynamics of two-cell inhibitory networks is relevant for a better understanding of the rhythmic dynamic activity produced by central pattern generators and other inhibitory circuits.

Here we describe the activity states in a network of two cells with type-I excitability coupled by reciprocal inhibition. Weak coupling analysis is very successful in the study of phase-locked activity in such a network, and predicts synchronous or anti-synchronous dynamics, depending on the time scale of inhibition and the intrinsic cell properties [3]. However, it is known that an increase in coupling strength can destabilize synchrony in many neural circuits [4]. In particular, recent work by Maran and Canavier [5] has shown that non-weak coupling leads to alternating-order firing (termed "leap-frog" spiking by G.B. Ermentrout) in an inhibitory network of two Wang-Buzsáki model neurons, whereby the order of spiking of the two cells alternates in each cycle of the oscillation. Here we show that such activity is a generic property of an inhibitory network of oscillators of type-I excitability class. In particular, we demonstrate that leap-frog spiking can also be obtained in a two-cell network of simpler Morris-Lecar model cells, as shown in Figure 1.

Figure 1
figure1

Alternating-order (leap-frog) activity in an inhibitory network of two Morris-Lecar cells with type-I excitability. The membrane potentials of the two model cells are shown as black and red traces, respectively.

We examine the phase-plane geometry of such order non-preserving dynamics, and find that it arises when the inhibition is sufficiently strong to allow a presynaptic cell to transiently suppress the postsynaptic cell below its excitability threshold (saddle-node on an invariant cycle bifurcation) in each cycle of the oscillation. Therefore, non-zero synaptic decay time is crucial for obtaining leap-frog spiking in a continuous system. However, we show that alternating-order spiking can also be obtained in an appropriately modified pulse-coupled integrate-and-fire network. Following the approach of Maran and Canavier [5], we show that the entire activity state bifurcation profile of the two-cell Morris-Lecar network can be completely characterized using the phase-resetting properties of each of the two cells, and describe the conditions on the phase resetting curve that lead to the loss of synchrony and the emergence of leap-frog spiking.

References

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    Marder E, Calabrese RL: Principles of rhythmic motor pattern generation. Physiol Rev. 1996, 76: 687-717.

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    White JA, Chow CC, Ritt J, Soto-Treviño C, Kopell N: Synchronization and oscillatory dynamics in heterogeneous, mutually inhibitory networks. J Computational Neurosci. 1998, 5: 5-16. 10.1023/A:1008841325921.

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    Ermentrout GB: Type I membranes, phase resetting curves, and synchrony. Neural Computation. 1996, 8: 979-1001. 10.1162/neco.1996.8.5.979.

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    Ermentrout GB, Kopell N: Multiple pulse interactions and averaging in systems of coupled neural oscillators. J Math Biol. 1991, 29: 195-217. 10.1007/BF00160535.

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    Maran SK, Canavier CC: Using phase resetting to predict 1:1 and 2:2 locking in two neuron networks in which firing order is not always preserved. J Comput Neurosci. 2008, 24: 37-55. 10.1007/s10827-007-0040-z.

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Acknowledgements

This work is partially supported by the National Science Foundation grant DMS-0417416

Author information

Correspondence to Victor Matveev.

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Open Access This article is published under license to BioMed Central Ltd. This is an Open Access article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Oh, M., Matveev, V. Loss of synchrony in an inhibitory network of type-I oscillators. BMC Neurosci 9, P149 (2008) doi:10.1186/1471-2202-9-S1-P149

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Keywords

  • Central Pattern Generator
  • Reciprocal Inhibition
  • Inhibitory Circuit
  • Inhibitory Network
  • Rhythmic Motor