## Abstract

The formation of oscillating phase clusters in a network of identical Hodgkin–Huxley neurons is studied, along with their dynamic behavior. The neurons are synaptically coupled in an all-to-all manner, yet the synaptic coupling characteristic time is heterogeneous across the connections. In a network of N neurons where this heterogeneity is characterized by a prescribed random variable, the oscillatory single-cluster state can transition—through [InlineEquation not available: see fulltext.] (possibly perturbed) period-doubling and subsequent bifurcations—to a variety of multiple-cluster states. The clustering dynamic behavior is computationally studied both at the detailed and the coarse-grained levels, and a numerical approach that can enable studying the coarse-grained dynamics in a network of arbitrarily large size is suggested. Among a number of cluster states formed, double clusters, composed of nearly equal sub-network sizes are seen to be stable; interestingly, the heterogeneity parameter in each of the double-cluster components tends to be consistent with the random variable over the entire network: Given a double-cluster state, permuting the dynamical variables of the neurons can lead to a combinatorially large number of different, yet similar “fine” states that appear practically identical at the coarse-grained level. For weak heterogeneity we find that correlations rapidly develop, within each cluster, between the neuron’s “identity” (its own value of the heterogeneity parameter) and its dynamical state. For single- and double-cluster states we demonstrate an effective coarse-graining approach that uses the Polynomial Chaos expansion to succinctly describe the dynamics by these quickly established “identity-state” correlations. This coarse-graining approach is utilized, within the equation-free framework, to perform efficient computations of the neuron ensemble dynamics.

Original language | English |
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Article number | 2 |

Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Journal of Mathematical Neuroscience |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - 1 Dec 2015 |

### Bibliographical note

Funding Information:The authors would like to thank M. Krupa for his helpful comments. The work of I.G.K. and K.R. was partially supported by the US Department of Energy, while the work of C.R.L. was supported by the Marsden Fund Council, administered by the Royal Society of New Zealand. The work of J.C. was supported by Fondecyt grant 1140143.

Funding Information:

Acknowledgements The authors would like to thank M. Krupa for his helpful comments. The work of I.G.K. and K.R. was partially supported by the US Department of Energy, while the work of C.R.L. was supported by the Marsden Fund Council, administered by the Royal Society of New Zealand. The work of J.C. was supported by Fondecyt grant 1140143.

Publisher Copyright:

© 2015, S.J. Moon et al.; licensee Springer.

## Keywords

- Clustering dynamics
- Heterogeneous coupling
- Polynomial chaos expansion