## Abstract

We investigate the modes of oscillation of heterogeneous ring networks of quadratic integrate-and-fire (QIF) neurons with nonlocal, space-dependent coupling. Perturbations of the equilibrium state with a particular wave number produce transient standing waves with a specific temporal frequency, analogously to those in a tense string. In the neuronal network, the equilibrium corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network's oscillatory modes, which reflect the interplay of episodes of synchronous spiking with the excitatory-inhibitory spatial interactions. In the thermodynamic limit, an exact low-dimensional neural field model describing the macroscopic dynamics of the network is derived. This allows us to obtain formulas for the Turing eigenvalues of the spatially homogeneous state and hence to obtain its stability boundary. We find that the frequency of each Turing mode depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity. The decay rate instead is identical for all oscillation modes as a consequence of the heterogeneity-induced desynchronization of the neurons. Finally, we numerically compute the spectrum of spatially inhomogeneous solutions branching from the Turing bifurcation, showing that similar oscillatory modes operate in neural bump states and are maintained away from onset.

Original language | English |
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Article number | 052407 |

Journal | Physical Review E |

Volume | 96 |

Issue number | 5 |

DOIs | |

Publication status | Published - 13 Nov 2017 |

Externally published | Yes |

### Funding

J.M.E.-A. and E.M acknowledge support by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant No. 642563. J.M.E.-A. and E.M. acknowledge the projects grants from the Spanish Ministry of Economics and Competitiveness, Grants No. PSI2016-75688-P and No. PCIN-2015-127. A.R. acknowledges a project grant from the Spanish ministry of Economics and Competitiveness, Grant No. BFU2012-33413. A.R. has been partially funded by the CERCA progam of the Generalitat de Catalunya. D.A. was partially supported by the EPSRC Grant No. EP/P510993/1 (United Kingdom). APPENDIX A:

Funders | Funder number |
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Horizon 2020 Framework Programme | 642563 |