# Jose-Marí Esnaola

## Modes of oscillation of a neural field model

We investigate the modes of oscillation of a large network of spiking neurons arranged in a one dimensional ring. Perturbations of the equilibrium state with a particular wave number produce standing waves with a specific frequency, in a similar fashion as in a tense string. In the neuronal network, the equilibrium state corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network's oscillatory modes, reflecting episodes of spike synchrony of the spatially distributed neuronal ensembles and excitatory-inhibitory spatial interactions. In the thermodynamic limit, a low-dimensional neural field model describing the macroscopic dynamics of the ring network is exactly derived. This allows us to analytically obtain the spectrum of the normal modes of oscillation of the network. We find that the frequency of each mode only depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity; the decay rate, instead, is exactly the same for all oscillation modes. We finally show that the spectrum of spatially inhomogeneous solutions has a continuous part off the real axis, indicating that similar oscillatory modes operate in neural bump states as well.

E. Montbrió, D. Pazó and A. Roxin, *Macroscopic Description for Networks of Spiking Neurons* Phys Rev X. 06/2015 5(2).