  Elucidating the neurophysiological mechanisms underlying neural pattern
formation remains an outstanding challenge in Computational Neuroscience. In
this paper, we address the issue of understanding the emergence of neural
patterns by considering a network of renewal neurons, a well-established class
of spiking cells. Taking the thermodynamics limit, the network's dynamics can
be accurately represented by a partial differential equation coupled with a
nonlocal differential equation. The stationary state of the nonlocal system is
determined, and a perturbation analysis is performed to analytically
characterize the conditions for the occurrence of Turing instabilities.
Considering neural network parameters such as the synaptic coupling and the
external drive, we numerically obtain the bifurcation line that separates the
asynchronous regime from the emergence of patterns. Our theoretical findings
provide a new and insightful perspective on the emergence of Turing patterns in
spiking neural networks. In the long term, our formalism will enable the study
of neural patterns while maintaining the connections between microscopic
cellular properties, network coupling, and the emergence of Turing
instabilities.
