  Neurons are often connected, spatially and temporally, in phenomenal ways
that promote wave propagation. Therefore, it is essential to analyze the
emergent spatiotemporal patterns to understand the working mechanism of brain
activity, especially in cortical areas. Here, we present an explicit
mathematical analysis, corroborated by numerical results, to identify and
investigate the spatiotemporal, non-uniform, patterns that emerge due to
instability in an extended homogeneous 2D spatial domain, using the excitable
Izhikevich neuron model. We examine diffusive instability and perform
bifurcation and fixed-point analyses to characterize the patterns and their
stability. Then, we derive analytically the amplitude equations that establish
the activities of reaction-diffusion structures. We report on the emergence of
diverse spatial structures including hexagonal and mixed-type patterns by
providing a systematic mathematical approach, including variations in
correlated oscillations, pattern variations and amplitude fluctuations. Our
work shows that the emergence of spatiotemporal behavior, commonly found in
excitable systems, has the potential to contribute significantly to the study
of diffusively-coupled biophysical systems at large.
