  The coexistence of coherent and incoherent domains, namely the appearance of
chimera states, is being studied extensively in many contexts of science and
technology since the past decade, though the previous studies are mostly built
on the framework of one-dimensional and two-dimensional interaction topologies.
Recently, the emergence of such fascinating phenomena has been studied in
three-dimensional (3D) grid formation while considering only the nonlocal
interaction. Here we study the emergence and existence of chimera patterns in a
three dimensional network of coupled Stuart-Landau limit cycle oscillators and
Hindmarsh-Rose neuronal oscillators with local (nearest neighbour) interaction
topology. The emergence of different types of spatiotemporal chimera patterns
is investigated by taking two distinct nonlinear interaction functions. We
provide appropriate analytical explanations in the 3D grid of network formation
and the corresponding numerical justifications are given. We extend our
analysis on the basis of Ott-Antonsen reduction approach in the case of
Stuart-Landau oscillators containing infinite number of oscillators.
Particularly, in Hindmarsh-Rose neuronal network the existence of
non-stationary chimera states are characterized by instantaneous strength of
incoherence and instantaneous local order parameter. Besides, the condition for
achieving exact neuronal synchrony is obtained analytically through a linear
stability analysis. The different types of collective dynamics together with
chimera states are mapped over a wide range of various parameter spaces.
