  Chimera state is defined as a mixed type of collective state in which
synchronized and desynchronized subpopulations of a network of coupled
oscillators coexist and the appearance of such anomalous behavior has strong
connection to diverse neuronal developments. Most of the previous studies on
chimera states are not extensively done in two-dimensional ensembles of coupled
oscillators by taking neuronal systems with nonlinear coupling function into
account while such ensembles of oscillators are more realistic from a
neurobiological point of view. In this paper, we report the emergence and
existence of chimera states by considering locally coupled two-dimensional
networks of identical oscillators where each node is interacting through
nonlinear coupling function. This is in contrast with the existence of chimera
states in two-dimensional nonlocally coupled oscillators with rectangular
kernel in the coupling function. We find that the presence of nonlinearity in
the coupling function plays a key role to produce chimera states in
two-dimensional locally coupled oscillators. We analytically verify explicitly
in the case of a network of coupled Stuart - Landau oscillators in two
dimensions that the obtained results using Ott-Antonsen approach and our
analytical finding very well matches with the numerical results. Next, we
consider another type of important nonlinear coupling function which exists in
neuronal systems, namely chemical synaptic function, through which the
nearest-neighbor (locally coupled) neurons interact with each other. In
numerical simulations, we consider two paradigmatic neuronal oscillators,
namely Hindmarsh-Rose neuron model and Rulkov map for each node which exhibit
bursting dynamics. The existence of chimera states is confirmed by
instantaneous angular frequency, order parameter and strength of incoherence.
