  We study the emergence of the traveling chimera state in a two-dimensional
network of Hindmarsh-Rose burst neurons with the mutual presence of local and
non-local couplings. We show that in the unique presence of the non-local
chemical coupling modeled by a nonlinear function, the traveling chimera
phenomenon occurs with a displacement in both directions of the plane of the
grid. The introduction of local electrical coupling shows that the mutual
influence of the two types of coupling can, for certain values, generate
traveling chimera, imperfect-traveling, traveling multi-clusters, and
alternating traveling chimera, ie the presence in the network under study, of
patterns of coherent elements interspersed by other incoherent elements in
movement and alternately changing their position over time. The confirmation of
the states of coherence is done by introducing the parameter of instantaneous
local order parameter in two dimensions. We extend our analysis through
mathematical tools such as the Hamilton energy function to determine the
direction of propagation of patterns in two dimensions.
