  We study the two-flavor Nambu--Jona-Lasinio model with the Polyakov loop
(PNJL model) in the presence of a strong magnetic field and a chiral chemical
potential $\mu_5$ which mimics the effect of imbalanced chirality due to QCD
instanton and/or sphaleron transitions. Firstly we focus on the properties of
chiral symmetry breaking and deconfinement crossover under the strong magnetic
field. Then we discuss the role of $\mu_5$ on the phase structure. Finally the
chirality charge, electric current, and their susceptibility, which are
relevant to the Chiral Magnetic Effect, are computed in the model.
