We have investigated copper electrodeposition from copper sulfate solutions in thin cells with small inter-electrode distance. At large enough current densities, when the concentration at the cathode approaches zero, we observe large concentration fluctuations in the direction parallel to the electrodes, indicating local fluctuations of the current density. These fluctuations are essentially observed in the vicinity of the anode. They occur much before the onset of dendritic growth at the cathode. A preliminary model is presented, which accounts for the observed behavior.