The cord is simply there to prevent the loss of the shako if it becomes dislodged. It started to disappear around 1805 when shakos began to be issued with chin scales. There were two main types. The figure from FN51 that is wiping his hair back has the simpler style that just loops around the neck. The other figure, with his hand on his sword balanced on end, has the more complicated fixture. It runs down his back, loops up under his right arm and is fastened to the front of his pelisse. It ends in a feature called a 'flounder', a flattened portion of braid with a tassle. You can see it hanging down just below his shoulder belt.
Considering these figures represent ADC's to various senior officers, the cord braid is probably gold.