The Brink is a point during the process of exsanguination, where the person being brought over can return from before the point of death. This moment is a point at which the Maker (vampire turning the future progeny) would give their blood in return so that the process of becoming a vampire can be completed. If there is not a trade of blood at the point of The Brink, the exsanguinated person will die.
Many people who are turned in the Odeum series witness moments of their lives mixed with those of their makers. The image above depicts a moment in book two, The Keeper, where Anna experiences her return from The Brink. In some cases, people who return from The Brink may suffer from temporary amnesia or a loss of self if they had imbibed on alcohol or were under the influence of a drug prior to their maker turning them. This can be seen in both book one, Odeum, when Jas is turned, and in book two, The Keeper, when Anna is turned.
During the point where an individual is at The Brink, their organs begin to shut down. They may experience the feeling of euphoria or bliss in this moment and are unresponsive, though they can hear any words spoken from their maker. At this point there is lower brain activity. However, once the blood of a vampire has been introduced into the system, the first organs that are "jump-started" are the brain, the heart, and the stomach. This begins the return from The Brink.