Jules Bianchi remains "critical but stable" in hospital, nearly four weeks after he suffered severe head injuries in a crash at the Japanese Grand Prix.
The Frenchman had surgery at the Mie General Medical Center in Yokkaichi where he is still receiving care.
His family said in a statement that the Frenchman "continues to fight".
They added: "Although we have no new information to give, we recognise that there are a huge number of people all around the world who are supporting Jules and willing him on in his fight.
"We owe it to his many fans to acknowledge the continued outpouring of messages, and to provide some information, however brief it may be."
Two weeks ago, Bianchi's father Philippe described the situation as "desperate" and added that the family hoped to transfer him, potentially to Tokyo, if his condition improved.
However, on Thursday the family said: "At this time his fight will continue here in Yokkaichi. We are taking things step by step."
Bianchi collided with a tractor vehicle that was recovering Adrian Sutil's Sauber, which had crashed on the previous lap.
Following an investigation into the crash, governing body the FIA is to trial a "virtual safety car" during first and second practice for this weekend's US Grand Prix.
The system, which forces drivers to slow in an accident zone, is aimed at eliminating the possibility of a repeat of the incident that left Bianchi in a critical condition.