Marussia's Jules Bianchi has faced "a number of medical challenges" after sustaining severe head injuries in a crash at the Japanese Grand Prix, his team and family said in a statement.
But the Frenchman's situation remains "challenging" and the 25-year-old is in a critical but stable condition.
Earlier, Bianchi's father Philippe had described the situation as "desperate".
"Every time the telephone goes, we know it could be the hospital to tell us that Jules is dead," he said.
"He will not give up, I'm sure of that. I can see it. I believe it."
He added in an interview with Italian newspaper La Gazzetta dello Sport: "I speak to him. I know he can hear me.
"His doctors have told us that this is already a miracle, no-one has ever survived such a serious accident. But Jules won't give up.
"His trainer Andrea says that if there is one person who can make it happen with his will, it's Jules."
Philippe said the family had been touched by the show of support from other Formula 1 drivers, who gathered at the front of the grid before the Russian Grand Prix, arms around each other's shoulders, forming a 'circle of solidarity' for Jules.
"I've never seen anything like that," Philippe said. "It touched us very deeply. We thank every one of them. So many of them have been in touch, written to me; [Fernando] Alonso, [Jean-Eric] Vergne, [Felipe] Massa have given strong messages.
"[Lewis] Hamilton wrote me a beautiful email in which he says that if there is anything he can do, he's there. [Valentino] Rossi and [Marc] Marquez from Moto GP too."
Philippe also said he is using the accident of seven-time world champion Michael Schumacher, who continues his recovery at home after suffering a head injury in a skiing accident in December, as a reference point.
"I was very sad when he got hurt," Philippe said. "I kept wondering, like every one else 'Why don't they tell us more about how he is?'.
"But now I'm in the same position I understand. Everyone keeps asking me how Jules is but I can't reply, there is no answer. It's very serious but he's stable.
"One day he seems a bit better, other days a bit worse. The doctors don't say. The damage from the accident is very bad but we don't know how it will evolve.
"Even with Schumacher it took months to come out of the coma. [FIA president] Jean Todt said he hopes Michael will one day be able to have a normal life. One day I hope we can say the same about Jules."
He added: "It's tough. In a week the life of this family has been destroyed. What are we doing here? Living a nightmare in a place very far from home.
"But when Jules gets a bit better we can transfer him, maybe to Tokyo and things will be a bit easier. But who knows when that will happen. If it will happen. We have no certainties, we just have to wait."