(CNN) -- Doctors treating Rep. Gabrielle Giffords said Sunday that while she remains in critical condition, early signs have made them hopeful.
Giffords is able to communicate with doctors by following simple commands, such as squeezing her hand, said Dr. Michael Lemole, Jr., chief of neurosurgery at University Medical Center in Tuscon, Arizona. "We are very encouraged by that," he said, adding, "I am cautiously optimistic."
Brain swelling is the biggest threat Giffords faces now, said Lemole.
Jared Lee Loughner allegedly shot the congresswoman at an event in Tucson, Arizona.
The bullet traveled through the left hemisphere of her brain and out of her head, doctors said. It did not cross from one brain hemisphere to the other, Lemole said. One of the things most concerning to doctors in the wake of a shot to the head is whether the bullet crossed "through the geometric center," he added.
CNN Chief Medical Correspondent Dr. Sanjay Gupta, a neurosurgeon, said when a bullet hits the bone of the skull, the bone moves with the bullet -- and fragments come shooting out, "like missiles moving into the brain." Surgeons worked to remove those fragments. "There's bleeding often associated with that," said Gupta. "That bleeding needs to be stopped."
Surgeons also removed tissue damaged by the impact, the doctors said, and removed more of the skull as well, giving the brain room to swell without being confined.
"When the brain starts to heal and the swelling goes down, that bone can subsequently be placed back," Gupta said.
Brain swelling generally takes a few days, said Gupta. Doctors monitor it, and after a week or so, "pretty reliably the brain swelling will start to go down."
Degeneration of brain cells or neurons is a longer term concern, and it's "very difficult to put a timetable" on degeneration and regeneration, said Gupta.
Giffords was under sedation, said Dr. Peter Rhee, medical director of the UMC trauma center. Doctors will turn off the sedation every so often and allow her brain to begin functioning and "see what her progress is."
Witnesses indicated the congresswoman had "communicated" at the scene somehow, but University Medical Center spokeswoman Darci Slaten said reports that Giffords had "talked" at the scene or at the hospital are inaccurate.