Doctors said that Giffords has remained in a medical coma since her two-hour surgery, but that they frequently wake her up to see what her progress is and to make sure something catastrophic hasn't occurred while she was asleep.
Giffords is on a ventilator and her eyes are closed but she is able to communicate non-verbally, doctors said.
At a press conference Sunday, Dr. Michael Lemole, the chief of neurosurgery at the University Medical Center in Arizona, said that during the surgery, the removed bone fragments caused by the bullet fracture to take pressure off the brain. They also removed some devitalized brain, but, Lemole said, "I'm happy to say we didn't have to do a whole lot of that."
Special Section: Tragedy in Tucson
Lemole said that it was good news that she was able to follow simple commands from the doctors and that the bullet did not cross from one hemisphere of the brain to another. Giffords can't speak yet or open her eyes.
"She is still in critical condition, brain swelling at any time can take a turn for the worse," Lemole said.
Lemole said that simple commands can range from squeezing a hand or showing fingers.
"One thing I'm going to emphasize here is we take those kind of simple commands for granted but they imply a very high level of functioning in the brain," Lemole said.
Lemole gave an account of how surgeons were able to get Giffords into the operating room within 38 minutes of the attack. He said they were quickly able to control the bleeding, which was not severe or excessive.
Dr. Peter Rhee, the trauma director at the University of Arizona, emphasized that Giffords wound was serious.
"This wasn't a little grazing wound through the brain, this was a devastating wound that traveled the entire length of the brain on the left side," Rhee said. "The neurosurgeons, Dr. Wan and Dr. Lemole saved her life. With the function she had before surgery tells us how much injury the bullet did. But the fact that she had a lot of preservation before surgery is what's making this situation as well as it can."
The bullet entered Giffords brain on the left side, moving from back to front. In most people, the left side of the brain controls their right-sided strength and sensation, as well as the ability to understand and speak - including the ability to understand commands, Lemole said.
Rhee said he was optimistic after seeing Giffords injury.
"This is about as good as it's going to get," Rhee said. "When you get shot in the head and the bullet goes through your brain, the chances of you living is very small, and the chances of you waking up and actually following commands is even much more - much smaller than that. So this so far has been a very good situation. Hopefully, it will stay that way."
Outside of the hospital, candles flickered at a makeshift memorial. Signs read "Peace + love are stronger," "God bless America and "We love you, Gabrielle." People also laid down bouquets of flowers, American flags and pictures of Giffords.
Doctors said the other living victims of the rampage are doing relatively well and had been transferred from the ICU to the ward - a lesser amount of care, which was a good indication of how well everyone is doing.
"The people who needed surgery are all recovering well, and so far we're extremely happy with the prognosis of all those other individuals," Rhee said.
Authorities said Gifford, 40, was targeted at a public gathering by a man with a semiautomatic weapon around 10 a.m. Saturday outside a busy Tucson supermarket. Arizona's chief federal judge and five others were killed and 13 people were wounded, including the Democrat lawmaker.
He also fired at her district director and shot indiscriminately at staffers and others standing in line to talk to the congresswoman, said Mark Kimble, a communications staffer for Giffords.
"He was not more than three or four feet from the congresswoman and the district director," Kimble said, describing the scene as "just complete chaos, people screaming, crying."
Pima County Sheriff Clarence Dupnik said the rampage ended only after two people tackled the gunman.
"He was definitely on a mission," said Villec, a former Giffords intern.
The Arizona Republic reports that Daniel Hernandez, and intern that had worked with Giffords for five days, ran towards the shots on Saturday. He checked pulses and then saw Giffords. Using his hand, he applied pressure to the entry wound on her forehead and pulled her onto his lap and held her upright so she wouldn't choke on her own blood.
Hernandez also instructed a bystander how to apply pressure to Ron Barber, Giffords' district director's wounds.
Barber told Hernandez "Make sure you stay with Gabby. Make sure you help Gabby," The Arizona Republic reported.
Hernandez then used meat smocks from Safeway to apply pressure to the entrance wound - unaware there was also an exit wound.
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Police say the shooter was in custody, and was identified by people familiar with the investigation as , 22. U.S. officials who provided his name to the AP spoke on condition of anonymity because they were not authorized to release it publicly.
His motivation was not immediately known, but Dupnik described him as mentally unstable and possibly acting with an accomplice.
The assassination attempt left Americans questioning whether divisive politics had pushed the suspect over the edge.
A shaken President Barack Obama called the attack "a tragedy for our entire country."