The family of the man accused of killing six people and injuring 14 others during a shooting rampage in Arizona expressed their condolences to the victims' families Tuesday.

Jared Loughner is accused of firing into a crowd outside a Safeway store Saturday morning, an attack that killed a judge and critically wounded Arizona Rep. Gabrielle Giffords.

Loughner's family, who a neighbour described as being devastated by the accusations against their son, expressed sorrow over the rampage in a statement handed to reporters outside their home.

"There are no words that can possibly express how we feel. We wish there were, so we could make you feel better," read the statement.

"We don't understand why this happened. It may not make any difference, but we wish that we could change the heinous events of Saturday. We care very deeply about the victims and their families. We are so very sorry for their loss."

Loughner, 22, faces five federal criminal counts in the shooting: the attempted assassination of a member of Congress; two counts of killing an employee of the federal government; and two counts of attempting to kill a federal employee.

He appeared in a Phoenix court on Monday in which he acknowledged that he understood the charges he faced. ABC News reporter Diana Alvear said Loughner addressed the court calmly during his appearance.

"At times he even appeared to smirk -- it was a little unsettling for many of the reporters who were present there," Alvear told CTV's Canada AM from Tucson Tuesday morning.

Alvear said none of Loughner's family or friends was present for the hearing.

About 600 people attended a memorial at the St. Odilia Catholic Church Tuesday for the victims of the shooting.

On Wednesday, U.S. President Barack Obama will travel to Arizona for a larger memorial, where he will deliver a message to the nation.

"In Tucson, the President and the First Lady will attend the memorial event ‘Together We Thrive: Tucson and America' at the University of Arizona to support and remember victims of the mass shooting in Tucson," the White House said in a statement.

Giffords' condition improves

Giffords, who was shot in the head at point blank range, is now breathing on her own according to her doctors, who updated reporters on her condition earlier Tuesday.

Her neurosurgeon, Dr. Michael Lemole Jr., said Giffords is able "to generate her own breaths" but a breathing tube remains in place to protect her from complications such as pneumonia.

"I'm happy to say that she's holding her own," Lemole said during a news conference Tuesday at University Medical Center in Tucson, Ariz. "Her status is the same as it was yesterday."

Five other people remain hospitalized after the shooting rampage. Three of the victims are in serious condition and two in fair condition.

Giffords, a 40-year-old, three-term Democrat, was rushed to hospital quickly after the shooting. Not long after, doctors treating her reported the congresswoman was conscious and responsive.

She underwent two hours of surgery Sunday in which doctors removed debris from the gunshot and a small amount of dead brain tissue. They also removed nearly half of Giffords' skull to accommodate swelling, preserving the skull bone for later replanting.

Giffords was being kept in a medically induced coma to give her brain time to rest. Doctors have since been able to ease her sedation, but said the timeline for her recovery remains uncertain.

"This is the phase of the care where it's so much up to her," Lemole said.

"She's going to take her recovery at her own pace," he said, adding that he's "very encouraged by the fact that she has done so well."

Death penalty possible

Federal prosecutors have not yet definitively said whether they will seek the death penalty in the case. Pima County Attorney Barbara LaWall said there are two factors that promote seeking execution: the death of a child and the fact that multiple victims died.

LaWall said her office would handle all other charges against Loughner, including those related to the four people killed who were not federal employees and those who were wounded.

Miami defence lawyer Neal Sonnett said the defence will have to push to dissuade federal prosecutors from pursuing the death penalty.

"That's the task of his lawyer in the first instance," Sonnett told the Associated Press.

Well-known Harvard Law School professor Alan Dershowitz also predicted "the prosecution will seek the maximum punishment in a case like this."

It will also be difficult for the defence to claim that Loughner could be innocent by reason of insanity. Congress raised the bar for insanity defences after John Hinckley shot former president Ronald Reagan in 1981.

A jury later found Hinckley was innocent by reason of insanity. Amid public outrage, Congress shifted the burden of proof over insanity to the defence, among other changes to the law.

Dershowitz said the rampage in Arizona "would be a clear case of insanity, because the pre-mediation would not be seen as undercutting insanity, it would be part of demonstrating insanity." But with the changes brought in after the Hinkley case, Dershowitz said "that's a very uphill battle."

Arizona also has modified its expectations for the insanity defence over the years. Pima County Attorney LaWall said if a convicted person "is held at a state mental hospital, and if sanity somehow comes back, he's transferred to prison, not just let go."

The team that represented Loughner in his Monday court appearance was led by Judy Clarke, a San Diego attorney who is a veteran of many high-profile death cases.

Clarke previously defended "Unabomber" Theodore Kaczynski, serial bomber Eric Rudolph and Susan Smith, a mother who drowned her two young sons. All three of these individuals avoided execution.

With files from The Associated Press