(CNN) Tubes and wires crisscross Israel Stinson's little body in the heart-wrenching video. Moments after the 2-year-old is tickled under his arm, Israel appears to move slightly on his bed in a Northern California hospital.

"Israel, you have to stop fooling everybody," his mother says in a video posted online by the Life Legal Defense Foundation.

"How long is it going to take? You make Mommy so excited. I know you're going to come out of this, baby. Whenever you're ready."

Israel's parents, Nathaniel Stinson and Jonee Fonseca, have been granted an extended restraining order to stop Kaiser Permanente Roseville Medical Center from taking the boy off a ventilator.

A U.S. District Court judge on Friday extended the order for one more week. Fonseca asked for the time so she and her attorney can take the case to the 9th Circuit U.S. Court of Appeals.

Israel was being treated for an asthma attack in early April when he went into cardiac arrest at a pediatric intensive care unit, according to court documents. A neurological exam showed he was brain-dead.

His parents transferred Israel to Kaiser in Roseville, the court documents said. Additional exams confirmed brain death, a declaration of death was issued and doctors decided to disconnect the boy from a machine making his heart beat.

'Are we pulling the plug too soon?'

The legal fight over Israel is unfolding in federal court, where attorneys for his family have argued that denying the child medical treatment is a violation of his rights.

The parents also say that removing Israel's breathing tube would violate their beliefs as Christians.

"The question we're asking is, are we pulling the plug too soon on these patients who have some chance for recovery down the road?" asked Alexandra Snyder, executive director of the Life Legal Defense Foundation, a nonprofit representing the parents. (The foundation's website says its mission is "to give innocent and helpless human beings of any age, particularly unborn children, a trained and committed defense against the threat of death.")

In a statement Wednesday night, Kaiser's chief of staff at Roseville, Dr. Chris Palkowski, repeated his earlier position: "Our hearts go out to this family as they cope with the irreversible brain death of their son. We continue to offer our support and compassion to the family during this sad time. We will continue to follow the court's directions."

Judge extends temporary restraining order

On May 2, a federal judge issued a temporary injunction against removing Israel from a ventilator. A state court had earlier ruled that Kaiser complied with necessary legal steps for determining brain death.

On Wednesday, Israel's parents pushed their challenge to the constitutionality of state law. Federal law, they argued, provides an exception for the religious belief that brain death does not exist.

In its decision Friday, the federal district court cast doubt on Fonseca's ability to win the case.

"While Ms. Fonseca's maternal instincts and moral position are completely understandable, the concerns reviewed here suggest she is unlikely to obtain the relief she seeks," the judge wrote.

After Friday's ruling, Kaiser resent Palkowski's previous statement when asked for comment.

California's law is modeled on the Uniform Determination of Death Act, which nearly all the states have adopted. According to the act, an individual is dead when he or she "has sustained either (1) irreversible cessation of circulatory and respiratory functions, or (2) irreversible cessation of all functions of the entire brain, including the brain stem."

That essentially means the brain, an extremely complex organ, no longer enables other organs to function.

Snyder said state law -- more than 40 years old -- fails to take into account the latest neurological research.

The parents said they hope to move Israel to a long-term care facility.

"We are mentally and physically drained, but spiritually we are stronger than ever because we know that Israel is still very much alive," Fonseca said in a statement before the hearing.

"Three weeks ago, Israel started moving his head and upper body in response to our voice and touch. He has kicked his legs. And he has taken some breaths on his own 'over' or in addition to the ventilator."

Medical experts believe a patient's condition can't improve if the person is truly brain-dead, because there's no way to come back to life once brain activity ceases.

Dr. Thomas Nakagawa, chief of the division of critical care medicine at Johns Hopkins All Children's Hospital in Florida, told CNN that patients must meet specific criteria -- including whether they're in a coma and are unable to breathe on their own -- for doctors to make a determination of neurologic death. The patients must also undergo a detailed physical examination.

"There are no reports of any person surviving after being declared dead by neurologic criteria," said Nakagawa, who helped write national guidelines for the determination of brain death in infants and children.

"Reports of people who have survived following a determination of neurologic death are the result of diagnostic error, and these patients did not meet initial criteria for neurologic death."

'No signs of recovery'

In court filings, Kaiser Permanente said it has merely complied with California law.

"Three different physicians have administered brain death examinations and each found the results to be consistent with brain death," according to papers filed by attorneys for the hospital.

Dr. Michael Myette, a Kaiser physician who treated Israel, said the boy's brain "is not telling his organs how to function" and he "demonstrates no signs of recovery," according to court documents. Israel's gastrointestinal system does not function; his only nutrition is dextrose, which he receives from one of the various tubes in his body.

As for the boy's occasional movements, that's a spinal reflex -- "a reflexive action mediated by cells in the spinal cord, bypassing the brain altogether," according to the doctor.

"Unfortunately, Israel's mother, family, and attorneys, all nonmedical professionals, interpret Israel's spinal reflex as a sign his brain may be functioning or even that he is recovering," Myette said.

"They are incorrect. The videos offered by Israel's mother merely show the single ... spinal reflex."

Dr. Paul Byrne, a pediatrician who examined the boy's medical records on behalf of his parents, said in court documents that Israel is "likely to continue to live" with proper treatment and "may find limited to full recovery of brain function, and may possibly regain consciousness."

'A lot of fight in him'

Israel's parents seek a court order for Kaiser doctors to perform a tracheotomy and administer a feeding tube to give their son "proper respiratory support and nutrition," the court documents said. They eventually want to transfer him to New Jersey, where a religious exemption allows patients to remain on ventilators.

Israel's case has drawn comparisons to the story of Jahi McMath , an Oakland teenager whom a judge and doctors declared brain-dead last year over her family's objections. The 13-year-old girl underwent tracheotomy and gastrostomy tube procedures and was eventually released from a hospital. McMath, who moved with her family to New Jersey, hasn't recovered.

Fonseca set up a GoFundMe page and raised more than $15,300 so far on a goal of $50,000 to cover the costs of Israel's care and transfer.

"God is telling me not to let go," she wrote on the site. "Israel means apple of God's eye and he truly is a child of God with a lot of fight in him."