UPDATE:
Thanks to all the people who saw this and pointed us in the right direction.
We ended up taking her to the children's hospital in Pittsburgh where a team of pediatricians and pediatric neurologist looked at her for a couple days. They came to the diagnoses of Benign Neonatal Sleep Myoclonus.
What this is, is shaking ONLY during sleep. The shaking does not stop when you try to hold your child but if you wake them up it should stop abruptly. But the only way to diagnose this is to completely rule out any other factor AND to make sure that there is no seizure activity in the brain when it is happening.
Here is the thing about this ... It's very important to have the testing to rule out seizures. It might be boring/traumatic and take a couple days but it is very important. If you think your newborn is doing this, take them to a children's hospital that has a Pediatric Neurologist. Go straight through the ER and tell them you need to make sure your child is not having seizures, otherwise you will have to wait 2 months to get an appointment.
They will most likely get blood, urine, and spinal fluid samples to send for cultures that will rule out any bacterial or viral causes. They will also run tests on the samples to rule out metabolic factors.
However the test you need is a 24-48 hour Video EEG. The key here is your child needs to have an episode while on the EEG to rule out the possibility of seizures.
Feel free to comment if you have questions about this.
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