Assuming all other variables are defined properly, there is still an error in your static Podio::authenticate() method.
Just strip the quotes from MY_APP_TOKEN:
So
Podio::authenticate('app', array('app_id' => MY_APP_ID, 'app_token' => 'MY_APP_TOKEN'));
should better be
Podio::authenticate('app', array('app_id' => MY_APP_ID, 'app_token' => MY_APP_TOKEN));
Cheers!