To set CURRENT_TIMESTAMP as default value for a column you should use DB::raw().

$table->timestamp('created_at')->default(DB::raw('CURRENT_TIMESTAMP'));

In Laravel 5.1.25 useCurrent() can be used to modify the default value.

$table->timestamp('created_at')->useCurrent();

On a mysql database, you can use 'CURRENT_TIMESTAMP ON UPDATE CURRENT_TIMESTAMP'

$table->timestamp('updated_at')->default(DB::raw('CURRENT_TIMESTAMP ON UPDATE CURRENT_TIMESTAMP'));

To update both created_at and updated_at fields:

$table->timestamp('created_at')->default(DB::raw('CURRENT_TIMESTAMP'));
$table->timestamp('updated_at')->default(DB::raw('CURRENT_TIMESTAMP on update CURRENT_TIMESTAMP'));