Manager Robbie Neilson has spoken of the club’s ambition to be challenging at the top of the table and putting pressure on the Glasgow giants.
It is an aim shared by his coaching staff.
McCulloch has expressed the need for constant progression, building on a strong start to the season with Hearts unbeaten in the league heading into Saturday's match with Ross County, aided by a stronger mentality.
"That's what this club is all about," he said. "It's about trying to split them and can we go above and beyond that.
"It's high expectations. It's a massive club and we need to deal and cope with the expectations. As a staff we've certainly done that and in the dressing room we've done that as well.
"It's that message, can the staff and players get better every day together and week by week.”
He added: “Happy with where we are but we want to kick on and keep progressing.
"Our message to the players is 'can they get better everyday' and for us is can we get better every week.”
McCulloch is well placed to know what it takes to reach the level of the Old Firm having played more than 300 games for Rangers, winning the league in three successive seasons.
Key to getting to that standard will be the home form.
“I've known since I started following football how hard it is going to Tynecastle to get a result and to play," he said. "I think we've brought that back.
"Before the manager came in the club won four league games that season so it speaks volumes of the management of Robbie Neilson and that change in mentality.
"It shows you the recruitment has been good from Joe Savage and Robbie Neilson. We're certainly heading in the right direction.
"I keep reiterating, it's about getting better and better and we've certainly not got complacency in the dressing room or management staff.
"We've had some big results this season but as soon as the results are finished it's onto the next game. That's the way a winning culture, winning mentality should be.”