There's no one-size-fits-all answer.
But here are a couple of scenario's...
The action is clearly part of your day-to--day job responsibilities...speak up clearly, claim the action and let the other participants know when you will have a result or an update.
You need the action completed in order to do your job (or your part of a project)...this is going to be in your best interest so take it on - ask for some assistance if need be. Make it clear to others why you are taking this on and how it links to what you need to do next.
It's a great opportunity to build your profile...do it! Just ensure that you have the time and resource and it's not going to have a negative impact on your workload.
It's your turn...to take on some extra work for the team. Fair's fair if everyone else has pitched-in at other times you need to do your share.
You are the only person with the skill or knowledge to do it...lend your skills and expertise to help find solutions. It may be that you have access to other resources who could assist or have alternate ideas that could get the job done (without you necessarily doing it!). If it is down to you...negotiate the time frames, resources and/or relief on your other accountabilities so you can get the job done.
No one else volunteers...unless it's part of your job, or meets one of the other criteria...think about it very carefully before you put your hand up to take it on. Your motivation to 'help out' may be admirable but think about the implications of volunteering for more work. Don't take on extra work just because you are seen as the person who will always take on the extra responsibilities. It can undermine the value of what your regular job is if you are always able to drop it to do the 'extra's'.
Don't let the peer pressure of silence get to you. Sit on your hands. Count to ten or 10,000. Whatever you need to do to resist the urge to volunteer just because no one else has taken it on.
So there's six possible possibilities for when actions are being assigned in a meeting.
Have you got some more? Let's chat...