There are some problems with the way that we currently implement representative democracy in parliamentary democracies in many countries, including the UK. Part of the issue with this is that your representative is not always going to align with your viewpoint. For instance, I am currently in a constituency with a Conservative MP who is very loyal to the party line and who I am unlikely to change the views of by writing to them. For instance, in a previous post I made, I advocated for proportional representation - however, my MP has come out in favour of FPTP. In some sense, that means that my representative cannot effectively represent my views because of an ideological divide.
Now when you look at direct democracy, where voters can vote on propositions/bills directly, that has some benefits in the sense that it directly acts on a voter's viewpoint. Although it does have some problems such as tyranny of the majority and people who do not vote on particular bills or don't have the time to keep up to date on every little detail of every issue (i.e. most people) end up not having their views represented properly.
The idea that one of my colleagues suggested to me was this: For a particular bill, an MP would have as many votes as the amount of people in their constituency - when an MP votes, they cast all of those votes on their constituents' behalf. However, if a constituent wishes to, they have the option of voting a different way to their MP. So for instance if your MP is voting to cut welfare, then you would have the option of voting the opposite way to them. This also means that if you aren't interested in a particular issue, then you're still represented by your MP. This may also help to balance out "tyranny of the majority" since the representatives are still likely to cast most of the votes on most issues.
The main argument against such a system seems to be that historically it would have been difficult to administer. These days, however, setting up an electronic software system to do it is easily within the realms of possibility. It seems like it would be a good candidate for giving citizens more of a say in the way that their country is run without some of the drawbacks of direct democracy and to help give people power to counter other forms of democratic deficit.
I've had trouble finding any systems that work like this in practice, so if you know of any, let me know! This hybrid democratic model may well be worth voting for.