Hwo to update a synonym settings with PUT command

I have created a index setting like this

PUT uk-2016.06.22
{
  "settings": {
    "analysis": {
      "filter": {
        "my_synonym_filter": {
          "type": "synonym", 
          "synonyms": [ 
            "british,uk,england,britain"
          ]
        }
      },
      "analyzer": {
        "my_synonyms": {
          "tokenizer": "standard",
          "filter": [
            "lowercase",
            "my_synonym_filter" 
          ]
        }
      }
    }
  }
}

But then I would like to also include United Kingdom into the synonym. But when I just simply add into it and run again, it gives me error

{
   "error": {
      "root_cause": [
         {
            "type": "index_already_exists_exception",
            "reason": "already exists",
            "index": "uk-2016.06.22"
         }
      ],
      "type": "index_already_exists_exception",
      "reason": "already exists",
      "index": "uk-2016.06.22"
   },
   "status": 400
}

How should I correctly update it?

You can't. You have to reindex.