Update checkbox List via URL querystring

I have a URL which have some default model,
example.com/selector/
when user type something it update the list

<input type="text" name="filter" placeholder="{{$lang.common_forms.field_filter_hint}}"/>

example.com/selector/?q=something

<div id="unik" class="button-group">
	{{foreach item="item" from=$data}}
		<div class="item">
			<input id="model_{{$item.model_id}}" type="checkbox" class="checkbox" value="{{$item.model_id}}"/>
			<label for="model_{{$item.model_id}}">{{$item.title}}</label>
		</div>
	{{/foreach}}
</div>

I want to update it, when user start typing input text box:

<div class="filter">
	<input type="text" name="filter" placeholder="{{$lang.common_forms.field_filter_hint}}"/>
</div>

Via ajax or javascript

var params = parsedUrl.param();
delete params["page"];

var newUrl = "?page=" + $(this).val() + "&" + $.param(params);