Did you try adding a OnSelectedIndexChanged="YOURSelection_SelectedIndexChange" event handler yet?
Here is our example:
<asp:RadioButtonList runat="server" Style="float: right;" ID="PeriodSelectionList" RepeatDirection="Horizontal" AutoPostBack="true" OnSelectedIndexChanged="PeriodSelectionList_SelectedIndexChange" ForeColor="White">
<asp:ListItem Text="Current Period" Value="CurrentPeriod"></asp:ListItem>
<asp:ListItem Text="Next Period" Value="NextUpcomingPeriod"></asp:ListItem>
</asp:RadioButtonList>
in Code behind:
protected void PeriodSelectionList_SelectedIndexChange(object sender, EventArgs e)
{
// ... do stuff
// tell Panel to open /display
}