The 28-year-old was sent off for kicking the Neymar in the 58th minute, a decision that Brown, along with manager Neil Lennon, deemed has harsh.
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"I realise that I made a mistake in getting sent off. It's something that happened in the heat of the moment but I shouldn't have acted in this way," told the club's official website.
"I still think it was a harsh decision. However, I recognise that I made a mistake and put myself and my team-mates in a difficult position. I need to live with that - and there is no one more disappointed than me about losing the match."
Neymar's role in the midfielder's dismissal has faced criticism from Lennon and Scotland manger Gordon Strachan, while Brown has questioned the Brazil star's reaction to the challenge.
"There was not a huge amount of contact at all, to be honest, but I have to accept responsibility for my actions - and I do."
Brown will now miss Ajax's visit to Celtic Park on October 22, but if his actions are deemed excessive by Uefa he could also be forced to sit out the return clash with the Eredivisie side and the home game against AC Milan.