And the same part of the brain may kick in to stop people acting on their feelings after a particularly bad break-up.
Professor Barbara Sahakian from the University of Cambridge, is currently testing brain training to stop compulsive behaviours in people suffering from mental illness.
She said: "The frontal lobes in the brain are so important for inhibitory control, for inhibiting responses.
"People don't realise that the frontal lobes exert control in many different situations, whether in a brain-training task or in stopping people ruminating on lost love.
"It is like exercising a muscle, and it might stop someone who is heartbroken from repeatedly texting their ex-partner. The brain would have the tools to put a stop to that.'
The principle is the same as for existing brain training for people with obsessive compulsive disorder.
Professor Sahakian said: "The thing about love is that it can affect the wrong systems in the brain.
"People can become quite compulsive about it all, which is important if you are going to marry someone and have children with them, so need strong feelings.
"However, if that person does not love you any more or has gone off with someone else, that becomes maladaptive."
Heartbreak involves the amygdala and limbic parts of the brain, but the mechanism which governs "inhibitory control", in the prefrontal cortex, can prevent us giving in entirely to our emotions.
Scientists from Rutgers University in Newark found in 2010 that lovelorn people can spend months struggling to cope with their emotions, as the brain still expects the "reward" of being in love with the person they are no longer with.
The academics reported people sobbing for hours, pleading for reconciliation and showing up at their previous partners' workplaces to make declarations of anger or love.
But brain training could help people get better at not acting on their feelings.
Previous research has already shown that the arrow task activates the parts of the brain involved in impulse control.
Professor Sahakian said: "It is a case of 'use it or lose it" when it comes to inhibitory control.
"If it follows other forms of brain training, eight hours a month could be enough to help."
The exercise involving following the arrows could also work because it distracts people from their newly single status.
She added: "People take their friends on holiday or a night out following a divorce or break-up because they know distraction works.
"The worst thing someone can do after getting their heart broken is to stay at home alone ruminating on the same cycle of thoughts about their happy memories from their relationship.
"We know from psychology studies and brain scans that it is hard to properly focus on things at once, and therefore focusing on exercise or enjoyable activities and going out with friends helps distract from focusing on thoughts on your previous partner.
"Inhibiting maladaptive thoughts and switching to adaptive thoughts is the key to getting over unrequited love."
The neuroscientist was speaking after appearing at Cheltenham Science Festival to talk about neuroimaging, based on her book Sex, Lies and Brain Scans.
The brain in love
The frontal cortex, vital to judgement, shuts down when we fall in love.
MRI scans show this de-activation occurs only when someone is shown a photo of the person they adore, causing them to suspend all criticism or doubt.
Semir Zeki, professor of neuro-aesthetics at University College London, says: "When you look at someone you are passionate about, some areas of the brain become active," he says. "But a large part is de-activated, the part that plays a role in judgement."
Prof Zeki believes the brain may behave in this way for "higher biological purposes" - it makes reproduction more likely.
If judgement is suspended, the most unlikely pair can get together and reproduce.
Someone in love will still be capable of making other major decisions in their lives, from striking a business deal to choosing a new mortgage.
And this sanity makes it harder for friends to convince them "they have taken leave of their senses" when it comes to an ill-advised affair.
Brain scans have also shown the area of the brain that controls fear, and another region involved in negative emotions, close down, explaining why people feel so happy with the world - and unafraid of what might go wrong - when they fall head over heels.
