Without the breath, there's no Yoga*.
Way before any neuroscience, the ancient yogis have found a correlation between the breath and the mind, and developed various practices to direct the breath to control the mind, which is the goal of Yoga.
The science of controlling the breath, is known as Pranayama. Prana is a Sanskrit word which means life force, vital life energy. Prana exists in all things having life. Our body is controlled and regulated by the force of prana, every cell in the body is controlled by prana. Prana is the universal energy, it is in the air, food, water and sunlight. It manifests itself as gravity, electricity and in the actions of the body, from the movement of your finger to a thought in your mind. If you think about it, prana drives and permeates EVERYTHING!
When we practise pranayama, we are practising controlling our breath AND the subtle prana that is affected by the breath. Our minds are directly affected by our breaths and in order to bring our mind (which runs on of a subtler prana) under control, we can start with controlling the grosser part of the prana which is the breath.
"Respiration being disturbed, the mind becomes disturbed. By restraining the respiration, the Yogi gains steadiness of mind." - Hatha Yoga Pradipika, Verse 2
Since thoughts are also a reflection of prana at work, when our breath slows down and becomes steady, our minds become steady with less fluctuation of thoughts... slowly becoming more single-pointed, more concentrated. Starting with breath awareness and moving gradually to breath control, yoga practitioners will slowly start to experience in the increase of prana in their body and mind.
Simple exercise:
Start with resonance breathing, i.e. same counts in breath and out breath (e.g. 4 counts in breath, 4 counts out breath) for 10 minutes everyday.
When you get comfortable with the breathing, start to lengthen the exhalation twice as long as a breath modification, i.e. in breath and out breath at a ratio of 1:2.
Maintain the full yogic breath (inhaling expanding abdomen, rib cage and chest) and allow natural pause between exhalation and the next inhalation. No holding of breath at this stage.