Deep breathing increases heart rate variability (HRV) - a marker of stress and sympathetic activation. Higher HRV is associated with a parasympathetic state and also has been shown to have mortality benefits. Long term deep breathing may lower blood pressure and has been shown to reduce anxiety, improve ratings of perceived stress, sleep quality, and exercise tolerance.
Newborns are deep diaphragmatic breathers naturally. However, factors such as temperature, noise, pollution, and anxiety cause adults to become shallow breathers. Shallow breathing may be linked to decreased cognitive function such as making judgment calls and emotional processing. It can also cause less efficient ventilation, worse cardiac output, and can harm sleep quality, which increases pain.
The vagus nerve serves an important purpose in proper breathing as it is the communication pathway between the brain and organs. When stimulated through deep breathing, it counteracts the effects of the sympathetic nervous system that may be taxed, resulting in increased anxiety and stress. It sends messages to the brain and heart based on what breathing indicates. The ideal breathing rate is around 6 breaths per minute as is when you maximize your HRV, a marker of parasympathetic tone.
The diaphragm is the most important muscle in respiration as it accounts for about 80% of total muscle activity during breathing. It impacts multiple organ systems, the autonomic and cardiovascular systems being the two most predominant. The practice of coordinated breathing is aimed at maximizing the use of your diaphragm and eliminating uncoordinated overuse of secondary breathing muscles in the chest such as the intercostals.
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https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4153063/
https://www.jneurosci.org/content/36/49/12448