Karate was once defined by a certain high level instructor I know as an endeavor that is both hard and painful - hard as in difficult, not the external or linear aspects. Painful is self-explanatory. Pain for me is not just getting walloped or incurring injuries but also a by-product of arduous training. Karate workouts tend to be very cardio/aerobic in nature. Speaking from experience, the feeling of fatigue brought on by oxygen debt is what, in the words of legendary football coach Vince Lombardi, "makes cowards of us all." Nothing pushes one's boundaries more than a brutal training session (shugyo). Shugyo "polishes the spirit" as Draeger puts it, and can even be undertaken solo sans a group setting, according to Forrest Morgan, author of the now classic Living The Martial Way.
The breath is closely related to the (controversial) concept of ki (chi/qi). Kiai (spirited shout) is a manifestation of this force that is not limited to karate. At a recent Japanese Spring festival I attended, a local kendo (sword way) school put on an exhibition match that featured players that liberally employed the kiai. I thought it was a bit much, but then I don't do kendo. Years ago, I took a jiu-jitsu seminar given by a renown Daito-ryu exponent who thought the kiai was "fake" if not downright silly. Shortly thereafter he went on to show a technique that was concurrent with a forceful exhalation. "Why exhale there?" I inquired. "It makes you stronger" was the response. A silent kiai, perhaps?
In karate, techniques that are performed with slow, deliberate muscular contractions are frequently executed alongside of forceful breathing. In Isshinryu the kata Seisan, Seiuchin and Sanchin use such methods. Sanchin is a unique form with no real bunkai (fighting applications) - at least none that I'm aware of - that dates back to antiquity. Supposedly it was Sanchin that the mytho-historical figure Bodhidharma brought from India to the Shaolin Temple in China to shock the lethargic monks out of meditative stupor. In reality, the Sanchin kata was probably a cognate of an earlier qigong form used to facilitate respiration and the circulation of blood.
One of my instructors would advise to "breathe through the heels" when performing Sanchin. Though not anymore, back in his day (the 60s) Sanchin was the last karate kata taught in Isshinryu. Naturally then, when testing for shodan (1st black belt) Sanchin was the last kata performed after numerous forms had been completed at full power and speed. The now heavily fatigued aspirant was then expected to fight everyone in the school! Talk about shugyo. And no uninspired renditions of kata for testing were allowed, as I once witnessed two candidates fail their 2nd dan promotions for not 'working up enough sweat' through a dozen or so forms. I must say, Sanchin produces a training effect like nothing else. A friend of mine that runs an Isshinryu school nearby does a version of Sanchin utilizing "Dragon Breathing" (i.e. very intense) that he says he learned from Advincula Sensei.
In early Chinese texts much is said about the power and magic of the breath. One Zhou dynasty (c. 500 BCE) inscription on the breath translates thus
When transforming the breath, the inhalation must be full to gather the magic. To gather the magic, fullness must be extended. When it is extended it can penetrate downward. When it can penetrate downward it is magic. When it descends it becomes calm, solidifies, and is both strong and firm.
When novices are preparing for a test or tourney I advise them to take a slow, deep breath a moment before a point match or performance of kata. It has been long known that emotions affect the rate of breathing, but the opposite is also true: controlled breathing regulates the emotions. Stan Grof, a pioneer in transpersonal psychology and LSD research, has developed something called holotropic breathing that reportedly brings about altered states of consciousness. Thanks, no. If I want to achieve euphoria or pain-induced ecstasy I'll stick with good ol' martial arts training.