The heart is its own pacemaker. It spontaneously generates electrical signals without any help from the nervous system. The nervous system's job, through autonomics (i.e. the system of nerves that you don't control consciously), is to speed up or slow down the heart, or increase or decrease force of contraction. But if we remove the heart completely from your body, disconnected from your nerves, it still beats. The heart's important function is basically beating blood around your body. Theoretically, if you change the heart into a robotic beating system, it's not going to change your personality, your spirituality, your thoughts, etc. Your brain is in your brain, not in your heart. When one receives a heart transplant, he receives flesh from another flesh, but his own spirit and thoughts are the same.
When someone says that heart has its own nervous system, that doesn't mean it has a brain. A nervous system is simply that which produces electrical signals (well, that's kinda oversimplifying it). All your muscles in your body produce electric signals. Your gut muscles also have its own nervous system and when disconnected from the body, it gyrates alone. Your skeletal muscle however (with very few exceptions) has to generate electricity when you want it to, i.e. consciously.
Ok, Thanks! Hey, would this have anything to do with muscle memory? Or is that something completely different?
Sorry I edited my post for accuracy's sake.
Anyways, ummm...yes and no. Your muscles can get stronger but really the memory "part" is procedural memory which exists in your brain. Procedural memory is strong form of memory. When you are repeating the motions, you are reinforcing nerve connections in your brain that allows you to be faster and more efficient at what you do.
Does the brain in our heart communicate with the brain in our brain?
First off, I think it's misleading to say the heart has a brain. Depending on how you define "nervous system," we pretty much say the heart muscle contains specialized muscle cells that conducts electricity on its own. Reason for this is that when you compare a nervous tissue to a pacemaker cell, they're structurally different, but the function is similar. And each type of heart conductive muscle cell has its own "pace." The cell with the faster pace overrides all the other cells with a slower pace.
There is communication though, directly and indirectly. When you are afraid, panicing, or angry, or just plain exercising, your brain sends signals through a system called the sympathetic system (the fight, flight fright system). Directly through this system and indirectly through your adrenal glands that pumps adrenaline in your blood. It speeds up your heart and increases force of contraction (increasing blood pressure). When you are relaxing and you just had a nice meal for instance, and you want to be a couch potato or read a book, you're acting your parasympathetic system (rest and digest system), which slows down your heart and decreases the force of contraction (decreasing blood pressure).
That's basically how there is communication with the heart. The sympathetics and parasympathetics together comprise the autonomic nervous system, that is the system that does things without you thinking about it, like digesting your food after a nice meal or dilating or constricting your pupils, or constricting or dilating blood vessels (decreasing or increasing blood flow in a certain area). Somatic nervous system is that which controls your skeletal muscle, like walking, moving your arms, breathing by your diaphragm, moving your eyes, etc.