Does your heart rate go up or down when you stand up?

When you stand up, you present the cardiovascular system with a problem. When supine, blood flowed almost effortlessly back to your heart, but an upright posture means that blood tends to pool (in your legs especially). The effect of this decreased venous return is that the heart's stroke volume (and thus cardiac output) is also reduced, which would tend to decrease your blood pressure, in accordance with the famous equation:

blood pressure = (stroke volume x heart rate) x total peripheral resistance

But the equation also suggests two ways to compensate for a decreased stroke volume: you can increase the peripheral arteriolar resistance (i.e. vasoconstrict) and you can increase the heart rate. Both of these responses are mediated by the autonomic nervous system, and the result is that the brain's perfusion is utterly unaffected by your upright posture.

Of course, that only applies if your autonomic nervous system is alright. In the elderly, and in diabetics in particular, for instance, this reflex can be slow and ineffective, resulting in light-headedness and even fainting if the patient stands up too quickly.