**Core Concept**
The automatic contractility of the heart is controlled by the autonomic nervous system, specifically the sympathetic and parasympathetic branches, through various mechanisms including the modulation of calcium influx into cardiac myocytes. However, the single most important factor in control of automatic contractility of the heart is the **sodium-calcium exchanger (NCX)**, which plays a crucial role in regulating the intracellular concentration of calcium ions.

**Why the Correct Answer is Right**
The NCX is a critical component in the regulation of cardiac contractility by controlling the amount of calcium ions available for contraction. The NCX uses the energy from the sodium gradient to remove excess calcium ions from the cytosol, thereby reducing the likelihood of calcium overload and subsequent cardiac arrhythmias. This process is essential for maintaining the normal functioning of the heart, especially during periods of increased sympathetic activity. The NCX is also influenced by the autonomic nervous system, with sympathetic stimulation increasing the activity of the NCX and parasympathetic stimulation decreasing it.

**Why Each Wrong Option is Incorrect**
* **Option A:** This option is not relevant to the control of automatic contractility of the heart.
* **Option B:** Although the autonomic nervous system plays a role in regulating cardiac contractility, it is not the single most important factor.
* **Option D:** This option is not a known mechanism for controlling cardiac contractility.

**Clinical Pearl / High-Yield Fact**
The sodium-calcium exchanger is a critical regulator of cardiac contractility, and its dysfunction has been implicated in various cardiac arrhythmias and diseases.

**Correct Answer:** C.