The His-Purkinje fiber system is a network of one type of specialized conducting cardiac cells that carry an action potential in the heart. There are other cells which perform this function in different parts of the heart. The initiation time, shape, and duration of the action potential are distinctive for different parts of the heart, reflecting their different functions. These distinctions arise because the myocytes in each region of the heart have a characteristic set of channels and anatomy. You can completely dissociate the nervous system from the heart, and the heart will still pace itself. Think of what happens with heart transplants. This completely eliminates neuronal influence. See the following brief excerpt which explains one type of cells conduction method. Please note there is no reference to neurons.
The positive charge that enters cell A not only depolarizes cell A but also produces a flow of positive charge to cell B—intracellular current. This flow of positive charge discharges the membrane capacitance of cell B, thereby depolarizing cell B and releasing extracellular positive charges that had been associated with the membrane. The movement of this extracellular positive charge from around cell B toward the extracellular region around cell A constitutes the extracellular current. The flow of intracellular current from cell A to cell B and the flow of extracellular current from around cell B to around cell A are equal and opposite. It is the flow of this extracellular current in the heart that gives rise to an instantaneous electrical vector, which changes with time. Each point on an electrocardiogram (ECG) is the sum of the many such electrical vectors, generated by the many cells of the heart.
This has been extensively studied, so much so that cardiac electrophysiologists can pinpoint the location of a small group of cells that are misfiring by the nature of the impulse they are giving off. The annual median salary for an electrophysiologist in the US is $480,000. :-( There is a lot of grant money in this area. It's a big deal.
For in-depth study of electrophysiology of cardiac cells, please see Electrophysiology of Cardiac Cells or any other of a number of reliable sources.