Heart arrhythmia is a pathological condition where the sequence of electrical impulses in the heart deviates from the normal rhythm. It is often associated with specific channelopathies in cardiac tissue, yet how precisely the changes in ionic channels affect the electrical activity of cardiac cells is still an open question. Even though sodium channel mutations that underlie cardiac syndromes like the Long-Q-T and the Brugada-syndrome are known to affect a number of channel parameters simultaneously, previous studies have predominantly focused on the persistent late component of the sodium current as the causal explanation for an increased risk of heart arrhythmias in these cardiac syndromes. A systematic analysis of the impact of other important sodium channel parameters is currently lacking. Here, we investigate the reduced ten-Tusscher-model for single human epicardium ventricle cells and use mathematical bifurcation analysis to predict the dependence of the cardiac action potential on sodium channel activation and inactivation time-constants and voltage dependence. We show that, specifically, shifts of the voltage dependence of activation and inactivation curve can lead to drastic changes in the action potential dynamics, inducing oscillations of the membrane potential as well as bistability. Our results not only demonstrate a new way to induce multiple co-existing states of excitability (biexcitability) but also emphasize the critical role of the voltage dependence of sodium channel activation and inactivation curves for the induction of heart-arrhythmias.