Electrical alternans, a beat-to-beat alternation in the electrocardiogram or electrogram, is frequently seen during the first few minutes of acute myocardial ischemia, and is often immediately followed by malignant cardiac arrhythmias such as ventricular tachycardia and ventricular fibrillation. As ischemia progresses, higher-order periodic rhythms (e.g., period-4) can replace the period-2 alternans rhythm. This is also seen in modelling work on a two-dimensional (2-D) sheet of regionally ischemic ventricular muscle. In addition, in the experimental work, ventricular arrhythmias are overwhelmingly seen only after the higher-order rhythms arise. We investigate an ionic model of a strand of ischemic ventricular muscle, constructed as a 3-cm-long 1-D cable with a centrally located 1-cm-long segment exposed to an elevated extracellular potassium concentration ([K(+)](o)). As [K(+)](o) is raised in this "ischemic segment" to represent one major effect of ongoing ischemia, the sequence of rhythms {1:1-->2:2 (alternans)-->2:1} is seen. With further increase in [K(+)](o), one sees higher-order periodic 2N:M rhythms {2:1-->4:2-->4:1-->6:2-->6:1-->8:2-->8:1}. In a 2N:M cycle, only M of the 2N action potentials generated at the proximal end of the cable successfully traverse the ischemic segment, with the remaining ones being blocked within the ischemic segment. Finally, there is a transition to complete block {8:1-->2:0-->1:0} (in an n:0 rhythm, all action potentials die out within the ischemic segment). Changing the length of the ischemic segment results in different rhythms and transitions being seen: e.g., when the ischemic segment is 2 cm long, the period-6 rhythms are not seen; when it is 0.5 cm long, there is a 3:1 rhythm interposed between the 2:1 and 1:0 rhythms. We discuss the relevance of our results to the experimental observations on the higher-order rhythms that presage reentrant ischemic ventricular arrhythmias.