In the primary visual cortex of monkeys, cats and ferrets, preferred orientations are arranged along the cortical surface in an orderly manner, forming orientation maps. Recently, 2-photon calcium imaging has revealed that preferred orientations are randomly distributed in the rodent visual cortex. This finding raised a question of why orientation maps exist in some species and not in others. To address this question, we examined the activity-dependent self-organization of orientation representation for different degrees of sparseness of intracortical excitatory connections in the visual cortex. We assumed that cortical cells send excitatory connections to neighboring cells with the connection probability p, whereas inhibitory connections are long-range and nonspecific. In addition, we normalized each synaptic weight to be proportional to 1/p, because without the normalization, small connection probability leads to relatively strong lateral inhibition and cortical cells tend to be silent. Computer simulations were conducted by changing the value of p. As a result, we found that for p<p*, arrangements of preferred orientations were random, whereas for p>p*, orderly orientation maps were formed. For any value of p, model cortical cells exhibited simple-cell-like receptive fields. Note that without the normalization of synaptic weight, receptive fields were disrupted. The value of p* depended on the "temperature" that is interpreted to indicate the magnitude of fluctuation in synaptic modification within our mathematical framework. Although we cannot determine the exact value of the temperature in real systems, a transition between an orderly map representation and a random representation occurred at p=p* for lower temperatures. Combining these theoretical results with species-dependent orientation representations, it is suggested that excitatory connections are sparse and each synaptic weight is large in the rodent visual cortex, whereas excitatory connections are relatively dense and each synaptic weight is small in the visual cortex of monkeys, cats and ferrets.