I am not sure if you are referring to a mechanochemical cycle of "stepping" of a myosin molecule along an actin filament, or a cycle of contraction and relaxation of a muscle cell as a whole. These take place on completely different timescales and control the binding and unbinding of the two proteins in entirely different ways.
In the mechanochemical cycle, there are no proteins directly involved other than actin and myosin themselves, and the breaking and re-formation of contacts is controlled by the nucleotide occupying the ATPase site of myosin. ATP binding leads to a low affinity (weak interaction) state and ADP binding leads to a high affinity (strong interaction) state. ATP hydrolysis and nucleotide binding/unbinding "power" this cycle.
The cycles of contraction and relaxation of the cell occurs on a much longer timescale--each contraction involves many repetitions of the actin/myosin "stepping" cycle across many actin and myosin molecules, while relaxation involves slowing or stopping this cycle. This is where calcium and the troponin I/C complex come in--they are a "switch" that allows the stepping cycle to happen or not. So when the calcium is low and the troponin complex is bound, the actin/myosin interaction is continuously disengaged, whereas when calcium is high and the troponin complex releases, the cross-bridges form and release in a cyclical manner as ATP binds to myosin and is hydrolyzed.
As an analogy, think of a person walking across a room full of chairs. As the person is walking, his feet alternately make and break contact with the ground, and this is due to the coordinated movement of the legs. If the person sits down in a chair, then his legs stop propelling him entirely for a while and are still, and this is like when calcium is low in a muscle cell.
