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As the interdisciplinary team of scientists state in the introduction of the paper, the ability of animals to coordinate their movements greatly relies upon two factors—the speed with which an animal can respond to stimuli and its ability to react appropriately to those stimuli. These factors are influenced by the length and density of nerves in the animal's body. Among larger animals there is a trade-off between the speed with which an animal can respond and their sensitivity to stimuli, and to gain a better understanding of this the scientists looked at the nerves of shrews and elephants.
Taken together, shrews and elephants represent the extremes of the animal world—elephants are the largest terrestrial mammals, and shrews are among the smallest. By experimentally stimulating the sciatic nerve in the hindlimb and figuring out the speed with which the signal caused a reaction in the medial gastrocnemius muscle, the scientists could determine how size affects an animal's ability to respond quickly and appropriately to its environment. When they compared the results, the scientists found that the speed with which the signals were transmitted through the nerves of both animals were essentially constant. Since the elephant is so much larger than a shrew, it took much longer for the nerve impulse to elicit a reaction from the muscle (100 milliseconds in the elephant versus 1 millisecond in the shrew). In other words, one of the reasons that an elephant can't run too fast is that its nervous system needs more time for information from the body to get to the brain, and if it ran ahead before these signals were processed it could trip itself up and become seriously hurt.
If the elephant had much larger neurons, it could transmit information faster. But the researchers also found that long neural projections called axons were only about twice as wide in elephants as in shrews. That's presumably because if neurons get too wide, fewer of them can project to a given part of the body, cutting down on how sensitive the nervous system can be to external stimuli.
Now, even though the scientists did not study Tyrannosaurus or any other dinosaur, these findings have implications for how the largest of dinosaurs would have moved. Just as it is with elephants, large body size among dinosaurs would have caused a trade-off between sensitivity to their environment and speed of nerve impulses which would have required them to be slower than their smaller relatives. As disappointing as it may be to some movie-goers, Tyrannosaurus may have been able to overtake a person on foot, but not keep up with a car at high speed. Given that Tyrannosaurus was after large prey, however, it would not have had to have been a speed demon in an absolute sense—it just had to be faster than its slowest prey.
More, H., Weber, D., Hutchinson, J., Aung, S., Collins, D., & Donelan, M. (2009). Scaling of sensorimotor control in terrestrial mammals Comparative Biochemistry and Physiology - Part A: Molecular & Integrative Physiology, 153 (2) DOI: 10.1016/j.cbpa.2009.04.510