The role of N-methyl-D-aspartate (NMDA) receptors in wind-up: a mathematical model

We present a mathematical model for the phenomenon of wind-up (Mendell, 1966, Exper. Neur. 16,316-22) which occurs in many neurons. We concentrate on its occurrence in the substantia gelatinosa of the dorsal horns of the spinal cord, where it is connected with certain pathological and nonpathological pain states. The model is a development of the model by Britton & Skevington (1989, J. Theor. Biol. 137, 91-105) for Melzack & Wall's gate control theory of pain (1965, Science, New York, 150, 971-9; 1982, The Challenge of Pain, Penguin: Harmondsworth), modified to take account of more recent information. Its variables are the electric potentials of various cells in the midbrain and the spinal cord. Britton & Skevington's original model simulated many of the phenomena observed in acute pain in humans, but not the wind-up mechanism. This is not surprising, since this model did not include the N-methyl-D-aspartate (NMDA) receptors that are now recognized as being crucial to the phenomenon. Here we rectify this omission, and obtain good agreement between the model and experimental data on wind-up. The positive feedback that NMDA receptors exhibit is shown to be the essential feature in producing wind-up. As an independent test of the model we simulate a completely different experimental set-up, and obtain good qualitative agreement with data there. Finally, we present a prediction of the model that has yet to be tested experimentally.