Skip to main content
SpringerLink
Log in

On the mathematical modelling of pain

  • Original Articles
  • Published:
Neurochemical Research Aims and scope Submit manuscript

Abstract

In this review a case is presented for the use of mathematical modelling in the study of pain. The philosophy of mathematical modelling is outlined and a recommendation is made for the use of modern nonlinear techniques and computational neuroscience in the modelling of pain. Classic and more recent examples of modelling in neurobiology in general and pain in particular, at three different levels—molecular, cellular and neural networks—are described and evaluated. Directions for further progress are indicated, particularly in plasticity and in modelling brain mechanisms. Major advantages of mathematical modelling are that it can handle extremely complex theories and it is non-invasive, and so is particularly valuable in the investigation of chronic pain.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Verrill, P. 1990. Does the gate control theory of pain supplant all others? Br. J. Hosp. Med. 43:325.

    PubMed  CAS  Google Scholar 

  2. Arbib, M. A. 1995. Handbook of Brain Theory and Neural Networks. MIT Press, Cambridge Mass.

    Google Scholar 

  3. Bower, J. M., and Beeman, D. 1995. The Book of Genesis. Springer, New York.

    Google Scholar 

  4. Eeckman, F. H., and Bower, J. M. (eds.) 1993. Computation and Neural Systems. Kluwer, Boston, Mass.

    Google Scholar 

  5. Koch, C. and Segev, I. (eds.) 1989. Methods in Neuronal Modelling. MIT Press, Cambridge Mass.

    Google Scholar 

  6. Koch, C., and Poggio, T. 1987. Biophysics of computation; neurons, synapses and membranes. In Edelman, G. M., Gall, W. E. and Cowan, W. M. (eds.) 1987. Synaptic Function, Wiley, New York. pp 637–698

    Google Scholar 

  7. Segev, I. 1992. Single neuron models: oversimple, complex and reduced. Trends in Neurosci. 15:414–421.

    Article  CAS  Google Scholar 

  8. Dubner, R., and Beitel, R. E. 1976. Neural correlates of escape behaviour in rhesus monkeys to noxious heat applied to the face. In Advances in Pain Research and Therapy Vol. 1, eds. J. J. Bonica and D. Albe-Fessard, Raven Press, New York pp 155–160.

    Google Scholar 

  9. Price, D. D., Barrell, J. J., and Gracely, R. H. 1980. A psychophysical analysis of experiential factors that selectively influence the affective dimension of pain. Pain 8:137–179.

    Article  PubMed  CAS  Google Scholar 

  10. Price, D. D., McGrath, P. A., Rafii, A., and Buckingham, B. 1983. The validation of visual analogue scales as ratio scale measures for chronic and experimental pain. Pain 17:45–56.

    Article  PubMed  CAS  Google Scholar 

  11. Price, D. D., and Harkins, S. W. 1987. The combined use of visual analogue scales and experimental pain in proving standardised assessment of clinical pain. Clin. J. Pain 3:1–8.

    Google Scholar 

  12. Hodgkin, A. L., and Huxley A. F. 1952. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117:500–544.

    PubMed  CAS  Google Scholar 

  13. Rinzel, J. 1990. Electrical excitability of cells, theory and experiment: review of the Hodgkin-Huxley foundation. Bull. Math. Biol. 52:5–23.

    Article  Google Scholar 

  14. Nelson, M., and Rinzel J. 1995. The Hodgkin-Huxley model. Bower J. and Beeman, D. (eds) The Book of Genesis. Springer, New York, Chapter 4, pp 29–52.

    Google Scholar 

  15. Davies, S. N., and Lodge, D. 1987. Evidence for the involvement of N-methyl-D-aspartate receptors in ‘wind-up’ of class 2 neurones in the dorsal horn of the rat. Brain Res. 424:402–406.

    Article  PubMed  CAS  Google Scholar 

  16. Dickenson, A. H., and Sullivan, A. F. 1987. Evidence for a role of the NMDA receptors in the frequency-dependent potentiation of deep dorsal horn neurons following C-fibre stimulation. Neuropharmacol. 26:1235–1238.

    Article  CAS  Google Scholar 

  17. Thompson, S. W. N., and Woolf, C. J. 1991. Primary afferentevoked prolonged potentials in the spinal cord and their central summation: role of the NMDA receptor. In Proc of the VIth World Congress on Pain, Adelaide. Eds M. R. Bond, J. E. Charlton and C. J. Woolf, Elsevier, Amsterdam.

    Google Scholar 

  18. Woolf, C. J., and Thompson, S. W. N. 1991. The induction and maintenance of central sensitisation is dependent on N-methyl-D-aspartic acid receptor activation; implications for the treatment of post-injury pain hypersensitivity states. Pain, 44:293–300.

    Article  PubMed  CAS  Google Scholar 

  19. Mendell, L. M. 1996. Physiological properties of unmyelinated fibre projections to the spinal cord. Exp. Neurol., 16:316–32.

    Article  Google Scholar 

  20. Wilcox, G. L. 1991. Excitatory neurotransmitters and pain. In Proc. of the VIth World Congress on Pain, Adelnide. Eds M. R. Bond, J. E. Charlton and C. J. Woolf, Elsevier, Amsterdam, 97–117.

    Google Scholar 

  21. Randall, A. D., Schofield, J. G., and Collingridge, G. L. 1990. Whole-cell, patch-clamp recordings of an NMDA receptor-mediated synaptic current in rat hippocampal slices. Neurosci. Lett. 114:191–196.

    Article  PubMed  CAS  Google Scholar 

  22. Watkins, J. C., and Collingridge, G. L. 1989. The NMDA Receptor, Oxford University Press, Oxford.

    Google Scholar 

  23. Zador, A., Koch, C., and Brown, T. H. 1990. Biophysical model of a Hebbian receptor channel kinetic behaviour. Proc. Nat. Acad. Sci. USA, 10:6718–6722.

    Article  Google Scholar 

  24. Ben-Yoseph, O., Bachelard, H. S., Bader-Goffer R., Dolin, S. J., and Morris, P. G. 1990. Effects of N-Methyl-D Aspartate on Ca2+, and the energy state in the by19F and31P-Nuclear Magnetic Resonance Spectroscopy. Journal of Neurochemistry, 55(4):1446–1449.

    Article  PubMed  CAS  Google Scholar 

  25. McCulloch, W. S., and Pitts, W. 1943. A logical calculus of ideas immanent in nervous activity. Bull. Math. Biophys. 5:115–133.

    Article  Google Scholar 

  26. Traub, R. D., Wong, R. K. F., Miles, R., and Michelson, H. 1991. A model of a CA3 hippocampal neurone incorporating voltage clamp data on intrinsic conductances. J. Neurophysiol. 66:635–650.

    PubMed  CAS  Google Scholar 

  27. Bachelard, H. S., Morris, P., Taylor, A., and Thatcher, N. 1995. High field MRS studies in brain slices. Magnetic Resonance Imaging 13(8):1223–1226.

    Article  PubMed  CAS  Google Scholar 

  28. Fors, U. G. H., Ahlquist, M., Skagerwall, R., Edwall, L., and Haegerstam, G. 1984. Relation between intradental nerve activity and estimated pain in man—a mathematical analysis. Pain 18: 397–408.

    Article  PubMed  CAS  Google Scholar 

  29. Fors, U., Ahlquist, M., Edwall, L., and Haegerstam, G. 1986. An evaluation of the usefulness of a mathematical model analyzing the relation between intradental nerve impulse activity and perceived pain in man. International Journal of Biomedical Computing 19:261–277.

    Article  PubMed  CAS  Google Scholar 

  30. Fors, U. G. H., Edwall, L. G. A., Haegerstam, G. A. T. 1988. The ability of a mathematical model to evaluate the effects of two pain modulating procedures on pulpal pain in man. Pain 33(2):253–264.

    Article  PubMed  CAS  Google Scholar 

  31. Britton, N. F., and Skevington, S. M. 1989. A mathematical model of the Gate Control theory of pain. Journal of Theoretical Biology 137:91–105.

    Article  PubMed  CAS  Google Scholar 

  32. Britton, N. F., Chaplain, M. A. J., and Skevington, S. M. 1996. The role of NMDA receptors in painful wind-up: mathematical model. IMA Journal of Maths Appl Med Biol. In press.

  33. Melzack, R., and Wall, P. D. 1965. Pain mechanisms: a new theory. Science, 150:971–979.

    Article  PubMed  CAS  Google Scholar 

  34. Melzack, R., and Wall, P. D. 1982. The Challenge of Pain, Penguin, Harmondsworth, Middlesex, UK.

    Google Scholar 

  35. Nathan, P. W., and Rudge, P. 1974. Testing the gate control theory of pain in man. Journal of Neurology, Neurosurgery and Psychiatry 37:1366–1372.

    Article  CAS  Google Scholar 

  36. McQuay, H., Carroll, D., and Moore, R. A. 1988. Post-operative orthopaedic pain—the effect of opiate pre-medication and local anaesthetic blocks. Pain 33(3):79–87.

    Article  Google Scholar 

  37. Bach S., Noreng, M. F., and Tjellden, N. U. 1988. Phantom limb pain in amputees durin their first 12 months following limb amputation after preoperative lumbar epidural blockade. Pain 33: 297–301.

    Article  PubMed  CAS  Google Scholar 

  38. Humphries, S. A., Johnson, M. H., and Long, N. R. 1993. An investigation of the Gate Control theory of pain by ramping-off the experimental pain stimulus of potassium iontophoresis. Abstracts of the 7th World Congress on Pain, Paris, Abs. 1261 International Association for the Study of Pain.

  39. Ermentrout, G. B., and Cowan, J. D. 1979. A mathematical theory of visual hallucination patterns. Biol. Cybemetics, 34:137–150.

    Article  CAS  Google Scholar 

  40. Hillman, P., and Wall, P. D. 1969. Inhibitory and excitatory factors influencing the receptive fields of lamina 5 spinal cord cells. Exp. Brain Res. 9:284–306.

    PubMed  CAS  Google Scholar 

  41. Price, D. D. 1988. Psychological and Neural Mechanisms of Pain, Raven Press, New York.

    Google Scholar 

  42. Bachelard, H. S., and Badar-Goffer, R. 1993. NMR spectroscopy in neurochemistry. J. Neurochemistry, 6(12):412–429.

    Google Scholar 

  43. Bachelard, H. S. 1988. General neurochemistry. Current Opinion in Neurology and Neurosurgery, 1:1091–1095.

    Google Scholar 

  44. Hsieh, J. C., Belfrage, M., Stone-Elander, S., Hansson, P., and Ingvar, M. 1995. Central representation of chronic ongoing neuropathic pain studied by positron emission tomography, Pain 63: 225–236.

    Article  PubMed  CAS  Google Scholar 

  45. Wilson, M. A., and Bower, J. M. 1989. The simulation of large scale neural networks. In Koch, C. and Segev, I. (eds) Methods in Neuronal Modelling. MIT Press Cambridge Mass. Chapter 9 pp 291–333.

    Google Scholar 

  46. Hebb, D. O. 1949. The Organization of Behaviour, Wiley New York.

    Google Scholar 

  47. Lynch, G. 1986. Synapses, circuts and the beginnings of memory. Bradford Books, M.I.T. Press, Cambridge Mass.

    Google Scholar 

  48. Stent, G. S. 1973. A physiological mechanism for Hebb's postulate of learning. Proc. Nat. Acad. Sci. 70:997–1001.

    Article  PubMed  CAS  Google Scholar 

  49. Marr, D. 1969. A theory of cerebellar cortex. J.. Physiol. 202:437–470.

    PubMed  CAS  Google Scholar 

  50. Marr, D. 1971. Simple memory: a theory for archicortex. Phil. Trans. Roy. Soc. B 262:23–81.

    CAS  Google Scholar 

  51. Woolf, C. J., and Thompson, S. W. N. 1991: Slow potentials, receptive field plasticity and pain. In Information Processing in the Somatosensory System. Eds O. Franzen and J. Westman, Macmillan, Chapter 32, 427-437.

  52. Campbell, J. N., and Meyer, R. A. 1991. Plasticity of the neural events related to pain. In Information Processing in the Somatosensory System. Eds O. Franzen and J. Westman, Macmillan, Chapter 33, 439–451.

  53. Sternbach, R. A. 1983. Ethical considerations in pain research in man. Pages 259–266,in R. Melzack, (ed). Pain Measurement and Assessment, Raven Press, New York.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Bath, Claverton Down, BA2 7AY, Bath, Avon, UK

    Nicholas F. Britton

  2. School of Social Sciences, University of Bath, Claverton Down, BA2 7AY, Bath, Avon, UK

    Suzanne M. Skevington

Authors
  1. Nicholas F. Britton
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suzanne M. Skevington
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Special issue dedicated to Dr. Herman Bachelard

Rights and permissions

Reprints and permissions

About this article

Cite this article

Britton, N.F., Skevington, S.M. On the mathematical modelling of pain. Neurochem Res 21, 1133–1140 (1996). https://doi.org/10.1007/BF02532424

Download citation

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02532424

Key Words

Search

Navigation