Asymptotics of conduction velocity restitution in models of electrical excitation in the heart

We extend a non-Tikhonov asymptotic embedding, proposed earlier, for calculation of conduction velocity restitution curves in ionic models of cardiac excitability. Conduction velocity restitution is the simplest non-trivial spatially extended problem in excitable media, and in the case of cardiac tissue it is an important tool for prediction of cardiac arrhythmias and fibrillation. An idealized conduction velocity restitution curve requires solving a non-linear eigenvalue problem with periodic boundary conditions, which in the cardiac case is very stiff and calls for the use of asymptotic methods. We compare asymptotics of restitution curves in four examples, two generic excitable media models, and two ionic cardiac models. The generic models include the classical FitzHugh–Nagumo model and its variation by Barkley. They are treated with standard singular perturbation techniques. The ionic models include a simplified “caricature” of Noble (J. Physiol. Lond. 160:317–352, 1962) model and Beeler and Reuter (J. Physiol. Lond. 268:177–210, 1977) model, which lead to non-Tikhonov problems where known asymptotic results do not apply. The Caricature Noble model is considered with particular care to demonstrate the well-posedness of the corresponding boundary-value problem. The developed method for calculation of conduction velocity restitution is then applied to the Beeler–Reuter model. We discuss new mathematical features appearing in cardiac ionic models and possible applications of the developed method

Asymptotics of conduction velocity restitution in models of electrical excitation in the heart.

Copyright © Springer 2011Journal ArticleThe original publication is available at www.springerlink.com - http://link.springer.com/article/10.1007/s11538-010-9523-6We extend a non-Tikhonov asymptotic embedding, proposed earlier, for calculation of conduction velocity restitution curves in ionic models of cardiac excitability. Conduction velocity restitution is the simplest non-trivial spatially extended problem in excitable media, and in the case of cardiac tissue it is an important tool for prediction of cardiac arrhythmias and fibrillation. An idealized conduction velocity restitution curve requires solving a non-linear eigenvalue problem with periodic boundary conditions, which in the cardiac case is very stiff and calls for the use of asymptotic methods. We compare asymptotics of restitution curves in four examples, two generic excitable media models, and two ionic cardiac models. The generic models include the classical FitzHugh-Nagumo model and its variation by Barkley. They are treated with standard singular perturbation techniques. The ionic models include a simplified "caricature" of Noble (J. Physiol. Lond. 160:317-352, 1962) model and Beeler and Reuter (J. Physiol. Lond. 268:177-210, 1977) model, which lead to non-Tikhonov problems where known asymptotic results do not apply. The Caricature Noble model is considered with particular care to demonstrate the well-posedness of the corresponding boundary-value problem. The developed method for calculation of conduction velocity restitution is then applied to the Beeler-Reuter model. We discuss new mathematical features appearing in cardiac ionic models and possible applications of the developed method

Facilitation of transmitter release at squid synapses

Facilitation is shown to decay as a compound exponential with two time constants (T1, T2) at both giant and non-giant synapses in squid steilate ganglia bathed in solutions having low extracellular calcium concentrations ([Ca++]o). Maximum values of facilitation (F~) were significantly larger, and T1 was significantly smaller in giant than non-giant synapses. Decreases in [Ca++]o or increases in [Mn++]o had variable effects on T1 and F1, whereas decreases in temperature increased T~ but had insignificant effects on/'1. The growth of facilitation during short trains of equal interval stimuli was adequately predicted by the linear summation model developed by Mallart and Martin (1967.J. Physiol. (Lond.). 193: 676-694) for frog neuromuscular junctions. This result suggests that the underlying mechanisms of facilitation are similar in squid and other synapses which release many transmitter quanta.This work was supported by National Science Foundation research grant GB-36949, National Research Council (Canada) and Grass Fellowships to Dr. Charlton, and a National Institutes of Health career award (NS-00070) to Dr. Bittner.Neuroscienc

Physiology of exercise in health and disease, with special reference to effort intolerance, training and thermoregulation in man

1. Davies, C.T.M., (1968) Limitdtions to the prediction of maximum oxygen intake from cardiac frequency measurements. J. Appl. Physiol. 24, 700 -706 || 2. Cotes, J.E., Davies, C.T.M., Edholm, G G., Healy, M.J.R., and Tanner, J.M., (1969). Factors relating to the aerobic capacity of 46 healthy British males and females, ages 18 -28 years. Proc. Roy. Soc. Lond. B 174, 91 -114 || 3. Davies, C.T.M., Tuxworth, W., and Young, J.M., (1970). Physiological effects of repeated exercise. Clin. Sci. 39, 247 -258 || 4. Di Prampero, P.E., Davies, C.T.M., Cerretelli, P., and Margaria R., (1970) An analysis of 02 debt contracted in submaximal exercise. J. Appl. Physiol. 29, 547 -551 || 5. Godfrey, S., and Davies, C.T.M., (1970) Estimates of arterial PCO2 and their effect on the calculated values of cardiac output and deadspace in exercise tests. Clin. Sci. 39, 529 -537 || 6. Davies, C.T.M., Kitchin, A.H., Knibbs, A.V., and Neilson, J.M., (1971) Computer Quantitation of ST segment response to graded exercise in untrained and trained normal subjects. Cardiovascular Research, 5, 201 -209 || 7. Godfrey, S., Davies, C.T.M., Wozniak, E., and Barnes, Carolyn A., (1971) Cardio -respiratory response to exercise in normal children. Clin. Sci. 40, 419 -431 || 8. Davies, C.T.M., (1972). The oxygen transporting system in relation to age. Clin. Sci. 42, 1 -13 || 9. Davies, C.T.M., Di Prampero, P.E., and Ceretelli, P., (1972) Kinetics of cardiac output and respiratory gas exchange during exercise and recovery. J. Appl. Physiol. 32, 618 -625 || 10. Edwards, R.H.T., Denison, D.M., Jones, G., Davies, C.T.M., and Campbell, E.J.M., (1972) Changes in mixed venous gas tensions at the start of exercise in man. J. Appl. Physiol. 32, 165 -169 || 11. Davies, C.T.M., and Barnes, Carolyn A., (1972) Plasma FFA in relation to maximum power output in man. Int. Z. Angew Physiol. 30, 247-257 || 12. Davies, C.T.M., Barnes Carolyn A., Fox, R.H., Ojikutu, R., Ola and Samueloff A.S., (1972). Ethnic difference in physical work capacity. J. Appl. Physiol. 33, 726 -732 || 13. Davies, C.T.M., (1973) Relationship of maximum aerobic power output to productivity and absenteeism of East African sugar cane workers. Brit. J. Ind. Med. 30, 146 -154 || 14. Davies, C.T.M., Chukweumeka, A.C., and Van Haaren, J.P.M., (1973) Iron deficiency anaemia: its effect on maximum aerobic power and responses to exercise in African males, aged 17 -40 years. Clin. Sci. 44, 555 -566 || 15. Cotes, J.E., Berry, G., Burkinshaw, L., Davies, C.T.M., Hall, A.M., Jones, P.R.M., and Knibbs, A.V., (1973). Cardiac frequency during submaximal exercise in young adults: relation to lean body mass, total body potassium and amount of leg muscle. Q.J.Expl.Physiol. 58, 239 -250 || 16. Davies, C.T.M., and Van Haaren, J.P.M., (1973) The effect of treatment the physiological responses to exercise in East African Industrial workers with iron deficiency anaemia. Brit. J. Ind. Med. 30, 335 -340. || 17. Sargeant, A.J., and Davies, C.T.M., (1973). Perceived exertion during rhythmic exercise involving different muscle masses. Human Ergology, 2, 3 -11 || 18. Davies, C.T.M., Sargeant, A.J., and Smith, B., (1974). The physiological responses to running downhill. Europ. J. Appl. Physiol. 32, 187 -194 || 19. Davies, C.T.M., and Sargeant, A.J., (1974) Physiological responses to standardised arm work. Ergonomics 17, 41 -49 || 20. Davies, C.T.M., and Sargeant, A.J., (1974). Exercise performance with one and two -legs breathing air and 45% oxygen. J. Appl. Physiol. 36, 142 -148 || 21. Davies, C.T.M., Few J.D., Foster, K.G., and Sargeant, A.J., (1974). Plasma catecholamine concentration during dynamic exercise involving different muscle groups. Eur. J. Appl. Physiol. 32, 195 -206 || 22. Davies, C.T.M., and Sargeant, A.J., (1974). Indirect determination of maximal aerobic power during work with one or two limbs. Europ. J. Appl. Physiol. 32, 207 -215 || 23. Davies, C.T.M., and Sargeant, A.J., (1974). Effects of hypoxic training on normoxic maximal aerobic power output. Europ. J. Appl. Physiol. 33, 227 -236 || 24. Davies, C.T.M., and Sargeant, A.J., (1975) Changes in physiological performance of the lower limb after fracture and subsequent rehabilitation Clin. Sci. & Mol. Med. 48, 107 -114 || 25. Davies, C.T.M., and Sargeant, A.J., (1975) physiological responses to 1 and 2 leg 377 -381 || 26. Davies, C.T.M., and Sargeant, A.J., (1975). exercise in patients following fracture J. Rehab. Med. 7, 45 -50. Effects of training on the work. J. Appl. Physiol. 38, Physiological responses to of the lower limb. Scand. || 27. Davies, C.T.M., Godfrey, S., Light, M., Sargeant, A.J., and Zeidifard, E.,. (1975) Cardiopulmonary responses to exercise in obese girls and young women. J. Appl. Physiol. 38, 373 -376 || 28. Davies, C.T.M., and Sargeant, A.J., (1975) Circadian variation in physiological responses to exercise on a stationary bicycle ergometer Brit. J. Ind. Med. 32, 110 -114 || 29. Collins, K.J. Brotherhood, J.R., Davies, C.T.M., Dore, Caroline, Hackett, J., Imms, F.J., Musgrove, J., Weiner, J.S., Amin, M.A., El Karim, M., Ismail, H11.M., Omer A.J.S., and Sukkar, M.Y., (1976). Physiological performance and work capacity of Sudanese can cutters with Schistosoma mansoni infection. Amer. J. Trop. Med. & Hyg. 25, 401,421 || 30. Nielsen, B., and Davies, C.T.M., (1976). Temperature regulation during exercise in water and air. Acta Physiol. Scand. 98, 500 -508 || 31. Davies, C.T.M., Brotherhood, J.R., Few,J.D., and Zeidifard, E., (1976) Effects of ß blockade and atropinisation on plasma catecholamine concentration during exercise. Eur. J. Appl. Physiol. 36, 49 -56 || 32. Davies, C.T.M., Brotherhood, J.R., and Zeidifard, E., (1976). Temperature regulation during severe exercise with some observations on the effects of skin wetting. J. Appl. Physiol. 41, 772 -776 || 33. Sargeant, A.J., Davies, C.T.M., Edwards, R.H.T., Maunder, C., and Young A., (1977). Functional and structural changes after disuse of human muscle. Clin. Sci. & Mol. Med. 52,337 -342 || 34. Sargeant, A.J., and Davies, C.T.M., (1977). Forces applied to cranks of a bicycle ergometer during one and two leg cycling. J. Appl. Physiol. 42, 514 -518 || 35. Sargeant, A.J., & Davies, C.T.M., (1977). The effect of disuse muscular atrophy on the forces generated in dynamic exercise. Clin. Sci. & Mol. Med. 53, 182 -188 || 36. Fohlin, L., Freyschuss, E., Bjarke, B., Davies, C.T.M., and Thoren, C., (1978). Function and Dimensions of the circulatory system in anorexia nervosa. Acta. Paed. Scand. 67, 11 -16 || 37. Davies, C.T.M., Von Dobeln, W. Fohlin, L., Freyschuss u., and Thoren, C., (1978). Total body potassium, fat free weight and maximal aerobic power in children with Anorexia Nervosa. Acta. Pediatr. Scand. 67, 229 -334 || 38. Davies, C.T.M., Brotherhood, J.P., and ZeidiFard, E., (1978). Effects of Atropine and (3-Blockade on Temperature Regulation and Performance during Prolonged Exercise. Europ. J. Appl. Physiol. 38, 225 -232 || 39. Zeidifard, E., and Davies, C.T.M., (1978). An Assessment of a N20 Rebreathing Method for the Estimation of Cardiac Output During Severe Exercise. Ergonomics, 21, 567 -572 || 40. Sargeant,A.J., Crawley, M.A., and Davies, C.T.M., (1979). Physiological Responses to Exercise in Myocardial Infarction Patients Following Residential Rehabilitation. Arch. Phys. Med. Rehabil. 60, 121 -125 || 41. Davies, C.T.M., (1979). The effects of different levels of heat production induced by diathermy and eccentric work on thermoregulation during exercise at a given skin temperature. Eur. J. Appl. Physiol. 40, 171 -180 || 42. Davies, C.T.M., and Thompson, M.W., (1979). Aerobic Performance of Female Marathon and Male Ultra- marathon Athletes. Eur. J. Appl. Physiol. 41, 233 -245 || 43. Davies, C.T.M., (1979). Thermoregulation during exercise in relation to sex and age. Eur. J. Appl. Physiol. 42, 71 -79 || 44. Davies, C.T.M., (1979). Influence of skin temperature on sweating and aerobic performance during severe work. J. Appl. Physiol. 47, 770 -777 || 45. Davies, C.T.M., and Sargeant, A.J., (1979). The effects of atropine and Practolol on the perception of exertion during treadmill exercise. Ergonomics 22, 1141 -1146 || 46. C.T.M., Fohlen L., and Temperature regulation in Anorexia nervosa patients during prolonged exercise. Acta. Med. Scand. 205, 257 -262 || 47. Davies, C.T.M., (1980) Influence of air flow and skin temperature on sweating during and following exercise. Ergonomics. 23, 559 -569 || 48. Davies, C.T.M., Fohlen L., and Thoren C., (1979) The effects of wind resistance on the forward motion of a runner. J. Appl. Physiol. 48, 702 -709 || 49. Davies, C.T.M., (1980) Metabolic cost of exercise and physical performance in children with some observations on external loading. Eur. J. Appl. 45, 95 -102 || 50. Davies, C.T.M., (1980) Effect of air resistance on the metabolic cost and performance of cycling. Eur. J. Appl. Physiol. 45, 245 -254