Abstract
A commonly accepted mathematical model for the slow-wave electrical activity of the gastro-intestinal tract of humans and animals comprises a set of interconnected nonlinear oscillators. Using a van der Pol oscillator with third-power conductance characteristics as the unit oscillator a number of structures have been analysed using a matrix Krylov-Bogolioubov method linearisation. The mode analysis of one-dimensional chains and two-dimensional arrays has been reported. In this paper the method has been extended to consider a tubular structure which is relevant to modelling small-intestinal rhythms. It is shown that this structure is capable of producing stable single models, non-resonant double modes and degenerated modes. General expressions are obtained for anm×n structure and examples given of two special conditions of 3×4 (i.e. odd numbers of oscillators in a ring) and 4×3 cases. The analytical results obtained for these two cases have been vertified experimentally using an electronic implementation of coupled van der Pol oscillators. Results obtained using fifth-power non-linear oscillators are summarised.
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Literature
Alian, S. E. D. 1981. “Mode Analysis and Stability for Populations of Mutually Coupled Nonlinear Oscillators.” Ph.D. Thesis, University of Sheffield.
Davies, M. L. 1976. “Electronic Modelling of the Human Colon.” M.Sc. Thesis, Department of Control Engineering, University of Sheffield, England.
Endo, T. and S. Mori. 1976a. “Mode Analysis of a Multimode Ladder Oscillator.”IEEE Trans. CAS-23, 100–113.
— and —. 1976b. “Mode Analysis of Two-dimensional Low-pass Multimode Oscillator.”IEEE Trans. CAS-23, 517–530.
— and —. 1978 “Mode Analysis of a Ring of a Large Number of Mutually Coupled van der Pol Oscillators.”IEEE Trans. CAS-25, 7–18.
Gantmacher, F. R. 1960.The Theory of Matrices. Chelsea, New York.
Linkens, D. A. 1974. “Analytical Solution of Large Numbers of Mutually Coupled Nearsinusoidal Oscillators.”IEEE Trans. CAS-21, 294–300.
— 1977. “The Stability of Entrainment Conditions for RLC Coupled van der Pol Oscillators.”Bull. math. Biol. 39, 359–372.
— 1979. “Regions of Attraction in Coupled Non-linear Oscillators.”IEEE Trans. CAS-26, 663–666.
— and S. P. Datardina. 1977. “Frequency Entrainment of Coupled Hodgin-Huxley type Oscillators for Modelling Gastro-intestinal Electrical Activity.”IEEE Trans. BME-24, 362–365.
— and R. I. Kitney. 1982. “Mode Analysis of Non-linear Oscillators Intercoupled with Pure Time Delays.”Bull. math. Biol. 44, 57–74.
—, I. Taylor and H. L. Duthie. 1976. “Mathematical Modeling of the Colorectal Myoelectrical Activity in Humans.”IEEE. Trans. BME-23, 101–110.
Nelsen, T. S. and J. C. Becker. 1968. “Simulation of the Electrical and Mechanical Gradient of the Small Intestine.”Am. J. Physiol. 114, 749–757.
Patton, R. J. and D. A. Linkens. 1977. “Phenomenological Investigation of a Distributed Parameter Model for Co-ordinating the Mechanical Activity of the Mammalian Gut.” IFAC 2nd Symposium on Control of Ditributed Parameter Systems, Warwick.
Sarna, S. K., E. E. Daniel and Y. J. Kingma. 1971. “Simulation of Slow-wave Electrical Activity of Small Intestine.”Am. J. Physiol. 221. 166–175.
— and —. 1972. “Simulation of the Electrical Control Activity of the Stomach by an Array of Relaxation Oscillators.”Am. J. Dig. Dis. 17, 299–310.
Scott, A. C. 1970. “Distributed Multimode Oscillators of One and Two Spatial Dimensions.”IEEE Trans. CT-17, 55–60.
Utkin, G. M. 1959. “Oscillator Synchronisation at Combination Frequency Stabilisation.”Radiotek. elek.,4, 286–194.
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Alian, S.E.D., Linkens, D.A. Mode analysis of a tubular structure of coupled non-linear oscillators for digestive-tract modelling. Bltn Mathcal Biology 47, 71–110 (1985). https://doi.org/10.1007/BF02459647
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DOI: https://doi.org/10.1007/BF02459647