On the rank of certain classes of cut-sets and tie-sets of a graph

Abstract

distributed throughout the plane. This point is relevant to smallintestinal modeling, since only one mode is known to exist in this organ, and thus capacitive coupling appears unlikely as the predominant form of coupling. Resistively coupled oscillators have normally been considered to give a single mode only, and this has been shown to be true for small nonlinearity and small asymmetry. Such a condition appears to exist in the small intestine where waveforms are not highly asymmetrical. Under conditions of high asymmetry a stable antiphase mode has been found which persists down to low coupling values. This mode is equivalent to a ‘stable alternation’ condition noticed by Lewis [lo] in neural modeling. Lewis observed this mode in an electronic model based on Hodgkin-Huxley equations. In this type of electronic model the mode could only be excited for space: mark ratios (i.e., ratio of negative-going period of waveform to its positive-going period) of greater than 3: 1, and similarly in the van der Pol model b had to be greater than 1.0 indicating a high space: mark ratio. For b =0.8 the space: mark ratio was about 3: 1 and stable alternation did not occur. The nonexistence of the mode can be explained by considering the rising portion of the negative section of the waveform (see Fig. 2). For wide pulse widths this portion rises relatively rapidly and the injected current from the adjacent oscillator is sufficient to drive the oscillator into positive pulse generation. Thus the oscillators are forced into inphase entrainment under all conditions. Such a condition does not occur in capacitive coupling since the feedthrough is a differentiated pulse and has equal positive and negative effects. For narrow pulsewidths, however, the feedthrough for resistive coupling is small and occurs during the slowly rising negative period of an oscillator output. In this case insufficient energy is fed through to force the oscillator into its pulse generation phase. The mode stabilizes at 180” because of symmetrical coupling, in that any other phase shift would mean one oscillator having injected energy nearer to the threshold point of pulse generation. This behavior is related to threshold and refractoriness properties of van der Pol and Hodgkin-Huxley oscillations [ 1 I], and has relevance to large-intestinal studies where apparent frequency doubling effects occur. The concept of feedthrough between oscillators explains why antiphase modes are not obtained for inductive coupling under high asymmetry. During the quiescent period of an oscillation, 180” entrainment would mean the injection of the integral of a pulse from the adjacent oscillator. This would not give steady-state conditions and thus the system is always driven to inphase entrainment for which there is no energy transfer between oscillators. For fifth-power van der Pol oscillators the major effect seen from the attraction planes was the increase in likelihood of the double mode as nonlinearity was increased. Greater effects on the relative ease of exciting inphase or antiphase modes would probably have been seen if the size of the zero mode region was reduced, but the values used, however, were representative of conditions existing in the human large intestine [7].