The phase response and state space of slow wave contractions in the small intestine
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NEW FINDINGS: What is the central question of this study? What are the dynamical rules governing interstitial cell of Cajal (ICC)-generated slow wave contractions in the small intestine, as reflected in their phase response curve and state space? What is the main finding and its importance? The phase response curve has a region of phase advance surrounding a phase delay peak. This pattern is important in generating a stable synchrony within the ICC network and is related to the state space of the ICC; in particular, the phase delay peak corresponds to the unstable equilibrium point that threads the ICC's limit cycle. Interstitial cells of Cajal (ICCs) generate electrical oscillations in the gut. Synchronization of the ICC population is required for generation of coherent electrical waves ('slow waves') that cause muscular contraction and thereby move gut content. The phase response curve (PRC) is an experimental measure of the dynamical rules governing a population of oscillators that determine their synchrony and gives an experimental window onto the state space of the oscillator, its dynamical landscape. We measured the PRC of slow wave contractions in the mouse small intestine by the novel combination of diameter mapping and single pulse electrical field stimulation. Phase change (τ) was measured as a function of old phase (ϕ) and distance from the stimulation electrode (d). Plots of τ(ϕ, d) showed an arrowhead-shaped region of phase advance enclosing at its base a phase delay peak. The phase change mirrored the perturbed pattern of contraction waves in response to a pulse. The (ϕ, d) plane is the surface of a displacement tube extending from the limit cycle through state space. To visualize the state space vector field on this tube, latent phase (ϕlat ) was calculated from τ. At the transition from advance to delay, isochrons made boomerang turns before tightening and winding around the phase delay peak corresponding to the unstable equilibrium point that threads the limit cycle. This isochron foliation had previously been observed in oscillator models such as the Fitzhugh-Nagumo but has not been demonstrated experimentally. The spatial extension of the PRC afforded by diameter mapping allows a better understanding of the dynamical properties of ICCs and how they synchronize as a population.
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