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<p>Various bursting patterns of coupled <i>β</i>-cells exhibit complex dynamics, which are related to secretion amount of insulin and diabetes, and the blood glucose concentration maintains a normal level for long burst duration. In the present paper, dynamics underlying burst with different durations and spiking synchronizations at different coupling strengths (<i>g</i><sub><i>c</i></sub>) are obtained in two coupled cells, modulated by two slow gating variables <i>z</i> and <i>s</i>. Then, two-parameter (<i>s</i> and <i>z</i>) bifurcations underlying bursting patterns are acquired using fast-slow analysis, and slow dynamics around <i>s</i>-nullcline are obtained. A short burst terminating at a homoclinic bifurcation curve for isolated cell with zero <i>g</i><sub><i>c</i></sub> is chosen as control. For middle <i>g</i><sub><i>c</i></sub> of coupled cells, bursting exhibits extended ranges in <i>z</i> and<i> s</i>, and manifests very long burst duration, which can be divided into 3 segments by different bifurcations. Segment 1 behaves as several spikes with in-phase synchronization, originating from a pitchfork bifurcation. Segment 2 transits to run along stable anti-phase synchronization (AP) spiking, beginning and terminating respectively via a former and latter torus bifurcation. Segment 3 changes to run along unstable AP spiking which is very near the <i>s</i>-nullcline. The latter torus bifurcation appears after the homoclinic bifurcation to a large extent and <i>s</i>-nullcline exhibits extra slow dynamics, presenting two causes for the very long burst. For small and large <i>g</i><sub><i>c</i></sub>, the burst duration is longer than that of isolated cell and shorter than that of middle <i>g</i><sub><i>c</i></sub>, which is determined by the bifurcation/phase via which the burst terminates and the distance to the <i>s</i>-nullcline. Later bifurcation/phase after the homoclinic bifurcation or closer distance induce longer burst duration. These results present complex nonlinear dynamics of different burst durations of coupled <i>β</i>-cells, especially the long burst for middle coupling strength, which is helpful in modulating the bursting to maintain normal levels of glucose concentration.</p>

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Two-parameter bifurcations and nullcline underlying bursting with different durations of coupled β-cells with two slow variables

  • 期刊论文

  • 发表日期：

    2026年1月26日

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  • Yilan Jiang,
  • Huaguang Gu,
  • Hongtao Hua,
  • Juntian Li

摘要

Various bursting patterns of coupled β-cells exhibit complex dynamics, which are related to secretion amount of insulin and diabetes, and the blood glucose concentration maintains a normal level for long burst duration. In the present paper, dynamics underlying burst with different durations and spiking synchronizations at different coupling strengths (gc) are obtained in two coupled cells, modulated by two slow gating variables z and s. Then, two-parameter (s and z) bifurcations underlying bursting patterns are acquired using fast-slow analysis, and slow dynamics around s-nullcline are obtained. A short burst terminating at a homoclinic bifurcation curve for isolated cell with zero gc is chosen as control. For middle gc of coupled cells, bursting exhibits extended ranges in z and s, and manifests very long burst duration, which can be divided into 3 segments by different bifurcations. Segment 1 behaves as several spikes with in-phase synchronization, originating from a pitchfork bifurcation. Segment 2 transits to run along stable anti-phase synchronization (AP) spiking, beginning and terminating respectively via a former and latter torus bifurcation. Segment 3 changes to run along unstable AP spiking which is very near the s-nullcline. The latter torus bifurcation appears after the homoclinic bifurcation to a large extent and s-nullcline exhibits extra slow dynamics, presenting two causes for the very long burst. For small and large gc, the burst duration is longer than that of isolated cell and shorter than that of middle gc, which is determined by the bifurcation/phase via which the burst terminates and the distance to the s-nullcline. Later bifurcation/phase after the homoclinic bifurcation or closer distance induce longer burst duration. These results present complex nonlinear dynamics of different burst durations of coupled β-cells, especially the long burst for middle coupling strength, which is helpful in modulating the bursting to maintain normal levels of glucose concentration.