Exploring the Interplay of Intrinsic Fluctuation and Complexity in Intracellular Calcium Dynamics

The concentration of intracellular calcium ion (Ca$^{2+}$) exhibits complex oscillations, including bursting and chaos, as observed experimentally. These dynamics are influenced by inherent fluctuations within cells, which serve as crucial determinants in cellular decision-making processes and fate determination. In this study, we systematically explore the interplay between intrinsic fluctuation and the complexity of intracellular cytosolic Ca$^{2+}$ dynamics using complexity measures such as permutation entropy (PE) and statistical complexity (SC). Using the chemical Langevin equation, we simulate the stochastic dynamics of cytosolic Ca$^{2+}$. Our findings reveal that PE and SC effectively characterize the diverse, dynamic states of cytosolic Ca$^{2+}$ and illustrate their interactions with intrinsic fluctuation. PE analysis elucidates that the chaotic state is more sensitive to intrinsic fluctuation than the other periodic states. Furthermore, we identify distinct states of cytosolic Ca$^{2+}$ occupying specific locations within the theoretical bounds of the complexity-entropy causality plane. These locations indicate varying complexity and information content as intrinsic fluctuation varies. When adjusting the permutation order, the SC for the different states exhibits peaks in an intermediate range of intrinsic fluctuation values. Additionally, we identify scale-free or self-similar patterns in this intermediate range, which are further corroborated by multifractal detrended fluctuation analysis. These high-complexity states likely correspond to optimal Ca$^{2+}$ dynamics with biological significance, revealing rich and complex dynamics shaped by the interplay of intrinsic fluctuation and complexity. Our investigation enhances our understanding of how intrinsic fluctuation modulates the complexity of intracellular Ca$^{2+}$ dynamics that play crucial roles in biological cells.