  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.
