  Fluctuations-driven complex oscillations are experimentally observed in
cellular systems such as hepatocytes, cardiac cells, neuronal cells, etc. These
systems are generally operating in regimes far from thermodynamic equilibrium.
To study nonequilibrium thermodynamic properties such as energy dissipation in
stochastic complex oscillations, we consider stochastic modeling of two
nonlinear biological oscillators, namely, the intracellular calcium (Ca$^{2+}$)
oscillation model and the Hindmarsh-Rose model for neuronal dynamics. These
models exhibit various types of complex oscillations like bursting and
quasi-periodic oscillations for various system parameter values. In this work,
we formulate open chemical reaction schemes for the two model systems driving
the systems far from thermodynamic equilibrium. We then analyze the
steady-state total entropy production rate (EPR) in the various types of
stochastic complex oscillations. Our results show higher values of steady-state
total EPR in stochastic complex oscillations than simple periodic oscillations.
Moreover, in the Hindmarsh-Rose neuronal model, we observe an order-to-disorder
transition from periodic (organized) bursts of spikes to chaotic (unorganized)
oscillations with distinct behaviors of steady-state total EPR. Our results
reveal that stochastic complex oscillations are produced at the cost of higher
energy consumption and that it requires a higher thermodynamic cost to maintain
the periodic bursts than chaotic oscillations. Our findings indicate that
complex cellular regulatory or signaling processes by Ca$^{2+}$ that help
perform complex tasks of the nervous system or rich information coding by
neurons involve a higher thermodynamic cost. The results deepen our
understanding of energy dissipation in nonlinear, nonequilibrium biological
systems with stochastic complex oscillatory dynamics.
