The potential and flux landscape, Lyapunov function and non-equilibrium thermodynamics for dynamic systems and networks with an application to signal-induced Ca²⁺ oscillation

In this review, we summarize our recent efforts in exploring the non-equilibrium potential and flux landscape for dynamical systems and networks. The driving force of non-equilibrium dynamics can be decomposed into the gradient of the non-equilibrium potential and the divergent free probability flux divided by the steady-state probability distribution. The potential landscape is linked to the probability distribution of the steady state. We found that the intrinsic potential landscape in the zero noise limit is a Lyapunov function. We have defined and quantified the entropy, energy and free energy of the non-equilibrium systems. These can be used for formulating the first law of non-equilibrium thermodynamics. The free energy of the non-equilibrium system is also a Lyapunov function. Therefore, we can use both the intrinsic potential landscape and the free energy to quantify the robustness and global stability of the system. The Lyapunov property provides the formulation for the second law of non-equilibrium thermodynamics. The non-zero probability flux breaks the detailed balance. The two driving forces from the gradient of intrinsic potential landscape and the probability flux are perpendicular to each other under the zero noise limit. We investigate the dynamics of a new biological example of signal-induced Ca²⁺ oscillation. We explored the underlying potential landscape which shows a Mexican hat shape attracting the system down to the oscillation ring and the flux which provides the driving force on the ring for coherent and stable oscillation. We explored how the landscape and flux topography change with respect to the system parameters and the relationship to the period of oscillations and how the non-equilibrium free energy changes with respect to different dynamic phases and phase transitions when the system parameters vary. These explain how the system becomes robust and stable under different conditions and can help guide the experiment.