Potential landscape and flux framework of nonequilibrium networks: Robustness, dissipation, and coherence of biochemical oscillations

Abstract

We established a theoretical framework for studying nonequilibrium networks with two distinct natures essential for characterizing the global probabilistic dynamics: the underlying potential landscape and the corresponding curl flux. We applied the idea to a biochemical oscillation network and found that the underlying potential landscape for the oscillation limit cycle has a distinct closed ring valley (Mexican hat-like) shape when the fluctuations are small. This global landscape structure leads to attractions of the system to the ring valley. On the ring, we found that the nonequilibrium flux is the driving force for oscillations. Therefore, both structured landscape and flux are needed to guarantee a robust oscillating network. The barrier height separating the oscillation ring and other areas derived from the landscape topography is shown to be correlated with the escaping time from the limit cycle attractor and provides a quantitative measure of the robustness for the network. The landscape becomes shallower and the closed ring valley shape structure becomes weaker (lower barrier height) with larger fluctuations. We observe that the period and the amplitude of the oscillations are more dispersed and oscillations become less coherent when the fluctuations increase. We also found that the entropy production of the whole network, characterizing the dissipation costs from the combined effects of both landscapes and fluxes, decreases when the fluctuations decrease. Therefore, less dissipation leads to more robust networks. Our approach is quite general and applicable to other networks, dynamical systems, and biological evolution. It can help in designing robust networks. entropy production ͉ stability ͉ attractor ͉ landscape ͉ nongradient force ͉ cellular network

Nonequilibrium landscape theory of neural networks

“…We found the dynamics of the neural networks is controlled by both the gradient of the potential landscape and the probability flux. Although the gradient force attracts the system down to the ring, the flux is responsible for the coherent oscillations on the ring (12,13). The probability flux is closely related to the asymmetric part of the driving force or the connections.…”

“…The barrier heights are shown to be associated with the escape time from one state to another. Therefore, the topography of the landscape through barrier height can provide a good measurement of global stability and function (12,(14)(15)(16)(17). Such probability landscape can be constructed through solving the corresponding probabilistic equation.…”

“…Studying the oscillations may give rise to new insights toward how the various neural networks in the nervous system work. Our studies will provide a way to explore the global properties of various dynamical systems, in particular the oscillations (12,13). We will apply our method to study asymmetric Hopfield circuits and their oscillatory behaviors, and tie them with the biological functions.…”

“…We see the dynamical driving force F of neural networks can be decomposed into a gradient of a nonequilibrium potential [the nonequilibrium potential U here is naturally defined as UðuÞ ¼ − lnðP ss ðuÞÞ related to the steady-state probability distribution, analogous to equilibrium systems where the energy is related to the equilibrium distribution through the Boltzman law] and a divergent free flux, modulo to the inhomogeneity of the diffusion, which can be absorbed and redefined in the total driving force (12)(13)(14)(15). The divergent free force J ss has no source or sink to go to or come out.…”

“…The curl flux can lead to spiral motion, the underlying network dynamics deviating from the gradient path, and even generate coherent oscillations, which are not possible under pure gradient force. We show a potential landscape as Lyapunov function of monotonically decreasing along the dynamics in time still exists even with asymmetric connections (12)(13)(14) for the neural networks. Our approach is based on the statistical probabilistic description of nonequilibrium dynamical systems.…”