The Potential Landscape And Flux Theory Of The Neural Network Dynamics

Understanding the functions of human brains is always a grand goal of modern science. Despite the significant efforts, the global and physical understandings of the function and behavior of the brain are still challenging. In this thesis, we meet the chanllenge by constructing a non-equilibrium landscape and flux theory for general neural networks and establishing the connections between theoretical predications and experiment observations.Learning and memory processes were previously globally quantified through equilibrium energy for symmetrically connected neural networks. The energy basins of attractions store memories, and the memory retrieval dynamics is determined by the energy gradient. However, in the realistic neural networks,neurons are always asymmetrically connected, and oscillations associated with physiological rhythm regulations cannot emerge from symmetrical neural networks. We first developed a non-equilibrium landscape and flux theory for general neural networks. We uncovered the underlying potential landscape associated with the steady state probability distributions and the corresponding Lyapunov function for quantifying the global stability and function. We found the oscillatory activities are determined not only by the landscape gradient but also by the flux. We found the flux originated from the asymmetrical connections of the network. The neural oscillation landscape shows a closed-ring attractor topology. After being attracted by the landscape gradient down to the ring, the flux drives the coherent oscillations as the main driving force. We suggested the flux may provide the driving force for associations among memories. Furthermore, we explored a decision-making neural network with our landscape approach. We quantified the decision-making processes with optimal paths from the undecided attractor states to the decided attractor states on the landscape. We explored the the tradeoffs among speed, accuracy and energy cost in decision-making. Furthermore, we explored the mechanisms of changes of mind in decision-making. We also applied our landscape and flux theory to the basal ganglia neural circuit to explore the mechanism of this motor modulation network in Parkinson?s disease. We uncovered the underlying landscape as a Mexican hat-shape closed ring valley where abnormal oscillations emerge under dopamine depletion. Our quantified landscape and flux can directly reflect how changes in underlying neural network regulatory wirings and external inputs influence the dynamics of the system. We quantitatively explored the therapeutic mechanism of the deep brain stimulation(DBS) in terms of the effective reduction of the synchronized oscillations in the circuit. Our approach provides a general way to quantitatively explore neural networks and may help for uncovering more efficacious therapies for movement disorders.