Analyzing Stability of Equilibrium Points in Neural Networks: A General Approach

Networks of coupled neural systems represent an important class of models in computational neuroscience. In some applications it is required that equilibrium points in these networks remain stable under parameter variations. Here we present a general methodology to yield explicit constraints on the coupling strengths to ensure the stability of the equilibrium point. Two models of coupled excitatory-inhibitory oscillators are used to illustrate the approach.Comment: 20 pages, 4 figure

Editorial [to] Analysis of nonlinear dynamics of neural networks

Fundação para a Ciência e a Tecnologia (FCT

Topological geometry analysis for complex dynamic systems based on adaptive control method

Several models had been proposed for dynamic systems, and different criteria had been analyzed for such models such as Hamiltonian, synchronization, Lyapunov expansion, and stability. The geometry criteria play a significant part in analyzing dynamic systems and some study articles analyze the geometry of such topics. The synchronization and the complex-network control with specified topology; meanwhile, the exact topology may be unknown. In the present paper, and by making use of the adaptive control method, a proposed control method is developed to determine the actual topology. The basic idea in the proposed method is to receive evolution of the network-node

Learning for System Identification of NDAE-modeled Power Systems

System identification through learning approaches is emerging as a promising strategy for understanding and simulating dynamical systems, which nevertheless faces considerable difficulty when confronted with power systems modeled by differential-algebraic equations (DAEs). This paper introduces a neural network (NN) framework for effectively learning and simulating solution trajectories of DAEs. The proposed framework leverages the synergy between Implicit Runge-Kutta (IRK) time-stepping schemes tailored for DAEs and NNs (including a differential NN (DNN)). The framework enforces an NN to cooperate with the algebraic equation of DAEs as hard constraints and is suitable for the identification of the ordinary differential equation (ODE)-modeled dynamic equation of DAEs using an existing penalty-based algorithm. Finally, the paper demonstrates the efficacy and precision of the proposed NN through the identification and simulation of solution trajectories for the considered DAE-modeled power system

Analysis of Nonlinear Dynamics of Neural Networks

[No abstract available]Publisher's Versio

Modelling and Contractivity of Neural-Synaptic Networks with Hebbian Learning

This paper is concerned with the modelling and analysis of two of the most commonly used recurrent neural network models (i.e., Hopfield neural network and firing-rate neural network) with dynamic recurrent connections undergoing Hebbian learning rules. To capture the synaptic sparsity of neural circuits we propose a low dimensional formulation. We then characterize certain key dynamical properties. First, we give biologically-inspired forward invariance results. Then, we give sufficient conditions for the non-Euclidean contractivity of the models. Our contraction analysis leads to stability and robustness of time-varying trajectories -- for networks with both excitatory and inhibitory synapses governed by both Hebbian and anti-Hebbian rules. For each model, we propose a contractivity test based upon biologically meaningful quantities, e.g., neural and synaptic decay rate, maximum in-degree, and the maximum synaptic strength. Then, we show that the models satisfy Dale's Principle. Finally, we illustrate the effectiveness of our results via a numerical example.Comment: 24 pages, 4 figure

Singular Perturbation via Contraction Theory

In this paper, we provide a novel contraction-theoretic approach to analyze two-time scale systems. In our proposed framework, systems enjoy several robustness properties, which can lead to a more complete characterization of their behaviors. Key assumptions are the contractivity of the fast sub-system and of the reduced model, combined with an explicit upper bound on the time-scale parameter. For two-time scale systems subject to disturbances, we show that the distance between solutions of the nominal system and solutions of its reduced model is uniformly upper bounded by a function of contraction rates, Lipschitz constants, the time-scale parameter, and the time variability of the disturbances. We also show local contractivity of the two-time scale system and give sufficient conditions for global contractivity. We then consider two special cases: for autonomous nonlinear systems we obtain sharper bounds than our general results and for linear time-invariant systems we present novel bounds based upon log norms and induced norms. Finally, we apply our theory to two application areas -- online feedback optimization and Stackelberg games -- and obtain new individual tracking error bounds showing that solutions converge to their (time-varying) optimizer and computing overall contraction rates.Comment: This paper has been submitted to IEEE Transactions on Automatic Contro