Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People's Republic of China.
Published: November 2010
AI Article Synopsis
Article Abstract
Hybrid dynamical systems combine evolution equations with state transitions. When the evolution equations are discrete-time (also called map-based), the result is a hybrid discrete-time system. A class of biological neural network models that has recently received some attention falls within this category: map-based neuron models connected by means of fast threshold modulation (FTM). FTM is a connection scheme that aims to mimic the switching dynamics of a neuron subject to synaptic inputs. The dynamic equations of the neuron adopt different forms according to the state (either firing or not firing) and type (excitatory or inhibitory) of their presynaptic neighbours. Therefore, the mathematical model of one such network is a combination of discrete-time evolution equations with transitions between states, constituting a hybrid discrete-time (map-based) neural network. In this paper, we review previous work within the context of these models, exemplifying useful techniques to analyse them. Typical map-based neuron models are low-dimensional and amenable to phase-plane analysis. In bursting models, fast-slow decomposition can be used to reduce dimensionality further, so that the dynamics of a pair of connected neurons can be easily understood. We also discuss a model that includes electrical synapses in addition to chemical synapses with FTM. Furthermore, we describe how master stability functions can predict the stability of synchronized states in these networks. The main results are extended to larger map-based neural networks.
The control properties of discrete-time switched linear systems (SLSs) with switching signals generated by logical dynamical systems are studied using the semitensor product (STP) approach. With the algebraic state-space representation (ASSR), the linear modes and the logical generators are aggregated as a system with hybrid states, leading to the criteria of reachability, controllability, observability, and reconstructibility of the SLSs. Algorithms for checking these properties are given.
This article focuses on solving the secure control problem by developing a novel resilient hybrid learning scheme for discrete-time Markov jump cyber-physical systems with malicious attacks. Within the zero-sum game framework, the secure control problem is converted into solving a set of game coupled algebraic Riccati equations. However, it contains the coupling terms arising from the Markov jump parameters, which are difficult to solve.
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