This study focuses on the qualitative and quantitative characterization of chaotic systems with the use of a symbolic description. We consider two famous systems, Lorenz and Rössler models with their iconic attractors, and demonstrate that with adequately chosen symbolic partition, three measures of complexity, such as the Shannon source entropy, the Lempel–Ziv complexity, and the Markov transition matrix, work remarkably well for characterizing the degree of chaoticity and precise detecting stability windows in the parameter space. The second message of this study is to showcase the utility of symbolic dynamics with the introduction of a fidelity test for reservoir computing for simulating the properties of the chaos in both models’ replicas. The results of these measures are validated by the comparison approach based on one-dimensional return maps and the complexity measures.
The purpose of this paper is to serve as an instructive resource and a reference catalog for biologically plausible modeling with i) conductance-based models, coupled with ii) strength varying slow synapse models, culminating in iii) two canonical pair-wise rhythm-generating networks. We document the properties of basic network components: cell models and synaptic models, which are prerequisites for proper network assembly. Using the slow-fast decomposition we present a detailed analysis of the cellular dynamics including a discussion of the most relevant bifurcations. Several approaches to model synaptic coupling are also discussed, and a new logistic model of slow synapses is introduced. Finally, we describe and examine two types of bicellular rhythm-generating networks: i) half center oscillators ii) excitatory-inhibitory pairs and elucidate a key principle - the network hysteresis underlying the stable onset of emergent slow bursting in these neural building blocks. These two cell networks are a basis for more complicated neural circuits of rhythmogenesis and feature in our models of swim central pattern generators.
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