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In this paper, we study an excitable and nonoscillating neuron on the basis of a technique of extending the concept of isochrons from oscillatory to excitable systems. The extended isochrons allow reduction of an excitable system described by possibly high dimensional differential equations to a simpler system. We analytically derive a one-dimensional model of an excitable neuron stimulated by instantaneous pulses with the technique of the extended isochrons and show its similarity to an isochronal map numerically obtained from the FitzHugh-Nagumo model. Response characteristics of the one-dimensional model to periodic impulsive stimulations are also analyzed numerically.

Abstract:We propose a statistical method to test whether time series data are quasiperiodic or periodic. A time series is defined to be quasiperiodic if its generating map is topologically conjugate to an irrational rotation. We present an algorithm to estimate the conjugacy map from time series data. We also show that a noisy irrational rotation is equivalent to a random walk model. Since there are general tests for a random walk, we can statistically evaluate quasiperiodicity of a time series by using the estimated conjugacy map. The proposed method is validated by asymmetric chaotic neural networks controlled to be quasiperiodic.

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