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Mode-locking behavior of Izhikevich neurons under periodic external forcing

Many neurons in the auditory system of the brain must encode amplitude variations of a periodic signal. These neurons under periodic stimulation display rich dynamical states including mode-locking and chaotic responses [1]. Periodic stimuli such as sinusoidal waves and amplitude modulated (AM) sounds can lead to various forms of n:m mode-locked states, similar to the mode-locking phenomenon in a LASER resonance cavity. Obtaining Arnold tongues provides useful insight into the organization of mode-locking behavior of neurons under periodic forcing. In this study we obtained the regions of existence of various mode-locked states on the frequency-amplitude plane, which are called Arnold tongues, for Izhikevich neurons (see Figure 1). This study is based on the model for neurons by Izhikevich (2003), which is a reduced model of a Hodgkin-Huxley neuron [2]. This model is much simpler in terms of the dimension of the coupled non-linear differential equations compared to other existing models, but excellent for generating the complex spiking patterns observed in real neurons [3]. Hence we can describe the construction of harmonic and sub-harmonic responses in the early processing stages of the auditory system, such as the auditory nerve and cochlear nucleus.

Figure 1
figure 1

The Arnold tongues diagram for a Class 1 Izhikevich neuron driven by an external sinusoidal forcing. This plot represents the mode-locked regions as a function of the amplitude and frequency of this sinusoidal forcing. Each colored region represents a different phase-locked regime in terms of an integer ratio. Here, an n:m ratio means the neuron produces n action potentials in response to every m cycles of the stimulus. For example, for stimulus amplitudes and frequencies corresponding to the yellow region, the neuron exhibits 3:1 mode-locking.


Different mode-locked regions that are shown in the Arnold tongues diagram are predictors of mode-locking of auditory system neurons to sound, which in turn predict the formation of harmonics and sub-harmonics of the sound in the brain.


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This work was supported by University of Connecticut College of Liberal Arts and Sciences.

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Correspondence to AmirAli Farokhniaee.

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver ( applies to the data made available in this article, unless otherwise stated.

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Farokhniaee, A., Large, E.W. Mode-locking behavior of Izhikevich neurons under periodic external forcing. BMC Neurosci 16 (Suppl 1), P140 (2015).

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