Analytical insights on theta-gamma coupled neural oscillators

Abstract : In this paper we study the dynamics of a quadratic integrate-and-fire neuron, spiking in the gamma (30-100 Hz) range, coupled to a delta/theta frequency (1-8 Hz) neural oscillator. Using analytical and semi-analytical methods we were able to derive characteristic spiking times for the system in two distinct regimes (depending on parameter values): one regime where the gamma neuron is intrinsically oscillating in the absence of theta input, and a second one in which gamma spiking is directly gated by theta input, i.e. windows of gamma activity alternate with silence periods depending on the underlying theta phase. In the former case we transform the equations such that the system becomes analogous to the Mathieu differential equation. By solving this equation we can compute numerically the time to the first gamma spike and then use singular perturbation theory to find successive spike times. On the other hand in the excitable condition we make direct use of singular perturbation theory to obtain an approximation of the time to first gamma spike, and then extend the result to calculate ensuing gamma spikes in a recursive fashion. We thereby give explicit formulas for the onset and offset of gamma spike burst during a theta cycle, and provide an estimation of the total number of spikes per theta cycle both for excitable and oscillator regimes.
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Submitted on : Thursday, July 18, 2013 - 1:51:06 PM
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Lorenzo Fontolan, Maciej Krupa, Alexandre Hyafil, Boris Gutkin. Analytical insights on theta-gamma coupled neural oscillators. Journal of Mathematical Neuroscience, BioMed Central, 2013, 3 (1), pp.16. ⟨10.1186/2190-8567-3-16⟩. ⟨hal-00846024⟩

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