Abstract
In this article, we describe and analyze the chaotic behavior of a conductance-based neuronal bursting model. This is a model with a reduced number of variables, yet it retains biophysical plausibility. Inspired by the activity of cold thermoreceptors, the model contains a persistent Sodium current, a Calcium-activated Potassium current and a hyperpolarization-activated current (Ih) that drive a slow subthreshold oscillation. Driven by this oscillation, a fast subsystem (fast Sodium and Potassium currents) fires action potentials in a periodic fashion. Depending on the parameters, this model can generate a variety of firing patterns that includes bursting, regular tonic and polymodal firing. Here we show that the transitions between different firing patterns are often accompanied by a range of chaotic firing, as suggested by an irregular, non-periodic firing pattern. To confirm this, we measure the maximum Lyapunov exponent of the voltage trajectories, and the Lyapunov exponent and Lempel-Ziv's complexity of the ISI time series. The four-variable slow system (without spiking) also generates chaotic behavior, and bifurcation analysis shows that this is often originated by period doubling cascades. Either with or without spikes, chaos is no longer generated when the Ih is removed from the system. As the model is biologically plausible with biophysically meaningful parameters, we propose it as a useful tool to understand chaotic dynamics in neurons.
Highlights
Chaotic behavior in neural systems has been observed for many years
The chaotic features disappear when the time constant is above 210 ms. In this Figure, we show the good correspondence between the maximal Lyapunov exponent (MLE) calculated from voltage traces and the Lyapunov exponent (LE) calculated from the interspike interval (ISI) sequences
In this article we have shown that a conductance-based model displays chaotic behavior in many biologically plausible regions of the explored parameter space
Summary
Chaotic behavior in neural systems has been observed for many years. Experimental observations of non-periodic responses range from molluscan neurons (Aihara et al, 1984) to rat sciatic nerves (Gu, 2013), including lobster CPGs (Abarbanel et al, 1996) and fish’s Mauthner cells (Faure et al, 2000) (for a review, see Korn and Faure, 2003). Most of the models reported to show chaotic activity present different types of bursting oscillations, that arise from the interaction between fast membrane voltage dynamics and a slower current or intracellular mechanism, such as Calcium dynamics (Chay and Rinzel, 1985; Canavier et al, 1990; Falcke et al, 2000). A simpler model, of only 3 variables, that produces burst firing and is known as the Hindmarsh-Rose model (Hindmarsh and Rose, 1984), presents chaotic dynamics for some parameter combinations Despite its simplicity, it shows a variety of aperiodic behaviors that are being actively studied by several groups (Holden and Fan, 1992; Abarbanel et al, 1996; Barrio and Shilnikov, 2011; Barrio et al, 2014), giving useful insight into the intrinsic mathematical mechanisms that drive bursting dynamics. A drawback of the Hindmarsh-Rose model is that some of its equations and parameters lack actual biophysical meaning, and its usefulness to interpret biological data is somewhat limited
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