Nonlinear Dynamics of Neuronal Excitability, Oscillations, and Coincidence Detection

Abstract

We review some widely studied models and firing dynamics for neuronal systems, both at the single cell and network level, and dynamical systems techniques to study them. In particular, we focus on two topics in mathematical neuroscience that have attracted the attention of mathematicians for decades: single-cell excitability and bursting. We review the mathematical framework for three types of excitability and onset of repetitive firing behavior in single-neuron models and their relation with Hodgkin’s classification in 1948 of repetitive firing properties. We discuss the mathematical dissection of bursting oscillations using fast/slow analysis and demonstrate the approach using single-cell and mean-field network models. Finally, we illustrate the properties of Type III excitability in which case repetitive firing for constant or slow inputs is absent. Rather, firing is in response only to rapid enough changes in the stimulus. Our case study involves neuronal computations for sound localization for which neurons in the auditory brain stem perform extraordinarily precise coincidence detection with submillisecond temporal resolution.

Soma-axon coupling configurations that enhance neuronal coincidence detection

“…The value of 517 this ratio increases with forward coupling strength. Phasic dynamics are closely 518 associated with temporally-precise neural coincidence detection [34,35]. This result 519 indicates that phasic dynamics in models with stronger forward coupling are robust, in 520 the sense that this coupling configuration can allow neurons to maintain phasic 521 dynamics over a larger range of g N a and input levels.…”

“…Voltage-gated 27 currents are also sources of dynamic negative feedback that contribute to the 28 remarkable coincidence detection capabilities of these neurons. In MSO neurons, 29 activation of low-threshold potassium (KLT) current and inactivation of sodium current 30 are two identified sources of dynamic negative feedback [33][34][35][36]. In response to, say, a 31 pair of brief excitatory inputs, these feedback mechanisms will transiently raise the 32 spiking threshold after the first input, and thereby reduce the chance that the neuron 33 will spike in response to the second input unless the inputs arrive nearly synchronously 34 PLOS 2/30 (within a coincidence detection "time window").…”

“…In response to, say, a 31 pair of brief excitatory inputs, these feedback mechanisms will transiently raise the 32 spiking threshold after the first input, and thereby reduce the chance that the neuron 33 will spike in response to the second input unless the inputs arrive nearly synchronously 34 PLOS 2/30 (within a coincidence detection "time window"). 35 We investigate the extent to which a "structural" specialization -namely, the 36 coupling between the input region (soma and dendrite) and the output region (axon and 37 axon initial segment) -can further optimize coincidence detection sensitivity. To do 38 this, we develop a two-compartment neuron model as a minimal description of 39 73 Nonetheless, our method for systematically exploring the parameter space of coupling 74 configurations can be applied to study the relationships between structure, dynamics, 75 and computation in other neurons that are well-described by a two-compartment 76 idealization [24, 45-50].…”

Analysis of Neural Oscillations on Drosophila’s Subesophageal Ganglion Based on Approximate Entropy

“…One of them is approximate entropy (ApEn), a simple and straightforward nonlinear index, which can be used to measure the complexity of signals and quantify statistics [15][16][17][18]. In the past few years, ApEn has been extensively applied in the analysis of regularity and complexity of short-term physiological time series such as heart rate, blood pressure, and electroencephalogram signals [19].…”

“…Mathematical computing tools have been widely used in the analysis of neural signals [13][14][15][16][17]. One of them is approximate entropy (ApEn), a simple and straightforward nonlinear index, which can be used to measure the complexity of signals and quantify statistics [15][16][17][18].…”