Recurrence-Based Synchronization Analysis of Weakly Coupled Bursting Neurons under External ELF Fields

We investigate the response characteristics of a two-dimensional neuron model exposed to an externally applied extremely low frequency (ELF) sinusoidal electric field and the synchronization of neurons weakly coupled with gap junction. We find, by numerical simulations, that neurons can exhibit different spiking patterns, which are well observed in the structure of the recurrence plot (RP). We further study the synchronization between weakly coupled neurons in chaotic regimes under the influence of a weak ELF electric field. In general, detecting the phases of chaotic spiky signals is not easy by using standard methods. Recurrence analysis provides a reliable tool for defining phases even for noncoherent regimes or spiky signals. Recurrence-based synchronization analysis reveals that, even in the range of weak coupling, phase synchronization of the coupled neurons occurs and, by adding an ELF electric field, this synchronization increases depending on the amplitude of the externally applied ELF electric field. We further suggest a novel measure for RP-based phase synchronization analysis, which better takes into account the probabilities of recurrences.Peer Reviewe

Classification of bursting patterns: A tale of two ducks

Bursting is one of the fundamental rhythms that excitable cells can generate either in response to incoming stimuli or intrinsically. It has been a topic of intense research in computational biology for several decades. The classification of bursting oscillations in excitable systems has been the subject of active research since the early 1980s and is still ongoing. As a by-product it establishes analytical and numerical foundations for studying complex temporal behaviors in multiple-timescale models of cellular activity. In this review, we first present the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable systems. We recall a complementary mathematical classification approach by Bertram et al., and then by Golubitsky et al., which together with the Rinzel-Izhikevich proposals provide the state-of-the-art foundations to these classifications. Beyond classical approaches, we review a recent bursting example that falls outside the previous classification systems. Generalizing this example leads us to propose an extended classification, which requires the analysis of both fast and slow subsystems of an underlying slow-fast model and allows the dissection of a larger class of bursters. Namely, we provide a general framework for bursting systems with both subthreshold and superthreshold oscillations. A new class of bursters with at least two slow variables is then added, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. Key to this mechanism are so-called canard or duck orbits, organizing the underpinning excitability structure. We describe the two main families of folded-node bursters, depending upon the phase (active/spiking or silent/non-spiking) of the bursting cycle during which folded-node dynamics occurs. We classify both families and give examples of minimal systems displaying these novel bursting patterns. Finally, we provide a biophysical example by reinterpreting a generic conductance-based episodic burster as a folded-node burster, showing that the associated framework can explain its subthreshold oscillations over a larger parameter region than the fast-subsystem approach

Synchronization in Neuronal Networks with Electrical and Chemical Coupling

Synchronized cortical activities in the central nervous systems of mammals are crucial for sensory perception, coordination, and locomotory function. The neuronal mechanisms that generate synchronous synaptic inputs in the neocortex are far from being fully understood. This thesis contributes toward an understanding of the emergence of synchronization in networks of bursting neurons as a highly nontrivial, combined effect of chemical and electrical connections. The first part of this thesis addresses the onset of synchronization in networks of bursting neurons coupled via both excitatory and inhibitory connections. We show that the addition of pairwise repulsive inhibition to excitatory networks of bursting neurons induces synchrony, in contrast to one’s expectations. Through stability analysis, we reveal the mechanism underlying this purely synergistic phenomenon and demonstrates that it originates from the transition between different types of bursting, caused by excitatory-inhibitory synaptic coupling. We also report a universal scaling law for the synchronization stability condition for large networks in terms of the number of excitatory and inhibitory inputs each neuron receives, regardless of the network size and topology. In the second part of this thesis, we show that similar effects are also observed in other models of bursting neurons, capable of switching from square-wave to plateau bursting. Finally, in the third part, we report a counterintuitive find that combined electrical and inhibitory coupling can synergistically induce robust synchronization in a range of parameters where electrical coupling alone promotes anti-phase spiking and inhibition induces anti-phase bursting. We reveal the underlying mechanism which uses a balance between hidden properties of electrical and inhibitory coupling to act together to synchronize neuronal bursting. We show that this balance is controlled by the duty cycle of the self-coupled system which governs the synchronized bursting rhythm. This work has potential implications for understanding the emergence of abnormal synchrony in epileptic brain networks. It suggests that promoting presumably desynchronizing inhibition in an attempt to prevent seizures can have a counterproductive effect and induce abnormal synchronous firing

Principles for Making Half-center Oscillators and Rules for Torus Bifurcation in Neuron Models

In this modelling work, we adopted geometric slow-fast dissection and parameter continuation approach to study the following three topics: 1. Principles for making the half-center oscillator, a ubiquitous building block for many rhythm-generating neural networks. 2. Causes of a novel electrical behavior of neurons, amplitude modulation, from the view of dynamical systems; 3. Explanation and predictions for two common types of chaotic dynamics in single neuron model. To make our work as general as possible, we used and built both exemplary biologically plausible Hodgkin-Huxley type neuron models and reduced phenomenological neuron models