Burst synchronization of electrically and chemically coupled map-based neurons

Abstract

Burst synchronization and burst dynamics of a system consisting of two map-based neurons coupled through electrical or chemical synapses are discussed. Some basic characteristic quantities are introduced to describe burst synchronization and burst dynamics of neurons. It is observed that excitatory coupling leads to in-phase burst synchronization but inhibitory coupling results in anti-phase one. By using the basic characteristics of burst dynamics, the effects of the intrinsic bursting properties and the coupling schemes on complex bursting behaviors are also presented for both inhibitory and excitatory couplings. The results are instructive to identify bursting behaviors through experimental data.

Introduction

Classical phenomena such as mutual synchronization, entrainment, and chaotic synchronization are now observed in many biological experiments, not only in vitro but also in vivo (see Ref. [1] for a review). The observation of synchronous neural activities in the central nervous system [2], [3] suggests that neural activity is a cooperative process of neurons and synchronization plays a vital role in mechanisms of information processing and information preface within different brain area [4]. It is also suggested that synchronization is the origin of neurological diseases such as epilepsy [5] and Parkinson’s disease [6].

There are two basic types of neural firing activities, bursting and tonic spiking, observed in experiments and model simulations. Bursting occurs when neural activity alternates between the quiescent state and fast repetitive spiking. Bursting is important since it is considered to enhance the reliability of communications between neurons by facilitating transmitter release [7]. Different types of bursting and mechanisms of their generation have been extensively studied [8], [9], [10], [11]. When coupled, neurons may exhibit different forms of synchronization, including spike synchronization [12], [13], burst synchronization [13], [14] when only the envelopes of the spikes become synchronized, and complete synchronization [15].

In studies on synchronization of neural networks, channel-based neuron models expressed by some ordinary differential equations (ODE) are most commonly used. Recently, map-based neuron models [10], [16] have also been introduced for their simplicity to reproduce characteristic behaviors of biological neurons. The coupling between neurons may occur via two different types of synapses, the electrical and chemical ones. In the former case, the coupling occurs through gap junctions and its strength depends linearly on the difference between the membrane potentials. In order to realize this synaptic connection the neurons must be very close to each other. In the chemical case, the synapse is mediated by neurotransmitters and the connection occurs between the dendrites and the axons, therefore, it allows long range connections, which generates complex network structures. Electrical synapses are found throughout the nervous system, yet are less common than chemical synapses.

As for a neural network, neural synchronization results from the interplay between the intrinsic properties of the individual neurons, the properties of the synaptic coupling, as well as the network topology. Each property may play an important role in shaping the emergent synchronous behaviors and it is important to determine the precise role each factor plays. This problem becomes more challenging when the neural network has a complex topology and is composed of both inhibitory and excitatory coupling. To gain insight into the rules governing pattern formation in complex networks of neurons, one should first investigate the rules underlying the emergence of cooperative rhythms in smaller network building blocks. In this article, we give an insight into the synchronization of the smallest network block — two coupled nonidentical neurons.

Relating the neural synchronization with the firing rhythms, spike synchronization (SS) and burst synchronization (BS) can be identified. Due to the importance of the bursting rhythm, BS, especially that of bursting neurons, has attracted more and more interests in recent years. By using a two-dimensional map describing the chaotic neuron, how synchronization in a group of chaotically bursting cells can lead to the onset of regular bursting was discussed in Ref. [17]. Ivanchenko et al. [18], [19] presented a transition to mutual chaotic phase synchronization occurring on the bursting time scale of ensembles of bursting map neurons, and found that the phase synchronization of burst neurons is not a singular function of coupling strength and can be break via spatiotemporal intermittency. In a system of diffusively coupled chaotic neurons, spatiotemporal chaos observed can be tamed into ordered BS patterns, and the transition from spatiotemporal chaos to BS patterns is studied in Refs. [20], [21]. Besides the BS of diffusively coupled chaotic neuron networks, the mechanism of BS of two coupled map-based neurons were also offered in Refs. [16], [22], [23]. In-phase and anti-phase BS of electrically coupled map-based neurons were found in Ref. [16] and its generation mechanism was presented in Ref. [23]. Bursting regimes in map-based neuron models coupled through reciprocal excitatory or inhibitory chemical synapses were discussed in Ref. [22], and parameter space was explored to determine the biological meaningful regimes and effects.

All of the results summarized above were based on the bursting neurons (chaotic or nonchaotic) as units of the networks. However, the effect of intrinsic properties of the individual neurons on synchronization are not fully understood, which is actually an important factor. In this study, we firstly define some characteristics of the burst dynamics of a single neuron, and then investigate the effects of the burst properties and the coupling schemes (electrical and chemical) on BS and the burst dynamics of the coupled neurons. In addition, coupled spiking neurons also can exhibit BS. We also give some results on BS and burst dynamics of coupled spiking neurons and compare them with those of bursting ones.

An outline of this article is given as follows. The map-based neuron model and coupled neuron system are introduced in Section 2. In Section 3, we define some basic characteristics of burst dynamics and give the meaning of them. The burst synchronization and burst dynamics of electrically and chemically coupled neurons are presented in Sections 4 Burst synchronization and burst dynamics of electrically coupled neurons, 5 Burst synchronization of coupled neurons with chemical coupling through fast threshold modulation. Finally, conclusion and discussion is given in Section 6.