Synchronous firing patterns of neuronal population with excitatory and inhibitory connections

Abstract

A stochastic model of neuronal population with excitatory and inhibitory connections is proposed, where excitatory synaptic dynamics is considered. Oscillatory synchronized firing patterns of a neuronal population by means of firing density are investigated. Numerical simulations using Fokker–Planck equation show that slow inhibitory connection contributes to oscillatory synchronized firing of the neuronal population, and synchronous activity is enhanced due to inhibitory connection. The effect of time delay on the oscillatory synchronized firing in the neuronal population using phase mode is explored. Numerical simulation indicates that short synaptic transmission delay can suppress oscillatory synchronized firing, but this suppression is instable.

Introduction

Oscillatory synchronized firing is a widespread phenomenon occurring in different parts of the mammalian cerebral cortex, e.g., in the visual cortex [1], olfactory bulb [2], the hippocampus [3], [4], and the auditory neocortex [5]. For instance, experimental evidences have shown that neurons in the cat visual cortex responding to two independent images of a bar on a serene fired asynchronously when two short bars are moving in different directions, but these neurons are fired synchronously when the same bars are moving together [6]. It seems likely that oscillatory synchronized firing is an essential mechanism for the binding of visual information [7]. The visual-binding problem can be stated as follows: how are the different attributes of an object brought together in a unified representation provided that its various features, e.g., edges, color, movement, shape, depth, and so on. One proposed solution is based on the idea that visual objects are coded by neuronal assemblies that fire synchronously [8], [9]. However, it is unknown whether the oscillations are caused by the properties of single neurons or by intra-cortical network interactions. As for the theory that the oscillation is caused by the properties of single neurons, it is suggested that chattering cells in visual cortex show periodic bursts of gamma frequency, which might be related to the generation of oscillatory responses [10]. On the other hand, many physiological evidences support the theory that the oscillations are generated by intra-cortical network interactions [7], [11]. In particular, in neocortical and hippocampal networks, recent experimental studies have suggested that GABA-ergic interneurons play an important role in the emergence of various types of synchronous oscillatory patterns of neuronal activity in the central nervous system [12], [13].

On the basis of oscillatory synchronized firing of neurons, it is justified that a neuron is modeled as a limit cycle oscillator, and a neuronal network is often modeled as a network of globally coupled oscillators. Moreover, the background noise in the nervous system is ubiquitous, e.g., ion channels in cell membranes randomly open and close in response to voltage and chemical changes in a cell's environment, the release of neurotransmitter is probabilistic, and so on. Therefore, the theory of stochastic phase resetting is often used in investigating the dynamical behaviors of the nervous systems [14], [15]. Winfree first proposed to approximate the dynamics of a population of limit cycle oscillators by means of the dynamics of a population phase oscillators [16]. The dynamics of a population of oscillators interacting uniformly via mean-field coupling were investigated by Kuromoto [17]. Since then, many theoretical studies and numerical simulations have been carried out on the phase model of neuronal oscillators [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. Synchronization in a population of neuronal oscillators has been observed using phase model [18], [20], [21], [22]. However, neocortical and hippocampal networks in vivo are composed of a mixture of excitatory and inhibitory neurons. Taking into account different types of neurons, which means modeling two different subpopulations of coupled neuronal oscillators. As yet, a theory investigating the dynamical properties of a neuronal population with excitatory and inhibitory connections is still lacking, although synchronizations in two interacting populations of heterogeneous neuronal oscillators have been investigated using the phase model, in which the coupling strengths were chosen as fixed constant [28], [29]. For real biological neuronal networks, synaptic coupling strength changes in an activity-dependent manner, and is modified by presynaptic and postsynaptic activities [30]. It is clear that we cannot understand neural coding and neural information processing without taking synaptic dynamics into account. In order to explore the mechanism for oscillatory synchronized firings in the neuronal network, we study the neuronal networks that are composed of excitatory neurons and inhibitory neurons, in which synaptic dynamics is taken into account.

An important property of neural interactions is that they involve time delay because of the finite propagation velocities of action potentials along neurites, synaptic transmission delays at chemical synapses and dendritic processing [31]. It is thus important to understand the effects of time delay on the synchronized firing [32]. This paper also investigates the effect of time delay in the interaction on the synchronized firing in the population of coupled neuronal oscillators using Fokker–Planck equations.