Computational Intelligence and Neuroscience

Volume 2016 (2016), Article ID 6023547, 8 pages

http://dx.doi.org/10.1155/2016/6023547

## Rhythmic Oscillations of Excitatory Bursting Hodkin-Huxley Neuronal Network with Synaptic Learning

Engineering Research Center of Digitized Textile & Apparel Technology, College of Information Science and Technology, Donghua University, Shanghai 201620, China

Received 27 November 2015; Accepted 24 February 2016

Academic Editor: Hiroki Tamura

Copyright © 2016 Qi Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Rhythmic oscillations of neuronal network are actually kind of synchronous behaviors, which play an important role in neural systems. In this paper, the properties of excitement degree and oscillation frequency of excitatory bursting Hodkin-Huxley neuronal network which incorporates a synaptic learning rule are studied. The effects of coupling strength, synaptic learning rate, and other parameters of chemical synapses, such as synaptic delay and decay time constant, are explored, respectively. It is found that the increase of the coupling strength can weaken the extent of excitement, whereas increasing the synaptic learning rate makes the network more excited in a certain range; along with the increasing of the delay time and the decay time constant, the excitement degree increases at the beginning, then decreases, and keeps stable. It is also found that, along with the increase of the synaptic learning rate, the coupling strength, the delay time, and the decay time constant, the oscillation frequency of the network decreases monotonically.

#### 1. Introduction

Neural systems could exhibit rhythmic oscillations, which are a type of synchronous state. Synchronization in neural systems is thought to be important for processing of sensory information and motor function [1], but the occurrence of synchronization in some specific areas of the brain may also be associated with some diseases, such as the epilepsy and Parkinson’s disease [2]. Because of the importance of synchronization in neural systems, it has been studied for a long time from many aspects in neuroscience research [3–7].

There are up to 10^{11} neurons in human brain, and each neuron is connected to approximately 10^{4} other neurons. Neurons are coupled with each other by electrical or chemical synapses, and chemical synapses are dominant in quantity. The chemical synapse is related to the exchange of neurotransmitters between neurons and can be inhibitory or excitatory. Neurons coupled by different types of synapses constitute different networks, in which the dynamical behaviors can be very different [8]. Han et al. [9] found that the synchronization for inhibitory neuronal systems is more robust and stable than that for excitatory neuronal systems, so they investigated robust synchronization for a globally coupled inhibitory neuronal network. However, they did not study dynamical behaviors for excitatory neuronal systems, which should also be explored to reveal the underlying mechanisms of rhythmic oscillations in neural systems.

Synaptic plasticity is a prevalent feature of biological neural systems and considered to be critical for memory and learning functions of brains. Synaptic efficacy could be regulated by the plasticity at a variety of time scales, like from milliseconds to minutes. To study how synaptic plasticity works in neural systems, many synaptic learning rules, such as Hebbian learning rule and STDP rule, are proposed. And accordingly, the dynamics of neuronal systems under the influence of synaptic plasticity has been explored [10–13]. For example, Han et al. [13] investigated the dynamical properties of Newman-Watts (NW) small-world neuronal networks with a short-term synaptic plasticity named Oja rule and got some interesting findings for electrically and chemically coupled neuronal networks, respectively. But the study is insufficient, because synapses have more detailed structure which influences the dynamics of the neural systems a lot, especially for chemical ones.

This paper aims to find out how the synaptic learning and chemical synaptic parameters influence rhythmic oscillations in excitatory neuronal network. It is organized as follows. The model of a globally coupled excitatory bursting Hodkin-Huxley (HH) neuronal network is presented in Section 2. The results of simulations on the excitatory neuronal network, including the effects of the coupling strength and the synaptic learning rate and the effects of chemical synaptic parameters, are presented in Section 3. The conclusions are given in Section 4.

#### 2. Model and Dynamics

The traditional Hodkin-Huxley (HH) model neuron only emits spikes. By incorporating a slow calcium ionic channel into the HH model, a modified model neuron which could emit bursts can be obtained [9]. In this paper, the modified HH model neuron is used to construct neuronal network.

##### 2.1. Model Neuron

The equations and parameters of a single modified HH model neuron can be described as follows (the membrane potential is measured in mV and time in ms):where the parameter is the membrane potential of the modified model neuron. In (1), the activation or inactivation variables , , , and are associated with Na^{+}, K^{+}, and Ca^{2+} voltage-dependent ion currents, respectively, which changes quickly and satisfies the following steady-state functions:

In (1), the activation variable is related to transient sodium current, which changes very fast and satisfies the steady-state function:

In (2a)–(2d), the related variables satisfy the following functions:

The values of parameters in (1), (2a), (2b), (2c), and (2d) are set as follows: = 1 *μ*F/cm^{2}, = 35 ms/cm^{2}, = 55 mV, = 9 ms/cm^{2}, = −90 mV, = 3 ms/cm^{2}, = 120 mV, = 0.1 ms/cm^{2}, and = −65 mV, .

In (1), is injected current (in *μ*A/cm). Neurons could exhibit different firing patterns for different values of . If belongs to , the modified HH neuron exhibits bursting behaviors (see Figure 1). The value of the injected current is set as −0.5 in this figure.