Phase synchronizing in Hindmarsh–Rose neural networks with delayed chemical coupling

Abstract

Although diffusive electrical connections in neuronal networks are instantaneous, excitatory/inhibitory couplings via chemical synapses encompass a transmission time-delay. In this paper neural networks with instantaneous electrical couplings and time-delayed excitatory/inhibitory chemical connections are considered and scaling of the spike phase synchronization with the unified time-delay in the network is investigated. The findings revealed that in both excitatory and inhibitory chemical connections, the phase synchronization could be enhanced by introducing time-delay. The role of the variability of the neuronal external current in the phase synchronization is also investigated. As individual neuron models, Hindmarsh–Rose model is adopted and the network structure of the electrical and chemical connections is considered to be Watts–Strogatz and directed random networks, respectively.

Introduction

Synchronous activity of neuronal populations has been suggested as a mechanism for information binding, i.e. integration of separately processed information, in the brain [1], [2], [3], [4]. Various brain disorders such as schizophrenia, Alzheimer's disease, Parkinson's, and epilepsy have been linked to the abnormal patterns of synchronization in the brain [5], [6], [7]. In order to understand the synchronization properties of neural systems, computer simulations using model neurons should be performed. In vast majority of simulations neuron models of Hodgkin–Huxley type are used [8], [9], [10], [11]. Furthermore, the individual model neurons are considered to be positioned on networks with realistic features such as small-world property [1], [12]. In this way, the synchronous activity of neuronal populations is linked to the properties of the individual neurons and the structural attributes of the underlying network [10], [13].

There are about 10¹¹ neurons in a human brain each having in average 1000 links in which many of them are chemical synapses. The connection efficiency of brain wirings could be explained after the seminal work by Watts and Strogatz [12] who proposed a model to construct networks having simultaneously the small-world property and high clustering coefficient. Although neurons are sparsely connected, they are within only a few synaptic steps from all other neurons and their underlying network has small-world property [1]. This property enables neurons in distal parts of the brain to harmonize their behavior and get into synchrony. Studies of neuronal synchronization based on different neuronal models can be separated into two categories; those using threshold models of integrate-and-fire type and those with conductance-based realization such as various Hodgkin–Huxley type models. Large networks of Hodgkin–Huxley are expensive to simulate, and thus, often reduced-ordered models are used. Hindmarsh–Rose neuron model is one of those with three first-order differential equations [14]. This model has been shown to be capable of producing many of observed neuronal behaviors such as tonic spiking, tonic bursting, spike frequency adaptation, subthershold oscillations, accommodation, and chaotic behavior [15].

Communication between neurons is of two types: electrical connections via gap junctions and excitatory/inhibitory connections via chemical synapses. These two types of connections are clearly distinguished. Electrical connection is bidirectional while the communication between two neurons through chemical synapses is unidirectional, from a presynaptic cell to the postsynaptic one. Another important difference between them is the transmission time-delay. Although electrical couplings can be instantaneous, excitatory/inhibitory chemical connections have some time-delay, usually in the range of a few milliseconds. Signaling of chemical synapses is through neurotransmitters from the presynaptic cell to the postsynaptic one. There are many voltage gated calcium channels in the axonal point of the presynaptic neurons and as the action potential reaches the axonal point, the membrane becomes depolarized and both sodium and calcium channels are opened [16]. This process causes a rapid arrival of calcium into the axonal points, which reconciles a release of the contents of the synaptic vesicles by exocytosis. As a result, the synaptic vesicles bind with the membrane of the axonal terminals and release neurotransmitter into the cleft [16]. The neurotransmitter spreads across the synaptic cleft and travels towards the postsynaptic axonal terminals. It takes some time for the neurotransmitter to be released, spread across the cleft, and bind to the corresponding in the receiving axonal terminals. This time is known as synaptic transmission time-delay [16]. The synaptic time-delay is typically in the range of a few milliseconds and may vary from species to species [16]. An important question in studying synchronization in time-delayed coupled oscillators is the dependence of the synchronization on the time-delay [17], [18], [19].

A number of studies have considered synchronization in delayed neural networks [20], [21], [22], [23]. It has been shown that in diffusively electrically coupled neural networks time-delay may enhance the synchronizability of the network, i.e. in neural networks with delayed electrical coupling the synchronization can be achieved at lower coupling strength compared to the ones with instantaneous coupling [21], [22]. Liang et al. [24] found that distributed time-delay in inhibitory neural networks can induce a variety of phase-synchronous behavior. They also showed that the critical time-delay for the emergence of synchronized oscillations is inversely proportional to both coupling strength and average degree but independent of network topology [24]. Time-delay has also been shown to change not only the synchronization behavior in coupled neural networks but also the qualitative behavior of individual neurons, e.g. changing from chaotic motion to periodic ones [25].

In this paper we study the synchronization behavior of delayed coupled neural networks. As neuron models, we consider the Hindmarsh–Rose model, coupled through both electrical and chemical synapses. The neurons are connected through electrical synapses whose network topology is of the Watts–Strogatz type showing small-world property. Moreover, they are randomly interconnected via excitatory/inhibitory chemical synapses. These chemical connections are not instantaneous and have a unified transmission time-delay, i.e. the signal reaches from the presynaptic neuron to the postsynaptic neuron with a delay. We perform computer simulations on networks with various sizes and find out that the time-delay in the chemical synapses may enhance spike phase synchronization. We also investigate the influence of the variability of the neurons' external input current on the level of synchrony and find out that the more the variability of the external currents the worse the phase synchronization.