Mathematical Models of Dynamic Behavior of Individual Neural Networks of Central Nervous System

Mathematical Models of Dynamic Behavior of Individual Neural Networks of Central Nervous System

Dimitra-Despoina Pagania (University of Patras, Greece), Adam Adamopoulos (University of Patras and Democritus University of Thrace, Greece) and Spiridon Likothanassis (University of Patras, Greece)
DOI: 10.4018/ijsbbt.2012040104
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Abstract

In this paper, the authors present mathematical models that describe individual neural networks of the Central Nervous System. Three cases are examined, varying in each case the values of the refractory period and the synaptic delay of a neuron. In the case where both the refractory period and the synaptic delay are bigger than one, the authors split the population of neurons into sub-groups with their own distinct synaptic delay. The proposed approach describes the neural regime of the network efficiently, especially in the case mentioned. Various examples with different network parameters are presented to investigate the network’s behavior.
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Introduction

In the scientific area that is called Computational Neuroscience questions are set that their goal is to provide with a deeper understanding of the neural system. This specialized scientific area can be defined as the theoretical study of the human brain that aims to reveal the development, organization, information processing and mental skills of the neural system.

Computational Neuroscience is a specialization of Neuroscience. Neuroscience itself is a scientific area with various faces. Its purpose is to understand the neural system and specifically the Central Nervous System (CNS) that is called human brain. Many researchers come from different specializations such as physiology, psychology, medicine, computer science, physics and mathematics. Neuroscience emerged since it was clear that through intersciplinary studies is the further understanding of the human brain possible. The human brain is one of the most complex systems to be ever observed in nature. How does the human brain work, how is it organized, which are the biological mechanisms that get involved, which are the principles of information processing that are being used to achieve complex goals - such as conception. Questions like that are constantly being set by the neuroscientists. Several techniques are being used by Neuroscience to answer problems like that. These techniques deal with genetic manipulation, in vivo and in vitro log of the cells’ activity, visual imaging and computer simulation, etc. Each of these techniques is complicated enough. This is why neurophysiologists, cognitive scientists and anatomy scientists are getting involved. However, it is crucial for every neuroscientist to develop basic knowledge of all these fields so that they can understand and use every aspect of them. The importance of every technique must be appreciated if all the problems and constraints are taken into account. Computational Neuroscience is gaining more ground and its basic understanding is becoming essential for all neuroscientists.

The easily recognized structure in neural system is the neuron that is specialized in the signal processing. Depending on the environmental conditions, it is possible for the electric potential that is used for information transmission, to be produced. The complexity of a single neuron or even the complexity of individual sub-cell mechanisms makes the computational study necessary. The theory of production of action potential that was developed by Hodgkin and Huxley is of equal importance to the Maxwell’s theory of electromagnetism. Neurons are special cells that enable certain mechanisms of information processing in human brain. In this article, the basic operation of neurons is overviewed while the mathematical models that are introduced are explained thoroughly. But first of all, a few major issues concerning the internal operations of a single neuron have to be discussed. Such issues are the mechanisms of information transmission in a single neuron, and also among different neurons, or the way the biological complex neurons can be sufficiently modeled. Many of the computational studies that we focus on are based on very simplified versions of mechanisms that exist in real biological neurons. There is a double reason for such simplifications. On the one hand, simplifications like that make the computations of a big number of neurons easier to be done. On the other hand, it might be wiser to detect the few essential elements that allow specific attributes in neural networks to be raised. Of course, it is part of every serious study to examine if the simplified hypotheses are appropriate under the specific question that is analyzed through the model.

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