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Our emotions, thoughts, realization of our selves and our environment, our decisions are all outputs of our brain, a physical structure. The methods and approaches used and developed by science to examine the physical systems will influence us to understand the operation of the brain and its outputs - the cognitive processes - as much as they help to define the movements of the planets or predict weather conditions. To develop a model which explains the emotional cognitive processes and especially drug addiction mechanism using mathematical approaches, and the mathematical analysis of this model will help us to explain the cognitive processes in the brain.

Computational neuroscience develops theories on the operation of the brain based on the information processing properties of the neural systems. It explains the activities of the neural structures and their functions, the representation of the physical world in the neural systems using the notions and technical tools of the computational sciences. The main goal of neuroscience is to understand the operation and computation dynamics of a neuron and to analyze the roles of neurons in large neural networks. Mathematical models and computer simulations are used to mimic and/or predict the behavior of the nervous system. Computational neuroscience has matured and become a separate research field in the past 30 years. There are many various applications of computational neuroscience research such as prosthesis organs, drugs designed by computer simulations instead of lab experiments, and so on (e.g.: A glance at the work done at Bernstein computational neuroscience network).

Cognitive science researches the biological foundations of cognition by focusing on the neural mechanisms of mental processes. Cognitive science tries to explain psychological processes and functions such as perception, behavior, memory, language, selective attention. Cognitive science utilizes the findings and hypothesis of various scientific research areas like psychology, neuropsychology, physiological psychology, neurology, cognitive psychology, linguistics, philosophy, computational modeling. Computational neuroscience gained its name as a separate research area after the Systems Development Foundation conference held in 1985 at California. Since then, the number and quality of research done has incremented exponentially due to the rapidly developing medical imaging techniques and computer software.

Computational neuroscience develops models mostly based on the findings and experiments of behavioral psychology and neurology while cognitive science tries to explain the behaviors and their neurological foundations based on the developed models. Therefore these two scientifica areas proceed by nourishing each other. Joseph LeDoux and Antonio Damasio are pioneers in psychological modeling of brain processes since mid-1970s. Stephen Grossberg (ART) and David Marr (first explanations of neural interactions) have given the first examples of mathematical models in this scientific area.

Mathematical models define the operation of a single neuron, a neural structure composed of several neurons, or a system composed of several neural substructures. The electrial circuitry correspondent of models of neuronal level show simple input-output functions. However, models of system level are more complex since they have to define the behaviors based on the neural activity in the brain and thus they use various approaches like nonlinear systems, rule-based structures, artificial neural networks, machine learning. The first studies in this field are based mostly on ANNs. Today it is preferred to model the neuron populations using dynamical system approach. The main difference of computational neuroscience approach from other approaches like machine learning, statistical learning, ANNs is that it tries to describe the exact function, physiology and dynamics of a physical neuron or neural system.