Bursting and Chaotic Activities in the Nonlinear Dynamics of FitzHugh Rinzel Neuron Model

Selection of an appropriate neuron model for neuroscience studies is a crucial task for researchers. Some of the neuron models are too simple to exhibit the complex dynamics of the neuron and others are very complex and cannot be used in a network as they result in computationally expensive analysis. Study of chaotic behavior and bursting phenomenon of biophysical neuron models is an important step towards analyzing the overall functioning of the brain. In depth analysis of bursting and chaotic behavior of FitzHughRinzel neuron model has been made in this paper work.  Biological Neuron, Time Response, Phase Portrait, FitzHugh Rinzel Model, Nonlinear Dynamics, Bursting, Chaotic Behavior.  The brain is one of the most complex objects in the universe. Although many attempts have been made to investigate and model the functionalities of the brain, the exact working of it is still unknown. The research in the field of computational neuroscience is aimed to know about the brain with more intricacy and to develop more realistic models of its constituents. These models are important tools for characterizing what nervous systems do, determining how they function, and understanding why they operate in particular ways. As most of these models are dynamical in nature, theory of dynamical systems is useful in gaining new insights into the operation of nervous system. The primary step for understanding the brain dynamics is to understand the dynamical behavior of mathematical models of individual neurons. The most important part of this study is the bifurcation analysis of the neurons and their networks. Certain bifurcations in the membrane potential result in neural excitability, spiking, and bursting. Revealing these bifurcations in neuron models helps in knowing various functions of the brain. Such types of studies include the analysis of chaotic behavior of neural systems. These neural systems can be individual neurons or their interconnections. The ongoing research in this regard is to examine the role of chaos in learning. Exploring dynamics of biological neuron models is helpful not only in neuroscience studies but also in neural network applications. In literature, different dynamical models are proposed to represent biophysical activities of neurons. Commonly used models for the study of spiking and bursting behaviors of neurons include integrate and fire model and its variants [5, 6, 18, 30] FitzHugh Nagumo model [7] Hindmarsh Rose model [15, 10] Hodgkin Huxley model [13, 11] and Morris Lecar model  A short review of these models is provided by Rinzel in [26, 27, 28] An excellent comparison of more than twenty neurocomputational properties of the most popular spiking and bursting models have been made in  Bifurcation phenomena in individual neuron models including the Hodgkin Huxley, Morris Lecar and FitzHugh Nagumo models have been investigated in the literature [15, 27, 4] Rinzel and Ermentrout [27] studied bifurcations in the Morris Lecar model by treating the externally applied direct current as a bifurcation parameter. From various experiments, it has been well established that neuronal activities show many characteristics of chaotic behavior. Some researchers believe that this sort of behavior is necessary for the brain to engage in continual learning categorizing a novel input into a novel category rather than trying to fit it into an existing category [29, 9, 1] Freeman developed a mathematical model for EEG signals generated by the olfactory system in rabbits [9] He suggested that the learning and recognition of novel odors, as well as the recall of familiar odors can be explained through chaotic dynamics of the olfactory cortex. Attempts have been made to represent the neurodynamics of biological neural networks in terms of artificial neural network type of structures with some extent to their intricacies. Chaotic dynamics based neural networks have also been proposed to capture some of the characteristics of learning in the brain [2] Nonlinear dynamics of various neuronal models has been investigated in [24, 19, 20] Chaos in firing rate recurrent neural network models have been investigated in  The effect of synaptic bombardment has been explored in the dynamics of various biological neuron models [21] Nonlinear dynamical analysis on coupled modified FitzHugh Nagumo neuron model has also been performed [23]  May June, 2016 ISSN 2091 2730 174 www.ijergs.org The brain takes the incoming sensory data as input, encodes them into various biophysical variables and performs a number of computations on these variables to extract relevant features from the input. Biophysical mechanisms responsible for these computations are dynamical in nature and lead to various types of learning. Therefore, there are two closely related issues in computational neuroscience nonlinear dynamics of different constituents of nervous system and the roles of various biophysical activities in learning. There have been many evidences in literature to experimentally explore the chaotic behavior of neuronal activities. Study of chaos and other phenomena of nonlinear dynamics in various levels of brain modeling can provide a significant help in investigating the learning mechanism. Nonlinear dynamical analysis of FitzHugh Rinzel neuron models is carried out to investigate different bifurcations and chaos. The same analysis has been carried out also for a firing rate recurrent neural network of three neurons and it is observed that its dynamical behavior becomes chaotic at some set of parametric values. This study supports the role of chaos in continual learning categorizing a novel input into a novel category rather than trying to fit it into an existent category. NEURON MODELS Neurons or nerve cells are the fundamental building blocks of the nervous system. These cells form the basis of the brain. Therefore, a sufficient in depth knowledge of neurons is necessary for study of the brain. A typical human brain consists of approximately 100 billion neurons, each neuron having at least 10,000 connections with other neurons. A typical neuron has three major regions: the soma, the axon and the dendrites. Dendrites form a dendritic tree which is a very fine bush of thin fibers around body of the neuron. Dendrites receive information from neurons through axons long fibers that serve as transmission lines. An axon is a long cylindrical connection that carries impulses from the neuron. The end part of an axon splits into a fine arborization which terminates in a small end bulb almost touching the dendrites of neighboring neurons. The connections between the ends of axons and the dendrites or cell bodies of other neurons are specialized structures called synapses. Electrochemical signals flow in neurons, originating at the dendrites or cell body in response to stimulation from other neurons and propagating along axon branches which terminate on the dendrites or cell bodies of thousands of other neurons  Various mathematical models for biological neurons have been proposed in literature [7, 15, 13, 25, 26, 16] to represent their biological activities. As it is generally believed that neurons communicate with each other via action potentials, almost all of these models represent neuronal behavior in terms of membrane potential and action potential. Some most popular models are Hodgkin Huxley [13] integrate and fire [18] FitzHugh Nagumo (FHN)  FitzHugh Rinzel (FHR) [26] Morris Lecar  WilsonCowan [31, 14] Izhikevich [16] and Hindmarsh Rose [10] models. These neuron models represent the characteristics of the responses of different types of real neurons with different levels of biological plausibility. Some widely used models of spiking and bursting neurons [7, 15, 13, 25, 26, 16] are reviewed in the remaining part of this section. These models can be expressed in the form of ordinary differential equations (ODE) It is discussed whether these models have biophysically meaningful and measurable parameters and can exhibit bursting and chaotic activities. FITZHUGH RINZEL MODEL Dendrites receive electrical signals through dendritic spines that consist of a spherical head and a stem connected to the dendrite. FitzHugh Rinzel (FHR) model describes the dynamics of dendritic spines  This model shows how the potential difference between the spine head and its surrounding medium vacillates between the silent phase and the active phase or bursting. The system switches phases depending on the strength of the slowly changing drift current in the dendrite. This model is a modification of the FHN model and incorporates a third state variable to model the bursting behavior of neurons. This model is represented by a set of three coupled nonlinear differential equations. It takes the following form