Mathematical Analysis of a Model for Generating

electroencephalograms (EEGs) and Evoked Potentials (EPs)

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Huda Mahdi, 3rd year, University of Exeter

BACKGROUND:

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The response of our brain is composed both stimulus-specific activity, representing the direct response to a specific event known as evoked potentials (EP), and a much larger ongoing electrical activity in the brain, occurring even in the absence of specific external stimuli, referred to as spontaneous or background electroencephalogram (EEG).

The origin of EEG and EP lies in extracellular current flow resulting from the summation of postsynaptic potentials (PSPs) generated in the cortex of neurons activated synchronously and oriented vertically. Both of these measurements can be used in a clinical setting to test for potential problems such as post-traumatic stress disorder (PTSD) and epilepsy seizures.

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Motivation: To understand how the brain responds to a stimulus, such as visual or auditory, is quite challenging. The study of neurophysiologically-based mathematical models has the potential to shed light on the mechanisms that underlie the generation of a particular response from the brain. Such Jansen Rit’s model of generation EEG and EP, that is a model of three coupled units representing average postsynaptic membrane potential (PSP) of three distinct neural populations, namely, main cell is excitatory pyramidal cells and inhibitory and excitatory population. The model is based on what Jansen & Rit considered a single cortical column. Originally, this model based on upon Lopes da Silva et al. (1976) and Freeman (1987) work.

This model simplified the activity of brain in cortical area that generated by interaction of three neural populations, as shown in the schematic below -

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METHODOLOGY:

To study the behaviour and changes in the mathematical model over time, known as model dynamics, I perform numerical bifurcation analysis using the software packages XPPAUTO and MATLAB, specifically the COCO toolbox.

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RESULTS:

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In this article, I present a computational analysis of the well-established mathematical model of whole brain activity (electroencephalogram or EEG) in a cortical column and event-related potentials (ERPs), which was developed by Jansen and Rit (1995). We are specifically investigating how the model switches from resting to oscillatory behaviour when we change the parameter that represents the average postsynaptic potentials (PSPs) amplitude ratio.

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Figure 1. An Illustration of Standard Periodic Solutions and Phase Space Projections in the scaled Jansen & Rit model, featuring varying ratios of postsynaptic potential (PSP) amplitudes between inhibitory and excitatory populations. The variables correspond to the PSPs for the pyramidal population, excitatory and inhibitory interneurons, "reps," and the half-max points of the sigmoid function. In the context of an electroencephalogram (EEG) signal, this is closely linked to the incoming firing rate of the pyramidal cell population.

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Fig 1. Schematic diagram of single cortical column, Neural populations interact through excitatory and inhibitory feedback loops.

FUTURE WORK:

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I intend to conduct a numerical bifurcation analysis incorporating a non-zero input. This investigation will consider an external input as a constant factor affecting the system. My focus is on interpreting this external input as a pulse and exploring its bifurcation characteristics.

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FUNDED BY:

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The Higher Committee for Education Development (HCED) in Iraq

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CONTACT:

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Huda Mahdi