# User:Jrm228/Benchmark I: Introduction

Author: James McGinnity

## Contents

## Introduction: Modeling Excitatory and Inhibitory Neurons

### The Problem

In the past few decades, the combination of massive leaps in technology and of an increase in the accessibility of scientific data has led to incredible strides in modeling neuronal networks. However, the complexity of pre-existing models such as the Hodgkins-Huxley model has made them computationally inefficient and prevented a more complete understanding of the brain from being developed ^{[1]}^{[2]}^{[3]}. One researcher, Dr. Eugene Izhikevich, created a computationally efficient system that would still be biologically plausible by reducing the number of parameters and state variables to only the most influential parts of the system ^{[4]}. This model can be run in real time for tens of thousands of spiking neurons, but there are a variety of differences between this model and conventional models, and so there is a concern that Izhikevich’s model is not accurate enough to provide realistic simulations of brain activity.

### Background and Hypothesis

There are two parts to demonstrating the accuracy of Izhikevich’s model that this paper will explore. The first part will involve reproducing the figures that he generated in his paper, which show the changes in the model for several different types of firing in neurons ^{[4]}. This will be compared to a Mathematica model of the Hodgkins-Huxley set of equations that is demonstrated in chapter 7 of our textbook ^{[5]}. The second part will expand upon both models by way of a bifurcation analysis, which can demonstrate the degree of difference between the two models to a much greater extent ^{[6]} The model proposed by Izhikevich does not lose a significant amount of data when compared to the Hodgkins-Huxley model, and the bifurcation analysis will demonstrate that the results of both models are similar.

### Summary of Results

Our models, shown in the next section, were fairly similar on the surface. Both showed the spike at the same time, and had the same post-spike dip, but the Hodgkins-Huxley model had a lot more little bumps and irregularities that are characteristic of a more accurate model. We were unable to make a significant conclusion on our hypothesis, so we moved to conduct our extension through a bifurcation analysis. We modeled the nullclines and equilibrium points of the membrane potential of both sets of equation. This analysis ultimately demonstrated that a substantial difference existed between the two models.

### References

- ↑
Hagan, Martin; Howard, Demuth; Beale, Mark; De Jess, Orlando (2014).
*Neural Network Design*(2nd ed.). New York: Martin Hagan. - ↑
Rojas, Raul (2013).
*Neural Networks: A Systematic Introduction*(1st ed.). Berlin: Springer. - ↑ John, Staddon; Grossberg, Stephen; Commons, Michael (1991).
*Quantitative Analyses of Behavior*(1st ed.). Hillsdale: Lawrence Elbaum Associates. - ↑
^{4.0}^{4.1}Izhikevich, Eugene. "Simple Model of Spiking Neurons".*IEEE Transactions on Neural Networks*.**14**(6): 1569–1572. - ↑ Chiel, Hillel J. (2017).
*Chapter 7*(18th ed.). Hillel J. Chiel. - ↑ Soleimani, Hamid; Bayandpour, Mohammad; Ahmadi, Arash; Abbott, Derek (April 2017). "Neural Dynamics and Bifurcation".
*IEEE*.