User:Jrm228/Benchmark III: Results
Author: James McGinnity
Results: Modeling Excitatory and Inhibitory Neurons
Basic Model of Spiking Neurons
We will go over the two different models for spiking neurons, with special attention given to the discrepancies between our model and the model that Izhikevich published in his paper. While the models have enough differences to comment on, the more extensive comparison will come with the bifurcation analysis following that section. For more information on how the figures were created, please consult the Mathematica notebook file at the following link; Basic Model
Izhikevich Model (Original Model)
The paper we are examining had several different graphs, but the one that drew the most attention was a graph with eight different types of neuron firing demonstrated by his code. This graph is shown below;
By displaying the effects that each parameter has in sync with each other, Izhikevich is able to demonstrate the full scope of the model that he claims is not only much more computationally efficient than the Hodgkins-Huxley model, but also just as biologically plausible. The hypothesis that we are testing is whether or not that claim is accurate, and that means we need to replicate Figure 1.
The first step to replicating Figure 1 is to make a representation of the equations for a single set of parameters. To this end, we used NDSolve to input the parameters and apply them to his equations, then evaluated the results and graphed them to produce Figure 2.
There are several discrepancies between his graphs and ours that should be addressed. First, in this representation, the membrane recovery variable u[t] is displayed alongside the membrane potential v[t]. This was done to represent the reset of the system after each spike, which is clearly shown with u[t]. In addition, there is only one spike here compared to several spikes in the original paper model. This was a result of the time scope that is examined in each. His model used 0.1 milliseconds, while ours only went for 0.08 milliseconds. The difference came from the desire to compare his model to the code represented in Chapter 7 for the Hodgkins-Huxley model, as well as concerns over the total runtime of the model. Finally, though it is not shown by the figures, the model that we used does not always reach 30 mV. As Izhikevich discusses in his paper, the original graphs were altered to automatically jump to 30 mV when they began to spike to show a uniform graph. This alteration does not show the accuracy of his model very well, which is the point of our hypothesis, so the cutoff value for our graphs was 20 mV to show the replication of the model without compromising our results.
Once we had created Figure 2, it was a simple matter to alter the inputted parameters to fit the seven other situations, then use GraphicsGrid to show all eight types of spiking at once in Figure 3.
While the previous discrepancies are once again present in this figure, we can say with confidence that the difference between each situation is as prevalent as in the original figure. The first parameter, the time scale of the membrane potential, is closely related to the second parameter, the sensitivity of the recovery variable. As both parameters increase, the ability of the system to return to resting state and spike again increases and allows for more spikes in the same time period as the regularly spiking neuron (such as in the fast spiking and low threshold spiking graphs). When the third parameter, the reset value of the membrane potential, is increased, the neuron also fires faster, but when the effect of the reset value for the recovery variable is simultaneously decreased, the neuron becomes more likely to fire in clusters (such as in the chattering neuron). These results are consistent with the conclusions that Izhikevich came to in his paper, so we can reasonably say that we have replicated his model. Below, we used a Manipulate function to fully illustrate the effect of each parameter on the behavior of the neuron, and invite the reader to follow the link to our code to see the effects firsthand.
Having created an accurate representation of Izhikevich's model for a spiking neuron, we now need something to compare it to. We turned to Chapter 7, where the Hodgkins-Huxley model was represented, to find a set of equations and graphs that could be reliably compared to the results we found above. In Figure 5, we have recreated the model using the exact code from Chapter 7, with the exception of an alteration to the injected current and timescale to fit the model from Izhikevich. We did not model all eight different situations because only one is necessary to properly perform a bifurcation analysis.
Basic Comparison of the Two Models
Clearly, the two graphs we have for individual neuron firing in a Regularly Spiking cortical neuron scenario are similar. However, the Hodgkins-Huxley graph has several small dips and hills leading to and coming from the spike that are not present in the very smooth model from the paper. This can be explained by the complexity of their equations; the Izhikevich equations are simplistic in nature because he was attempting to make the model computationally efficient, while the Hodgkins-Huxley equations were designed to be as accurate as possible and had many more parameters as a result. On the surface, it would appear that our hypothesis has been proved. The graphs are remarkably similar, and no significant information appears to have been lost. However, the dips and hills previously observed are concerning, and so we shall also perform a bifurcation analysis on the models to test our hypothesis further.
Extension: Bifurcation Analysis
This section has not yet been completed. However, it will include graphs which demonstrate the phase portraits, nullclines, and bifurcations of both the Izhikevich and Hodgkins-Huxley models, as well as an analysis on the differences between the two and the implications for our hypothesis.