User:Jrm228/Benchmark II: Model Description

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Author: James McGinnity

Model Description: Modeling Excitatory and Inhibitory Neurons

State Variables and Parameters [1]

State Variables

The following table lists the two state variables in the model. The initial values are for the first neuron, and are subject to change for every other neuron.

Variable Description Initial Value
v membrane potential -65 mV
u membrane recovery variable 0

Parameters

The table below defines the five parameters present in the model. The first two, a and b, will influence the initial values of the state variables, and, together with the last, I, will influence the first derivatives. The other two, c and d, will effect the reset value when the spike occurs.

Variable Description Default Value RS IB CH FS LTS RZ
a timescale of membrane recovery variable 0.02 0.02 0.02 0.02 0.1 0.02 0.1
b sensitivity of recovery variable to fluctuations in membrane potential 0.2 0.2 0.2 0.2 0.2 0.25 0.25
c after-spike reset value of potential -65 mV -65 mV -55 mV -50 mV -65 mV -65 mV -65 mV
d after-spike reset value of recovery 2 8 4 2 2 2 2
I injected or synaptic current 10 10 10 10 10 10 10
  • Note: RS: Regular Spiking; IB: Intrinsically bursting; CH: Chattering; FS: Fast Spiking; LTS: Low Threshold Spiking; RZ: Resonator.

Assumptions [1]

There are several key assumptions that must be made in order to model the neuron with this set of variables and equations, as well as to make the extension;

  1. With regards to all five parameters, it is assumed that they accurately represent the cortical neuron as it would appear in a natural state.
  2. The models can be reasonably compared by the membrane potential. Both models have other state variables, but they are not consistent between the models, which makes them impossible to compare.
  3. A bifurcation analysis of regular spiking behavior is sufficient in demonstrating the properties of both models. The Hodgkins-Huxley changes the state variables substantially between different types of behavior, but the paper model only alters parameters.

Inputs to Model

The following are two equations describing the system and two conditions that are imposed upon the equations. After each, there is a description of what each component means in the context of the mathematical model[1].

Equation 1:

 \frac{dv}{dt} = 0.04v^{2} + 5v + 140 - u + I

This equation is representative of the membrane potential of the cortical neuron, which is the primary tool of sending out the waves we are interested in. The first three components are grouped together as a quadratic equation for the membrane potential itself, indicating that as the membrane potential grows, it speeds its rate of growth. The fourth term, u, describes the recovery variable that is shown in equation 2. It limits the rate of change of the membrane potential slightly, but not enough to stop it. Finally, the term I is representative of the injected or synaptic current in the cortical neuron. If there is a greater current present, then the potential will increase as well.

Equation 2:

 \frac{du}{dt} = a ( bv - u )

The membrane recovery variable is shown above as u, and is determined by this equation. The terms a and b are the parameters presented in the Parameters section above, and determine the rate at which the recovery variable slows growth. As u increases, the rate of change decreases, but the membrane potential increases so much more rapidly that the recovery variable continues to grow over time.

Condition 1:

If  v \ge 30 , then  v = c

This condition allows for a reset after each spike so that the signal can be sent out again. The parameter c represents the reset value for the membrane potential to return to after each spike, and the reset is triggered when the potential goes above 30 mV.

Condition 2:

If  v \ge 30 , then  u = u + d

At the same time, the value for the recovery variable must be changed. However, while the change to the membrane potential in the other condition always vastly decreases the value of v, this condition increases the value of u so that it can aid in swiftly returning v to its original, pre-spike level.

Simulation Method

In order to conduct our model, we had to first demonstrate that a simpler version of the Mathematica code would run correctly. We started by creating a model that would plot the first graph of Figure 2 of our paper by using the above equations and conditions with the parameter values of a Regularly Spiking neuron. This used an NDSolve to determine the equations for v and u, which we then plotted.

After confirming this, we then set out to create a similar model from Chapter 7 through use of the Hodgkins-Huxley equations in order to compare our model to a biologically accurate model. We used the code from the text, then altered the parameters until it approximated our model results.

Finally, in order to test our hypothesis and extend the original paper's model, we performed a bifurcation analysis on both sets of data. For the paper model, we used a Solve with the two equations in a quadratic form and without the WhenEvents that allowed for our conditional spike event. Chapter 7 was slightly trickier, as it was a four-dimensional system, but after applying a Solve and using our altered parameters, we were able to create nullclines and equilibrium points for it as well. We compared the two sets of results to each other to determine the validity of our hypothesis.

References

  1. 1.0 1.1 1.2 Izhikevich, Eugene. "Simple Model of Spiking Neurons". IEEE Transactions on Neural Networks. 14 (6): 1569–1572.