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4.1.4 Regular Neuron

Update Alert!

I have now finished work on a much more advanced version of the insect simulator named AnimatLab. AnimatLab is a software tool that combines biomechanical simulation and biologically realistic neural networks. You can build the body of an animal, robot, or other machine and place it in a virtual world where the physics of its interaction with the environment are accurate and realistic. You can then design a nervous system that controls the behavior of the body in the environment. The software currently has support for simple firing rate neuron models and leaky integrate and fire spiking neural models. In addition, there a number of different synapse model types that can be used to connect the various neural models to produce your nervous system. On the biomechanics side there is support for a variety of different rigid body types, including custom meshes that can be made to match skeletal structures exactly. The biomechanics system also has hill-based muscle and muscle spindle models. These muscle models allow the nervous system to produce movements around joints. In addition, there are also motorized joints for those interested in controlling robots or other biomimetic machines. This allows the user to generate incredibly complicated artificial lifeforms that are based on real biological systems. Best of all AnimatLab is completely free and it includes free C++ source code!

The page that corresponds to this one on the AnimatLab site is " Firing rate neural model"

1. Neuron Properties

The regular neuron is the most basic type of neuron in the simulator system. All other neurons are derived from this one and have the basic functionality that this one displays. Figure 1 shows the dialog box and all of the basic variables that are associated with a regular neuron. A name and a description can be entered along with a number of other display attributes like the color, shape and font to use when drawing the neuron in the design window. Also, the (x,y,z) position of this neuron can be set. All neurons in the brain of an insect have a specific three dimensional location. Within the configuration file for the insect it specifies what the valid value ranges are where neurons can reside. However, the most important parameters are these:

Regular neuron Dialog

Figure 1. This is a dialog for setting the properties for a regular neuron.

2. Neuron Output

Regular neuron Output Sample

Figure 2. This is a graph of the output from a regular neuron with externally applied input currents of -4 na, 2 na, 4 na, 6 na, and 8 na in 100 ms intervals with the following neuron properties: Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz Gain = 70

Figure 2 gives a demonstration of what the output looks like for a regular neuron when it is injected with different levels of current. When the graph initially begins there is no net input current into the neuron and the membrane voltage (V_n) is at 0. At 100 ms an external current of -4 na is applied and the membrane voltage dips to around -8 mV. This does not cause the neuron to fire because it will only fire if V_n >= V_th. Then at intervals an increasing amount of current is injected into the neuron and the voltage rises and falls, and the neuron fires once V_n passes the threshold. Also notice on the last current injection that the firing frequency saturated. It attempted to go above 1, but this is not allowed and the firing frequency peaked at this value and stayed there until membrane voltage fell.

3. Modifying C_n

Regular Neuron Output Sample. Cn Changed

Figure 3. This is a graph of the output from a regular neuron with externally applied input currents of 2 na, 4 na, 6 na, and 8 na in 1 second intervals with the following neuron properties: Cn = 50 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz Gain = 70

In Figure 3 the capacitance of the neuron was increased from 10 nf to 50 nf. A very important property of RC circuits is the time constant = R_n * C_n = C_n / G_n. This value tells the time it will take for the voltage to increase by 63% when charging. This is demonstrated in figure 4. So as the capacitance is increased the time it takes to charge that capacitor increases. For instance, in the first example the circuit had G_n = 0.5 uS and C_n = 10 nf. This gives it a = 20 ms. Looking at figure 2 it  can be approximated that this was the correct value. But the second example had G_n = 0.5 uS and C_n = 50 nf. Giving that circuit a = 100 ms. Again, this can roughly be seen by looking at figure 3. Also, it can be seen that while it took figure 3 almost 200 ms to charge completely, it only took figure 2 around 80 ms. This is the major affect that can be seen from changing the capacitance of the neuron. Increasing the capacitance increases the time needed to charge / discharge the capacitor, and this acts like a simple kind of memory. Decreasing the capacitance reduces the affect of the capacitor and makes the conductor much more important so that the current changes in the neuron have a more immediate affect on the membrane voltage.

Time Constant Illustration

Figure 4. This graph illustrates the time constant of an RC circuit. The first time constant location is shown with the dotted red line. Later time constants are shown with dots on the curve.

4. Modifying G_n

Regular neuron Output Sample. Gn Changed

Figure 5. This is a graph of the output from a regular neuron with externally applied input currents of -4 na, 2 na, 4 na, 6 na, and 8 na in 100 ms intervals with the following neuron properties: Cn = 10 nf Gn = 0.8 uS Vth = 0 mv Fmin = 0 Hz Gain = 70

In figure 5 the conductance was increased from 0.5 uS to 0.8 uS relative to figure 2. Changing the conductance of a neuron has two major affects. First, it changes the time constant. The time constant for figure 2 is 20 ms, while = 12.5 ms for figure 5. This means that increasing the conductance decreases the amount of time it takes for the capacitor of the RC circuit to charge. The second major affect is related to the final steady state voltage. The steady state voltage is the voltage that results once the capacitor has been fully charged. The final membrane voltage is directly related to the input current and the conductance of the equation V = I * R. The capacitance has no relation to this final value. It only affects how long it takes to actually reach the value. In figure 2, an input current of 4 na causes a steady state voltage of 8 mV. In figure 5, an input current of 4 na causes a steady state voltage of 5 mV. A difference of 3 mV. This can easily be seen by comparing the amplitudes of both the membrane voltage and the firing frequencies from figures 2 and 5.

5. Modifying V_th

Regular neuron Output Sample. Vth Changed

Figure 6. This is a graph of the output from a regular neuron with externally applied input currents of -4 na, 2 na, 4 na, 6 na, and 8 na in 100 ms intervals with the following neuron properties: Cn = 10 nf Gn = 0.5 uS Vth = 4 mv Fmin = 0 Hz Gain = 70

In figure 6 the threshold voltage was increased from 0 mV to 4 mV relative to figure 2. Right off the bat it can be seen that this caused the final firing frequency to be lower for all input currents. It also meant that the neuron did not even start firing until after it was injected with 4 na or higher. The reason for this is because the 2 na current injection was no longer sufficient to produce a membrane voltage over the 4 mV level. This means it did not exceed the threshold and thus the neuron did not fire. The reason that the final amplitude of the firing frequency for the other current injections were lower is because the firing frequency is based on the difference from the threshold, not the difference from the 0 mV level. So even though an injection of 4 na caused a membrane voltage of 8 mV, this was only 4 mV above the threshold and so it was treated as if it was 4 mV.

6. Modifying F_min

Regular neuron Output Sample. Fmin Changed

Figure 7. This is a graph of the output from a regular neuron with externally applied input currents of -4 na, 2 na, 4 na, 6 na, and 8 na in 100 ms intervals with the following neuron properties: Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0.2 Hz Gain = 70

In figure 7 the minimum firing frequency was increased from 0 Hz to 0.2 Hz relative to figure 2. The first noticeable change is that the neuron is firing at 0.2 Hz even though no current has been injected and the membrane voltage is at 0 mV. This is because the firing threshold is set at 0 mV. Once the negative current is injected and the membrane potential falls below zero it can be seen that the firing frequency is clamped back down to zero. Next, once the positive currents are injected it is as if the output from figure 2 was taken and simply shifted up by 0.2 Hz. This is in affect what has been done. When the capacitor discharges it returns not to a zero firing frequency as before, but instead falls to a 0.2 Hz firing frequency. This is the major affect caused by changing the minimum firing frequency. By playing with this value and the threshold simultaneously it is possible to build a neuron that spontaneously, and continuously fires at a given rate unless it is actively inhibited.

7. Modifying Gain

Regular neuron Output Sample. Gain Changed

Figure 8. This is a graph of the output from a regular neuron with externally applied input currents of -4 na, 2 na, 4 na, 6 na, and 8 na in 100 ms intervals with the following neuron properties: Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz Gain = 100

In figure 8 the gain was increased from 70 to 100 relative to figure 2. The gain is basically just the slope of the line of the function that relates membrane voltage to firing frequency. Increasing the gain means increasing the firing frequency of the neuron for the same membrane potential. And decreasing the gain does the exact opposite. This can be seen by comparing figures 8 and 2. The steady state firing frequency for the 2 na input current is around 0.3 Hz for figure 2, but it is around 0.40 Hz for figure 8. Otherwise the graphs are the same. So gain makes the firing frequency of the neuron more or less sensitive to the membrane voltage.

8. Neuron Property Overview

The preceding sections have demonstrated some of the affects that can be obtained by modifying each of the different properties of the model neuron. This was done by using a base neuron with standard parameters and then modifying one of the values to see what affect this had on the output. Once these basic affects are understood it is then possible to begin putting together multiple changes to try and produce neurons that will behave in the desired manner. Without this understanding it will be very difficult for the experimenter to understand what parameters need to be tuned in order to get a specific behavior of the insect to work correctly. So a good insight into what each of these properties do is critical to really beginning to understand what is happening in the overall network of neurons.