Normal Neuron

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Free C++ source code for the firing rate neuron model is available.

Neuron Properties

Figure 1. Normal Neuron Properties

The normal firing rate 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 properties toolbar and all of the basic variables that are associated with a normal neuron.

All the neurons in this module are derived from the Node object and share the properties of that item. For a description of these properties please see the text that discusses the Node properties.

Neuron Output

Figure 2. This is a graph of the output from a normal 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: Cm = 3 nf, Gm = 100 nS, Vth = 0 mv, Fmin = 0 Hz, Gain = 15 V^-1

Figure 2 gives a demonstration of what the output looks like for a normal 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_m) 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_m >= 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_m 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.

Modifying C_m

Figure 3. This is a graph of the output from a normal 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:

Figure 4. This is the data for the chart in figure 3.

In Figure 3 the capacitance of the neuron was increased from 3 nf to 10 nf. A very important property of RC circuits is the time constant = R_m * C_m = C_m / G_m. 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_m = 100 nS and C_m = 3 nf. This gives it a = 30 ms. For the new graph where we reduced C_m this value is = 3 nf/100 nS = 10 ms. If you are repeating these experiments in AnimatLab then right click on the chart to display the pop-up menu and select the "View Data" item. This will display the dialog from figure 4 that lists all the data points in the chart. First, if we look at 800 ms you will see that the max voltage for the 6 na current is 60 mv. So according to our value the voltage should be V = 0.63 * 0.06 = 0.0378 = 38 mv at 10 ms. Our data does not actually stop at 10 ms, but by looking at the highlighted values in the data we can see that at 8 ms it is 35 mv and at 12 ms it is 44 ms. So it does go through 37 mv in the correct period.

Also, you can seen that even after 100 ms figure 2 is just reaching its steady state voltage level, but since we reduced the capacitance in figure 3 it was able to charge much faster and reached its steady state quickly. This is the major effect that can be seen from changing the capacitance of the neuron. Increasing the capacitance increases the time needed to charge / discharge the capacitor. This changes the responsiveness of a neuron.

Figure 5. 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.

Modifying G_m

Figure 6. This is a graph of the output from a normal 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: Cm = 3 nf, Gm = 200 nS, Vth = 0 mv, Fmin = 0 Hz, Gain = 15 V^-1

In figure 6 the conductance was increased from 100 nS to 200 nS relative to figure 2. Changing the conductance of a neuron has two major effects. First, it changes the time constant. The time constant for figure 2 is 30 ms, while = 15 ms for figure 6. 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 effect 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 effects how long it takes to actually reach the value. In figure 2, an input current of 4 na causes a steady state voltage of 40 mV. In figure 6, an input current of 4 na causes a steady state voltage of 20 mV. This makes sense because since we doubled the conductance we halved the steady state voltage. This can easily be seen by comparing the amplitudes of both the membrane voltage and the firing frequencies from figures 2 and 6.

Modifying V_th

Figure 7. This is a graph of the output from a normal 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: Cm = 3 nf, Gm = 100 nS, Vth = 20 mv, Fmin = 0 Hz, Gain = 15 V^-1

In figure 7 the threshold voltage was increased from 0 mV to 20 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 20 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 40 mV, this was only 20 mV above the threshold and so it was treated as if it was 20 mV.

Modifying F_min

Figure 8. This is a graph of the output from a normal 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: Cm = 3 nf, Gm = 100 nS, Vth = 0 mv, Fmin = 0 Hz, Gain = 15 V^-1

In figure 8 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. It does not return back to the minimum firing until the 2 na current is injected because the voltage is still discharging back to its 0 v level. 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 effect 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.

Modifying Gain

Figure 9. This is a graph of the output from a normal 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: Cm = 3 nf, Gm = 100 nS, Vth = 0 mv, Fmin = 0 Hz, Gain = 20 V^-1

In figure 9 the gain was increased from 15 V^-1 to 20 V^-1 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 9 and 2. The steady state firing frequency for the 4 na input current is around 0.6 Hz for figure 2, but it is around 0.80 Hz for figure 9. Otherwise the graphs are the same. So gain makes the firing frequency of the neuron more or less sensitive to the membrane voltage.

Neuron Property Overview

The preceding sections have demonstrated some of the effects 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 effect this had on the output. Once these basic effects 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 organism 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.