4.1.5 Pacemaker Neuron

1. Pacemaker Properties

Research into the details of locomotion and the rhythmic properties of some neural systems has shown that there is usually a subsystem that is able to produce rhythmic bursting to drive those behaviors. For example, the motion to move a leg would be controlled by one of these pacemaker units. Another good example would be in the heart, where the pacemaker would control the contraction of the heart muscle to pump blood. The pacemaker is usually a group of neurons that interact in a really complex way to produce this bursting property. However, this behavior can be modeled by a single neuron. Kandel ^(4.1.5.1) described five properties for pacemakers that have been discovered through research. These properties are:

Dr. Randall Beer^(4.1.5.2) then used these findings to come up with a set of rules that could produce that behavior in a neuron using two intrinsic currents. I_h is the high, depolarizing current and would tend to pull the membrane potential above threshold. I_l is the low, hyperpolarizing current and would pull the membrane potential below threshold. The rules that use these two currents to produce the pacemaker behavior are:

The pacemaker neuron has all of the properties associated with a regular neuron: Cn, Gn, Vth, Fmin, and Gain. For a description of them please see the text that discusses the regular neuron. The properties that are unique for the pacemaker are listed below with a description of each.

The steady state voltages mentioned above are calculated using V_ss = (I_input / G_n)  (1). I_input is the input and external currents only. It does not include the intrinsic currents themselves. Also, The length of time that the I_l current remains active is determined by this equation. T_l = (M_tl * V_ss) + B_tl  (2). So increasing B_tl will increase the amount of time between pulses irregardless of the steady state voltage. M_tl determines in what way, and how much, the steady state voltage affects the length of time between pulses. What is desired is that by increasing the input current it will increase the pulse frequency. So if V_ss is positive then M_tl should be negative in order to decrease the time that I_l is active, and thus the length between pulses. Also, if V_ss is negative then it means that the pacemaker neuron is being inhibited. If it is being inhibited strongly enough that V_ss < V_ssm then the pacemaker should be kept from firing at all. However, if enough depolarizing current is input into the pacemaker then the I_l intrinsic current will no longer be strong enough to be able to pull the membrane voltage down enough to disable firing. This will mean that the neuron will fire continuously at that point until the input current is lessened.

Pacemaker neuron Dialog

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

2. Neuron Output

Pacemaker neuron Output Sample

Figure 2. This is a graph of the output from a pacemaker neuron with the following neuron properties: Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV Mtl = -5 Btl = 200 ms Th = 100 ms

Figure 2 shows the output of a simple pacemaker cell that has no input current. It can be seen that the time between bursts is 200 ms and that the time of the burst is 100 ms. This matches with the values for B_tl and T_h that were used for this example. Also notice the way the intrinsic current oscillates between 10 na and -10 na. This is what is causing the rhythmic bursting that is characteristic of the pacemaker neuron.

3. Increasing The Bursting Frequency

Pacemaker neuron Output Sample. Faster Bursting

Figure 3. This is a graph of the output from a pacemaker neuron that is being injected with 10 na of current. It has the following neuron properties: Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV Mtl = -5 Btl = 200 ms Th = 100 ms

Figure 3 shows the output for the same neuron as in figure 2, except it is being injected with 10 na of current. This causes the steady state voltage to be Vss = (10 na / 0.5 uS) = 20 mV. Plugging this into equation 2 gives T_l = (-5 * 20 mV) + 200 ms = 100 ms. And by looking at the oscillation of the intrinsic current, it can be seen that this is roughly what the output is doing. So by injecting 10 na of current into the cell it has increased the bursting frequency from 3 bursts / 1.1 seconds to 5 bursts / 1.1 seconds.

4. Decreasing The Bursting Frequency

Pacemaker neuron Output Sample. Slower Bursting

Figure 4. This is a graph of the output from a pacemaker neuron that is being injected with -10 na of current. It has the following neuron properties: Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV Mtl = -5 Btl = 200 ms Th = 100 ms

Figure 4 shows the output for the same neuron as in figure 2, except it is being injected with -10 na of current. This causes the steady state voltage to be Vss = (-10 na / 0.5 uS) = -20 mV. Plugging this into equation 2 gives T_l = (-5 * -20 mV) + 200 ms = 300 ms. And by looking at the oscillation of the intrinsic current, it can be seen that this is roughly what the output is doing. So by injecting -10 na of current into the cell it has decreased the bursting frequency from 3 bursts / 1.1 seconds to 2 bursts / 1.1 seconds.

5. Resetting The Bursting Frequency

Pacemaker neuron Output Sample. Resetting Bursting High

Figure 5. This is a graph of the output from a pacemaker neuron that is hit with a brief injection of 40 na of current for 20 ms. It has the following neuron properties: Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV Mtl = -5 Btl = 200 ms Th = 100 ms

Pacemaker neuron Output Sample. Resetting Bursting Low

Figure 6. This is a graph of the output from a pacemaker neuron that is hit with a brief injection of -40 na of current for 20 ms. It has the following neuron properties: Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV Mtl = -5 Btl = 200 ms Th = 100 ms

Figure 5 shows what happens when a depolarizing current enters the neuron in between bursts. As the membrane voltage goes above the V_th threshold it causes the neuron to turn on the I_h intrinsic current and reset itself. Figure 6, on the other hand, shows what happens if a hyperpolarizing current is injected during a burst. It causes the membrane voltage to drop and turns on I_l. This resets the pacemaker neuron. So when a pacemaker neuron is injected with a depolarizing current between bursts, it causes a new burst and resets the timing. But when a pacemaker is injected with a hyperpolarizing current during a pulse it terminates the pulse and resets the timing.

6. Inhibiting Bursting

Pacemaker neuron Output Sample. Inhibiting Bursting

Figure 7. This is a graph of the output from a pacemaker neuron that is being injected with -20 na of current. It has the following neuron properties: Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV Mtl = -5 Btl = 200 ms Th = 100 ms

Figure 7 shows the output for the same neuron as in figure 2, except it is being injected with -20 na of current after the 400 ms mark. This pulls the membrane voltage down low enough so that the I_h current is not strong enough to bring the potential back above the V_th threshold value and cause the neuron to fire. So the bursting property of the pacemaker is shutdown and it does not fire when sufficiently hyperpolarized.

7. Continuous Bursting

Pacemaker neuron Output Sample. Continuous Bursting

Figure 8. This is a graph of the output from a pacemaker neuron that is being injected with 20 na of current. It has the following neuron properties: Cn = 10 nf Gn = 0.5 uS Vth = 0 mv Fmin = 0 Hz Gain = 100 Il =-15 na Ih = 15 na Vssm = -50 mV Mtl = -5 Btl = 200 ms Th = 100 ms

Figure 8 shows the output for the same neuron as in figure 2, except it is being injected with 20 na of current after the 400 ms mark. This causes the neuron to fire, which triggers the I_h intrinsic current to activate. That current stays on for T_h seconds and then the intrinsic current is reset back to the I_l value. Normally, an injected current sufficiently large to insure that it would continue firing even with I_l active would be required. However, in this instance something else happened so that this was not necessary. The input current was large enough that the length of time that I_l is active was zero. That is why the intrinsic current drops in spikes in figure 8. It is being reset and then immediately being set back to I_h in the next iteration. This was a fluke, but it was in here to demonstrate that these systems are pretty complex and sometimes even the people who build them are surprised by what they find it doing. Also, it would have been a real pain in the butt to go back and redo the parameters for all of the pacemaker graphs just because the numbers ended up odd in this instance.

8. Pacemaker Overview

The pacemaker neuron is the most complex neuron that is contained within the library of model neurons. It has a variety of complex behaviors that were quite difficult to try and model even after reading Dr. Beer's descriptions on how he was able to do so. However, this neuron, and all of it's main features play a key role in allowing the simulated insect be able to move its legs in a synchronized manner that allows it to walk, instead of simply flailing its legs about in opposition to one another.