Exemplary Discussion Draft 2

Author: Catherine Osborn

Discussion

We have been able to replicate all of the figures produced with Purvis and Butera's study using mathematical modeling to explain how variations in ionic current densities can relate to differences in hypoglossal motoneuron action potential firing pattern dynamics seen between neonatal and postnatal rats. However, the working model we used varied slightly from their model. When we first ran our model following the given initial conditions, we saw that the peak produced did not resemble the action potential seen in Figure 1 of their paper. If the system was stable as expected, the removal of the applied stimulus should have resulted in a voltage that maintained its given initial value, or the membrane's resting potential. Other papers have found that this resting potential is around -72 mV . Without a stimulus, we saw that the cell did not have a constant membrane potential, but that instead, the voltage decreased sharply from the given initial voltage (-71.847 mV) and eventually rose again after 150 ms to the expected resting potential.

When using the given initial conditions, it would appear that the time constants associated with the calcium currents and the calcium-dependent potassium (I_SK) currents are much longer than those of the other ionic currents, such that these currents require enough time to settle and allow for a constant voltage without stimulation. Our solution was to run the model and plot each of the state variables. We selected a time at which all of the 16 variables were constant (t = 6000 ms), found the values of the functions at this time, and used those as our new initial conditions. It is likely that the Purvis and Butera model contained a clause of code which ensured that stimulation occurred only after the first derivative of the voltage (without stimulation) was equal to zero, thus allowing the calcium currents and the I_SK current to stabilize before attempting to elicit an action potential. Although our methods were different, we were able to reach the same results. The action potential produced by both models shows depolarization and repolarization phases of the spike, the fast-afterhyperpolarization phase (fAHP), the after-depolarization phase (ADP), and the medium-afterhyperpolarization phase afterwards (mAHP) (Figure 1).

The second figure we reproduced examined the change in conductivity of the various ion currents within the model. During an action potential, the sodium channels are the first to be activated by the stimulation and positive sodium ions rush into the cell (Figure 2A), which results in the depolarization phase of the action potential .This rise in voltage immediately results in the activation of the voltage-gated potassium channels, which are responsible for the repolarization of the action potential (Figure 2C). The calcium currents (Figure 2B) are activated by the increase in cell voltage caused by the influx of positive sodium ions, and the opening of these channels results in a second increase in cell voltage as positive calcium ions flow into the cell. The secondary wave of positive ions into the motoneuron are responsible for the after-depolarization phase of the action potential . The activation of the calcium currents initiates the calcium-dependent potassium current, which allows potassium ions to leave the cell, resulting in the medium-duration afterhyperpolarizationphase (a decrease in voltage to below that of the resting potential). The calcium-dependent potassium channel does not inactivate, but instead is dependent upon the amount of internal calcium within the cell .

We next examined whether our model was able to reproduce various action potential dynamics seen in experimental studies. Like Purvis and Butera, we saw that increasing the conductance of the low-voltage T-type calcium channel, g_T, from 0.1 μS to 0.15 μS resulted in an increase of the ADP amplitude. Increasing the calcium conductance allows for a greater influx of calcium into the cell and a greater increase in voltage during the ADP. Additionally, increasing the amplitudes of prepulses administered to lower the membrane potential, thus hyperpolarizing the cell, before stimulation, increased the amplitude of the ADP. The prepulses remove the inactivation of the T-type and N-type calcium channels, meaning that calcium flows into the cell longer than without the prepulses. Both of these results have been demonstrated experimentally .

This model is able to replicate the effects of apamin treatment on the hypoglossal motoneurons seen in experimental data . Apamin, a toxin in bee venom, blocks the calcium-dependent potassium channels. When the I_SK current is blocked using this toxin, the mAHP is removed from the action potential. This is because there is no outflow of potassium in response to the built up intracellular calcium that causes the cell to become hyperpolarized. For both the Purvis and Butera model and ours, the I_SK current was removed by setting g_SK to 0 μS, resulting in an action potential without an mAHP. Additionally, the ADP height is increased, also seen experimentally . When we increased g_SK to 0.03 μS (a value which is 10 times less than given in the parameters table) and administered a stimulus with an amplitude of 0.33 nA, the firing frequency was initially increased and the motoneuron exhibited an adaptive firing pattern (i.e. the firing slowed over time). This adaptation occurred because of the slow inactivation of the persistent sodium current, I_NaP. .The persistent sodium current-mediated slow firing can still occur when the mAHP is blocked by apamin. In this case, the fAHP seems to be sufficient to deactivate the I_NaP after each spike, and the same I_NaP-mediated ramp and acceleration depolarization follows after each fAHP. Thus slow firing continues with the same underlying persistent Na oscillations during a held current but with the mAHP no longer interposed between the spike and I_NaP activation. In apamin, the firing that is mediated by the I_NaP oscillations is not restricted to very slow rates but occurs at faster rates with interspike intervals much less than the usual mAHP duration. Thus, I_NaP oscillations likely participate in determining the interspike interval during repetitive firing at all rates .

Manipulating the stimulus amplitude, or the maximum conductances, g_T and g_N, of the T and N-type resulted in the changes in action potential firing from adaptation to acceleration. When we administered a held stimulus of 0.22 nA, which is very near the threshold for the hypoglossal motoneuron, the model produced a train of action potentials that showed adaptation and eventual cessation. Again, the persistent sodium current's slow inactivation slows down the firing rate, and finally stops the firing entirely. Such a stimulus also results in delayed excitation of the first spike. This delay is a result of the large I_A current initially triggered by the low stimulus amplitude . Increasing the stimulus to 2.0 nA, we were were able to show how manipulation of the maximum conductances could elicit these differences in firing behavior seen between neonatal and postnatal rats. Purvis and Butera showed in their Figure 6 that a held stimulus of amplitude 2.0 nA would show initial adaptation of firing and then relatively steady-state firing. When working with our model, we only saw these results when we set the g_T parameter to 0.01 μS. This value is 10 times less than the value given in the parameter table. Once this value was changed, we saw that there was rapid adaptation from the first to the second interspike interval. When the density of g_T is low, as seen in neonatal rats, a relatively low calcium inflow only partially activates the the SK current during the first amplitude, resulting in a shortened afterhyperpolarization period (AHP). The level of internal calcium increases during the second action potential and causes summation of the SK current, resulting in a longer AHP, and thus rapid adaption. Computer simulations based on motoneuron models indicate that this process of AHP temporal summation appears to contribute to the initial phase of spike-frequency adaptation . Temporal summation of the AHP across successive interspike intervals presumably reflects the fact that the calcium concentration does not return to its resting level by the end of the interspike interval . Such increases in calcium concentration might reflect the saturation of intracellular calcium sequestering systems . When the g_T value was increased to 0.1 μS, as given in the parameters table, and we administered a held stimulus of 2.0 nA, the large calcium influx results in a fully activated SK current. However, the calcium currents inactivate after the first spike, so that less calcium enters into the cell during the following spikes. As the calcium levels of the cell decrease, the SK current becomes less activated, resulting in decreased AHP's, allowing the subsequent spikes to occur more rapidly. This results in an accelerating pattern of action potential firing, which is seen experimentally in older rat hypoglossal motoneurons . When we repeated this process comparing low g_N values to high g_N values, there was a change from adaptation to acceleration. This work shows how variations in the densities of these currents can be in part responsible for the age-dependent changes in the firing of rat hypoglossal motoneurons.

There are, of course, limitations to this model. It does not take into account the three-dimensional organization of the neuron, or how changes in the anatomy over development may be partially responsible for the age-dependent changes found experimentally . Additionally, this model did not take into account the P-Type calcium current whose density may change, thus impacting firing patterns within the cell . Additionally, our model represents a single-calcium compartment. This assumes that the entrance of calcium into the cell is instantaneous and neglects spatial heterogeneities of calcium concentration .

Notes

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