Exemplary Final Term Paper 2

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Author: Catherine Osborn

Introduction

The hypoglossal motoneuron innervates the intrinsic muscles of the tongue and is involved in a variety of behaviors including breathing, chewing, and swallowing. [1] The rat hypoglossal motoneuron shares many properties with the human neuron. Both have relatively high resting potentials and require relatively high stimulus amplitudes to reach firing threshold. [2] Thus, the rat motoneuron has been used as a model system in studying how the various ionic currents respond to stimuli and pharmacological treatments. [3] Dysfunction of these cells results in a condition termed sleep apnea, when serious obstruction to breathing develops periodically in infants or adults. [4] [5] Several ionic currents have been associated with the motoneuron’s response to stimulation [6],[7]. Understanding how variations in these currents impact action potential firing and, on a larger scale, the physiological aspects of breathing, is vital in the creation of treatment plans for such dangerous respiratory conditions.

The model we will be recreating, described by Purvis and Butera [8], was constructed to explain the age-dependent changes in action potential firing of the hypoglossal motoneuron. These motoneurons have several different types of currents, some of which that are affected by aging. [9]. Changes in the density of ionic channels, as well as alternations in the anatomy of the motoneuron throughout development have been documented within the literature [10], [11], [12]. While our model does not consider the structure of the neuron, we can show how such variations in channel density may partially explain the differences in firing pattern displayed in neonatal and adult rats [13],[14] . Changes in the current density are also associated with changes in other membrane properties, including membrane resistance, firing threshold, and action potential duration and action potential firing patterns [15], [16], [17]. In addition to the sodium (fast and persistent) and potassium (delayed-rectifying and fast transient) currents, this model is composed of several different types of calcium channels as well as a calcium-dependent potassium channel [18]. The calcium currents we will replicate include the N- and P-type high-voltage-activated calcium currents and the low-voltage-activated T-type current. There also exists L-type channels that carry only about 5–7% of the total calcium current measured in neonatal rats, which will not be included, although the density of this channel is known to increase with age. [19] [20]. A final component contributing to the action potentials of the rat hypoglossal motoneuron is the hyperpolarization-activated current, whose density also increases with age. [21]


These age-dependent changes in current density have been associated with changes in the action potential firing patterns exhibited by adult rats. The overall hypothesis of this paper is that changes in the densities of the ionic currents within the rat hypoglossal are responsible for the age-dependent changes seen in the excitability of the neuron, such as the switch from a decrementing firing pattern to an incrementing firing pattern [22] and decreases in the duration of the medium-duration afterhyperpolarization phase of the action potentials. [23] We have also extended the model to ascertain whether the age-dependent changes in the density of the fast-transient potassium current, IA could be responsible for the changes seen in action potential duration or firing patterns seen in this model. Neonatal rats do not have this current, and instead exhibit burst firing. [24] Upon finding that our model is able to reproduce this result, we have tested if increasing the density of the A-type channels results in a switch from an decrementing to an incrementing firing pattern. We have found that increasing the density of IA changes the firing pattern from adaptation to acceleration and includes a sharp change in the level of adaptation. We will look at the graphs of ion conductivity in assessing the possible mechanisms behind this abrupt shift.


Studying these age-dependent changes can be important in understanding how such differences may be responsible for conditions including sleep apnea or SIDS. For example, the drug, Clonidine, has been associated with reduction of carbon dioxide ventilatory response and increases occurrence of obstructive sleep apnea [25], by blocking the hyperpolarization-activated current of the rat hypoglossal motoneuron. The decreased gH from clonidine results in a firing pattern resembling that of neonatal rats. [26] Thus, understanding the mechanisms behind hypoglossal and its age-dependent changes could be useful in the development of pharmaceuticals that treat sleep apnea and in understanding the side effects of others.


The Purvis and Butera model of the rat hypoglossal uses the Hodgkin-Huxley formulation of a single-cell compartment that neglects the motoneuron's spatial structure and focuses entirely on how its various ionic currents contribute to subthreshold behavior and spike generation. [27] [28] The change in intercellular calcium concentration in this model follows Michalis-Menten mechanics, and the parameters used in this model were determined experimentally. We can use the model to reproduce the neuron’s response apamin, a neurotoxic component of bee venom, which blocks the calcium-dependent potassium channels, [29] responses to currents near threshold, and can vary the densities of the included currents to elicit the specific action potential dynamics produced by neonatal and adult rat hypoglossal motoneurons.


Model Plan

Simulations

I used Wolfram's Mathematica 8 to model the currents composing an action potential within the hypoglossal motoneuron of a rat.

There are 10 currents included in the neuron's action potential modeled in this paper. Eight of these current are represented by a steady-state voltage-dependent activation and/or inactivation function and a voltage-dependent time constant. These functions describe the openness of the specific channel types that regulate the flow of ions into or out of the cell. The steady-state voltage-dependent activation or inactivation functions take the form:

 x_\infty (V) = \frac {1}{1+e^{(V-\theta_x)/\sigma_x}}

where x(V) is a the sigmoidal function representing the (in)activation function, θx is the value of the half-(in)activation point of the function, and σx is the sloping factor. The voltage-dependent time constants take the form of the function:

 \tau_x (V) = \frac {A}{e^{((V-\theta_1)/\sigma_1)}-e^{(-(V-\theta_2)/\sigma_2)}} + B

where τx is the bell-shaped curve representing the voltage-dependent time constant, θ1 and θ2 are half-(in)activation constants, and σ1 and σ2 are sloping factors. If there is not enough experimental data on the time constants or if their addition to the model does not result in a significant amount of time, then the time constant is represented only by a parameter, B, found in Table 1 below. I will now go through the steady-state voltage-dependent activation and inactivation functions and the voltage-dependent time constants for the eight different currents included in this model that are dependent upon voltage.



Steady-State Voltage-Dependent (In)Activation Functions

The Fast Sodium Current: INa

This current is responsible for the initial rush of sodium ions into the cell, and thus, blocking this current prevents the formation of action potentials.

 m_{\infty}(V) = \frac {1}{1+e^{-(V-36)/8.5}} voltage-dependent activation function

 h_{\infty}(V) = \frac {1}{1+e^{(V-44.1)/7}} voltage-dependent activation function

 \tau_h(V) = \frac {3.5}{e^{((V-35)/4)}-e^{(-(V-35)/25)}} + 1 voltage-dependent time constant for inactivation




The Persistent Sodium Current: INaP

A sodium current that activates more slowly that the fast sodium current.

 m_{NaP\infty}(V) = \frac {1}{1+e^{-(V-47.1)/4.1}} voltage-dependent activation function

 h_{NaP\infty}(V) = \frac {1}{1+e^{(V-65)/5}} voltage-dependent activation function




Delayed-Rectifier Current: IK

The model does not distinguish between the activation and inactivation of this potassium current.

 n_{\infty}(V) = \frac {1}{1+e^{-(V-30)/25}} voltage-dependent activation function

 \tau_n(V) = \frac {2.5}{e^{((V-30)/40)}-e^{(-(V-30)/50)}} + 0.1 voltage-dependent time constant for inactivation




Low-Voltage-Activated Calcium Current: IT

 m_{T\infty}(V) = \frac {1}{1+e^{-(V-38)/5}} voltage-dependent activation function

 \tau_{mT}(V) = \frac {5}{e^{((V-28)/25)}-e^{(-(V-28)/70)}} + 2 voltage-dependent time constant for activation

 h_{T\infty}(V) = \frac {1}{1+e^{(V-70.1)/7}} voltage-dependent inactivation function

 \tau_{hT}(V) = \frac {20}{e^{((V-70)/65)}-e^{(-(V-70)/65)}} + 0.1 voltage-dependent time constant for inactivation




High-Voltage-Activated Calcium Current: IN

 m_{N\infty}(V) = \frac {1}{1+e^{-(V-30)/6}} voltage-dependent activation function

 h_{N\infty}(V) = \frac {1}{1+e^{(V-70)/3}} voltage-dependent inactivation function




High-Voltage-Activated Calcium Current: IP

This is a non-inactivating, high-voltage-activated current.

 m_{P\infty}(V) = \frac {1}{1+e^{-(V-17)/3}} voltage-dependent activation function




Fast-Transient Potassium Current: IA

 m_{A\infty}(V) = \frac {1}{1+e^{-(V-27)/16}} voltage-dependent activation function

 \tau_{mT}(V) = \frac {1}{e^{((V-40)/5)}-e^{(-(V-74)/7.5)}} + 0.37 voltage-dependent time constant for activation

 h_{N\infty}(V) = \frac {1}{1+e^{(V-80)/11}} voltage-dependent inactivation function




Hyperpolarization-activated current: IH

The hyperpolarization current activates much more slowly than the other currents in the hypoglossal motoneuron.

 m_{H\infty}(V) = \frac {1}{1+e^{-(V-79.8)/5.3}} voltage-dependent activation function

 \tau_{mT}(V) = \frac {475}{e^{((V-70)/11)}-e^{(-(V-70)/11)}} + 50 voltage-dependent time constant for activation



Calcium-Dependent Potassium Current, ISK

In addition to the eight currents listed above, there are two other currents of hypoglossal motoneuron modeled in this paper. The calcium-dependent potassium current, ISK, is dependent upon the concentration of calcium within the cell, rather than the voltage of the cell. This current is not modeled by the steady-state voltage-dependent activation and inactivation functions, but can be represented instead by the Michaelis-Menten function:

 skz_\infty([Ca^{2+}]_i) = \frac {1}{1+(\frac {0.003}{[Ca^{2+}]_i})^2}


where [Ca2+]i is the intercellular concentration of calcium. One of the assumptions of our model is in representing this system as a single-compartment cell as formulated by Hodgkin and Huxley.




Differential Equations using the Steady-State Voltage-Dependent (In)Activation Functions

To understand the change of conductance over time, the steady-state voltage-dependent activation and inactivation functions were manipulated by the equation:

 \frac {dx}{dt} = \frac {x_\infty (V) -x(t)}{\tau_x(V)}

where x(V) is the steady-state voltage dependent activation or inactivation function for each current, τx(V) is the voltage-dependent time constant, x is the conductance in terms of time. The initial conditions of the fourteen state variables (x(0)) from the manipulations of the activation and inactivation equations from the above equations are listed in Table 2.


Table 1: Parameters
Parameter Value
gNa 0.7000 μS
gNaP 0.0500 μS
gK 1.3000 μS
gleak 0.0005 μS
gT 0.1000 μS
gN 0.0500 μS
gP 0.0500 μS
gSK 0.3000 μS
gA 1.0000 μS
gH 0.005 μS
ENa 60.00 mV
EK -80.00 mV
Eleak -50.00 mV
ECa 40.00 mV
EH -38.80 mV
K1 -0.0005 M/nC
K2 0.0400 ms-1
Cm 0.0400 nF
τm(V) 0.1 ms
τmNaP(V) 0.1 ms
τhNaP(V) 150 ms
τmN(V) 5 ms
τhN(V) 25 ms
τmP(V) 10 ms
τzSK([Ca2+]i) 1 ms
τhA(V) 20 ms


Table 2: Initial Conditions
V[0] = -71.847 mV
m[0] = 0.015
h[0] = 0.981
mNaP[0] = 0.002
hNaP[0] = 0.797
n[0] = 0.158
mT = 0.001
hT = 0.562
mP = 0
mN = 0.001
hN = 0.649
zSK = 0
mA = 0.057
hA = 0.287
mH = 0.182
[Ca2+]i[0] = 0.0604 μM



Current Equations

The results of integrating the state variables describing the conductance dynamics are used to find each of the currents. Each current, I, is described by a specific equation. These are listed in Table 3. In these equations, V is the voltage (mV), the E parameters are the equilibrium reversal potential (mV) for each ion type, and the g parameters are the maximum conductances of the specific currents. These parameters are listed above in Table 1. All ten currents are listed below: the eight voltage-dependent current, the calcium-dependent current, and the last to mention, the leak current.


Table 3: Current Equations
INa = gNam(t)3h(t)(V(t)-ENa)
INaP = gNaPmNaP(t) hNaP(t) (V(t)-ENa)
IK = gKn(t)4(V(t)-EK)
Ileak = gleak(V(t)-Eleak)
IT = gT mT(t) hT(t) (V(t)-ECa)
IN = gN mN(t) hN(t) (V(t)-ECa)
IP = gP mP(t) (V(t)-ECa)
ISK = gSK zSK(t)2(V(t)-EK)
IA = gA mA(t) hA(t) (V(t)-EK)
IH = gH mH(t) (V(t)-EH)


These ion currents are added together to affect the change in voltage. The change in voltage is represented by the equation:


 \frac {dV}{dt} = \frac {1}{C_m} (- \Sigma  I_{ionic} + I_{stim})


where V is the membrane potential (mV), Cm is the whole cell capacitance (nF), t is the time (ms), Istim is the applied stimulus current (nA) and Iionic are the currents listed above. The initial condition of voltage is found in Table 2. In addition to the change in voltage, the change in calcium concentration, [Ca2+]i is included in our model because of its import to the calcium-dependent potassium current, ISK. This equation is represented by the equation:


 \frac {d[Ca^{2+}]_i}{dt} = K_1 I_{Ca} - K_2 [Ca^{2+}]_i


where K1 is an accumulation factor and K2 is the decay rate constant of calcium within the cell. These two parameters are found in Table 1. ICa is the sum of the three calcium currents in the model, IN, IT, and IP.




Assumptions of the Model

This Hodgkin-Huxley style model represents a single-compartment that measures only how the ionic currents contribute to resting potential and spike generation. Such a model does not include the neuron's spatial structure during calculations, thus localization of ion channels and variations in membrane conductance or potential are not considered. The calcium influx into the cell is similarly considered to enter instantaneously and homogeneously into the cell. This model does not include the L-type calcium channels because it considers these to have no significant effect on the firing dynamics. However because these channels change density with age, later of the addition the L-type channel may be useful in determining the mechanisms to the alteration of firing patterns.



Assembling the Model

In reproducing this model, we first initialized all of the parameter (26), listed in Table 1. We then created functions out of the steady-state voltage-dependent (in)activation functions (14) and the voltage-dependent time constants that are not included in the parameter list (5). Additional functions were created to administer the injection of current which allowed for a variety of stimulus durations and amplitudes. We created functions for the currents, listed in Table 3. NDSolve was used to solve the 16 differential equations -- one for the change in voltage, one for the change in calcium concentration, and the other 14 state variables, the initial variables listed in Table 2. The current functions were added together in the voltage differential equation. The results of the NDSolve, either voltage or current time plots, were graphed to recreate the figures. Additionally, FindMaximum was used to locate the times of the action potential peaks when calculating the interspike intervals (ISIs) used for the frequency graphs (Figures 12B, 13B, 14B, 15b, 16, 17) . FindRoot was used to find time points where the value of voltage returned to the same voltage as immediately before stimulation. The time of the action potential (found using FindMaximum was subtracted from the FindRoot solution to be used for the mAHP duration graphs (Figure 18, 19).

Results

Producing an Action Potential

When we first ran our model following the given initial conditions, we saw that the peak produced did not resemble the action potential in the Purvis and Butera 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 [30] [31]. 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 (ISK) 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. These new initial conditions are shown in Table 4. 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 ISK 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).We first ran the model using the given initial conditions with an injected current of 1 nA for 1 ms. Upon providing this injection, the shape of the output was not that of an action potential (Figure 1). We decided to examine the voltage of the hypoglossal motoneuron without providing an injection. We found that the voltage was not stable with the given initial conditions. Without stimulating the cell, we examined all of the state variables individually to determine the time at which all were stable. This point occurred when time was equal to 6000 ms. We solved all of the state variables at this time point and used those values as the new initial conditions.


alt text
Figure 1. Action potential simulation after injection of 1 nA for 1 ms using the provided initial condition.


Table 4: New Initial Conditions
V[0] = -71.630 mV
mNa[0] = 0.015
hNa[0] = 0.981
mNaP[0] = 0.0025
hNaP[0] = 0.790
n[0] = 0.159
mT[0] = 0.0012
hT[0] = 0.554
mP[0] = 1.235 x 10-8
mN[0] = 0.00097
hN[0] = 0.633,
zSK[0] = 0.002
mA[0] = 0.0579
hA[0] = 0.318
mH[0] = 0.176
[Ca+2]i[0] = 0.000135




When we reran the model with the new initial conditions for the state variables, the output was the same as produced by Purvis and Butera (Figure 2). The action potential we produced is shown in Figure 3. The stimulus is required to elicit an action potential in the hypoglossal motor neuron. When the stimulus is detected, sodium channels are opened (INa and INaP are activated), allowing for a rush of sodium into the cell and a rapid depolarization (the upward climb of the action potential spike). The repolarization is caused by sodium current inactivation and activation of the potassium currents (IK and IA), which causes potassium ions to rush out of the cell (the descent of the action potential spike). Both figures display the fast after-hyperpolarization period (fAHP), the the slight after-depolarization peak (ADP), and the medium duration after-hyperpolarization period (mAHP). The fAHP is caused by the INa, INaP, IK, IA, and the calcium currents (IN, IT, and IP). The large depolarization produced by the sodium currents activates the calcium currents. These inward currents cause calcium to rush into the motoneuron and depolarize the membrane after the cell has repolarized, which leads to the ADP. The mAHP is attributed to the calcium-dependent potassium current ISK. The buildup of internal calcium caused by the inward calcium currents activated during the action potential activates the outward SK current. The SK current in the model does not show inactivation, so it remains activated until the internal calcium concentration is decreased.



Purvis and Butera Results Our Results
Figure 1 from Purvis and Butera
An action potential stimulated after an injection of 1 nA for 1 ms.
Figure 2. Purvis and Butera. Action potential after stimulation of 1 nA for 1 ms. Figure 3. Group Simulation. Action potential after 1 nA for 1 ms pulse.

Ionic Currents during an Action Potential

We have been able to replicate all of the figures produced with Purvis and Butera's [32] 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. The change in conductivity of the various ion currents shape the action potentials produced (Figure 4 and Figure 5). The sodium channels are the first to be activated by the stimulation and positive sodium ions rush into the cell (Figure 4,5A), which results in the depolarization phase of the action potential [33].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 4,5C). The calcium currents (Figure 4,5B) 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 [34]. 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 [35].


Purvis and Butera Results Our Results
Fig. 4
Fig. 5
Figure 4. Purvis and Butera. Ionic currents during an action potential Figure 5, A and C. A) Sodium currents. INa is blue, INaP is red. C) Potassium currents. IK is blue, IA is red, Ileak is green.

Figure 5, B and D. B) Calcium currents. ICa is blue, IT is green, IN is red, and IP is orange. D) Calcium-dependent potassium current, ISK.

The ADP is both Calcium- and Voltage-Dependent

The Purvis and Butera model also replicates experimental data that shows how the ADP is both calcium- (Figure 6A) and voltage-dependent (Figure 6B). We were able to recreate this with varying levels of success. Our calcium-dependent comparison of the ADP (Figure 7A) was produced by varying the gT parameter. The red line is when gT = 0.15 μS, the green line is when gT = 0.125 μS, and the blue line is when gT = 0.1 μS. Increasing the gT parameter raises the amplitude of the ADP. Our Figure 7B was an attempt to recreate Purvis and Butera's Figure 6B. As In their paper, we gave pre-pulses of 0 nA (blue line), -0.1 nA (green line), or -0.2nA (red line) to hyperpolarize the cell to -72 mV, -75 mV, or -78 mV respectively. At these voltages, we stimulated the cell with a current of 1 nA for 1 ms. Purvis and Butera superimposed the graphs so as to only compare the change in amplitude from the starting voltage (-72 mV, -75 mV, or -78 mV), to the ADP by making the beginnings of the action potentials appear as if they all start at the same voltage and the ADP occur at different voltages. We have chosen to show these voltage-dependent ADP comparisons differently. Figure 7B shows that the traces start at 0 ms with an initial voltage of about -72 mV and, in the case of the -0.1 nA and -0.2 nA pre-pulses, hyperpolarize the cell for 5 ms before the action potential was stimulated. Because the cell voltage when this action potential was stimulated varies between the trials, and the ADP appears to occur at the same voltages, one can see that the starting voltage-to-ADP amplitude increased as the cell was hyperpolarized.


Like Purvis and Butera, we saw that increasing the conductance of the low-voltage T-type calcium channel, gT 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 [36] [37].


Figure 6. Purvis and Butera Results Figure 7. Our Results
Fig. 2 from Purvis and Butera
Fig. 7A
Fig. 7B
Figure 6. A) Calcium-Dependent ADP. B) Voltage-Dependent ADP. Figure 7. A) Calcium-Dependent ADP. gT = 0.15 μS (red), gT = 0.125 μS (green), and gT = 0.1 μS (blue). B) Voltage-Dependent ADP. Pre-pulses: 0 nA (blue), -0.1 nA (green), and -0.2nA (red).



Apamin Treatment Block the ISK Current

This model is able to replicate the effects of apamin treatment on the hypoglossal motoneurons seen in experimental data [38] [39]. Apamin, a toxin in bee venom, blocks the calcium-dependent potassium channels. When the ISK current is blocked using this toxin, the mAHP is removed from the action potential (Figures 8,9B). 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 ISK current was removed by setting gSK to 0 μS, resulting in an action potential without an mAHP. Additionally, the ADP height is increased, also seen experimentally [40]. When we increased gSK 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) (Figures 8,9B). This adaptation occurred because of the slow inactivation of the persistent sodium current, INaP. [41].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 INaP after each spike, and the same INaP-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 INaP activation. In apamin, the firing that is mediated by the INaP oscillations is not restricted to very slow rates but occurs at faster rates with interspike intervals much less than the usual mAHP duration. Thus,INaP oscillations likely participate in determining the interspike interval during repetitive firing at all rates [42].


Figure 8. Purvis and Butera Results Figure 9. Our Results
Fig. 4 from Purvis and Butera
Fig. 9A
Fig. 9B
Figure 8. A) Comparison of action potential from single 1.0 nA pulse when gSK = 0 (dotted) or gSK = 0.3 μS (bold). B) Held 0.33 nA current when gSK = 0.03 μS. Figure 9. A) Comparison of action potential from single 1.0 nA pulse when gSK = 0 (red) or gSK = 0.3 μS (blue). B) Held 0.33 nA current when gSK = 0.03 μS.




Near-Threshold Firing

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 (Figures 10,11A). 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 (Figure 10,11B). This delay is a result of the large IA current initially triggered by the low stimulus amplitude [43] [44].

Figure 10. Purvis and Butera Results Figure 11. Our Results
Fig. 10 from Purvis and Butera
Fig. 5A
Fig. 5B
Figure 10. A) Action potentials stimulated by 2500 ms of 0.22 nA current. B) Close up showing delayed excitation before the first peak. Figure 11. A) Action potentials stimulated by 2500 ms of 0.22 nA current. B) Close up showing delayed excitation before the first peak.




Adaptation is Stimulus Dependent: Low Values of gT

Purvis and Butera went on to examine how stimuli above threshold may result in adaption (Figure 12). Figure 12A shows action potentials from a 250-ms held current of 2.0 nA when gT is = 0.01 μS. Such densities of the T-type calcium current are seen in neonatal rats. Adaptation is visible between the first and second peaks. We were able to recreate this figure (Figure 13). Smaller values of gT are associated with greater levels of adaptation of firing. These values are associated with lower levels of calcium conductance, which are only able to partially activate the ISK current. This partial activation results in no mAHP, thus allowing the next action potential to occur rapidly. More calcium enters the cell and the SK channels fully activate, resulting in a long mAHP. This long mAHP after the second action potential causes the third action potential to be delayed. As seen in Figures 12B and Figures 13B, this rapid adaptation is not seen at lower stimulus amplitudes.


Purvis and Butera Results Our Results
Purvis and Butera. Istim = 2.0 nA for 250 ms. gT = 0.1
Cco14 AdaptationGraph.jpg
Figure 12. Purvis ad Butera. Istim = 2.0 nA for 250 ms. gT = 0.01. Figure 13. Group Stimulation. Istim = 2.0 nA for 250 ms. gT = 0.01.



Acceleration is Stimulus Dependent: High Values of gT

Purvis and Butera went on to examine how stimuli above threshold may result in acceration (Figure 14). Figure 14A shows action potentials from a 250-ms held current of 2.0 nA when gT is = 0.1 μS. Such densities of the T-type calcium current are seen in older rats. Acceleration is visible between the first and second peaks. We were able to recreate this figure (Figure 15). Larger values of gT are associated with greater levels of acceleration. At these values, calcium conductance is high, resulting in a large ADP. The SK channels are fully activated by the high calcium concentration, resulting in a long mAHP after the first peak. The second action potential is thus delayed. The T and N currents inactivate slowly, so the calcium influx during the second action potential is not as large as during the first (note that the second ADP is not as large as the first). This decreased calcium influx does not fully active the SK channels, so the second mAHP is shorter than the first, thus the third action potential occurs more quickly. As seen in Figures 14B and Figures 15B, this rapid accleration is not seen at lower stimulus amplitudes.


Purvis and Butera Results Our Results
Purvis and Butera. Istim = 2.0 nA for 250 ms. gT = 0.1
Cco14 Acceleration.jpg
Figure 14. Purvis ad Butera. Istim = 2.0 nA for 250 ms. gT = 0.1 μS. Figure 14. Group Stimulation. Istim = 2.0 nA for 250 ms. gT = 0.1 0μS.




Adaptation vs. Acceleration: Changes in Current Density

The biological hypothesis of this model was to explore whether the age-dependent changes seen in the rats could be attributed to varying the densities of gT and gN to account for the switch from an adapting to an accelerating firing pattern of action potentials. Purvis and Butera calculated the interspike intervals between the first and second, and then the second and third action potentials. When the second ISI is higher than the first ISI, that means that the firing pattern is adapting (firing is slowing down). If the first ISI is higher than the second ISI, the firing pattern is accelerating. Purvis and Butera found that varying the gT and gN parameters resulted in the switch in firing patterns (Figure 16A,B). Our data replicates this finding (Figure 17A,B). When the parameter being manipulated was excluded from the calcium equation (Figure 16C,D and Figure 17C,D), only adaptation occurred. This is because the large value of calcium conductance (represented by an increased gT or gN value was not included in the internal calcium concentration, [Ca2+]i, and thus had no effect on the activation of the ISK current. Thus, the first ISI was always shorter than the second ISI.


Purvis and Butera Results Our Results
Purvis and Butera. Istim = 2.0 nA for 250 ms. gT = 0.1
Cco14 Group11 Fig8.jpg
Figure 16. Purvis ad Butera. Age-dependent changes related to variations in gT and gN. Figure 17. Group Stimulation. The first interspike intervals are red and the second interspike intervals are shown in blue. Calculated during a 1 nA held current. A: Varying gT when ICa = IT+IN+IP. Switches from adaptation to acceleration. B: Varying gN when ICa = IT+IN+IP. Switches from adaptation to acceleration. C: Varying gT when ICa = IN+IP. Only adaptation. D:Varying gN when ICa = IT+IP. Only adaptation.

Effect of IH on mAHP Duration

Increasing the value of gH resulted in decreasing the duration of the mAHP and increasing the resting potential. We measured the mAHP duration from the voltage immediately before stimulation to time at which the voltage returned to that value. The deactivation kinetics of IH in the hypoglossal motorneuons are significantly slower than the duration of the action potential. The activation of these channels in the subthreshold voltage range provides a non-inactivating inward current that typically influences the value of the resting potential. Purvis and Butera's model (Figure 18) shows that neonatal rats with gH values of 0.005 μS have much longer mAHP (about 110 ms) than adult rats with gH ten times larger (about 77 ms). We were only able to replicate the trend in mAHP decrease with our model (Figure 19). Our values range from about 111 ms in neonates and about 69 ms for adults. This error could have been due to differences in mAHP duration measurement methodology.


Purvis and Butera Results Our Results
Purvis and Butera. Istim = 2.0 nA for 250 ms. gT = 0.1
Cco14 mAHPgraph.jpg
Figure 18. Purvis and Butera. Effect of varying gH on mAHP duration. Figure 19. Group Stimulation. Effect of varying gH on mAHP duration.

Extension: Neonatal Bursting from the Removal of IA

When the IA is removed from the model, we see the repetitive bursting pattern seen in the motoneurons neonatal rats. Figure 20 shows a trace of the voltage when no stimulus applied. A rhythmic pattern of bursting displays consistent ionic current conductances Figure 21. The SK current (magenta) is fully activated for each action potential. This activation greatly influences the normal repetitive firing of neonatal hypoglossal neurons. The outflow of potassium through the IK channels (orange) and the influx of sodium through the INa (blue) have compound peaks unlike those during stimulated action potentials when IA is included in the model. The expression of a calcium current with a ow threshold for a activation underlies bursting behavior and rebound depolarizations characteristic of neonatal motoneurons.

Voltage Currents
"Figure 20.' Voltage during bursting without stimulus. gH= 0 μS
Figure 21. Currents during neonatal bursting. Blue: INa, Red: INaP, Orange: IK, Purple: ICa, and Magenta: ISK. gH= 0 μS
Figure 20. Bursting without stimulus. gH= 0 μS Figure 21. Currents during neonatal bursting. Blue: INa, Red: INaP, Orange: IK, Purple: ICa, and Magenta: ISK. gH= 0 μS.



Extension: Varying IA, Adaptation to Acceleration

The original hypothesis of this model was that varying the densities of the ionic currents would result in the switch from an adapting to an accelerating model of firing, mimicking the change seen during rat development [45]. We varied gA from 0.5 μS to 1.10 μS (Figure 22). All of the other parameters were consistent with those in Table 1 and the injected pulse amplitude was 1 nA. We saw that at low values, there was a rapid adaptation. The first interspike intervals (red) were significantly shorter than the second interspike intervals (blue). At a point between 0.70 μS and 0.72 μS, there was an abrupt shift. While the neuron exhibited adaptation, the first and second interspike intervals were much more similar than before. Further increasing the gA parameter resulted in a slight but steady shift from adaptation to acceleration. When gA = 0.72 μS, the IK, IA, and ISK, peak conductances are larger than when gA = 1.10 μS. There is no difference in the amplitude of INaP as gA changes from 0.72 to 1.10 μS. The only channel conductances that increases during this shift of gA are the INa and the ICa. This means that when gA values are high, there is an increased flow of sodium into the cell. The potassium currents are depressed during repolarization. The calcium channels have sustained activity, resulting in activation of the SK channels during the first action potential and resulting in a long mAHP. The calcium channels decrease in activation, the activation of the SK channels decreases, and the mAHP of the second action potential is shorter than the first. [46]


ISI graph
"Figure 22.'
Figure 22. Graph of interspike intervals (ISI) as gA is increased. The red points are the first ISIs and the blue points are the second ISIs. Stimulus amplitude = 1 nA.


I chose to investigate the possible currents underlying the transition from the firing pattern showing rapid adaptation at gA = 0.70 μS to that of a near steady-state firing at gA = 0.72 μS Figure 23A,C. Figure 23B shows how conductances of INaP and IA interact to result in the pattern of firing. The lower gA results in a slower repolarization. The green A current does not fully inactivate. The red NaP current is activated quickly and the second action potential resumes. Figure 23D shows how there is increased inactivation of the A current, which cannot entirely activate the NaP current, resulting in the small "hump" of INaP. This increased influx of sodium results in the elevated ADP seen in Figure 23C. Although not shown, the SK channels are not activated until the second action potential in Figure 23A and Figure 23C.


Voltage and Current Graphs
"Figure 23.'
Figure 23. A: Action potentials during held current. gA = 0.70 μS. Stimulus amplitude = 1 nA. B: Selected currents during action potentials in A. Green = IA. Red = INaP. C: Action potentials during held current. gA = 0.72 μS. Stimulus amplitude = 1 nA. D: Selected currents during action potentials in D. Green = IA. Red = INaP.

Discussion

As hypothesized, we were were able to show that manipulation of the maximum conductances, gT and gN, of the T and N-type could elicit the differences in firing behavior seen between neonatal and postnatal rats. When the density of gT is low, as seen in neonatal rats, [47] 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 [48]. 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 [49] [50]. Such increases in calcium concentration might reflect the saturation of intracellular calcium sequestering systems [51]. When the gT 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 [52]. When we repeated this process comparing low gN values to high gN 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.


The extensions we performed have allowed us a better understanding of the complex mechanisms underlying the age-dependent changes seen between neonatal and adult rats. We have shown that alterations in the densities of ion currents can result in a change from an adapting to an accelerating firing pattern and the change in action potential duration. Our extension first tested whether or not our model could explain other phenomena witnessed in the hypoglossal. Experimental treatments with adult rats used 4-AP blocked the IA channels and resulted in repetitive burst firing seen in neonatal rats. [53] This success allowed us to further our extension. When we varied the gA parameter we found that there is switch from adaptation to acceleration. From 0.70 to 0.72 μS, there is an abrupt shift from a large initial adaptation to a much smaller adaptation. This small adaptation is likely because at higher values of gA, the repolarization phase is shorter and the IA current deactivates more quickly. Thus, the persistent potassium current, INaP, is not fully activated and a second action potential is not immediately initiated. As gA is further increased, the sodium current conductance, INa, was also increased. There is an increased calcium conductance results in full activation of the SK channels during the first action potential and initiates a long mAHP. The calcium conductances inactivate quickly and the second mAHP is shorter than the first. This work provides another possible mechanism by which older rats adopt an accelerating pattern of firing.


Our use of current density manipulation is not likely to fully represent the changes that occur within the hypoglossal motoneurons of rats. It is likely a combination of alterations on many levels, including those we have investigated here, and in other conditions -- such as changes in the density of serotonin receptors -- that may result in the differences seen between neonatal and adult rat motoneuron firing. [54] 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 [55]. Future work should seek to unite this model with the physical changes during the development and combine changes in the densities of the particular currents addressed here. Additionally, this model did not take into account the L-Type calcium current whose density may change, thus impacting firing patterns within the cell [56]. Inclusion of this current may help in explaining the age-dependent changes seen experimentally. 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 [57]. Another possibility in improving the model lies in incorporating a more realistic model of calcium entrance into the cell. Because the calcium channels we evaluated here seemed to play a pivotal role in changing the firing from adapting to accelerating, creating a model that fully considers calcium mechanics would serve as an ideal next step to improve our understanding of the age-dependent changes of rat hypoglossal motoneurons.


Mathematica Notebook

Notes

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