Exemplary Model Description Draft 2

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Author: Catherine Osborn

Model Plan: Model of Ionic Currents of Hypoglossal Motorneuron


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 is 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 data on 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.

Assembling the Model

In reproducing this model, we:

1) Initialize the parameters (26), listed in Table 1.

2) 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).

3) Created functions for the currents, listed in Table 3.

4) Used NDSolve to solve 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.

5) The current functions were added together in the voltage differential equation.

6) Plotted the voltage function to create the figures.