Computational Modeling of Channelrhodopsin-2 Photocurrent Characteristics in Relation to Neural Signaling

Abstract

Channelrhodopsins-2 (ChR2) are a class of light sensitive proteins that offer the ability to use light stimulation to regulate neural activity with millisecond precision. In order to address the limitations in the efficacy of the wild-type ChR2 (ChRwt) to achieve this objective, new variants of ChR2 that exhibit fast mon-exponential photocurrent decay characteristics have been recently developed and validated. In this paper, we investigate whether the framework of transition rate model with 4 states, primarily developed to mimic the biexponential photocurrent decay kinetics of ChRwt, as opposed to the low complexity 3 state model, is warranted to mimic the mono-exponential photocurrent decay kinetics of the newly developed fast ChR2 variants: ChETA (Gunaydin et al., Nature Neurosci. 13:387–392, 2010) and ChRET/TC (Berndt et al., Proc. Natl. Acad. Sci. 108:7595–7600, 2011). We begin by estimating the parameters of the 3-state and 4-state models from experimental data on the photocurrent kinetics of ChRwt, ChETA, and ChRET/TC. We then incorporate these models into a fast-spiking interneuron model (Wang and Buzsaki, J. Neurosci. 16:6402–6413, 1996) and a hippocampal pyramidal cell model (Golomb et al., J. Neurophysiol. 96:1912–1926, 2006) and investigate the extent to which the experimentally observed neural response to various optostimulation protocols can be captured by these models. We demonstrate that for all ChR2 variants investigated, the 4 state model implementation is better able to capture neural response consistent with experiments across wide range of optostimulation protocol. We conclude by analytically investigating the conditions under which the characteristic specific to the 3-state model, namely the monoexponential photocurrent decay of the newly developed variants of ChR2, can occur in the framework of the 4-state model.

Appendix

1.1 A.1 Analytical Solution of the 3-State Model

The equations describing the model are:

$$\begin{aligned} \begin{aligned}[c] \dot{o} &= P(1-o-d) - G_{d}o \\ \dot{d} &= G_{d}o - G_{r}d \end{aligned} \end{aligned}$$

(24)

The photocurrent is

$$ I = Vg_{1}o $$

(25)

The equivalent theoretical solution is

$$ I = C_{1}e^{-\lambda_{1}t} + C_{2}e^{-\lambda_{2}t} + I_{\mathrm{plat}} $$

(26)

where

$$\begin{aligned} & \lambda_{1} = \alpha- \beta; \qquad\lambda_{2} = \alpha+ \beta \end{aligned}$$

(27)

$$\begin{aligned} & \begin{aligned}[c] &\alpha= \frac{P+G_{d}+G_{r}}{2}; \qquad\beta= \sqrt{\frac {(P-G_{d}-G_{r})^{2}}{2} - G_{d}G_{r}}; \\ & I_{\mathrm{plat}} = \frac{P(\lambda_{1} + \lambda_{2} - P - G_{d})}{\lambda_{1}\lambda _{2}} \end{aligned} \end{aligned}$$

(28)

and

$$\begin{aligned} \begin{aligned}[c] C_{1} &= \frac{D_{0}P\lambda_{1} + (\lambda_{2} -P -G_{d})(P - O_{0}\lambda_{1})}{\lambda_{1}(\lambda_{1}-\lambda_{2})} \\ C_{2}&=O_{0} - C_{1} + \frac{P(P+G_{d}-\lambda_{1}-\lambda _{2})}{\lambda_{1}\lambda_{2}} \end{aligned} \end{aligned}$$

(29)

When ideal initial conditions are satisfied (D ₀=0; O ₀=0), the solution presented in Berndt et al. 2011 is recovered.

1.2 A.2 Derivation of Formula (8) from the Main Manuscript

We start with the expression of the first eigenvalue given by Eq. (27) above:

$$ \lambda_{1} = \frac{P+G_{d}+G_{r}}{2} - \frac{\sqrt {(P-G_{d}-G_{r})^{2} - 4G_{d}G_{r}}}{2} $$

(30)

We multiply the equation above with 2 and rearrange the terms to obtain:

$$ P+G{d}+G_{r} - 2\lambda_{1} = \sqrt{(P-G_{d}-G_{r})^{2} - 4G_{d}G_{r}} $$

(31)

we raise the equation above to the second power and rearrange the terms to obtain

$$ \lambda_{1}^{2} - \lambda_{1}(G_{d}+G_{r}) +P(G_{d}+G_{r} - \lambda _{1}) +G_{d}G_{r} = 0 $$

(32)

which finally gives

$$ P = \lambda_{1} + \frac{G_{r}G_{d}}{\lambda_{1} - G_{r} - G_{d}} $$

(33)

1.3 A.3 Evaluation of Necessary Initial Conditions for the 3-State Model

We can find the initial conditions necessary to obtain a photocurrent, which will exhibit the appropriate I _peak/I _plat ratio by use of the following conditions:

• 1. We first find the coefficients of the homogeneous solution (C ₁,C ₂), which will satisfy the experimental data by solving the following system of equations:

  $$\begin{aligned} \begin{aligned}[c] \lambda_{1}C_{1}e^{-\lambda_{1}t_{p}} + \lambda_{2}C_{2}e^{-\lambda _{2}t_{p}} &= 0 \\ \frac{C_{1}e^{-\lambda_{1}t_{p}}}{I_{\mathrm{plat}}} + \frac {C_{2}e^{-\lambda_{2}t_{p}}}{I_{\mathrm{plat}}} + 1 &= \frac {1}{R} \end{aligned} \end{aligned}$$

  (34)

  where the first equation represents the condition that the derivative of the photocurrent function is zero for the I=I _peak and the second equation instantiate the condition that the ration I _peak/I _plat must match the ratio $\frac{1}{R}$ provided by the experimental data.

• 2. With C ₁ and C ₂ determined above, we can find the initial conditions (o ₀,d ₀) by solving Eqs. (34) given in the previous section; then C ₀=1−o ₀−d ₀.

The solutions for the all of the above equations have been evaluated symbolically in Matlab by using the function solve and then numerically by allowing the parameters to take the appropriate values in the symbolic solution.

1.4 A.4 Experimental Results—Additional Information

1.4.1 A.4.1 Evaluation of the Activation Time Constant τ _rise

The evaluation of the time constant of the rising phase of the photocurrent from zero to peak has been performed by approximating the photocurrent curve with a mono-exponential function. Thus, we can write

$$ I(t) = I_{\max}\bigl(1-e^{-\frac{t}{\tau_{\mathrm{rise}}}}\bigr) $$

(35)

when the photocurrent reaches maximum (at t=t _p ) we can approximate

$$ \frac{I}{I_{\max}}\simeq0.99999 $$

(36)

which leads to

$$ \tau_{\mathrm{rise}} = -\frac{t_{p}}{\ln(0.00001)} $$

(37)

1.4.2 A.4.2 Comparison Between the Photocurrent Induced by Continuous 1 s and Brief 2 ms Optostimulation in Cell Expressing ChRwt and the Fast ChRET/TC Variant

We present in Fig. 10a comparison between the ChR wt and ChRET/TC photocurrent elicited by 1 s and 2 ms continuous optostimulation.

Fig. 10

Comparison example of the ChR2 photocurrent elicited by 1 s and 2 ms optostimulation. (A1) and (B1) Empirical data profile constructed for ChRwt and ChRET/TC variant using Eqs. (11) and (12) from the main paper as well as the experimental data provided in Table 1 is displayed for a continuous 1s optostimulation (A1) and for a brief 2 ms optostimulation (B1) The two normalized experimental photocurrent profiles are matching the experimental results reported in Berndt et al. (2011), Fig. 3c and Fig. 2d. (A2) and (B2) Photocurrent generated by the 3-state model starting from ideal initial conditions ((I): C(0)=1; O(0)=0; D(0)=0) for ChRwt and ChRET/TC variant for 1 s (A2), respectively, 2 ms (B2) optostimulation. (A3) and (B3) Photocurrent generated for ChRwt in comparison with ChR ET/TC, by the same 3-state model but starting from special initial conditions ((II): ChRwt: C(0)=0.0041; O(0)=0.0037; D(0)=0.9922; ChRET/TC: C(0)=0.00156; O(0)=0.0098; D(0)=0.9746) evaluated in Appendix A.3. (A4) and (B4) Photocurrent generated for the same ChR2 variants using the 4-state model with ideal initial conditions ((I): C ₁(0)=1; O ₁(0)=0; O ₂=0; C ₂(0)=0)

1.4.3 A.4.3 Dependence Between the Excitation Rate (P) and Light Intensity (I)

$$ P = \epsilon F = \epsilon\frac{\sigma_{\mathrm{ret}}\phi}{w_{\mathrm{loss}}} = \frac {\epsilon\sigma_{\mathrm{ret}}\lambda}{w_{\mathrm{loss}}hc}I $$

(38)

where ϵ is the quantum efficiency of photon absorbtion (a typical value for rhodopsin is ϵ≃0.5 Berndt et al. (2011)), F=σ _ret ϕ/w _loss is the number of photons absorbed by ChR2 molecule per unit time, σ _ret is the retinal cross-section (σ _ret≃1.2×10⁻²⁰ m² Berndt et al. 2011), w _loss is the measure of the loss of incidental photons due to scattering and absorbtion phenomena, ϕ=λI/hc is the photon flux per unit area, λ≃480 nm is the wave length of the light used in the stimulation protocol, I (mW/mm²) is the light intensity, $h = 6.626\times10^{-34}\ \mathrm{J\,s}$ is the Planck’s constant and c=3×10⁸ m/s is the speed of light in vacuum.

1.5 A.5 Derivation of the Semi-analytical Solution for the Light on Condition in the 4-State Model

The 4 state model can be written as follows:

$$\begin{aligned} \begin{aligned}[c] \dot{o_{1}} &= P_{1}(1-c_{2}-o_{1}-o_{2}) - (G_{d1}+e_{12})o_{1} + e_{21}o_{2} \\ \dot{o_{2}} &= P_{2}c_{2} + e_{12}o_{1} - (G_{d2}+e_{21})o_{2} \\ \dot{c_{2}} &= G_{d2}o_{2} - (P_{2}+G_{r})c_{2} \end{aligned} \end{aligned}$$

(39)

With the notation y=[o ₁ o ₂ c ₂]^T, Eq. (39) can be than expressed as follows:

$$\begin{aligned} \dot{y} =& \left [ \begin{matrix} -(P_{1}+G_{d1}+e_{12})& e_{21}-P_{1} & -P_{1} \\ e_{12} & -(G_{d2}+e_{21}) & P_{2} \\ 0 & G_{d2} & -(P_{2}+G_{r}) \end{matrix} \right ]y + \left [ \begin{matrix} P_{1}\\ 0\\ 0 \end{matrix} \right ] \\ =& Ay+P \end{aligned}$$

(40)

The general solution that needs to be evaluated can be written as

$$ y = y_{c}+y_{p} $$

(41)

where y _c of the homogeneous (complementary) solution and y _p is the particular solution of the system of Eqs. (39). In the following, we will evaluate both components.

1.5.1 A.5.4 Finding the Eigenvalues

The characteristic equation is

$$ \det(A - \lambda I) = 0 $$

(42)

or

$$ \left| \begin{matrix} -(P_{1}+G_{d1}+e_{12}) - \lambda& e_{21}-P_{1} & -P_{1} \\ e_{12} & -(G_{d2}+e_{21}) - \lambda& P_{2} \\ 0 & G_{d2} & -(P_{2}+G_{r}) - \lambda \end{matrix} \right| =0 $$

(43)

which leads to

$$\begin{aligned} & {-}\bigl[(P_{1}+G_{d1}+e_{12}+\lambda) (G_{d2}+e_{21}+\lambda)\bigr] - P_{1}e_{12}G_{d2} \\ &\quad {}+P_{2}G_{d2}(P_{1}+G_{d1}+e_{12}+ \lambda) + e_{12}(e_{21}-P_{1}) (P_{2}+G_{r}+\lambda) = 0 \end{aligned}$$

(44)

This equation is solved symbolically in Matlab using the commend solve, which gives the expressions for the solutions: λ ₁,λ ₂,λ ₃. The actual expressions are very elaborated, therefore they will not be included here. The numerical evaluation of these eigenvalues has been performed in Matlab using the function eval and the parameter values provided for each variant in the main paper, Table 3.

1.5.2 A.5.5 Finding the Eigenvectors

The characteristic equation is

$$ Av = \lambda v $$

(45)

or

$$ \left [ \begin{matrix} -(P_{1}+G_{d1}+e_{12})& e_{21}-P_{1} & -P_{1} \\ e_{12} & -(G_{d2}+e_{21}) & P_{2} \\ 0 & G_{d2} & -(P_{2}+G_{r}) \end{matrix} \right ] \left [ \begin{matrix} v_{1}\\ v_{2}\\ v_{3} \end{matrix} \right ] = \lambda_{1}\left [ \begin{matrix} v_{1}\\ v_{2}\\ v_{3} \end{matrix} \right ] $$

(46)

Then the eigenvectors satisfying this equation are

$$ \begin{aligned}[c] &v = \left [ \begin{matrix} 1\\ \frac{(P_{2}+G_{r}+\lambda_{1})(\lambda _{1}+P_{1}+G_{d1}+e_{12})}{(e_{21}-P_{1})(P_{2}-G_{r}+\lambda _{1})-P_{1}G_{d2}}\\ \frac{G_{d2}(\lambda _{1}+P_{1}+G_{d1}+e_{12})}{(e_{21}-P_{1})(P_{2}-G_{r}+\lambda_{1})-P_{1}G_{d2}} \end{matrix} \right ];\qquad u= \left [ \begin{matrix} 1\\ \frac{(P_{2}+G_{r}+\lambda_{2})(\lambda _{2}+P_{1}+G_{d1}+e_{12})}{(e_{21}-P_{1})(P_{2}-G_{r}+\lambda _{2})-P_{1}G_{d2}}\\ \frac{G_{d2}(\lambda _{2}+P_{1}+G_{d1}+e_{12})}{(e_{21}-P_{1})(P_{2}-G_{r}+\lambda_{2})-P_{1}G_{d2}} \end{matrix} \right ]; \\ & w= \left [ \begin{matrix} 1\\ \frac{(P_{2}+G_{r}+\lambda_{3})(\lambda _{3}+P_{1}+G_{d1}+e_{12})}{(e_{21}-P_{1})(P_{2}-G_{r}+\lambda _{3})-P_{1}G_{d2}}\\ \frac{G_{d2}(\lambda _{3}+P_{1}+G_{d1}+e_{12})}{(e_{21}-P_{1})(P_{2}-G_{r}+\lambda_{3})-P_{1}G_{d2}} \end{matrix} \right ]. \end{aligned} $$

(47)

1.5.3 A.5.6 The Complementary Solution

The complementary solution can then be written as

$$ y_{c} = C_{1}e^{\lambda_{1}t}v + C_{2}e^{\lambda_{2}t}u + C_{3}e^{\lambda_{3}t}w $$

(48)

or

$$\begin{aligned} y_{c} &= \left[ \begin{matrix} e^{\lambda_{1}t}&e^{\lambda_{2}t}\\ \frac{(P_{2}+G_{r}+\lambda_{1})(\lambda _{1}+P_{1}+G_{d1}+e_{12})}{(e_{21}-P_{1})(P_{2}-G_{r}+\lambda _{1})-P_{1}G_{d2}}e^{\lambda_{1}t}& \frac{(P_{2}+G_{r}+\lambda_{2})(\lambda _{2}+P_{1}+G_{d1}+e_{12})}{(e_{21}-P_{1})(P_{2}-G_{r}+\lambda _{2})-P_{1}G_{d2}}e^{\lambda_{2}t} \\ \frac{G_{d2}(\lambda _{1}+P_{1}+G_{d1}+e_{12})}{(e_{21}-P_{1})(P_{2}-G_{r}+\lambda _{1})-P_{1}G_{d2}}e^{\lambda_{1}t}& \frac{G_{d2}(\lambda _{2}+P_{1}+G_{d1}+e_{12})}{(e_{21}-P_{1})(P_{2}-G_{r}+\lambda _{2})-P_{1}G_{d2}}e^{\lambda_{2}t} \end{matrix} \right. \\ & \left.\begin{matrix} e^{\lambda_{3}t}\\ \frac{(P_{2}+G_{r}+\lambda_{3})(\lambda _{3}+P_{1}+G_{d1}+e_{12})}{(e_{21}-P_{1})(P_{2}-G_{r}+\lambda _{3})-P_{1}G_{d2}}e^{\lambda_{3}t}\\ \frac{G_{d2}(\lambda _{3}+P_{1}+G_{d1}+e_{12})}{(e_{21}-P_{1})(P_{2}-G_{r}+\lambda _{3})-P_{1}G_{d2}}e^{\lambda_{3}t} \end{matrix}\right] \left [ \begin{matrix} C_{1}\\ C_{2}\\ C_{3} \end{matrix} \right ] \end{aligned}$$

(49)

or, by notation

$$ y_{c} = A_{c}C $$

(50)

1.5.4 A.5.7 Finding the Particular Solution

The following analytical steps are performed in Matlab:

• (a) We evaluate the inverse of the complementary matrix $A_{c}^{-1}$;

• (b) We evaluate the product:

  $$ Z = A_{c}^{-1}\cdot P = A_{c}^{-1} \left [ \begin{matrix} P_{1}\\ 0\\ 0 \end{matrix} \right ] $$

  (51)

• (c) We integrate symbolically Z using the commend: R=int(Z,t);

• (d) We evaluate the particular solution: y _p =A _c R.

1.5.5 A.5.8 Finding Coefficients C ₁,C ₂,C ₃ by use of Initial Conditions

To coefficients of the homogeneous solution can be found by considering the following condition:

$$ A_{c}(0)C +y_{p}(0) = y_{0} $$

(52)

The above system of equations cannot be solved in Matlab straightforward as the function solve is unable to find a solution when all three equations are presented simultaneously; the system can be solved, however, if the analytical expression of two of the coefficients is given and only one equation is solved symbolically. For this purpose, we derive the equations for the first and second coefficient and solve symbolically in Matlab only for the last coefficient. We expand the above equation as follows:

$$ \left [ \begin{matrix} A_{c_{11}}(0)&A_{c_{12}}(0)&A_{c_{13}}(0)\\ A_{c_{21}}(0)&A_{c_{22}}(0)&A_{c_{23}}(0)\\ A_{c_{31}}(0)&A_{c_{32}}(0)&A_{c_{33}}(0) \end{matrix} \right ]\left [ \begin{matrix} C_{1}\\ C_{2}\\ C_{3} \end{matrix} \right ] + \left [ \begin{matrix} y_{p_{1}}(0)\\ y_{p_{2}}(0)\\ y_{p_{3}}(0) \end{matrix} \right ] = \left [ \begin{matrix} 0\\ 0\\ 0 \end{matrix} \right ] $$

(53)

or

$$\begin{aligned} & A_{c_{11}}(0)C_{1} + A_{c_{12}}(0)C_{2} + A_{c_{13}}(0)C_{3} +y_{p_{1}}(0) = 0 \\ &A_{c_{21}}(0)C_{1} + A_{c_{22}}(0)C_{2} + A_{c_{23}}(0)C_{3} +y_{p_{2}}(0) = 0 \\ &A_{c_{31}}(0)C_{1} + A_{c_{32}}(0)C_{2} + A_{c_{33}}(0)C_{3} +y_{p_{3}}(0) = 0 \end{aligned}$$

(54)

We get

$$ C_{1} = \bigl(-y_{p_{1}}(0) - A_{c_{12}}(0)C_{2} - A_{c_{13}}(0)C_{3} \bigr)/A_{c_{11}}(0) $$

(55)

and

$$ C_{2} = \frac{A_{c_{21}}(0)y_{p_{1}}(0) - A_{c_{11}}(0)y_{p_{2}}(0) - C_{3}(A_{c_{23}}(0)A_{c_{11}}(0) - A_{c_{21}}(0)A_{c_{13}})(0)}{A_{c_{22}}(0)A_{c_{11}}(0) - A_{c_{21}}(0)A_{c_{12}}(0)} $$

(56)

We introduce Eqs. (55) and (56) in the last Eq. (54) and then solve symbolically this equation in Matlab for C ₃; we then evaluate C ₁ and C ₂.

We are now able to evaluate the full theoretical solution:

$$\begin{aligned} \begin{aligned}[c] o_{1} &= C_{1}e^{\lambda_{1}t} +C_{2}e^{\lambda _{2}t}+C_{3}e^{\lambda_{3}t}+y_{p_{1}} \\ o_{2} &= A_{c_{21}}(0)C_{1}e^{\lambda_{1}t} + A_{c_{22}}(0)C_{2}e^{\lambda_{2}t}+A_{c_{23}}(0)C_{3}e^{\lambda_{3}t} + y_{p_{2}} \end{aligned} \end{aligned}$$

(57)

The photocurrent elicited in light on condition will be

$$ I = Vg_{1}(o_{1}+\gamma o_{2}) $$

(58)

or

$$\begin{aligned} I &= Vg_{1}\bigl[C_{1}\bigl(1+\gamma A_{c_{21}}(0)\bigr)e^{\lambda_{1}t} + C_{2}\bigl(1+\gamma A_{c_{22}}(0)\bigr)e^{\lambda_{2}t} + C_{3}\bigl(1+\gamma A_{c_{23}}(0)\bigr)e^{\lambda_{3}t}\bigr] \\ &\quad+ Vg_{1}(y_{p1}+ \gamma y_{p2}) \end{aligned}$$

(59)

The reader should note that all the eigenvalues above are negative. Therefore, we can write

$$ \lambda_{1} = -\varLambda_{1};\qquad \lambda_{2} = -\varLambda_{2}; \qquad\lambda_{3} = -\varLambda_{3} $$

(60)

where Λ ₁,Λ ₂,Λ ₃>0.

We evaluate the time constants associated with these eigenvalues to be

$$ \tau_{1} = \frac{1}{\varLambda_{1}}; \qquad\tau_{2} = \frac {1}{\varLambda_{2}}; \qquad\tau_{3} = \frac{1}{\varLambda_{3}} $$

(61)

The smallest time constant is associated with the rise phase (from zero to peak) of the photocurrent (the beginning of the optostimulation); the other two controls the fast and slow component of the photocurrent decay from peak to steady state. With the parameters found in the manuscript Table 3, we identify:

$$\begin{aligned} & \tau_{\mathrm{rise}(\mathrm{on})} = \tau_{2};\qquad\tau_{s(\mathrm{on})} = \tau_{1};\qquad \tau_{f(\mathrm{on})} = \tau_{3} \end{aligned}$$

(62)

$$\begin{aligned} & \begin{aligned}[c] &A_{\mathrm{rise}(\mathrm{on})} = C_{2}\bigl(1+\gamma A_{c_{22}}(0)\bigr); \qquad A_{s(\mathrm{on})} = C_{1}\bigl(1+ \gamma A_{c_{21}}(0)\bigr); \\ & A_{f(\mathrm{on})} = C_{3} \bigl(1+\gamma A_{c_{23}}(0)\bigr) \end{aligned} \end{aligned}$$

(63)

$$\begin{aligned} & \begin{aligned}[c] &I_{\mathrm{rise}(\mathrm{on})} = Vg_{1}A_{\mathrm{rise}(\mathrm{on})}; \qquad I_{s(\mathrm{on})} = Vg_{1}A_{s(\mathrm{on})};\\ & I_{f(\mathrm{on})} = Vg_{1}A_{f(\mathrm{on})}; \qquad I_{\mathrm {plat}} = Vg_{1}(y_{p1} + \gamma y_{p2}) \end{aligned} \end{aligned}$$

(64)

$$\begin{aligned} & I_{\mathrm{on}}(t) = I_{\mathrm{rise}(\mathrm{on})}e^{-\varLambda_{\mathrm{rise}(\mathrm{on})} t} + I_{s(\mathrm{on})}e^{-\varLambda_{s(\mathrm{on})} t} + I_{f(\mathrm{on})}e^{-\varLambda_{f(\mathrm{on})} t} +I_{\mathrm{plat}} \end{aligned}$$

(65)

For convenience, we reproduce here the analytical solution for light off condition provided in Gunaydin et al. 2010:

$$\begin{aligned} & \varLambda_{1} = b-c; \qquad\varLambda_{2} = b+c; \end{aligned}$$

(66)

$$\begin{aligned} & b = \frac{G_{d1}+G_{d2}+e_{12}+e_{21}}{2}; \qquad c = \sqrt {b^{2}-(G_{d1}G_{d2}+G_{d1}e_{21}+G_{d2}e_{12})} \end{aligned}$$

(67)

$$\begin{aligned} & I_{\mathrm{off}} = I_{s(\mathrm{off})}e^{-\varLambda_{s(\mathrm{off})} t} + I_{f(\mathrm{off})}e^{-\varLambda_{f(\mathrm{off})} t}; \end{aligned}$$

(68)

where

$$ I_{s(\mathrm{off})} = Vg_{1}A_{s(\mathrm{off})};\qquad I_{f(\mathrm{off})} = Vg_{1}A_{f(\mathrm{off})} $$

(69)

and

$$\begin{aligned} \begin{aligned}[c] &A_{s(\mathrm{off})} = \frac{[\varLambda_{2} - (G_{d1}+(1-\gamma)e_{12})]O_{10} + [(1-\gamma)e_{21} + \gamma(\varLambda_{2}-G_{d2})]O_{20}}{\varLambda _{2}-\varLambda_{1}} \\ &A_{f(\mathrm{off})} = \frac{[G_{d1} + (1-\gamma)e_{12} - \varLambda_{1}]O_{10} + [\gamma(G_{d2}-\varLambda_{1}) - (1-\gamma)e_{21}]O_{20}}{\varLambda _{2}-\varLambda_{1}} \end{aligned} \end{aligned}$$

(70)