Appendix B3: Sigworth 1983/1995 Rs compensation¶

Appendix B discusses and compares patch clamp model equations

In this notebook, we take a detailed look at the $R_s$ compensation and slow capacitance transient cancellation scheme in figures 18 and 19 of Sigworth 1995a, and re-derive the equations found in [REF-TO-PREPRINT-WITH-SUPERCHARGING].

First, we name the voltage after the $T_\text{sc}$ block $V_\text{est}$. To find an equation, we detransform the transfer function:

$$\begin{align} T_\text{sc} = \frac{A_2}{R_2C_2s + 1} \end{align}$$

Looking back a few pages, we see that $C_2$ is fixed, but $R_2$ is chosen so that $R_2C_2 \approx R_sC_m$. Translating to our own notation, and ignoring the true electronic implementation, we write

$$\begin{align} R_2C_2 = R_s^*C_m^* \end{align}$$

Similarly, $A_2$ was chosen so that $A_2C_i \approx C_m$, where $C_i$ is the fixed capacitance used for slow transient cancellation. Although this may be a seperate estimate in practice, we will simplify by writing

$$\begin{align} A_2C_i = C_m^* \end{align}$$

Going back to the transfer function, we find

$$\begin{align} T_\text{sc} = \frac{A_2}{R_s^*C_m^*s + 1} \quad \rightarrow \quad V_\text{est} + R_s^*C_m^*\dot{V}_\text{est} = A_2 V'_c \end{align}$$

for

$$\begin{align} \dot{V}_\text{est} = \frac{A_2 V'_c - V_\text{est}}{R_s^*C_m^*} \end{align}$$

In words, $V_\text{est}$ is $A_2$ times larger than $V'_c$, and lags behind it with a time constant set by our estimates of $R_s$ and $C_m$.

The updated command voltage¶

To find the voltage $V'_c$ we write the equation for the $\alpha R_s s C_i$ block, using "$\beta$" instead of Sigworth's "$\alpha$" and "$R_s^*$" instead of "$R_s$":

$$\begin{align} V'_c &= V_c + \beta R_s^*C_i\dot{V}_\text{est} \\ &= V_c + \beta R_s^*C_i \frac{A_2 V'_c - V_\text{est}}{R_s^*C_m^*} \\ (1 - \beta) V'_c &= V_c - \beta\frac{C_i}{C_m^*}V_\text{est} \\ V'_c &= \frac{V_c - \beta\frac{C_i}{C_m^*}V_\text{est}}{(1 - \beta)} \end{align}$$

Substituting this into the equation for $\dot{V}_\text{est}$ we find

$$\begin{align} R_s^*C_m^* \dot{V}_\text{est} &= A_2V'_c - V_\text{est} \\ (1 - \beta)R_s^*C_m^* \dot{V}_\text{est} &= A_2V_c - \beta\frac{A_2C_i}{C_m^*}V_\text{est} - (1 - \beta)V_\text{est} \\ &= A_2V_c - (\beta + 1 - \beta)V_\text{est} \end{align}$$

for

$$\begin{align} \dot{V}_\text{est} &= \frac{A_2V_c - V_\text{est}}{(1 - \beta)R_s^*C_m^*} \end{align}$$

In words, $V_\text{est}$ is $A_2$ times larger than $V_c$, and lags behind it with a time constant set by our estimates of the membrane capacitance and the uncompensated fraction of series resistance $(1-\beta)R_s^*$.

As a result, the term fed back into $V'_c$ is $\beta R_s^*C_m^*\dot{V}_\text{est}$ which can be understood as $\beta R_s^*C_m^* \dot{V'}_\text{c-with-lag}$.

Slow transient cancellation¶

Slow transient cancellation is implemented by feeding $\dot{V}_\text{est}$ into a capacitor $C_i$, leading to a term

$$\begin{align} I_\text{SC} = A_2C_i\dot{V}_\text{est} = C_m^*\dot{V}_\text{est} \end{align}$$

Series resistance compensation¶

The original series resistance compensation takes the updated command potential as input:

$$\begin{align} \dot{V}_\text{ref} = \frac{V'_c + \alpha R_s^*I_\text{obs} - V_\text{ref}}{\tau_\text{sum}} \end{align}$$

Simplification¶

We can omit the "implementation details" involving $A_2$ to write:

$$\begin{align} \dot{V}_\text{est} &= \frac{V_c - V_\text{est}}{(1 - \beta)R_s^*C_m^*} && \text{Estimate of }V_m \end{align}$$ $$\begin{align} V'_c &= V_c + \beta R_s^*C_m^*\dot{V}_\text{est} && \text{Prediction} \end{align}$$ $$\begin{align} V_\text{ref} &= V'_c + \alpha R_s^* I_\text{obs} && \text{Correction} \end{align}$$ $$\begin{align} I_\text{SC} = C_m^* \dot{V}_\text{est} && \text{Slow capacitance correction} \end{align}$$ $$\begin{align} I_\text{FC} = C_p^*\dot{V}_\text{ref} && \text{Fast capacitance correction} \end{align}$$

Although we used $V_c'$ in the derivation of $\dot{V}_\text{est}$, it only appears in the equation for $V_\text{ref}$ in the final model, and so we can simplify further by writing

$$\begin{align} V_\text{ref} &= V_c + \alpha R_s^* I_\text{obs} + \beta R_s^* C_m^* \dot{V}_\text{est} && \text{Correction-Prediction} \end{align}$$

A schematic for this set-up is shown below

A schematic showing series resistance prediction and correction and separate pathways for fast and slow capacitance correction.