Unify Delta rheobase and normalized delta rheobase throughout methods and results by creating new commands \drheo and \ndAUC

Figures/firing_characterization-jb.py

@@ -227,7 +227,7 @@ def plot_diff_sqrt(ax, a=1, b=0.2, c=100, d=0, a2=1, b2=0.2, c2=100, d2=0):
 def plot_quadrant(ax):
     ax.spines['left'].set_position('zero')
     ax.spines['bottom'].set_position('zero')
-    ax.text(1.2, 0.05, '$\\Delta$ rheobase', ha='right')
+    ax.text(1.2, -0.15, '$\\Delta$ rheobase', ha='right')
     ax.text(-0.03, 0.7, '$\\Delta$ AUC', ha='right', rotation=90)
     ax.tick_params(length=0)
     ax.set_xlim(-1, 1)

manuscript.tex

@@ -68,6 +68,11 @@
 \usepackage{xspace}
 \newcommand{\Kv}{\(\textrm{K}_{\textrm{V}}\textrm{1.1}\)\xspace}
 \newcommand{\IKv}{\(\textrm{I}_{\textrm{K}_{\textrm{V}}\textrm{1.1}}\)\xspace}
+\newcommand{\drheo}{\(\Delta\)rheobase\xspace}
+\newcommand{\ndAUC}{normalized \(\Delta\)AUC\xspace}
+
+
+
 \DeclareFloatingEnvironment[fileext=lop]{Extended Data}
 
 \newcommand{\beginsupplement}{
@@ -218,7 +223,7 @@ All modelling and simulation was done in parallel with custom written Python 3.8
     % Linux 3.10.0-123.e17.x86_64.
     
 \subsection*{Different Cell Models}
-    A group of neuronal models representing the major classes of cortical and thalamic neurons including regular spiking pyramidal (RS pyramidal), regular spiking inhibitory (RS inhibitory), and fast spiking (FS) cells were used \citep{pospischil_minimal_2008}. To each of these models, a \Kv current (\IKv; \citealt{ranjan_kinetic_2019}) was added. A cerebellar stellate cell model from \citet{alexander_cerebellar_2019} is used (Cb stellate). This model was also used with a \Kv current \citep{ranjan_kinetic_2019} in addition to the A-type potassium current (Cb stellate +\Kv) or replacing the A-type potassium current (Cb stellate \(\Delta\)\Kv). A subthalamic nucleus neuron model as described by \citet{otsuka_conductance-based_2004} are used (STN) and with a \Kv current (\IKv; \citealp{ranjan_kinetic_2019}) in addition to the A-type potassium current (STN +\Kv) or replacing the A-type potassium current (STN \(\Delta\)\Kv). The properties and conductances of each model are detailed in \Cref{tab:g} and the gating properties are unaltered from the original Cb stellate and STN models. For comparability to typical electrophysiological data fitting reported and for ease of further gating curve manipulations, a modified Boltzmann function
+    A group of neuronal models representing the major classes of cortical and thalamic neurons including regular spiking pyramidal (RS pyramidal), regular spiking inhibitory (RS inhibitory), and fast spiking (FS) cells were used \citep{pospischil_minimal_2008}. To each of these models, a \Kv current (\IKv; \citealt{ranjan_kinetic_2019}) was added. A cerebellar stellate cell model from \citet{alexander_cerebellar_2019} is used (Cb stellate). This model was also used with a \Kv current \citep{ranjan_kinetic_2019} in addition to the A-type potassium current (Cb stellate +\Kv) or replacing the A-type potassium current (Cb stellate \(\Delta\)\Kv). A subthalamic nucleus neuron model as described by \citet{otsuka_conductance-based_2004} are used (STN) and with a \Kv current (\IKv; \citealp{ranjan_kinetic_2019}) in addition to the A-type potassium current (STN +\Kv) or replacing the A-type potassium current (STN \(\Delta\)\Kv). The properties and maximal conductances of each model are detailed in \Cref{tab:g} and the gating properties are unaltered from the original Cb stellate and STN models. For comparability to typical electrophysiological data fitting reported and for ease of further gating curve manipulations, a modified Boltzmann function
     \begin{equation}\label{eqn:Boltz}
          x_\infty = {\left(\frac{1-a}{1+{\exp\left[{\frac{V-V_{1/2}}{k}}\right]}} +a\right)^j} 
     \end{equation}
@@ -246,17 +251,17 @@ with slope \(k\), voltage for half-maximal activation or inactivation (\(V_{1/2}
     Properties of ionic currents common to all models (\(\textrm{I}_{\textrm{Na}}\), \(\textrm{I}_{\textrm{K}}\), \(\textrm{I}_{\textrm{A}}\)/\IKv, and \(\textrm{I}_{\textrm{Leak}}\)) were systematically altered in a one-factor-at-a-time sensitivity analysis for all models. The gating curves for each current were shifted (\(\Delta V_{1/2}\)) from -10 to 10\,mV in increments of 1\,mV. The voltage dependence of the time constant associated with the shifted gating curve was correspondingly shifted. The slope (\(k\)) of the gating curves were altered from half to twice the initial slope. Similarly, the maximal current conductance (\(g\)) was also scaled from half to twice the initial value. For both slope and conductance alterations, alterations consisted of 21 steps spaced equally on a \(\textrm{log}_2\) scale.
     
     \subsection*{Model Comparison}
-    Changes in rheobase (\(\Delta rheobase\)) are calculated in relation to the original model rheobase. The contrast of each AUC value (\(AUC_i\)) was computed in comparison to the AUC of the unaltered wild type model (\(AUC_{wt}\))
+    Changes in rheobase (\drheo) are calculated in relation to the original model rheobase. The contrast of each AUC value (\(AUC_i\)) was computed in comparison to the AUC of the unaltered wild type model (\(AUC_{wt}\))
     \begin{equation}\label{eqn:AUC_contrast}
-        AUC_{contrast} = \frac{AUC_i - AUC_{wt}}{AUC_{wt}}
+       \textrm{normalized }  \Delta \textrm{AUC} = \frac{AUC_i - AUC_{wt}}{AUC_{wt}}
     \end{equation}
 
-    To assess whether the effects of a given alteration on \(AUC_{contrast}\) or \(\Delta rheobase\) are robust across models, the correlation between \(AUC_{contrast}\) or \(\Delta rheobase\) and the magnitude of the alteration of a current property was computed for each alteration in each model and compared across alteration types.
+    To assess whether the effects of a given alteration on \ndAUC  or \drheo are robust across models, the correlation between \ndAUC or \drheo and the magnitude of the alteration of a current property was computed for each alteration in each model and compared across alteration types.
     
     The Kendall's \(\tau\) coefficient, a non-parametric rank correlation, is used to describe the relationship between the magnitude of the alteration and AUC or rheobase values. A Kendall \(\tau\) value of -1 or 1 is indicative of monotonically decreasing and increasing relationships respectively.
     
     \subsection*{KCNA1/\Kv Mutations}\label{subsec:mut}
-    Known episodic ataxia type~1 associated KCNA1 mutations and their electrophysiological characterization reviewed in \citet{lauxmann_therapeutic_2021}. The mutation-induced changes in \IKv amplitude and activation slope (\(k\)) were normalized to wild type measurements and changes in activation \(V_{1/2}\) were used relative to wild type measurements. The effects of a mutation were also applied to \(\textrm{I}_{\textrm{A}}\) when present as both potassium currents display prominent inactivation. In all cases, the mutation effects were applied to half of the \Kv or \(\textrm{I}_{\textrm{A}}\) under the assumption that the heterozygous mutation results in 50\% of channels carrying the mutation. Frequency-current curves for each mutation in each model were obtained through simulation and used to characterize firing behaviour as described above. For each model the differences in mutation AUC to wild type AUC were normalized by wild type AUC (\(AUC_{contrast}\)) and mutation rheobases are compared to wild type rheobase values (\(\Delta rheobase\)). Pairwise Kendall rank correlations (Kendall \(\tau\)) are used to compare the the correlation in the effects of \Kv mutations on AUC and rheobase between models.
+    Known episodic ataxia type~1 associated KCNA1 mutations and their electrophysiological characterization reviewed in \citet{lauxmann_therapeutic_2021}. The mutation-induced changes in \IKv amplitude and activation slope (\(k\)) were normalized to wild type measurements and changes in activation \(V_{1/2}\) were used relative to wild type measurements. The effects of a mutation were also applied to \(\textrm{I}_{\textrm{A}}\) when present as both potassium currents display prominent inactivation. In all cases, the mutation effects were applied to half of the \Kv or \(\textrm{I}_{\textrm{A}}\) under the assumption that the heterozygous mutation results in 50\% of channels carrying the mutation. Frequency-current curves for each mutation in each model were obtained through simulation and used to characterize firing behaviour as described above. For each model the differences in mutation AUC to wild type AUC were normalized by wild type AUC (\ndAUC) and mutation rheobases are compared to wild type rheobase values (\drheo). Pairwise Kendall rank correlations (Kendall \(\tau\)) are used to compare the the correlation in the effects of \Kv mutations on AUC and rheobase between models.
 
 
 
@@ -297,20 +302,20 @@ Changes in firing as characterized by \(\Delta\)AUC and \(\Delta\)rheobase occup
 
 Neuronal firing is a complex phenomenon and a quantification of firing properties is required for comparisons across cell types and between conditions. Here we focus on two aspects of firing: rheobase, the smallest injected current at which the cell fires an action potential, and the shape of the frequency-current (fI) curve as quantified by the area under the curve (AUC) for a fixed range of input currents above rheobase (\Cref{fig:firing_characterizaton}A). \notejb{this way we make the AUC independent from changes in rheobase} \notenk{How about ``This enables a AUC measurement independent from rheobase.''}The characterization of firing by rheobase and AUC allows to characterize both a neuron's excitability in the sub-threshold regime (rheobase) and periodic firing in the super-threshold regime (AUC). Note that AUC is essentially quantifying the slope of a neuron's fI curve.
 
-Using these two measures we quantify the effects a changed property of an ionic current has on neural firing by the differences in both rheobase, $\Delta$rheobase, and in AUC, $\Delta$AUC, relative to the wild type neuron. $\Delta$AUC is in addition normalized to the AUC if the wild type neuron, see Eq.~\eqref{eqn:AUC_contrast}. Each fI curve resulting from an altered ionic current is a point in a two-dimensional coordinate system spanned by $\Delta$rheobase and \notejb{normalized $\Delta$AUC} (\Cref{fig:firing_characterizaton}B). An fI curve similar to the one of the wild type neuron is marked by a point close to the origin. In the upper left quadrant, fI curves become steeper (positive difference of AUC values: \(+\Delta\)AUC) and are shifted to lower rheobases (negative difference of rheobases: \(-\Delta\)rheobase), unambigously indicating an increased firing that clearly might be classified as a GOF of neuronal firing. The opposite happens in the bottom right quadrant where the slope of fI curves decreases (\(-\Delta\)AUC) and the rheobase is shifted to higher currents (\(+\Delta\)rheobase), indicating a decreased, LOF firing. In the lower left (\(-\Delta\)AUC and \(-\Delta\)rheobase) and upper right (\(+\Delta\)AUC and \(+\Delta\)rheobase) quadrants, the effects on firing are less clear-cut, because the changes in rheobase and AUC have opposite effects on neuronal firing. Changes in a neuron's fI curves in these two quadrants cannot uniquely be described as a gain or loss of excitability. In these cases it depends on the regime the neuron is operating in. If it is in its excitable regime and only occasionaly generates an action potential, then the effect on the rheobase matters more. If it is firing periodically with high rates, then the change in AUC might be more relevant.
+Using these two measures we quantify the effects a changed property of an ionic current has on neural firing by the differences in both rheobase, \drheo, and in AUC, \(\Delta\)AUC, relative to the wild type neuron. \(\Delta\)AUC is in addition normalized to the AUC if the wild type neuron, see Eq.~\eqref{eqn:AUC_contrast}. Each fI curve resulting from an altered ionic current is a point in a two-dimensional coordinate system spanned by \drheo and \notejb{normalized \(\Delta\)AUC} (\Cref{fig:firing_characterizaton}B). An fI curve similar to the one of the wild type neuron is marked by a point close to the origin. In the upper left quadrant, fI curves become steeper (positive difference of AUC values: \(+\Delta\)AUC) and are shifted to lower rheobases (negative difference of rheobases: \(-\)\drheo), unambigously indicating an increased firing that clearly might be classified as a GOF of neuronal firing. The opposite happens in the bottom right quadrant where the slope of fI curves decreases (\(-\Delta\)AUC) and the rheobase is shifted to higher currents (\(+\)\drheo), indicating a decreased, LOF firing. In the lower left (\(-\Delta\)AUC and \(-\)\drheo) and upper right (\(+\Delta\)AUC and \(+\)\drheo) quadrants, the effects on firing are less clear-cut, because the changes in rheobase and AUC have opposite effects on neuronal firing. Changes in a neuron's fI curves in these two quadrants cannot uniquely be described as a gain or loss of excitability. In these cases it depends on the regime the neuron is operating in. If it is in its excitable regime and only occasionaly generates an action potential, then the effect on the rheobase matters more. If it is firing periodically with high rates, then the change in AUC might be more relevant.
 
 \subsection*{Sensitivity Analysis}
 Sensitivity analyses are used to understand how input model parameters contribute to determining the output of a model \citep{Saltelli2002}. In other words, sensitivity analyses are used to understand how sensitive the output of a model is to a change in input or model parameters. One-factor-a-time sensitivity analyses involve altering one parameter at a time and assessing the impact of this parameter on the output. This approach enables the comparison of given alterations in parameters of ionic currents across models.
 
-For example, when shifting the half-activation voltage \(V_{1/2}\) of the potassium current in the FS +\Kv model to more depolarized values, then the rheobase of the resulting fI curves shifts to lower currents \(-\Delta\)rheobase), making the neuron more sensitive to weak inputs, but at the same time the slope of the fI curves is reduced (\(-\Delta\)AUC), reducing firing rate (\Cref{fig:AUC_correlation}~A). \notejb{Refer to Fig 2B} Plotting the corresponding changes in AUC against the change in \(V_{1/2}\) results in a monotonously falling curve (thick orange line in \Cref{fig:AUC_correlation}~B). For each of the many models we get a different relation between the changes in AUC and the shifts in \(V_{1/2}\) (thin lines in \Cref{fig:AUC_correlation}~B). To further summarize these different dependencies of the various models we characterize each of these curve by a single number, the \( \text{Kendall} \tau \) correlation coefficient. A monotonously increasing curve results in a \( \text{Kendall} \tau \) close to $+1$, a monotounsly decreasing curve in \( \text{Kendall} \tau \approx -1 \), and a non-monotonous, non-linear relation relation in \( \text{Kendall} \tau \) close to zero (compare lines in \Cref{fig:AUC_correlation}~B with dots in black box in panel C).
+For example, when shifting the half-activation voltage \(V_{1/2}\) of the potassium current in the FS +\Kv model to more depolarized values, then the rheobase of the resulting fI curves shifts to lower currents \(-\)\drheo), making the neuron more sensitive to weak inputs, but at the same time the slope of the fI curves is reduced (\(-\)\ndAUC), reducing firing rate (\Cref{fig:AUC_correlation}~A). \notejb{Refer to Fig 2B} Plotting the corresponding changes in AUC against the change in \(V_{1/2}\) results in a monotonously falling curve (thick orange line in \Cref{fig:AUC_correlation}~B). For each of the many models we get a different relation between the changes in AUC and the shifts in \(V_{1/2}\) (thin lines in \Cref{fig:AUC_correlation}~B). To further summarize these different dependencies of the various models we characterize each of these curve by a single number, the \( \text{Kendall} \ \tau \) correlation coefficient. A monotonously increasing curve results in a \( \text{Kendall} \ \tau \) close to \(+1\), a monotounsly decreasing curve in \( \text{Kendall} \ \tau \approx -1 \), and a non-monotonous, non-linear relation relation in \( \text{Kendall} \ \tau \) close to zero (compare lines in \Cref{fig:AUC_correlation}~B with dots in black box in panel C).
 
-Changes in gating \(V_{1/2}\) and slope factor \(k\) as well as the maximum conductance affect AUC (\Cref{fig:AUC_correlation}), but how exactly AUC is affected usually depends on the specific models. Increasing, for example, the slope factor of the \Kv activation curve, increases the AUC in all models (\( \text{Kendall} \tau \approx +1\)), but with different slopes (\Cref{fig:AUC_correlation}~B). Similar consistent positive correlations can be found for shifts in A-current activation \(V_{1/2}\). Changes in \Kv activation \(V_{1/2}\) and in A-current conductance result in negative correlations with the AUC in all models (\( \text{Kendall} \tau \approx -1\)).
+Changes in gating \(V_{1/2}\) and slope factor \(k\) as well as the maximum conductance \(g\) affect AUC (\Cref{fig:AUC_correlation}), but how exactly AUC is affected usually depends on the specific models. Increasing, for example, the slope factor of the \Kv activation curve, increases the AUC in all models (\( \text{Kendall} \ \tau \approx +1\)), but with different slopes (\Cref{fig:AUC_correlation}~B). Similar consistent positive correlations can be found for shifts in A-current activation \(V_{1/2}\). Changes in \Kv activation \(V_{1/2}\) and in maximal A-current conductance result in negative correlations with the AUC in all models (\( \text{Kendall} \ \tau \approx -1\)).
 
-Qualitative differences can be found, for example, when increasing the conductance of the delayed rectifier (\Cref{fig:AUC_correlation}~C). In some model neurons this increased AUC (\( \text{Kendall} \tau \approx +1\)), whereas in others AUC is decreased (\( \text{Kendall} \tau \approx -1\)). In the STN +\Kv model, AUC depends in a non-linear way on the conductance of the delayed rectifier, resulting in an \( \text{Kendall} \tau \) close to zero. Even more dramatic qualitative differences between models result from shifts of the activation curve of the delayed rectifier, as discussed already above (\Cref{fig:AUC_correlation}~A). Some model neurons do almost not depend on changes in K-current activation \(V_{1/2}\) or show strongly non-linear dependencies, both resulting in \( \text{Kendall} \tau\) close to zero. Many model neurons show strongly negative correlations, and a few show positive correlations with shifting the activation curve of the delayed rectifier.
+Qualitative differences can be found, for example, when increasing the maximal conductance of the delayed rectifier (\Cref{fig:AUC_correlation}~C). In some model neurons this increased AUC (\( \text{Kendall} \ \tau \approx +1\)), whereas in others AUC is decreased (\( \text{Kendall} \ \tau \approx -1\)). In the STN +\Kv model, AUC depends in a non-linear way on the maximal conductance of the delayed rectifier, resulting in an \( \text{Kendall} \ \tau \) close to zero. Even more dramatic qualitative differences between models result from shifts of the activation curve of the delayed rectifier, as discussed already above (\Cref{fig:AUC_correlation}~A). Some model neurons do almost not depend on changes in K-current activation \(V_{1/2}\) or show strongly non-linear dependencies, both resulting in \( \text{Kendall} \ \tau\) close to zero. Many model neurons show strongly negative correlations, and a few show positive correlations with shifting the activation curve of the delayed rectifier.
 
 \notejb{the slope factor has a name (``slope factor''), but \(V_{1/2}\) not. How is this called, ``midpoint potential''? ``half activation potential/voltage''}\notenk{How about ``half-maximal potential''?}
 
-\notejb{``maximum conductance''}
+\notejb{``maximum conductance''} \notenk{fixed this in results section}
 
 
 \begin{figure}[tp]
@@ -324,7 +329,7 @@ Qualitative differences can be found, for example, when increasing the conductan
 %        \\\notejb{second column: x-label are wrong, i.e. $k/k_{WT}$, $g/g_{WT}$}\notenk{done}
 %        \\\notejb{Make subplot size exactly like in Figure 4}\notenk{done}
         \linespread{1.}\selectfont
-        \caption[]{Effects of altered channel kinetics on AUC in various neuron models. The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in \(V_{1/2}\) and AUC, slope factor k and AUC as well as current conductances and AUC for each model are shown on the right in (A), (B) and (C) respectively. The relationships between AUC and \(\Delta V_{1/2}\), slope (k) and conductance (g) for the Kendall \(\tau\) coefficients highlights by the black box are depicted in the middle panel. The fI curves corresponding to one of the models are shown in the left panels.}
+        \caption[]{Effects of altered channel kinetics on AUC in various neuron models. The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in \(V_{1/2}\) and AUC, slope factor k and AUC as well as maximal current conductances and AUC for each model are shown on the right in (A), (B) and (C) respectively. The relationships between AUC and \(\Delta V_{1/2}\), slope (k) and maximal conductance (g) for the Kendall \(\tau\) coefficients highlights by the black box are depicted in the middle panel. The fI curves corresponding to one of the models are shown in the left panels.}
         \label{fig:AUC_correlation}
 \end{figure}
 
@@ -336,7 +341,7 @@ The rheobase is also affected by changes in channel kinetics (\Cref{fig:rheobase
         \includegraphics[width=\linewidth]{Figures/rheobase_correlation.pdf}
 	\\\notenk{Re-work legend to be in line with new figure}
         \linespread{1.}\selectfont
-        \caption[]{Effects of altered channel kinetics on rheobase. The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in \(V_{1/2}\) and rheobase, slope factor k and AUC as well as current conductances and rheobase for each model are shown on the right in (A), (B) and (C) respectively. The relationships between rheobase and \(\Delta V_{1/2}\), slope (k) and conductance (g) for the Kendall \(\tau\) coefficients highlights by the black box are depicted in the middle panel. The fI curves corresponding to one of the models are shown in the left panels.}
+        \caption[]{Effects of altered channel kinetics on rheobase. The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in \(V_{1/2}\) and rheobase, slope factor k and AUC as well as maximal current conductances and rheobase for each model are shown on the right in (A), (B) and (C) respectively. The relationships between rheobase and \(\Delta V_{1/2}\), slope (k) and maximal conductance (g) for the Kendall \(\tau\) coefficients highlights by the black box are depicted in the middle panel. The fI curves corresponding to one of the models are shown in the left panels.}
         \label{fig:rheobase_correlation}
 \end{figure}
 
@@ -357,7 +362,7 @@ Mutations in \Kv are associated with episodic ataxia type~1 (EA1) and have been
 
 \section*{Discussion (3000 Words Maximum - Currently 2145)}
 % \textit{The discussion section should include a brief statement of the principal findings, a discussion of the validity of the observations, a discussion of the findings in light of other published work dealing with the same or closely related subjects, and a statement of the possible significance of the work. Extensive discussion of the literature is discouraged.}\\
-Using a set of diverse conductance-based neuronal models, the effects of changes to properties of ionic currents and conductances on firing were determined to be heterogenous for the AUC of the steady state fI curve but more homogenous for rheobase. For a known channelopathy, episodic ataxia type~1 associated \Kv mutations, the effects on rheobase is consistent across model cell types, whereas the effect on AUC depends on cell type. Our results demonstrate that LOF and GOF on the biophysical level cannot be uniquely transfered to the level of neuronal firing.
+Using a set of diverse conductance-based neuronal models, the effects of changes to properties of ionic currents and maximal conductances on firing were determined to be heterogenous for the AUC of the steady state fI curve but more homogenous for rheobase. For a known channelopathy, episodic ataxia type~1 associated \Kv mutations, the effects on rheobase is consistent across model cell types, whereas the effect on AUC depends on cell type. Our results demonstrate that LOF and GOF on the biophysical level cannot be uniquely transfered to the level of neuronal firing.
 
 \subsection*{Validity of Neuronal Models}
 \notels{should we move this to a less prominent position? How much of this part could be counted as common knowledge and be left out?, for example model complexity in terms of currents and compartments, I just think that this part might be too harsh on the models, even if the criticism doesn't apply for the main points of the paper}
@@ -448,9 +453,9 @@ Accordingly, for accurate modelling and predictions of the effects of mutations
 
 \captionof{figure}{Diversity in Neuronal Model Firing. Spike trains (left), frequency-current (fI) curves (right) for Cb stellate (A), RS inhibitory (B), FS (C), RS pyramidal (D), RS inhibitory +\Kv (E), Cb stellate +\Kv (F), FS +\Kv (G), RS pyramidal +\Kv (H), STN +\Kv (I), Cb stellate \(\Delta\)\Kv (J), STN \(\Delta\)\Kv (K), and STN (L) neuron models. Black marker on the fI curves indicate the current step at which the spike train occurs. The green marker indicates the current at which firing begins in response to an ascending current ramp, whereas the red marker indicates the current at which firing ceases in response to a descending current ramp.}
 
-\captionof{figure}{The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in \(V_{1/2}\) and AUC, slope factor k and AUC as well as current conductances and AUC for each model are shown on the right in (A), (B) and (C) respectively. The relationships between AUC and \(\Delta V_{1/2}\), slope (k) and conductance (g) for the Kendall \(\tau\) coefficients highlights by the black box are depicted in the middle panel. The fI curves corresponding to one of the models are shown in the left panels.}
+\captionof{figure}{The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in \(V_{1/2}\) and AUC, slope factor k and AUC as well as current conductances and AUC for each model are shown on the right in (A), (B) and (C) respectively. The relationships between AUC and \(\Delta V_{1/2}\), slope (k) and maximal conductance (g) for the Kendall \(\tau\) coefficients highlights by the black box are depicted in the middle panel. The fI curves corresponding to one of the models are shown in the left panels.}
 
-\captionof{figure}{The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in \(V_{1/2}\) and rheobase, slope factor k and AUC as well as current conductances and rheobase for each model are shown on the right in (A), (B) and (C) respectively. The relationships between rheobase and \(\Delta V_{1/2}\), slope (k) and conductance (g) for the Kendall \(\tau\) coefficients highlights by the black box are depicted in the middle panel. The fI curves corresponding to one of the models are shown in the left panels.}
+\captionof{figure}{The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in \(V_{1/2}\) and rheobase, slope factor k and AUC as well as current conductances and rheobase for each model are shown on the right in (A), (B) and (C) respectively. The relationships between rheobase and \(\Delta V_{1/2}\), slope (k) and maximal conductance (g) for the Kendall \(\tau\) coefficients highlights by the black box are depicted in the middle panel. The fI curves corresponding to one of the models are shown in the left panels.}
 
 \captionof{figure}{Effects of episodic ataxia type~1 associated \Kv mutations on firing. Effects of \Kv mutations on AUC (\(AUC_{contrast}\)) and rheobase (\(\Delta\)rheobase) compared to wild type for RS pyramidal +\Kv (A), RS inhibitory +\Kv (B), FS +\Kv (C), Cb stellate (D), Cb stellate +\Kv (E), Cb stellate \(\Delta\)\Kv (F), STN (G), STN +\Kv (H) and STN \(\Delta\)\Kv (I) models V174F, F414C, E283K, and V404I mutations are highlighted in color for each model. Pairwise Kendall rank correlation coefficients (Kendall \(\tau\)) between the effects of \Kv mutations on rheobase and on AUC are shown in J and K respectively.}