added statistical_table.tex and relevant superscripts in results

manuscript.tex

@@ -243,6 +243,7 @@ The Kendall's \(\tau\) coefficient, a non-parametric rank correlation, is used t
 The code/software described in the paper is freely available online at [URL redacted for double-blind review]. The code is available as Extended Data.
 % The type of computer and operating system on which the code was run to obtain the results in the manuscript must be stated in the Materials and Methods section.\\
 
+\input{statistical_table}
 
 \section*{Results}
 % \textit{The results section should clearly and succinctly present the experimental findings. Only results essential to establish the main points of the work should be included.\\
@@ -278,11 +279,11 @@ Using these two measures we quantify the effects a changed property of an ionic
 \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 delayed rectifier 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). As a result the effect of a depolarizing shift in the delayed rectifier potassium current half activation \(V_{1/2}\) in FS neurons is in the bottom left quadrant of \Cref{fig:firing_characterization}~B and characterization as LOF or GOF in excitability is not possible. Plotting the corresponding changes in AUC against the change in half activation potential \(V_{1/2}\) results in a monotonically 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 half maximal potential \(V_{1/2}\) (thin lines in \Cref{fig:AUC_correlation}~B). To further summarize these different dependencies of the various models we characterize each of these curves by a single number, the \( \text{Kendall} \ \tau \) correlation coefficient. A monotonically 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 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 delayed rectifier 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). As a result the effect of a depolarizing shift in the delayed rectifier potassium current half activation \(V_{1/2}\) in FS neurons is in the bottom left quadrant of \Cref{fig:firing_characterization}~B and characterization as LOF or GOF in excitability is not possible. Plotting the corresponding changes in AUC against the change in half activation potential \(V_{1/2}\) results in a monotonically 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 half maximal potential \(V_{1/2}\) (thin lines in \Cref{fig:AUC_correlation}~B). To further summarize these different dependencies of the various models we characterize each of these curves by a single number, the \( \text{Kendall} \ \tau \) correlation coefficient\textcolor{red}{\textsuperscript{a}}. A monotonically increasing curve results in a \( \text{Kendall} \ \tau \) close to \(+1\)\textcolor{red}{\textsuperscript{a}}, a monotonously decreasing curve in \( \text{Kendall} \ \tau \approx -1 \)\textcolor{red}{\textsuperscript{a}}, and a non-monotonous, non-linear relation in \( \text{Kendall} \ \tau \) close to zero\textcolor{red}{\textsuperscript{a}} (compare lines in \Cref{fig:AUC_correlation}~B with dots in black box in panel C).
 
-Changes in gating half activation potential \(V_{1/2}\) 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}~D,E,F). Similar consistent positive correlations can be found for shifts in A-current activation \(V_{1/2}\). Changes in \Kv half activation \(V_{1/2}\) and in maximal A-current conductance result in negative correlations with the AUC in all models (\( \text{Kendall} \ \tau \approx -1\)).
+Changes in gating half activation potential \(V_{1/2}\) 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 (\( \text{Kendall} \ \tau \approx +1\)\textcolor{red}{\textsuperscript{a}}), but with different slopes (\Cref{fig:AUC_correlation}~D,E,F). Similar consistent positive correlations can be found for shifts in A-current activation \(V_{1/2}\). Changes in \Kv half activation \(V_{1/2}\) and in maximal A-current conductance result in negative correlations with the AUC in all models (\( \text{Kendall} \ \tau \approx -1\)\textcolor{red}{\textsuperscript{a}}).
 
-Qualitative differences can be found, for example, when increasing the maximal conductance of the delayed rectifier (\Cref{fig:AUC_correlation}~G,H,I). 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,B,C). Some model neurons do almost not depend on changes in K-current half 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}~G,H,I). In some model neurons this increased AUC (\( \text{Kendall} \ \tau \approx +1\)\textcolor{red}{\textsuperscript{a}}), whereas in others AUC is decreased (\( \text{Kendall} \ \tau \approx -1\)\textcolor{red}{\textsuperscript{a}}). 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\textcolor{red}{\textsuperscript{a}}. 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,B,C). Some model neurons do almost not depend on changes in K-current half activation \(V_{1/2}\) or show strongly non-linear dependencies, both resulting in \( \text{Kendall} \ \tau\) close to zero\textcolor{red}{\textsuperscript{a}}. Many model neurons show strongly negative correlations, and a few show positive correlations with shifting the activation curve of the delayed rectifier.
 
 
 \begin{figure}[tp]
@@ -293,7 +294,7 @@ Qualitative differences can be found, for example, when increasing the maximal c
         \label{fig:AUC_correlation}
 \end{figure}
 
-Changes in gating half activation potential \(V_{1/2}\) and slope factor \(k\) as well as the maximum conductance \(g\) affect rheobase (\Cref{fig:rheobase_correlation}), however, in contrast to AUC, qualitatively consistent effects on rheobase across models are observed. Increasing, for example, the maximal conductance of the leak current in the Cb stellate model increases the rheobase (\Cref{fig:rheobase_correlation}~G). When these changes are plotted against the change in maximal conductance a monotonically increasing relationship is evident (thick teal line in \Cref{fig:rheobase_correlation}~H). This monotonically increasing relationship is evident in all models (\( \text{Kendall} \ \tau \approx +1\)), but with different slopes (thin lines in \Cref{fig:rheobase_correlation}~H). Similarly, positive correlations are consistently found across models for maximal conductances of delayed rectifier K, \Kv, and A type currents, whereas the maximal conductance of the sodium current consistently is associated with negative correlations (\( \text{Kendall} \ \tau \approx -1\); \Cref{fig:rheobase_correlation}~I), i.e. rheobase is decreased with increasing maximum conductance in all models.
+Changes in gating half activation potential \(V_{1/2}\) and slope factor \(k\) as well as the maximum conductance \(g\) affect rheobase (\Cref{fig:rheobase_correlation}), however, in contrast to AUC, qualitatively consistent effects on rheobase across models are observed. Increasing, for example, the maximal conductance of the leak current in the Cb stellate model increases the rheobase (\Cref{fig:rheobase_correlation}~G). When these changes are plotted against the change in maximal conductance a monotonically increasing relationship is evident (thick teal line in \Cref{fig:rheobase_correlation}~H). This monotonically increasing relationship is evident in all models (\( \text{Kendall} \ \tau \approx +1\)\textcolor{red}{\textsuperscript{a}}), but with different slopes (thin lines in \Cref{fig:rheobase_correlation}~H). Similarly, positive correlations are consistently found across models for maximal conductances of delayed rectifier K, \Kv, and A type currents, whereas the maximal conductance of the sodium current consistently is associated with negative correlations (\( \text{Kendall} \ \tau \approx -1\)\textcolor{red}{\textsuperscript{a}}; \Cref{fig:rheobase_correlation}~I), i.e. rheobase is decreased with increasing maximum conductance in all models.
 
 Although changes in half maximal potential \(V_{1/2}\) and slope factor \(k\) generally correlate with rheobase similarly across model there are some exceptions. Changing the slope factor of Na-current inactivation, \Kv-current inactivation, and A-current activation affect rheobase both with positive and negative correlations in different models (\Cref{fig:rheobase_correlation}~F). Departures from monotonic relationships also occur in some models as a result of K-current activation  \(V_{1/2}\) and slope factor \(k\), \Kv-current inactivation slope factor \(k\), and A-current activation slope factor \(k\) in some models. Thus, identical changes in current gating properties such as the half maximal potential \(V_{1/2}\) or slope factor \(k\) can have differing effects on firing depending on the model in which they occur. 
 
@@ -307,7 +308,7 @@ Although changes in half maximal potential \(V_{1/2}\) and slope factor \(k\) ge
 \end{figure}
 
 \subsection*{\Kv Mutations}
-Mutations in \Kv are associated with episodic ataxia type~1 (EA1) and have been characterized biophysically \citep{lauxmann_therapeutic_2021}. They are used here as a case study in the effects of various ionic current environments on neuronal firing and on the outcomes of channelopathies. The changes in AUC and rheobase from wild type values for reported EA1 associated \Kv mutations are heterogenous across models containing \Kv, but generally show decreases in rheobase (\Cref{fig:simulation_model_comparision}A-I).  Pairwise non-parametric Kendall \(\tau\) rank correlations between the simulated effects of these \Kv mutations on rheobase are highly correlated across models (\Cref{fig:simulation_model_comparision}J) indicating that EA1 associated \Kv mutations generally decrease rheobase across diverse cell-types. However, the effects of the \Kv mutations on AUC are more heterogenous as reflected by both weak and strong positive and negative pairwise correlations between models (\Cref{fig:simulation_model_comparision}K), suggesting that the effects of ion-channel variant on super-threshold neuronal firing depend both quantitatively and qualitatively on the specific composition of ionic currents in a given neuron.
+Mutations in \Kv are associated with episodic ataxia type~1 (EA1) and have been characterized biophysically \citep{lauxmann_therapeutic_2021}. They are used here as a case study in the effects of various ionic current environments on neuronal firing and on the outcomes of channelopathies. The changes in AUC and rheobase from wild type values for reported EA1 associated \Kv mutations are heterogenous across models containing \Kv, but generally show decreases in rheobase (\Cref{fig:simulation_model_comparision}A-I).  Pairwise non-parametric Kendall \(\tau\) rank correlations\textcolor{red}{\textsuperscript{a}} between the simulated effects of these \Kv mutations on rheobase are highly correlated across models (\Cref{fig:simulation_model_comparision}J) indicating that EA1 associated \Kv mutations generally decrease rheobase across diverse cell-types. However, the effects of the \Kv mutations on AUC are more heterogenous as reflected by both weak and strong positive and negative pairwise correlations between models (\Cref{fig:simulation_model_comparision}K), suggesting that the effects of ion-channel variant on super-threshold neuronal firing depend both quantitatively and qualitatively on the specific composition of ionic currents in a given neuron.
 
 \begin{figure}[tp]
         \centering

statistical_table.tex

@@ -0,0 +1,20 @@
+
+\begin{table}[ht!]
+    \centering
+    \linespread{1.5}\selectfont
+    
+    \fontsize{10pt}{12pt}\selectfont{
+    \begin{tabular}{cccc}
+     	\Xhline{1\arrayrulewidth}
+	& Data Structure &  Type of test & Power  \\
+	\Xhline{1\arrayrulewidth}   
+     	a & Non-normal distribution & Kendal \(\tau\) rank correlation & ---  \\
+     	\Xhline{1\arrayrulewidth}   
+	 
+    \end{tabular}}
+    
+	    \caption[Statistical Table]{Statistical Table. Descriptive statistics including non-parametric Kendall \(\tau\) rank correlations are used. Statistical hypothesis tests are not used.}
+	     
+    \label{tab:stats}
+\end{table}
+