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\title{Loss or Gain of Function? Neuronal Firing Effects of Ion Channel Mutations Depend on the Cell Type}
\vspace{-1em}
\date{}
 



%\section*{Titlepage for eNeuro - will be put into Word file provided for submission}
%\subsection{Manuscript Title (50 word maximum)}
%Loss or Gain of Function? Neuronal Firing Effects of Ion Channel Mutations Depend on the Cell Type
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%\subsection{Abbreviated Title (50 character maximum)}
%Effects of Ion Channel Mutation Depend on Cell Type
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%\subsection{List all Author Names and Affiliations in order as they would appear in the published article}
%Nils A. Koch\textsuperscript{1,2}, Lukas Sonnenberg\textsuperscript{1,2}, Ulrike B.S. Hedrich\textsuperscript{3}, Stephan Lauxmann\textsuperscript{1,3}, Jan Benda\textsuperscript{1,2}
%
%\textsuperscript{1}Institute for Neurobiology, University of Tuebingen, 72072 Tuebingen, Germany\\
%\textsuperscript{2}Bernstein Center for Computational Neuroscience Tuebingen, 72076 Tuebingen, Germany\\
%\textsuperscript{3} Department of Neurology and Epileptology, Hertie Institute for Clinical Brain Research, University of Tuebingen, 72076 Tuebingen, Germany\\
%
%\subsection{Author Contributions - Each author must be identified with at least one of the following: Designed research, Performed research, Contributed unpublished reagents/ analytic tools, Analyzed data, Wrote the paper.}
%\notenk{Adjust as you deem appropriate}\\
%NK, LS, UBSH, SL, JB Designed Research;
%NK Performed research;
%NK, LS Analyzed data;
%NK, LS,UBSH, SL,  JB Wrote the paper
%
%\subsection{Correspondence should be addressed to (include email address)}
%\ \notenk{Nils oder Jan?}
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%\subsection{Acknowledgements}
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%\subsection{Conflict of Interest}
%Authors report no conflict of interest.
%\\\textbf{A.} The authors declare no competing financial interests.
%\subsection{Funding sources}
%\notenk{Add as appropriate - I don't know this information}\notejb{SmartStart}
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Nils A. Koch\textsuperscript{1,2}, Lukas Sonnenberg\textsuperscript{1,2}, Ulrike B.S. Hedrich\textsuperscript{3}, Stephan Lauxmann\textsuperscript{1,3}, Jan Benda\textsuperscript{1,2}

\textsuperscript{1}Institute for Neurobiology, University of Tuebingen, 72072 Tuebingen, Germany\\
\textsuperscript{2}Bernstein Center for Computational Neuroscience Tuebingen, 72076 Tuebingen, Germany\\
\textsuperscript{3} Department of Neurology and Epileptology, Hertie Institute for Clinical Brain Research, University of Tuebingen, 72076 Tuebingen, Germany

\section*{Abstract (250 Words Maximum - Currently )}
%\textit{It should provide a concise summary of the objectives, methodology (including the species and sex studied), key results, and major conclusions of the study.}

Ion channels determine neuronal excitability and disruption in ion channel properties caused by mutations can result in neurological disorders called channelopathies. Often, mutations within one gene are associated with a specific channelepothy and the effects of these mutations  on channel function, i.e. the gating current of the affected ion channel, are generally determined using heterologous expression systems. Nevertheless, the impact of such mutations on neuronal firing is essential not only for determining brain function, but also for selecting personalized treatment options for the affected patient. The effect of ion channel mutations on firing in different cell types has been mostly neglect and it is unclear whether the effect of a given mutation on firing can simply be inferred from the effects identified at the current level. Here we use a diverse collection of computational neuronal models to determine that ion channel mutation effects at the current level cannot be indiscriminantly used to infer firing effects without consideration of cell-type. In particular, systematic simulation and evaluation of the effects of changes in ion current properties on firing properties in different neuronal types as well as for mutations in the \textit{KCNA1} gene encoding the \Kv potassium channel subtype associated with episodic ataxia type~1 (EA1) was performed. The effects of changes in ion current properties generally and due to mutations in the \Kv channel subtype on the firing of a neuron depends on the ionic current environment, or the neuronal cell type, in which such a change occurs in. Thus, while characterization of ion channel mutations as loss or gain of function is useful at the level of the ionic current, this characterization should not be extended to the level of neuronal excitability as the effects of ion channel mutations on the firing of a cell is dependent on the cell type and the composition of different ion channels and subunits therein. For increased efficiency and efficacy of personalized medicine approaches in channelopathies, the effects of ion channel mutations must be examined in the context of the appropriate cell types in which these mutations occur.
%Using a diverse collection of computational neuronal models, the effects of changes in ion current properties on firing properties of different neuronal types were simulated systematically and for mutations in the \textit{KCNA1} gene encoding the \Kv potassium channel subtype associated with episodic ataxia type~1 (EA1). The effects of changes in ion current properties or changes due to mutations in the \Kv channel subtype on the firing of a neuron depends on the ionic current environment, or the neuronal cell type, in which such a change occurs in. Characterization of ion channel mutations as loss or gain of function is useful at the level of the ionic current. However, the effects of mutations causing channelopathies on the firing of a cell is dependent on the cell type and thus on the composition of different ion channels and subunits. To further the efficacy of personalized medicine in channelopathies, the effects of ion channel mutations must be examined in the context of the appropriate cell types in which these mutations occur.
\par\null

\section*{Significant Statement (120 Words Maximum - Currently )}
%\textit{The Significance Statement should provide a clear explanation of the importance and relevance of the research in a manner accessible to researchers without specialist knowledge in the field and informed lay readers. The Significance Statement will appear within the paper below the abstract.}

Ion channels determine neuronal excitability and mutations that alter ion channel properties result in neurological disorders called channelopathies. Although the genetic nature of such mutations as well as their effects on the biophysical properties of an ion channel are routinely assessed experimentally, determination of the role in altering neuronal firing is more difficult. In particular, cell-type dependency of ion channel mutations on firing has been observed experimentally, and should be accounted for. In this context, computational modelling bridges this gap and demonstrates that the cell type in which a mutation occurs is an important determinant in the effects of neuronal firing. As a result, classification of ion channel mutations as loss or gain of function is useful to describe the ionic current but should not be blindly extend to classification at the level of neuronal firing.
\par\null


\section*{Introduction (750 Words Maximum  - Currently )} 
%\textit{The Introduction should briefly indicate the objectives of the study and provide enough background information to clarify why the study was undertaken and what hypotheses were tested.}

%\textcolor{red}{Over the last years, new technology such as next generation sequencing has led to the identification of a growing number of associated de novo genetic variants, providing the basis for subsequent pathophysiological studies. The ongoing joint effort of many research groups continues to contribute to an increasingly better understanding of underlying disease mechanisms. At the same time, the unfolding complexity of the pathophysiological landscape of DEE emerges as one likely reason for limited therapeutic success experienced with standard care.
%Sodium channelopathies have initially been identified as one of the most frequent causes of genetic forms of epilepsy and more recent studies demonstrate their implication in DEE . SCN8A for example – the gene encoding the human NaV1.6 voltage gated sodium channel – gene has been recognized as an epilepsy-associated gene in 2012 (Veeramah et al., 2012).
%  }

Voltage-gated ion channels are vital in determining neuronal excitability, action potential generation and firing patterns \citep{bernard_channelopathies_2008, carbone_ion_2020}. In particular, the properties and combinations of ion channels and subunits and their resulting currents determine the firing properties of a neuron \citep{rutecki_neuronal_1992, pospischil_minimal_2008}. However, ion channel function can be disturbed, for instance through genetic alterations, resulting in altered ionic current properties and altered neuronal firing behavior \citep{carbone_ion_2020}.

In recent years, next generation sequencing has led to an increasing number of clinically relevant genetic mutations and has provided the basis for pathophysiological studies of genetic epilepsies, pain disorders, dyskinesias, intellectual disabilities, myotonias, and periodic paralyses \citep{bernard_channelopathies_2008, carbone_ion_2020}. Ongoing efforts of many research groups have contributed to the current understanding of underlying disease mechanism in channelopathies, however a complex pathophysiological landscape has emerged for many channelopathies and is likely a reason for limited therapeutic success with standard care.              

% Ion channel mutations are the most common cause of such channelopathies  and are often associated with hereditary clinical disorders including ataxias, epilepsies, pain disorders, dyskinesias, intellectual disabilities, myotonias, and periodic paralyses \citep{bernard_channelopathies_2008, carbone_ion_2020}.

The effects of mutations in ion channel genes on ionic current kinetics are frequently assessed using heterologous expression systems which do not express endogenous currents  \citep{Balestrini1044, Noebels2017, Dunlop2008}, and are frequently classified as either a loss of function (LOF) or a gain of function (GOF) with respect to changes in gating of the altered ion channels \citep{Musto2020, Kullmann2002, Waxman2011, Kim2021}. This classification of the effects on ionic currents is often directly used to predict the effects on neuronal firing \citep{Niday2018, Wei2017, Wolff2017}, which in turn is important for understanding the pathophysiology of these disorders and for identification of potential therapeutic targets \citep{Orsini2018, Yang2018, Colasante2020, Yu2006}. Genotype-phenotype relationships are complex and the understanding of the relationships between these is still evolving \citep{Wolff2017, johannesen_genotype-phenotype_2021}. Experimentally, the effects of channelopathies on neuronal firing can be assessed using primary neuronal cultures  \citep{Scalmani2006, Smith2018, Liu2019}  or \textit{in vitro} recordings from slices of transgenic mouse lines \citep{Mantegazza2019, Xie2010,Lory2020, Habib2015,  Hedrich2019}.

However, the effect of a given channelopathy on the firing behavior of different neuronal types across the brain is often unclear and not feasible to experimentally obtain. Different neuron types differ in their composition of ionic currents \citep{yao2021taxonomy, Cadwell2016, BICCN2021, Scala2021} and therefore likely respond differently to changes in the properties of a single ionic current. The expression level of an affected gene can correlate with firing behavior in the simplest case \citep{Layer2021}. However, if gating kinetics are affected this can have complex consequences on the firing behavior of a specific cell type and the network activity within the brain.

For instance, altering relative amplitudes of ionic currents can dramatically influence the firing behavior and dynamics of neurons \citep{rutecki_neuronal_1992, pospischil_minimal_2008,Kispersky2012, golowasch_failure_2002, barreiro_-current_2012, Noebels2017, Layer2021}, however other properties of ionic currents impact neuronal firing as well. Cell-type specific effects one firing can occur for instance increases inhibitory interneuron but not pyramidal neuron firing with R1648H mutation in \textit{SCN1A} \citep{Hedrich14874}. In extreme cases, a mutation can have opposite effects on different neuron types. For example, the R1627H  \textit{SCN8A} mutation is associated which increased firing in interneurons, but decreases pyramidal neuron excitability \citep{makinson_scn1a_2016}.

Despite this evidence of cell-type specific effects of ion channel mutations on firing, the dependence of firing outcomes of ion channel mutations is generally not known. Cell-type specificity is likely vital for successful precision medicine treatment approaches. For example, Dravet syndrome was identified as the consquence of LOF mutations in \textit{SCN1A} \citep{Claes2001,Fujiwara2003,Ohmori2002}, however limited succes in treatment of Dravet syndrome persisted \citep{Claes2001,Oguni2001}. Once it became evident that only inhibitory interneurons and not pyramidal neurons had altered excitability as a result of LOF \textit{SCN1A} mutations alternative approaches, based on this understanding such as gene therapy, began to show promise \citep{Colasante2020, Yu2006}. Due to the high clinical relevance of understanding cell-type dependent effects of channelopathies, we use computationaly modelling approaches to assess the impacts of altered ionic current properties on firing behavior, bridging the gap between changes in the biophysical properties induced by mutations, firing and clinical symptoms. Conductance-based neuronal models enable insight into the effects of ion channel mutations with specific effects of the resulting ionic current as well as enabling \textit{in silico} assessment of the relative effects of changes in biophysical properties of ionic currents on neuronal firing. Furthermore, modelling approaches enable predictions of the effects of specific mutation and drug induced biophysical property changes \citep{Layer2021,Liu2019,johannesen_genotype-phenotype_2021, lauxmann_therapeutic_2021}.

%Computational modelling approaches can be used to assess the impacts of altered ionic current properties on firing behavior, bridging the gap between changes in the biophysical properties induced by mutations, firing and clinical symptoms. Conductance-based neuronal models enable insight into the effects of ion channel mutations with specific effects of the resulting ionic current as well as enabling \textit{in silico} assessment of the relative effects of changes in biophysical properties of ionic currents on neuronal firing. Furthermore, modelling approaches enable predictions of the effects of specific mutation and drug induced biophysical property changes \citep{Layer2021,Liu2019,johannesen_genotype-phenotype_2021, lauxmann_therapeutic_2021}. 

In this study, we therefore investigated how the outcome of ionic current kinetic changes on firing depend on neuronal cell type by simulating the response of a repertoire of different neuronal models to changes in single current parameters as well as to more complex changes as they were observed for specific mutations. For this task we chose mutations in the \textit{KCNA1} gene, encoding  the potassium channel subunit \Kv, that are associated with episodic ataxia type~1 \citep{Browne1994, Browne1995, lauxmann_therapeutic_2021}.
\par\null

\section*{Materials and Methods}
% \textit{The materials and methods section should be brief but sufficient to allow other investigators to repeat the research (see also Policy Concerning Availability of Materials). Reference should be made to published procedures wherever possible; this applies to the original description and pertinent published modifications. }
\par\null

All modelling and simulation was done in parallel with custom written Python 3.8 software, run on a Cent-OS 7 server with an Intel(R) Xeon (R) E5-2630 v2 CPU.
    % @ 2.60 GHz  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; model D), regular spiking inhibitory (RS inhibitory; model B), and fast spiking (FS; model C) cells were used \citep{pospischil_minimal_2008}. Additionally,  a \Kv current (\IKv; \citealt{ranjan_kinetic_2019}) was added to each of these models (RS pyramidal +\Kv; model H, RS inhibitory +\Kv; model E, and FS +\Kv; model G respectively). A cerebellar stellate cell model from \citet{alexander_cerebellar_2019} is used (Cb stellate; model A) in this study. This cell model was also extended by a \Kv current \citep{ranjan_kinetic_2019}, either in addition to the A-type potassium current (Cb stellate +\Kv; model F) or by replacing the A-type potassium current (Cb stellate \(\Delta\)\Kv; model J). A subthalamic nucleus (STN; model L) neuron model as described by \citet{otsuka_conductance-based_2004} was also used. The STN cell model (model L) was  additionally extended by a \Kv current \citep{ranjan_kinetic_2019}, either in addition to the A-type potassium current (STN +\Kv; model I) or by replacing the A-type potassium current (STN \(\Delta\)\Kv; model K). Model letter naming corresponds to panel lettering in \Cref{fig:diversity_in_firing}. 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 (model A)  and STN (model L) models \citep{alexander_cerebellar_2019, otsuka_conductance-based_2004}. For enabling the comparison of models with the typically reported 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}
with slope \(k\), voltage for half-maximal activation or inactivation (\(V_{1/2}\)), exponent \(j\), and persistent current \(0 \leq a \leq 1\) were fitted to the original formulism for RS pyramidal (model D), RS inhibitory (model B) and FS (model C) models from \citet{pospischil_minimal_2008}. The properties of \IKv  were fitted to the mean wild type biophysical parameters of \Kv described in \citet{lauxmann_therapeutic_2021}. Each of the original single-compartment models used here can reproduce physiological firing behavior of the neurons they represent (\Cref{fig:diversity_in_firing}; \citealt{pospischil_minimal_2008, alexander_cerebellar_2019, otsuka_conductance-based_2004}) and capture key aspects of the dynamics of these cell types. 

    
\input{g_table}

\input{gating_table}
    
\subsection*{Firing Frequency Analysis} 
The membrane responses to 200 equidistant two second long current steps were simulated using the forward-Euler method with a \(\Delta \textrm{t} = 0.01\)\,ms from steady state. Current steps ranged from 0 to 1\,nA  (step size 5\,pA) for all models except for the RS inhibitory neuron models where a range of 0 to 0.35\,nA (step size 1.75\,pA) was used to ensure repetitive firing across the range of input currents. For each current step, action potentials were detected as peaks with at least 50\,mV prominence, or the relative height above the lowest contour line encircling it, and a minimum interspike interval of 1\,ms. The interspike interval was computed and used to determine the instantaneous firing frequencies elicited by the current step.  

To ensure accurate firing frequencies at low firing rates and reduced spike sampling bias, steady-state firing was defined as the mean firing frequency in a 500\,ms window in the last second of the current steps starting at the inital action potential in this last second.
Firing characterization was performed in the last second of current steps to ensure steady-state firing is captured and adaptation processes are neglected in our analysis. Alteration in current magnitudes can have different effects on rheobase and the initial slope of the fI curve \citep{Kispersky2012}.
For this reason, we quantified neuronal firing using the rheobase as well as the area under the curve (AUC) of the initial portion of the fI curve as a measure of the initial slope of the fI curve \Cref{fig:firing_characterization}A.

The smallest current at which steady state firing occured was identified and the current step interval preceding the occurrence of steady state firing was simulated at higher resolution (100 current steps) to determine the current at which steady state firing began. Firing was simulated with 100 current steps from this current upwards for 1/5 of the overall current range. Over this range a fI curve was constructed and the integral, or area under the curve (AUC), of the fI curve over this interval was computed with the composite trapezoidal rule and used as a measure of firing rate independent from rheobase. 
    
To obtain the rheobase at a higher current resolution than the fI curve, the current step interval preceding the occurrence of action potentials was explored at higher resolution with 100 current steps spanning the interval (step sizes of 0.05 pA and 0.0175 pA respectively). Membrane responses to these current steps were then analyzed for action potentials and the rheobase was considered the lowest current step for which an action potential was elicited.
 
All models exhibited tonic steady-state firing with default parameters. In limited instances, variations of parameters elicited periodic bursting, however these instances were excluded from further analysis.

    
\subsection*{Sensitivity Analysis and Comparison of Models}
    
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. We neglected the variation of time constants for the practical reason that estimation and assessment of time constants and changes to them is not straightforward \citep{Clerx2019, Whittaker2020}.



 \subsection*{Model Comparison}
Changes in rheobase (\drheo) were 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}
\textrm{normalized }  \Delta \textrm{AUC} = \frac{AUC_i - AUC_{wt}}{AUC_{wt}}
\end{equation}

To assess whether the effects of a given alteration on \ndAUC  or \drheo were 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*{\textit{KCNA1} Mutations}\label{subsec:mut}
 Known episodic ataxia type~1 associated \textit{KCNA1} mutations and their electrophysiological characterization have been 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. Although initially described to lack fast activation, \Kv displays prominent inactivation at physiologically relevant temperatures \citep{ranjan_kinetic_2019}. The effects of a mutation were also applied to \(\textrm{I}_{\textrm{A}}\) when present as both potassium currents display 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 behavior 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 were compared to wild type rheobase values (\drheo). Pairwise Kendall rank correlations (Kendall \(\tau\)) were used to compare the correlation in the effects of \Kv mutations on AUC and rheobase between models.



\subsection*{Code Accessibility}
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.\\
% Authors must provide detailed information for each analysis performed, including population size, definition of the population (e.g., number of individual measurements, number of animals, number of slices, number of times treatment was applied, etc.), and specific p values (not > or <), followed by a superscript lowercase letter referring to the statistical table provided at the end of the results section. Numerical data must be depicted in the figures with box plots.}

To examine the role of cell-type specific ionic current environments on the impact of altered ion channel properties on firing behavior a set of neuronal models was used and properties of channels common across models were altered systematically one at a time. The effects of a set of episodic ataxia type~1 associated \textit{KCNA1} mutations on firing was then examined across different neuronal models with different ionic current environments.

\begin{figure}[tp]
        \centering
        \includegraphics[width=\linewidth]{Figures/diversity_in_firing.pdf}
        \linespread{1.}\selectfont
        \caption[]{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 markers 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 (see \Cref{fig:ramp_firing}).}
        \label{fig:diversity_in_firing}
\end{figure}

\subsection*{Variety of model neurons}
%Neuronal firing is heterogenous across the CNS and a set of neuronal models with heterogenous firing due to different ionic currents is desirable to reflect this heterogeneity. The set of single-compartment, conductance-based neuronal models used here has considerable diversity as evident in the variability seen across neuronal models both in spike trains and their fI curves (\Cref{fig:diversity_in_firing}). The models chosen for this study all fire tonically and do not exhibit bursting (see methods for details and naming of the models). Some models, such as Cb stellate and RS inhibitory models, display type I firing, whereas others such as Cb stellate \(\Delta\)\Kv and STN models exhibit type II firing. Type I firing is characterized by continuous fI curves (i.e. firing rate increases from 0 in a continuous fashion) whereas type II firing is characterized by a discontinuity in the fI curve (i.e. a jump occurs from no firing to firing at a certain frequency) \cite{ermentrout_type_1996, Rinzel_1998}. The other models used here lie on a continuum between these prototypical firing classifications. Most neuronal models exhibit hysteresis with ascending and descending ramps eliciting spikes at different current thresholds. However, the STN +\Kv, STN \(\Delta\)\Kv, and Cb stellate \(\Delta\)\Kv models have large hysteresis (\Cref{fig:diversity_in_firing}, \Cref{fig:ramp_firing}). Different types of underlying current dynamics are known to generate these different firing types and hysteresis \cite{ERMENTROUT2002, ermentrout_type_1996, Izhikevich2006}.

Neuronal firing is heterogenous across the CNS and a set of neuronal models with heterogenous firing due to different ionic currents is desirable to reflect this heterogeneity. The set of single-compartment, conductance-based neuronal models used here has considerable diversity as evident in the variability seen across neuronal models both in spike trains and their fI curves (\Cref{fig:diversity_in_firing}). The models chosen for this study all fire tonically and do not exhibit bursting (see methods for details and naming of the models). Some models, such as models A and B, display type I firing, whereas others such as models J and L exhibit type II firing. Type I firing is characterized by continuous fI curves (i.e. firing rate increases from 0 in a continuous fashion) whereas type II firing is characterized by a discontinuity in the fI curve (i.e. a jump occurs from no firing to firing at a certain frequency) \cite{ermentrout_type_1996, Rinzel_1998}. The other models used here lie on a continuum between these prototypical firing classifications. Most neuronal models exhibit hysteresis with ascending and descending ramps eliciting spikes at different current thresholds. However, the models  I, J, and K have large hysteresis (\Cref{fig:diversity_in_firing}, \Cref{fig:ramp_firing}). Different types of underlying current dynamics are known to generate these different firing types and hysteresis \cite{ERMENTROUT2002, ermentrout_type_1996, Izhikevich2006}.

\subsection*{Characterization of Neuronal Firing Properties}
\begin{figure}[tp]
        \centering
	\includegraphics[width=0.5\linewidth]{Figures/firing_characterization_arrows.pdf}
        \linespread{1.}\selectfont
        \caption[]{Characterization of firing with AUC and rheobase. (A) The area under the curve (AUC) of the repetitive firing frequency-current (fI) curve. (B) 
Changes in firing as characterized by \(\Delta\)AUC and \(\Delta\)rheobase occupy four quadrants separated by no changes in AUC and rheobase. Representative schematic fI curves in red with respect to a reference (or wild type) fI curve (blue) depict the general changes associated with each quadrant.}
        \label{fig:firing_characterization}
\end{figure}

Neuronal firing is a complex phenomenon, and a quantification of firing properties is required for comparisons across cell types and between different 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_characterization}A). The characterization of the firing properties of a neuron by using 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) by two independent measures. 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, \drheo, and in AUC, \(\Delta\)AUC, relative to the wild type neuron. \(\Delta\)AUC is in addition normalized to the AUC of 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 \ndAUC (\Cref{fig:firing_characterization}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 gain of function (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, loss of function (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. 

% \notenk{Moved to discussion section ``Firing Frequency Analysis} \textit{In these cases, it depends on the regime the neuron is operating in. If it is in its excitable regime and only occasionally 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. \textcolor{red}{\noteuh{Das sind ja eigentlich schon Hypothesen, sollte eher in die Diskussion oder dort zumindest noch mal aufgegriffen werden. Kommt ja vermutlich noch ;-)}}  \textcolor{red}{\notenk{Add to dicussion? As intro and explanation as to why we characterize firing?}}}

\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 shifted to lower currents \(-\)\drheo, making the neuron more sensitive to weak inputs, but at the same time the slope of the fI curves was reduced (\(-\)\ndAUC), which resulted in a reduced 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 got 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 characterized each of these curves by a single number, the \( \text{Kendall} \ \tau \) correlation coefficient\textsuperscript{a}. A monotonically increasing curve resulted in a \( \text{Kendall} \ \tau \) close to \(+1\)\textsuperscript{a}, a monotonously decreasing curve in \( \text{Kendall} \ \tau \approx -1 \)\textsuperscript{a}, and a non-monotonous, non-linear relation in \( \text{Kendall} \ \tau \) close to zero\textsuperscript{a} (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 model G to more depolarized values, then the rheobase of the resulting fI curves shifted to lower currents \(-\)\drheo, making the neuron more sensitive to weak inputs, but at the same time the slope of the fI curves was reduced (\(-\)\ndAUC), which resulted in a reduced 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 model C 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 got 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 characterized each of these curves by a single number, the \( \text{Kendall} \ \tau \) correlation coefficient\textsuperscript{a}. A monotonically increasing curve resulted in a \( \text{Kendall} \ \tau \) close to \(+1\)\textsuperscript{a}, a monotonously decreasing curve in \( \text{Kendall} \ \tau \approx -1 \)\textsuperscript{a}, and a non-monotonous, non-linear relation in \( \text{Kendall} \ \tau \) close to zero\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 \(g\) affected the AUC (\Cref{fig:AUC_correlation}), but how exactly the AUC was affected usually depended on the specific neuronal model. Increasing the slope factor of the \Kv activation curve for example increased the AUC in all models (\( \text{Kendall} \ \tau \approx +1\)\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 resulted in negative correlations with the AUC in all models (\( \text{Kendall} \ \tau \approx -1\)\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 (\( \text{Kendall} \ \tau \approx +1\)\textsuperscript{a}), whereas in others AUC was decreased (\( \text{Kendall} \ \tau \approx -1\)\textsuperscript{a}). In the STN +\Kv model, AUC depended in a non-linear way on the maximal conductance of the delayed rectifier, resulting in a \( \text{Kendall} \ \tau \) close to zero\textsuperscript{a}. Even more dramatic qualitative differences between models resulted from shifts of the activation curve of the delayed rectifier, as discussed already above (\Cref{fig:AUC_correlation}~A,B,C). Some model neurons did almost not depend on changes in K-current half activation \(V_{1/2}\) or showed strong non-linear dependencies, both resulting in \( \text{Kendall} \ \tau\) close to zero\textsuperscript{a}. Many model neurons showed strongly negative correlations, and a few displayed 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 (\( \text{Kendall} \ \tau \approx +1\)\textsuperscript{a}), whereas in others AUC was decreased (\( \text{Kendall} \ \tau \approx -1\)\textsuperscript{a}). In model I, AUC depended in a non-linear way on the maximal conductance of the delayed rectifier, resulting in a \( \text{Kendall} \ \tau \) close to zero\textsuperscript{a}. Even more dramatic qualitative differences between models resulted from shifts of the activation curve of the delayed rectifier, as discussed already above (\Cref{fig:AUC_correlation}~A,B,C). Some model neurons did almost not depend on changes in K-current half activation \(V_{1/2}\) or showed strong non-linear dependencies, both resulting in \( \text{Kendall} \ \tau\) close to zero\textsuperscript{a}. Many model neurons showed strongly negative correlations, and a few displayed positive correlations with shifting the activation curve of the delayed rectifier.


\begin{figure}[tp]
        \centering
        \includegraphics[width=\linewidth]{Figures/AUC_correlation.pdf}
        \linespread{1.}\selectfont
%        \caption[]{Effects of altered channel kinetics on AUC in various neuron models. The fI curves corresponding to shifts in FS \(+\)\Kv model delayed rectifier K half activation \(V_{1/2}\) (A), changes \Kv activation slope factor \(k\) in the FS \(+\)\Kv model (D), and changes in maximal conductance of delayed rectifier K current in the STN \(+\)\Kv model (G) are shown. The fI curves from the smallest (grey) to the largest (black) alterations are seen for (A,D, and G) in accordance to the greyscale of the x-axis in B, E, and H. The \ndAUC of fI curves is plotted against delayed rectifier K half activation potential (\(\Delta V_{1/2}\); D), \Kv activation slope factor \(k\) (k/\(\textrm{k}_{WT}\); E) and maximal conductance \(g\) of the delayed rectifier K current (g/\(\textrm{g}_{WT}\); H) for all models (thin lines) with relationships from the fI curve examples (A, D, G respectively) highlighted by thick lines with colors corresponding to the box highlighting each set of fI curves. The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in half maximal potential \(V_{1/2}\) and \ndAUC (C), slope factor k and \ndAUC (F) as well as maximal current conductances and \ndAUC (I) for each model and current property is computed. The relationships between \(\Delta V_{1/2}\), k/\(\textrm{k}_{WT}\), and g/\(\textrm{g}_{WT}\)  and \ndAUC for the Kendall rank correlations highlighted in the black boxes are depicted in (B), (E) and (H) respectively. }
	\caption[]{Effects of altered channel kinetics on AUC in various neuron models. The fI curves corresponding to shifts in model G delayed rectifier K half activation \(V_{1/2}\) (A), changes \Kv activation slope factor \(k\) in model G (D), and changes in maximal conductance of delayed rectifier K current in the  model I (G) are shown. The fI curves from the smallest (grey) to the largest (black) alterations are seen for (A,D, and G) in accordance to the greyscale of the x-axis in B, E, and H. The \ndAUC of fI curves is plotted against delayed rectifier K half activation potential (\(\Delta V_{1/2}\); D), \Kv activation slope factor \(k\) (k/\(\textrm{k}_{WT}\); E) and maximal conductance \(g\) of the delayed rectifier K current (g/\(\textrm{g}_{WT}\); H) for all models (thin lines) with relationships from the fI curve examples (A, D, G respectively) highlighted by thick lines with colors corresponding to the box highlighting each set of fI curves. The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in half maximal potential \(V_{1/2}\) and \ndAUC (C), slope factor k and \ndAUC (F) as well as maximal current conductances and \ndAUC (I) for each model and current property is computed. The relationships between \(\Delta V_{1/2}\), k/\(\textrm{k}_{WT}\), and g/\(\textrm{g}_{WT}\)  and \ndAUC for the Kendall rank correlations highlighted in the black boxes are depicted in (B), (E) and (H) respectively. }
        \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 \(g\) affected rheobase (\Cref{fig:rheobase_correlation}). However, in contrast to AUC, qualitatively consistent effects on rheobase across models could be observed. An increasing of the maximal conductance of the leak current in the Cb stellate model increased the rheobase (\Cref{fig:rheobase_correlation}~G). When these changes were plotted against the change in maximal conductance a monotonically increasing relationship was evident (thick teal line in \Cref{fig:rheobase_correlation}~H). This monotonically increasing relationship was evident in all models (\( \text{Kendall} \ \tau \approx +1\)\textsuperscript{a}), but with different slopes (thin lines in \Cref{fig:rheobase_correlation}~H). Similarly, positive correlations were consistently found across models for maximal conductances of delayed rectifier K, \Kv, and A type currents, whereas the maximal conductance of the sodium current was consistently associated with negative correlations (\( \text{Kendall} \ \tau \approx -1\)\textsuperscript{a}; \Cref{fig:rheobase_correlation}~I), i.e. rheobase 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 \(g\) affected rheobase (\Cref{fig:rheobase_correlation}). However, in contrast to AUC, qualitatively consistent effects on rheobase across models could be observed. An increasing of the maximal conductance of the leak current in the model A increased the rheobase (\Cref{fig:rheobase_correlation}~G). When these changes were plotted against the change in maximal conductance a monotonically increasing relationship was evident (thick teal line in \Cref{fig:rheobase_correlation}~H). This monotonically increasing relationship was evident in all models (\( \text{Kendall} \ \tau \approx +1\)\textsuperscript{a}), but with different slopes (thin lines in \Cref{fig:rheobase_correlation}~H). Similarly, positive correlations were consistently found across models for maximal conductances of delayed rectifier K, \Kv, and A type currents, whereas the maximal conductance of the sodium current was consistently associated with negative correlations (\( \text{Kendall} \ \tau \approx -1\)\textsuperscript{a}; \Cref{fig:rheobase_correlation}~I), i.e. rheobase decreased with increasing maximum conductance in all models.

%Although changes in half maximal potential \(V_{1/2}\) and slope factor \(k\) generally correlated with rheobase similarly across model there were some exceptions. Changing the slope factor of Na-current inactivation, \Kv-current inactivation, and A-current activation affected the rheobase both with positive and negative correlations in different models \textcolor{red}{\noteuh{Würde diese hier noch mal benennen, damit es klar wird. }}\notenk{Ich mache das ungern, weil ich für jedes (Na-current inactivation, \Kv-current inactivation, and A-current activation) 2 Liste habe (+ und - rheobase Aenderungen} (\Cref{fig:rheobase_correlation}~F). Departures from monotonic relationships also occurred 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 \textcolor{red}{\noteuh{Auch hier die unterschiedlcihen betroffenen cell type models benennen, einfach in Klammer dahinter.}}\notenk{Hier mache ich das auch ungern, für ähnlichen Gründen}. 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. 

Although changes in half maximal potential \(V_{1/2}\) and slope factor \(k\) generally correlated with rheobase similarly across model there were some exceptions. Rheobase was affected with both with positive and negative correlations in different models as a result of changing slope factor of Na-current inactivation (positive: models A-H and J; negative: models I, K and L), \Kv-current inactivation (positive: models I and K; negative: models E-G, J, H), and A-current activation (positive: models A,F and L; negative: model I; \Cref{fig:rheobase_correlation}~F). Departures from monotonic relationships also occurred in some models as a result of K-current activation  \(V_{1/2}\) (e.g. model J) and slope factor \(k\) (models F and G), \Kv-current inactivation slope factor \(k\) (model K), and A-current activation slope factor \(k\) (model L). 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. 


\begin{figure}[tp]
        \centering
        \includegraphics[width=\linewidth]{Figures/rheobase_correlation.pdf}
        \linespread{1.}\selectfont
%        \caption[]{Effects of altered channel kinetics on rheobase.  The fI curves corresponding to shifts in FS \(+\)\Kv model \Kv activation \(V_{1/2}\) (A), changes \Kv inactivation slope factor \(k\) in the Cb stellate \(+\)\Kv model (D), and changes in maximal conductance of the leak current in the Cb stellate model (G) are shown. The fI curves from the smallest (grey) to the largest (black) alterations are seen for (A,D, and G) in accordance to the greyscale of the x-axis in B, E, and H. The \drheo of fI curves is plotted against \Kv half activation potential (\(\Delta V_{1/2}\); B), \Kv inactivation slope factor \(k\) (k/\(\textrm{k}_{WT}\); E) and maximal conductance \(g\) of the leak current (g/\(\textrm{g}_{WT}\); H) for all models (thin lines) with relationships from the fI curve examples (A, D, G respectively) highlighted by thick lines with colors corresponding to the box highlighting each set of fI curves. The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in half maximal potential \(V_{1/2}\) and \drheo (C), slope factor k and \drheo (F) as well as maximal current conductances and \drheo (I) for each model and current property is computed. The relationships between \(\Delta V_{1/2}\), k/\(\textrm{k}_{WT}\), and g/\(\textrm{g}_{WT}\)  and \drheo for the Kendall rank correlations highlighted in the black boxes are depicted in (B), (E) and (H) respectively.}
        \caption[]{Effects of altered channel kinetics on rheobase.  The fI curves corresponding to shifts in model G \Kv activation \(V_{1/2}\) (A), changes \Kv inactivation slope factor \(k\) in model F (D), and changes in maximal conductance of the leak current in model A (G) are shown. The fI curves from the smallest (grey) to the largest (black) alterations are seen for (A,D, and G) in accordance to the greyscale of the x-axis in B, E, and H. The \drheo of fI curves is plotted against \Kv half activation potential (\(\Delta V_{1/2}\); B), \Kv inactivation slope factor \(k\) (k/\(\textrm{k}_{WT}\); E) and maximal conductance \(g\) of the leak current (g/\(\textrm{g}_{WT}\); H) for all models (thin lines) with relationships from the fI curve examples (A, D, G respectively) highlighted by thick lines with colors corresponding to the box highlighting each set of fI curves. The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in half maximal potential \(V_{1/2}\) and \drheo (C), slope factor k and \drheo (F) as well as maximal current conductances and \drheo (I) for each model and current property is computed. The relationships between \(\Delta V_{1/2}\), k/\(\textrm{k}_{WT}\), and g/\(\textrm{g}_{WT}\)  and \drheo for the Kendall rank correlations highlighted in the black boxes are depicted in (B), (E) and (H) respectively.}
        \label{fig:rheobase_correlation}
\end{figure}

\subsection*{\textit{KCNA1} Mutations}
Mutations in \textit{KCNA1} are associated with episodic ataxia type~1 (EA1) and have been characterized biophysically (as reviewed by \citet{lauxmann_therapeutic_2021}). Here they were used as a test case 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 \textit{KCNA1} mutations were heterogeneous across models containing \Kv, but generally showed decreases in rheobase (\Cref{fig:simulation_model_comparision}A-I).  Pairwise non-parametric Kendall \(\tau\) rank correlations\textsuperscript{a} between the simulated effects of these \Kv mutations on rheobase were highly correlated across models (\Cref{fig:simulation_model_comparision}J) indicating that EA1 associated \textit{KCNA1} mutations generally decrease rheobase across diverse cell-types. However, the effects of the \Kv mutations on AUC were 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
        \includegraphics[width=\linewidth]{Figures/simulation_model_comparison.pdf}
        \linespread{1.}\selectfont
%        \caption[]{Effects of episodic ataxia type~1 associated \textit{KCNA1} mutations on firing.  Effects of \textit{KCNA1} mutations on AUC (percent change in normalized \(\Delta\)AUC) 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. All \textit{KCNA1} Mutations are marked in grey with the V174F, F414C, E283K, and V404I \textit{KCNA1} mutations highlighted in color for each model. Pairwise Kendall rank correlation coefficients (Kendall \(\tau\)) between the effects of \textit{KCNA1} mutations on rheobase and on AUC are shown in J and K respectively. Marker shape is indicative of model/firing type, and grey dashed lines denote the quadrants of firing characterization (see \Cref{fig:firing_characterization}).}
	\caption[]{Effects of episodic ataxia type~1 associated \textit{KCNA1} mutations on firing.  Effects of \textit{KCNA1} mutations on AUC (percent change in normalized \(\Delta\)AUC) and rheobase (\(\Delta\)Rheobase) compared to wild type for model H (A), model E (B), model G (C), model A (D), model F (E), model J (F), model L (G), model I (H) and model K (I). All \textit{KCNA1} Mutations are marked in grey with the V174F, F414C, E283K, and V404I \textit{KCNA1} mutations highlighted in color for each model. Pairwise Kendall rank correlation coefficients (Kendall \(\tau\)) between the effects of \textit{KCNA1} mutations on rheobase and on AUC are shown in J and K respectively. Marker shape is indicative of model/firing type, and grey dashed lines denote the quadrants of firing characterization (see \Cref{fig:firing_characterization}).}
        \label{fig:simulation_model_comparision}
\end{figure}


\section*{Discussion (3000 Words Maximum - Currently )}
% \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.}\\
%Changes to single ionic current properties, as well as known episodic ataxia type~1 associated \textit{KCNA1} mutations showed consistent effects on the rheobase across cell types, whereas the effects on AUC of the steady-state fI-curve depended on the cell type. Our results demonstrate that loss of function (LOF) and gain of function (GOF) on the biophysical level cannot be uniquely transferred to the level of neuronal firing. Thus the effects caused by different mutations depend on the properties of the other ion channels expressed in a cell and are therefore depend on the channel ensemble of a specific cell type.

To compare the effects ion channel mutations on neuronal firing of different neuron types, a diverse set of conductance-based models was used and the effect of changes in individual channel properties across conductance-based neuronal models and the effects of episodic ataxia type~1 associated (EA1) \textit{KCNA1} mutations were simulated. Changes to single ionic current properties, as well as known EA1 associated \textit{KCNA1} mutations showed consistent effects on the rheobase across cell types, whereas the effects on AUC of the steady-state fI-curve depended on the cell type. Our results demonstrate that loss of function (LOF) and gain of function (GOF) on the biophysical level cannot be uniquely transferred to the level of neuronal firing. Thus the effects caused by different mutations depend on the properties of the other ion channels expressed in a cell and are therefore depend on the channel ensemble of a specific cell type. 

\subsection*{Firing Frequency Analysis}
Although, firing differences can be characterized by an area under the curve of the fI curve for fixed current steps this approach characterizes firing as a mixture of key features: rheobase and the initial slope of the fI curve. By probing rheobase directly and using an AUC relative to rheobase, we disambiguate these features and enable insights into the effects on rheobase and initial fI curve steepness. This increases the specificity of our understanding of how ion channel mutations alter firing across cells types and enable classification as described in \Cref{fig:firing_characterization}. Importanty, in cases when ion channel mutations alter rheobase and initial fI curve sleepness in ways that opposing effects on firing (upper left and bottom right quadrants of \Cref{fig:firing_characterization}B) this disamgibuation is important for understanding the outcome of the mutation. In these cases, the regime the neuron is operating in is vital in determining the cells firing outcome. If it is in its excitable regime and only occasionally generates an action potential, then the effect on the rheobase is more important. If it is firing periodically with high rates, then the change in AUC might be more relevant. 

\subsection*{Modelling Limitations}
The models used here are simple and while they all capture key aspects of the firing dynamics for their respective cell, they fall short of capturing the complex physiology and biophysics of real cells. However, for the purpose of understanding how different cell-types, or current environments, contribute to diversity in firing outcomes of ion channel mutations, the fidelity of the models to the physiological cells they represent is of a minor concern and the variety in currents and dynamics across models is of utmost importance. With this context, models are labelled as models A-L here to highlight that the physiological cells they represent is not of chief concern, but rather that the collection of models with different attributes results in heterogenous firing responses to the same perturbation. Additionally, the development of more realistic models is a high priority and will enable cell-type specific predictions that may aid in precision medicine approaches. Thus, weight should not be put on any single predicted firing outcome here in a specific model, but rather on the differences in outcomes that occur across the cell-type spectrum the models used here represent. 

\subsection*{Neuronal Diversity}
The nervous system consists of a vastly diverse and heterogenous collection of neurons with variable properties and characteristics including diverse combinations and expression levels of ion channels which are vital for neuronal firing dynamics.

Advances in high-throughput techniques have enabled large-scale investigation into single-cell properties across the CNS \citep{Poulin2016} that have revealed large diversity in neuronal gene expression, morphology and neuronal types in the motor cortex \citep{Scala2021}, neocortex \cite{Cadwell2016, Cadwell2020}, GABAergic neurons in the cortex and retina \citep{Huang2019, Laturnus2020}, cerebellum \citep{Kozareva2021}, spinal cord \citep{Alkaslasi2021}, visual cortex \citep{Gouwens2019} as well as the retina \citep{Baden2016, Voigt2019, Berens2017,  Yan2020a, Yan2020b}. 

Diversity across neurons is not limited to gene expression and can also be seen electrophysiologically \citep{Tripathy2017, Gouwens2018, Tripathy2015, Scala2021, Cadwell2020, Gouwens2019, Baden2016, Berens2017} with correlations existing between gene expression and electrophysiological properties \citep{Tripathy2017}. At the ion channel level, diversity exists not only between the specific ion channels the different cell types express but heterogeneity also exists in ion channel expression levels within cell types \citep{marder_multiple_2011,  goaillard_ion_2021,barreiro_-current_2012}.  As ion channel properties and expression levels are key determinents of neuronal dynamics and firing \citep{Balachandar2018, Gu2014, Zeberg2015, Aarhem2007, Qi2013, Gu2014a, Zeberg2010, Zhou2020, Kispersky2012} neurons with different ion channel properties and expression levels display different firing properties.

To capture the diversity in neuronal ion channel expression and its relevance in the outcome of ion channel mutations, we used multiple neuronal models with different ionic currents and underlying firing dynamics here.

   
\subsection*{Ionic Current Environments Determine the Effect of Ion Channel Mutations}

To our knowledge, no comprehensive evaluation of how ionic current environment and cell type affect the outcome of ion channel mutations have been reported. However, comparisons between the effects of such mutations between certain cell types were described. For instance, the R1648H mutation in SCN1A does not alter the excitability of cortical pyramidal neurons, but causes hypoexcitability of adjacent inhibitory GABAergic neurons \citep{Hedrich14874}. In the CA3 region of the hippocampus, the equivalent mutation in \textit{SCN8A}, R1627H, increases the excitability of pyramidal neurons and decreases the excitability of parvalbumin positive interneurons \cite{makinson_scn1a_2016}. Additionally, the L858H mutation in \(\textrm{Na}_\textrm{V}\textrm{1.7}\), associated with erythermyalgia, has been shown to cause hypoexcitability in sympathetic ganglion neurons and hyperexcitability in dorsal root ganglion neurons \citep{Waxman2007, Rush2006}. The differential effects of L858H \(\textrm{Na}_\textrm{V}\textrm{1.7}\) on firing is dependent on the presence or absence of another sodium channel, namely the \(\textrm{Na}_\textrm{V}\textrm{1.8}\) subunit \citep{Waxman2007, Rush2006}.  These findings, in concert with our findings emphasize that the ionic current environment in which a channelopathy occurs is vital in determining the outcomes of the channelopathy on firing.

Cell type specific differences in ionic current properties are important in the effects of ion channel mutations. However within a cell type heterogeneity in channel expression levels exists and it is often desirable to generate a population of neuronal models and to screen them for plausibility to biological data in order to capture neuronal population diversity \citep{marder_multiple_2011}. The models we used here are originally generated by characterization of current gating properties and by fitting of maximal conductances to experimental data \citep{pospischil_minimal_2008, ranjan_kinetic_2019, alexander_cerebellar_2019, otsuka_conductance-based_2004}. This practice of fixing maximal conductances based on experimental data is limiting as it does not reproduce the variability in channel expression and neuronal firing behavior of a heterogeneous neuron population \citep{verma_computational_2020}. For example, a model derived from the mean conductances in a neuronal sub-population within the stomatogastric ganglion, the so-called "one-spike bursting" neurons fire three spikes instead of one per burst due to an L-shaped distribution of sodium and potassium conductances \citep{golowasch_failure_2002}.  
Multiple sets of conductances can give rise to the same patterns of activity also termed degeneracy and differences in neuronal dynamics may only be evident with perturbations \citep{marder_multiple_2011, goaillard_ion_2021}. 
The variability in ion channel expression often correlates with the expression of other ion channels \citep{goaillard_ion_2021} and neurons whose behavior is similar may possess correlated variability across different ion channels resulting in stability in the neuronal phenotype \citep{lamb_correlated_2013, soofi_co-variation_2012, taylor_how_2009}. 
The variability of ionic currents and degeneracy of neurons may account, at least in part, for the observation that the effect of toxins within a neuronal type is frequently not constant \citep{khaliq_relative_2006, puopolo_roles_2007, ransdell_neurons_2013}.
% \textcolor{red}{\notenk{add temperature sensitivity-> within cell-type heterogeneity exists - Marder paper?}}
\subsection*{Effects of \textit{KCNA1} Mutations}
Changes in delayed rectifier potassium currents, analogous to those seen in LOF \textit{KCNA1} mutations, change the underlying firing dynamics of the Hodgkin Huxley model result in reduced thresholds for repetitive firing and thus contribute to increased excitability \citep{hafez_altered_2020}. Although the Hodgkin Huxley delayed rectifier lacks inactivation, the increases in excitability observed by \citet{hafez_altered_2020} are in line with our simulation-based predictions of the outcomes of \textit{KCNA1} mutations.  LOF  \textit{KCNA1} mutations generally increase neuronal excitability, however the varying susceptibility on rheobase and different effects on AUC of the fI-curve of KCNA1 mutations across models are indicative that a certain cell type specific complexity exists. Increased excitability is seen experimentally with \Kv null mice \citep{smart_deletion_1998, zhou_temperature-sensitive_1998}, with pharmacological \Kv block \citep{chi_manipulation_2007, morales-villagran_protection_1996} and by \citet{hafez_altered_2020} with simulation-based predictions of \textit{KCNA1} mutations. Contrary to these results,  \citet{zhao_common_2020} predicted \textit{in silico} that the depolarizing shifts seen as a result of \textit{KCNA1} mutations broaden action potentials and interfere negatively with high frequency action potential firing. However, they varied stimulus duration between different models and therefore comparability of firing rates is lacking in this study.     

In our simulations, different current properties alter the impact of  \textit{KCNA1} mutations on firing as evident in the differences seen in the impact of \(\textrm{I}_\textrm{A}\) and \IKv in the Cb stellate and STN model families on  \textit{KCNA1} mutation firing. This highlights that not only knowledge of the biophysical properties of a channel but also its neuronal expression and other neuronal channels present is vital for the holistic understanding of the effects of a given ion channel mutation both at the single cell and network level. 

\subsection*{Loss or Gain of Function Characterizations Do Not Fully Capture Ion Channel Mutation Effects on Firing}
The effects of changes in channel properties depend in part on the neuronal model in which they occur and can be seen in the variance of correlations (especially in AUC of the fI-curve) across models for a given current property change. Therefore, relative conductances and gating properties of currents in the ionic current environment in which an alteration in current properties occurs plays an important role in determining the outcome on firing. The use of loss of function (LOF) and gain of function (GOF) is useful at the level of ion channels to indicate whether a mutation results in more or less ionic current. However, the extension of this thinking onto whether mutations induce LOF or GOF at the level of neuronal firing based on the ionic current LOF/GOF is problematic due to the dependency of neuronal firing changes on the ionic channel environment. Thus, the direct leap from current level LOF/GOF characterizations to effects on firing without experimental or modelling-based evidence, although tempting, should be refrained from and viewed with caution when reported. This is especially relevant in the recent development of personalized medicine for channelopathies, where a patient's specific channelopathy is identified and used to tailor treatments \citep{Weber2017, Ackerman2013, Helbig2020, Gnecchi2021, Musto2020, Brunklaus2022, Hedrich2021}. However, in these cases the effects of specific ion channel mutations are often characterized based on ionic currents in expression systems and classified as LOF or GOF to aid in treatment decisions \citep{johannesen_genotype-phenotype_2021, Brunklaus2022, Musto2020}.  Although positive treatment outcomes occur with sodium channel blockers in patients with GOF \(\textrm{Na}_{\textrm{V}}\textrm{1.6}\) mutations, patients with both LOF and GOF \(\textrm{Na}_{\textrm{V}}\textrm{1.6}\) mutations can benefit from treatment with sodium channel blockers \citep{johannesen_genotype-phenotype_2021}, suggesting that the relationship between effects at the level of ion channels and effects at the level of firing and therapeutics is not linear or evident without further contextual information. 

Therefore, this approach should be used with caution and the cell type which expresses the mutant ion channel may provide valuable insight into the functional consequences of an ion channel mutation. Where experimental assessment of the effects of a patient's specific ion channel mutation \textit{in vivo} is not feasible at a large scale, modelling approaches investigating the effects of patient specific channelopathies provides an alternative bridge between characterization of changes in biophysical properties of ionic currents and the firing consequences of these effects. In both experimental and modelling investigation into the effects of ion channel mutations on neuronal firing the specific cell-type dependency should be considered.


The effects of altered ion channel properties on firing is generally influenced by the other ionic currents present in the cell. In channelopathies the effect of a given ion channel mutation on neuronal firing therefore depends on the cell type in which those changes occur \citep{Hedrich14874, makinson_scn1a_2016, Waxman2007, Rush2006}. Although certain complexities of neurons such as differences in cell-type sensitivities to current property changes, interactions between ionic currents, cell morphology and subcellular ion channel distribution are neglected here, it is likely that this increased complexity \textit{in vivo} would contribute to the cell-type dependent effects on neuronal firing. The complexity and nuances of the nervous system, including cell-type dependent firing effects of channelopathies explored here, likely underlie shortcomings in treatment approaches in patients with channelopathies and accounting for cell-type dependent firing effects provide an opportunity to further the efficacy and precision in personalized medicine approaches.



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\section*{References}\sloppy
% \textit{Only published references should appear in the reference list at the end of the paper. The latest information on in-press references should be provided. In the case of in-press references (i.e., accepted for publication in a specific journal or book) the paper, which must be relevant for reviewers to see in order to make a well-informed evaluation should be included as a separate document text file along with the submitted manuscript. In this case, the authors recognize the loss of anonymity. “Submitted” references should be cited only in text and in the following form: (unpublished observations). If the paper is accepted, the authors can then add their names: A. B. Smith, C. D. Johnson, and E. Green, unpublished observations). The form for personal communications is similar: (F. G. Jackson, personal communication). Authors are responsible for all personal communications and must obtain written approval from persons cited before submitting the paper to eNeuro. Proof of such approval may be requested by eNeuro.

% References should be cited in the text as follows: “The procedure used has been described elsewhere (Green, 1978),” or “Our observations are in agreement with those of Brown and Black (1979) and of White et al. (1980),” or, with multiple references in chronological order: “Earlier reports (Brown and Black, 1979, 1981; White et al., 1980; Smith, 1982, 1984) ...”

% Papers should be given in alphabetical order according to the surname of the first author. In two-author papers with the same first author, the order is alphabetical by the second author’s name. In three-or-more-author papers with the same first author, the order is chronological. The name of the author(s) should be followed by the date in parentheses, the full title of the paper as it appeared in the original together with the source of the reference, the volume number, and the first and last pages. Do not number or bullet the references. If the author list for a paper in the references exceeds 20, the paper should be cited as Author A et al. The following illustrate the format to be used:
% Journal article

%     Hamill OP, Marty A, Neher E, Sakmann B, Sigworth F (1981) Improved patch-clamp techniques for high-resolution current recordings from cells and cell free membrane patches. Pflugers Arch 391:85–100.
%     Hodgkin AL, Huxley AF (1952a) The components of membrane conductance in the giant axon of Loligo. J Physiol (Lond) 116:473–496.
%     Hodgkin AL, Huxley AF (1952b) The dual effect of membrane potential on sodium conductance in the giant axon of Loligo. J Physiol (Lond) 116:497–506.

% Book

%     Hille B (1984) Ionic channels of excitable membranes. Sunderland, MA: Sinauer.

% Chapter in a book

%     Stent GS (1981) Strength and weakness of the genetic approach to the development of the nervous system. In: Studies in developmental neurobiology: essays in honor of Viktor Hamburger (Cowan WM, ed), pp288–321. New York: Oxford UP.

% Abbreviations of journal titles should follow those listed in the Index Medicus. Responsibility for correct references lies with the authors. All references on the reference list must have at least one corresponding in-text citation. References must be double-spaced, and no bullets, numbers, or other listing formats should be used.}
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\section*{Figures}
% \textit{Figures must be numbered independently of tables and multimedia and cited in the manuscript. Do not duplicate data by presenting it both in the text and in a figure.
% A title should be part of the legend and not lettered onto the figure. A legend must be included in the manuscript document after the reference list, and should include enough detail to be intelligible without reference to the text. Specific individuals’ contributions to data acquisition, analysis, or other responsibility resulting in a figure may be included at the end of each legend. Please use the heading “Figure Contributions” and state each contribution with the author’s full name.
% Figure Contributions: John Smith performed the experiments; Jane Jones analyzed the data.
% Figures must be submitted as separate files in TIFF or EPS format and be submitted at the size they are to appear: 1 column (maximum width 8.5 cm), 1.5 columns (maximum width 11.6 cm) or 2 columns (maximum width 17.6 cm). They should be the smallest size that will convey the essential scientific information.
% Illustrations should be prepared so that they are accessible to color-blind readers and color should only be used if it is necessary to accurately convey the information being presented by the image. Grayscale generally provides a more faithful representation when a single quantity is displayed. Use textures or different line types rather than colors in bar plots or graphs. Figures with red and green are particularly problematic and should generally be converted to magenta and green. If no suitable combination can be found, consider presenting separate monochrome images for the different color channels. For line drawings that require color, consider redundant coding by adding different textures or line types to the colors.
% Color figures should be in RGB format and supplied at a minimum of 300 dpi. Monochrome (bitmap) images must be supplied at 1200 dpi. Grayscale must be supplied at a minimum of 300 dpi. For figures in vector-based format, all fonts should be converted to outlines and saved as EPS files to ensure that they are reproduced correctly.
% Remove top and right borderlines that to not contain measuring metrics from all graph/histogram figure panels (i.e., do not box the panels in). Do not include any two-bar graphs/histograms; instead state those values in the text.
% All illustrations documenting results must include a bar to indicate the scale. All labels used in a figure should be explained in the legend. The migration of protein molecular weight size markers or nucleic acid size markers must be indicated and labeled appropriately (e.g., “kD”, “nt”, “bp”) on all figure panels showing gel electrophoresis.}

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%\begin{figure}[tp]
%        \centering
%        \includegraphics[width=\linewidth]{Figures/diversity_in_firing.pdf}
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%        \caption[]{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 (see \Cref{fig:ramp_firing}).}
%        \label{fig:diversity_in_firing}
%\end{figure}
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%
%\begin{figure}[tp]
%        \centering
%	\includegraphics[width=0.5\linewidth]{Figures/firing_characterization_arrows.pdf}
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%        \caption[]{Characterization of firing with AUC and rheobase. (A) The area under the curve (AUC) of the repetitive firing frequency-current (fI) curve. (B) 
%Changes in firing as characterized by \(\Delta\)AUC and \(\Delta\)rheobase occupy 4 quadrants separated by no changes in AUC and rheobase. Representative schematic fI curves in red with respect to a reference fI curve (blue) depict the general changes associated with each quadrant.}
%        \label{fig:firing_characterization}
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%
%\begin{figure}[tp]
%        \centering
%        \includegraphics[width=\linewidth]{Figures/AUC_correlation.pdf}
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%        \caption[]{Effects of altered channel kinetics on AUC in various neuron models. The fI curves corresponding to shifts in FS \(+\)\Kv model delayed rectifier K half activation \(V_{1/2}\) (A), changes \Kv activation slope factor \(k\) in the FS \(+\)\Kv model (B), and changes in maximal conductance of delayed rectifier K current in the STN \(+\)\Kv model (C) are shown. The \ndAUC of fI curves is plotted against delayed rectifier K half activation potential (\(\Delta V_{1/2}\); B), \Kv activation slope factor \(k\) (k/\(\textrm{k}_{WT}\); E) and maximal conductance \(g\) of the delayed rectifier K current (g/\(\textrm{g}_{WT}\); H) for all models (thin lines) with relationships from the fI curve examples (A, D, G respectively) highlighted by thick lines with colors corresponding to the box highlighting each set of fI curves. The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in half maximal potential \(V_{1/2}\) and \ndAUC (C), slope factor k and \ndAUC (F) as well as maximal current conductances and \ndAUC (I) for each model and current property is computed. The relationships between \(\Delta V_{1/2}\), k/\(\textrm{k}_{WT}\), and g/\(\textrm{g}_{WT}\)  and \ndAUC for the Kendall rank correlations highlighted in the black boxes are depicted in (B), (E) and (H) respectively.}
%        \label{fig:AUC_correlation}
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%\begin{figure}[tp]
%        \centering
%        \includegraphics[width=\linewidth]{Figures/rheobase_correlation.pdf}
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%        \caption[]{Effects of altered channel kinetics on rheobase.  The fI curves corresponding to shifts in FS \(+\)\Kv model \Kv activation \(V_{1/2}\) (A), changes \Kv inactivation slope factor \(k\) in the Cb stellate \(+\)\Kv model (B), and changes in maximal conductance of the leak current in the Cb stellate model (C) are shown. The \drheo of fI curves is plotted against \Kv half activation potential (\(\Delta V_{1/2}\); B), \Kv inactivation slope factor \(k\) (k/\(\textrm{k}_{WT}\); E) and maximal conductance \(g\) of the leak current (g/\(\textrm{g}_{WT}\); H) for all models (thin lines) with relationships from the fI curve examples (A, D, G respectively) highlighted by thick lines with colors corresponding to the box highlighting each set of fI curves. The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in half maximal potential \(V_{1/2}\) and \drheo (C), slope factor k and \drheo (F) as well as maximal current conductances and \drheo (I) for each model and current property is computed. The relationships between \(\Delta V_{1/2}\), k/\(\textrm{k}_{WT}\), and g/\(\textrm{g}_{WT}\)  and \drheo for the Kendall rank correlations highlighted in the black boxes are depicted in (B), (E) and (H) respectively..}
%        \label{fig:rheobase_correlation}
%\end{figure}
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%\begin{figure}[tp]
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%        \includegraphics[width=\linewidth]{Figures/simulation_model_comparison.pdf}
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%        \caption[]{Effects of episodic ataxia type~1 associated \Kv mutations on firing.  Effects of \Kv mutations on AUC (percent change in normalized \(\Delta\)AUC) 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. Marker shape is indicative of model/firing type, and grey dashed lines denote the quadrants of firing characterization (see \Cref{fig:firing_characterization}).}
%        \label{fig:simulation_model_comparision}
%\end{figure}


%\captionof{figure}{Characterization of firing with AUC and rheobase. (A) The area under the curve (AUC) of the repetitive firing frequency-current (fI) curve. (B) Changes in firing as characterized by \(\Delta\)AUC and \(\Delta\)rheobase occupy 4 quadrants separated by no changes in AUC and rheobase. Representative schematic fI curves in blue with respect to a reference fI curve (black) depict the general changes associated with each quadrant.}
%
%\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 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 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.}

%\newpage
\section*{Tables}
% \textit{All tables must be numbered independently of figures, multimedia, and 3D models and cited in the manuscript. Do not duplicate data by presenting it both in the text and in a table.
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\section*{Extended Data}
\beginsupplement
\begin{figure}[tp]%described
        \centering
        \includegraphics[width=\linewidth]{Figures/ramp_firing.pdf}
        \linespread{1.}\selectfont
	\vspace{-2cm}
        \caption[]{Diversity in Neuronal Model Firing Responses to a Current Ramp. Spike trains 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 in response to a slow ascending current ramp followed by the descending version of the current ramp (bottom). The current at which firing begins in response to an ascending current ramp and the current at which firing ceases in response to a descending current ramp are depicted on the frequency current (fI) curves in \Cref{fig:diversity_in_firing} for each model.}
        \label{fig:ramp_firing}
\end{figure}

% \textit{A legend for the code file, labeled as “Extended Data 1,” should be at the end of the manuscript.\\}
% The code files must be packaged into a single ZIP file, uploaded to the submission system as a “Multimedia/Extended Data” file type.}
\captionof{Extended Data}{Code in zip file. Description needs to be added once code is ready.}
\end{document}

