sensitivity analysis

manuscript.tex

@@ -273,7 +273,7 @@ The code/software described in the paper is freely available online at [URL reda
 To examine the role of cell-type specific ionic current environments on the impact of altered ion channel properties on firing behaviour 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 \Kv mutations on firing was then examined across different neuronal models with different ionic current environments.
 
 \subsection*{Characterization of Neuronal Firing Properties}
-\begin{figure}[t]
+\begin{figure}[tp]
         \centering
         \includegraphics[width=0.5\linewidth]{Figures/firing_characterization.pdf}
         \\\notejb{Das mit den LOF, GOF und ? in B ist mal ein Vorschlag, der erstens noch verbessert werden kann, aber der auch gerne wieder rueckgaengig gemacht werden kann.}\notenk{Ich wurde LOF, GOF  und ? rueckgaengig machen weil wir argumentieren dass LOF und GOF fuer da Feuerverhalten nicht so geeignet sind} \notejb{Ja, und genau die Fragezeichen unterstreichen das schon an dieser Stelle. Spaeter bei den Ergebnissen gibt Beispiele, die genau diese Problem haben (Fig 3.A links), da wuerde ich dann darauf hinweisen.}
@@ -285,7 +285,7 @@ Changes in firing as characterized by \(\Delta\)AUC and \(\Delta\)rheobase occup
 
 Neuronal firing is a complex phenomenon and a quantification of firing properties is required for comparisons across cell types and between conditions. Here we focus on two aspects of firing: rheobase (smallest injected current at which the cell fires an action potential) and the initial shape of the frequency-current (fI) curve as quantified by the area under the curve (AUC) for a fixed range of input currents above rheobase (\Cref{fig:firing_characterizaton}A). The characterization of firing by rheobase and AUC allows to characterize both a neuron's excitability in the sub-threshold regime (rheobase) and periodic firing in the super-threshold regime (AUC). Note that AUC is essentially quantifying the slope of a neuron's fI curve.
 
-In the upper left quadrant of \Cref{fig:firing_characterizaton}B, fI curves become steeper (positive difference of AUC values: \(+\Delta\)AUC) and are shifted to lower rheobases (negative difference of rheobases: \(-\Delta\)rheobase), unambigously indicating an increased firing that clearly might be classified as a GOF of neuronal firing. The opposite happens in the bottom right quadrant where the slope of fI curves decreases (\(-\Delta\)AUC) and the rheobase is shifted to higher currents (\(+\Delta\)rheobase), indicating a decreased, LOF firing. In the lower left (\(-\Delta\)AUC and \(-\Delta\)rheobase) and upper right (\(+\Delta\)AUC and \(+\Delta\)rheobase) quadrants, the effects on firing are less clear-cut, because the changes in rheobase and AUC have opposite effects on neuronal firing. Changes in a neuron's fI curves in these two quadrants cannot uniquely be described as a gain or loss of excitability. Im these cases it depends on the regime the neuron is operating in. If it is in its excitable regime and only occasionaly generates an action potential, then the effect on the rheobase matters more. If it is firing periodically with high rates, then the change in AUC might be more relevant.
+In the upper left quadrant of \Cref{fig:firing_characterizaton}B, fI curves become steeper (positive difference of AUC values: \(+\Delta\)AUC) and are shifted to lower rheobases (negative difference of rheobases: \(-\Delta\)rheobase), unambigously indicating an increased firing that clearly might be classified as a GOF of neuronal firing. The opposite happens in the bottom right quadrant where the slope of fI curves decreases (\(-\Delta\)AUC) and the rheobase is shifted to higher currents (\(+\Delta\)rheobase), indicating a decreased, LOF firing. In the lower left (\(-\Delta\)AUC and \(-\Delta\)rheobase) and upper right (\(+\Delta\)AUC and \(+\Delta\)rheobase) quadrants, the effects on firing are less clear-cut, because the changes in rheobase and AUC have opposite effects on neuronal firing. Changes in a neuron's fI curves in these two quadrants cannot uniquely be described as a gain or loss of excitability. In these cases it depends on the regime the neuron is operating in. If it is in its excitable regime and only occasionaly generates an action potential, then the effect on the rheobase matters more. If it is firing periodically with high rates, then the change in AUC might be more relevant. \notejb{These latter sentences could also go into the discussion. On the other hand they pave the way for the reader.}
 
 \begin{figure}[tp]
         \centering
@@ -295,10 +295,14 @@ In the upper left quadrant of \Cref{fig:firing_characterizaton}B, fI curves beco
         \label{fig:diversity_in_firing}
 \end{figure}
 
-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 all fire tonically and do not exhibit bursting. Bursting would add a third dimension to the characterization of neuronal firing properties, in addition to rheobase and AUC. 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 have type II firing. Type I firing is characterized by continuous fI curves (i.e. firing rate increases from 0 in a continuous fashion) generated through a saddle-node on invariant cycle bifurcation. 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) due to sub-critical Hopf bifurcation \cite{ERMENTROUT2002, ermentrout_type_1996}. The other models used here lie on a continuum between these prototypical firing classifications. \notejb{The STN models could be a homoclinic bifurcation (long delay but type 2 like firing), maybe cite Izhikevic book for this.} 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}).\notejb{This hysteresis also supports the homoclinic bifurcation. It is like Type II but with a saddle producing long delays and low rates like Type I. Just adding homoclinic to the Hopf is not quite right... I'll think about it. \citep{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 all fire tonically and do not exhibit bursting. Bursting would add a third dimension to the characterization of neuronal firing properties, in addition to rheobase and AUC. 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 have type II firing. Type I firing is characterized by continuous fI curves (i.e. firing rate increases from 0 in a continuous fashion) generated through a saddle-node on invariant cycle bifurcation. 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) due to sub-critical Hopf bifurcation \cite{ERMENTROUT2002, ermentrout_type_1996}. 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}). This prominent hysteresis despite a continues but steep fI curve, together with the long delays of the first spike hint at a homoclinic bifurcation for the STN models \citep{Izhikevich2006} \notejb{add paper by Susanne Schreiber and Jan-Hendrik Schleimer}.
 
 \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. Changes in gating \(V_{1/2}\) and slope factor \(k\) as well as the maximum conductance affect AUC (\Cref{fig:AUC_correlation} A, B and C). Heterogeneity in the correlation between gating and conductance changes and AUC occurs across models for most ionic currents. In these cases some of the models display non-monotonic relationships or no relationship (\( |\text{Kendall} \tau | \approx 0\)). However, shifts in A-current activation \(V_{1/2}\), changes in \Kv activation \(V_{1/2}\) and slope factor \(k\), and changes in A-current conductance display consistent monotonic relationships across models  (\( |\text{Kendall} \tau | \ne 0\)). The impact of a similar change in \(V_{1/2}\), slope factor \(k\), or conductance of different currents will impact firing behaviour differently not just within but also between models.
+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.
+
+Changes in gating \(V_{1/2}\) and slope factor \(k\) as well as the maximum conductance affect AUC (\Cref{fig:AUC_correlation}), but how exactly AUC is affected usually depends on the specific models. Increasing, for example, the slope factor of the \Kv activation curve, increases the AUC in all models (\( \text{Kendall} \tau \approx +1\)), but with different slopes ((\Cref{fig:AUC_correlation}~B). Similar consistent positive correlations can be found for shifts in A-current activation \(V_{1/2}\). Changes in \Kv activation \(V_{1/2}\) and in A-current conductance result in negative correlations with the AUC in all models (\( \text{Kendall} \tau \approx -1\)).
+
+Qualitative differences can be found, for example, when increasing the conductance of the delayed rectifier ((\Cref{fig:AUC_correlation}~C). In some model neurons this increased AUC (\( \text{Kendall} \tau \approx +1\)), whereas in others AUC is decreased (\( \text{Kendall} \tau \approx -1\)). In the STN +\Kv model, AUC depends in a non-linear way on the conductance of the delayed rectifier, resulting in an \( \text{Kendall} \tau \) close to zero. Even more dramatic qualitative differences between models result from shifts of the activation curve of the delayed rectifier ((\Cref{fig:AUC_correlation}~A). Some model neurons do almost not depend on changes in K-current activation \(V_{1/2}\) or show strongly non-linear dependencies, both resulting in \( \text{Kendall} \tau\) close to zero. Many model neurons show strongly negative correlations, and a few show positive correlations with shifting the activation curve of the delayed rectifier.
 
 \notejb{the slope factor has a name (``slope factor'', but \(V_{1/2}\) not. How is this called, ``midpoint potential''?}\notenk{How about ``half-maximal potential''?}
 
@@ -312,7 +316,7 @@ Sensitivity analyses are used to understand how input model parameters contribut
         \label{fig:AUC_correlation}
 \end{figure}
 
-Alterations in gating \(V_{1/2}\) and slope factor \(k\) as well as the maximum conductance also play a role in determining rheobase (\Cref{fig:rheobase_correlation} A, B and C).  Shifts in half activation of gating properties are similarly correlated with rheobase across models, however Kendall \(\tau\) values departing from \(-1\) indicate non-monotonic or no relationships between K-current \(V_{1/2}\) and rheobase in some models (\Cref{fig:rheobase_correlation}A). Changes in Na-current inactivation, \Kv-current inactivation, and A-current activation affect rheobase with positive and negative correlations in different models (\Cref{fig:rheobase_correlation}B). Departures from monotonic relationships occur in some models as a result of K-current activation, \Kv-current inactivation, and A-current activation in some models. Maximum conductance affects rheobase similarly across models (\Cref{fig:rheobase_correlation}C). However, identical changes in current gating properties such as activation or inactivation \(V_{1/2}\) or slope factor \(k\) can have differing effects on firing depending on the model in which they occur.
+The rheobase is also affected by changing the channel kinetics \Cref{fig:rheobase_correlation}). In contrast to AUC, most alterations result in similar changes of rheobase in all models, but there are some noteable 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}B). Departures from monotonic relationships occur in some models as a result of K-current activation, \Kv-current inactivation, and A-current activation in some models \notejb{which parameters are changed?}. Maximum conductance affects rheobase similarly across models (\Cref{fig:rheobase_correlation}C). However, identical changes in current gating properties such as activation or inactivation \(V_{1/2}\) or slope factor \(k\) can have differing effects on firing depending on the model in which they occur. \notejb{This we just have said...}
 
 \begin{figure}[tp]
         \centering