abstract

thesis/Masterthesis.tex

@@ -91,24 +91,14 @@ Außerdem erkläre ich, dass die eingereichte Arbeit weder vollständig noch in
 
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-\section*{Not to forget: TODO}
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-\item check time form in text results present!
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-\section{Zusammenfassung}
-% Abstract in deutsch
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 \section{Abstract}
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+The environment contains vital information for the survival of an organism. Thus it is critical for the senses of an organism to encode this information. This encoding process needs to be efficient to gather the most information possible while at the same time filtering and removing noise and irrelevant information.
+The active electric sense used by the electric fish \AptLepto is a well defined model system for adaptive signal processing. The population of P-Unit neurons, the electro sensory afferents, display strong heterogeneity. The proposed model is able to reproduce this heterogeneic population allow further research into the encoding of such heterogeneous neuron populations.
 
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@@ -149,7 +139,7 @@ Further it could allow researchers gain a better picture how higher brain areas
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+\newpage
 \section{Materials and Methods}
 
 \subsection{Cell Recordings}
@@ -483,7 +473,7 @@ The cells' burstiness distribution has two peaks: the continuously firing cells
 
 \begin{figure}[H]
 \includegraphics{figures/fit_burstiness_comparison.pdf}
-\caption{\label{fig:comp_burstiness}  Comparison of baseline firing properties between cells and their corresponding fits. The histograms on top compare the distributions of the n= 54 cells in blue and their respective models in orange. The scatter plot at the bottom directly compares them. Points on the identity line (grey) indicate perfect model predictions. The points are colored by the cell's burstiness. \textbf{A}: The model values for the burstiness agree well with the values of the model but again show a tendency that the higher the value of the cell the more the model value is below it. \textbf{B}: The CV also shows the problem of the burstiness but the values drift apart more slowly starting around 0.6. The  points are colored according to the correlation between burstiness and CV (shown below). \todo{}}
+\caption{\label{fig:comp_burstiness}  Comparison of baseline firing properties between cells and their corresponding fits. The histograms on top compare the distributions of the n= 54 cells in blue and their respective models in orange. The scatter plot at the bottom directly compares them. Points on the identity line (grey) indicate perfect model predictions. The points are colored by the cell's burstiness. \textbf{A}: The model values for the burstiness agree well with the values of the model but again show a tendency that the higher the value of the cell the more the model value is below it. \textbf{B}: The CV also shows the problem of the burstiness but the values drift apart more slowly starting around 0.6. The colouring of the points indicates the correlation between burstiness and CV (shown below).}
 \end{figure}
 
 
@@ -495,8 +485,7 @@ The fit of the onset response characterized by the slope of the Boltzmann functi
 \caption{\label{fig:comp_adaption}  Comparison of adaption properties between cells and their corresponding fits. The histograms on top compare the distributions of the n=54 cells in blue and their respective models in orange. The scatter plot at the bottom directly compares them. Points on the identity line (grey) indicate perfect model predictions. \textbf{A}: The $f_\infty$ slope pairs show good agreement with mostly low scattering in both direction. \textbf{B}: The $f_0$ values show a higher spread and for steeper slopes the models have more often too flat slopes.}
 \end{figure}
 
-\todo{clearer}
+Given the differences between the cell firing properties and the ones of the model the correlations were calculated and show differences (fig. \ref{fig:behavior_correlations}). Of the seven correlations found in the data set the fitted models show a correlation for all but the correlation between the VS and baseline firing rate, but the models also show four additional correlations. These are between the base rate and the $f_0$ slope, base rate and burstiness, base rate and SC and finally between SC and $f_\infty$ slope.  
-Given the differences between the cell firing properties and the ones of the model the correlations  were calculated and show differences (fig. \ref{fig:behavior_correlations}). Of the seven correlations found in the data set the fitted models show a correlation for all but the correlation between the VS and baseline firing rate, but the models also show four additional correlations. These are between the baseline firing rate and the $f_0$ slope, base rate and burstiness again base rate and SC and between SC and $f_\infty$ slope.  
 
 Before the parameter distributions (fig. \ref{fig:parameter_distributions}) and correlations (fig. \ref{fig:parameter_correlations}) of the model parameters were closer investigated, the potential influence of the different EOD frequencies was removed by scaling the time dependent parameters for all models. This was done by calculating the factor between the fish's EOD frequency and the chosen EOD frequency of 800\,Hz and then multiplying all time parameters appropriately to their dependence with the factor. These scaled parameter distributions are shown in figure \ref{fig:parameter_distributions}. With these scaled distributions the correlations between the parameters were computed giving the matrix in figure \ref{fig:parameter_correlations}, it shows extensive correlations between most parameters. The correlations indicate that the parameter can compensate for each other and that the model can produce similar firing properties for different parameter sets. A notable exception is the refractory period $t_{ref}$ which is independent of all other parameters and could as such be the only variable influencing the burstiness in this model. 
 
@@ -567,8 +556,7 @@ Even with these differences the firing property distributions (fig. \ref{fig:dra
 \section{Discussion}
 
 In this thesis a simple model based on the leaky integrate-and-fire (LIF) model was developed to allow the simulation of a neuron population that correctly represents the heterogeneity of P-units in the electrosensory pathway of the electric fish \textit{A. leptorhynchus}. The LIF model was extended by an adaption current, a refractory period and simulated the input synapses by rectifying and low-pass filtering the input current, building on the model proposed by \cite{walz2013Phd}. This model was then fit to in vivo recordings of single P-units characterized by seven firing properties and the resulting models were compared to their respective reference cell. Additionally estimates of the distributions and covariances of the model parameters were used to draw random parameter sets. Simulations of these generated populations were compared with the data.
-The previously proposed model by \cite{walz2013Phd} was limited to only non bursting P-units, but the extension allows now also the simulation of bursting neurons. The model proposed by \cite{chacron2001simple} also models the P-units with a extended LIF neuron but uses a dynamic threshold to add the negative ISI correlation, instead of an adaption current as used here. This causes the the neuron to display divisive adaption \citep{benda2010linear} instead of the substractive adaption shown in P-units \citep{benda2005spike}. Their model was also only shown for one representative neuron of bursting non-bursting cells. Lastly there is the model proposed by \cite{kashimori1996model}. Which is an model based on the physiological and anatomical properties of the P-units. \todo{read paper...}
+The previously proposed model by \cite{walz2013Phd} was limited to only non bursting P-units, but the extension allows now also the simulation of bursting neurons. The model proposed by \cite{chacron2001simple} also models the P-units with a extended LIF neuron but uses a dynamic threshold to add the negative ISI correlation, instead of an adaption current as used here. This causes the the neuron to display divisive adaption \citep{benda2010linear} instead of the substractive adaption shown in P-units \citep{benda2005spike}. Their model was also only shown for one representative neuron of bursting non-bursting cells. 
-
 
 \subsection{Model fit}
 
@@ -586,7 +574,7 @@ The parameters of the fitted models also show extensive correlations between eac
 
 \subsection{Heterogeneous Population}
 
-\todo{}The correlations and the estimated parameter distributions were used form of their covariances to draw random parameter sets from a multivariate normal distribution. The drawn parameters show the expected distributions but different correlations. That could mean that the 54 models used to calculate them were to few to give enough statistical power for the correct estimation of all correlations. Drawing more models and compensating for the increase in power showed that the involved correlations stay inconsistent, which points to an uncertainty already in the measured covariance matrix of the data. This could be further investigated with a robustness analysis estimating the reliability of the computed covariances. 
+The correlations and the estimated parameter distributions were used in form of their covariances to draw random parameter sets from a multivariate normal distribution. The drawn parameters show the expected distributions but different correlations. That could mean that the 54 models used to calculate them were to few to give enough statistical power for the correct estimation of all correlations. Drawing more models and compensating for the increase in power showed that the involved correlations stay inconsistent, which points to an uncertainty already in the measured covariance matrix of the data. This could be further investigated with a robustness analysis estimating the reliability of the computed covariances. 
 
 The firing behavior shown by the drawn models on the other hand fits the ones of the data quite well except for the VS, where it is consistently underestimating the VS of the data.