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alexanderott
2020-08-31 19:20:47 +02:00
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thesis/Masterthesis.aux
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\@writefile{toc}{\contentsline {section}{\numberline {1}Zusammenfassung}{4}{section.1}}
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\@writefile{toc}{\contentsline {section}{\numberline {3}Introduction}{4}{section.3}}
@@ -41,7 +38,7 @@
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces {\color {red}(TODO: place right in text)}On the left: The averaged response of a cell to a step in EOD amplitude. The beginning (at 0\tmspace +\thinmuskip {.1667em}s) and end (at 1\tmspace +\thinmuskip {.1667em}s) of the stimulus are marked by the gray lines. The detected values for the onset ($f_0$) and steady-state ($f_{inf}$) response are marked in {\color {red}(TODO: color)}. $f_0$ is detected as the highest deviation from the mean frequency before the stimulus while $f_{inf}$ is the average frequency in the 0.1\tmspace +\thinmuskip {.1667em}s time window, 25\tmspace +\thinmuskip {.1667em}ms before the end of the stimulus. On the right: The fi-curve visualizes the onset and steady-state response of the neuron for different stimuli contrasts. In {\color {red}(TODO: color)} the detected onset responses and the fitted Boltzmann, in {\color {red}(TODO: color)} the detected steady-state response and the linear fit.}}{8}{figure.3}}
\newlabel{fig:f_point_detection}{{3}{8}{\todo {place right in text}On the left: The averaged response of a cell to a step in EOD amplitude. The beginning (at 0\,s) and end (at 1\,s) of the stimulus are marked by the gray lines. The detected values for the onset ($f_0$) and steady-state ($f_{inf}$) response are marked in \todo {color}. $f_0$ is detected as the highest deviation from the mean frequency before the stimulus while $f_{inf}$ is the average frequency in the 0.1\,s time window, 25\,ms before the end of the stimulus. On the right: The fi-curve visualizes the onset and steady-state response of the neuron for different stimuli contrasts. In \todo {color} the detected onset responses and the fitted Boltzmann, in \todo {color} the detected steady-state response and the linear fit}{figure.3}{}}
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\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Overview about all variables of the model that are fitted.}}{11}{table.1}}
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\begin{thebibliography}{}
\bibitem[Benda and Herz, 2003]{benda2003universal}
Benda, J. and Herz, A.~V. (2003).
\newblock A universal model for spike-frequency adaptation.
\newblock {\em Neural computation}, 15(11):2523--2564.
\bibitem[Benda et~al., 2010]{benda2010linear}
Benda, J., Maler, L., and Longtin, A. (2010).
\newblock Linear versus nonlinear signal transmission in neuron models with
adaptation currents or dynamic thresholds.
\newblock {\em Journal of Neurophysiology}, 104(5):2806--2820.
\bibitem[Gao and Han, 2012]{gao2012implementing}
Gao, F. and Han, L. (2012).
\newblock Implementing the nelder-mead simplex algorithm with adaptive
parameters.
\newblock {\em Computational Optimization and Applications}, 51(1):259--277.
\bibitem[Todd and Andrews, 1999]{todd1999identification}
Todd, B.~S. and Andrews, D.~C. (1999).

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@@ -279,24 +279,17 @@ Afterwards a Boltzmann was fitted to the onset response and a rectified line was
% add info about simulation by euler integration and which time steps!
% show voltage dynamics with resistance :
also show function with membrane resistance before explaining that is unknown and left out: $ \tau_m \frac{dV}{dt} = -V + I$
% explain subthreshold behaviour first then add V_{th} and adaption etc
% explain modeling of the adaption current see Benda2010
% table with explanation of variables ?
\todo{add an introduction into models Pif - LIF - LIFAC}
\todo{restructure and rewrite sounds horrible}
The above described cell characteristics need to be reproduced by a simple and efficient model to be able to simulate bigger populations in a reasonable time. The simplest commonly used neuron model is the perfect integrate-and-fire (PIF) model. It's voltage can be described in one equation: $\tau_m \frac{dV}{dt} = \frac{I}{R_m}$ with $I$ the stimulus current, $R_m$ the membrane resistance and a voltage threshold $V_\theta$. In this model $I$ is integrated and when this threshold is reached the voltage is reset to zero and a spike is recorded (see fig. \ref{fig:model_comparison} PIF). The model is useful for basic simulations but cannot reproduce the richer behavior of the p-units, as it has neither a memory of previous spikes that could cause the negative serial correlation between successive spikes nor can it show any adaption behavior.
The next slightly more complex model is the leaky integrate-and-fire (LIF) model. As the name suggests it adds a leakage current to the PIF and as follows the equation \ref{eq:basic_voltage_dynamics} (fig. \ref{fig:model_comparison} LIF). The leakage current adds sub threshold behavior to the model but still cannot reproduce the adaption or serial correlation.
\begin{equation}
\tau_m \frac{dV}{dt} = -V + \frac{I}{R_m}
\tau_m \frac{dV}{dt} = -V + IR_m
\label{eq:basic_voltage_dynamics}
\end{equation}
To reproduce this behavior the model needs some form of memory of previous spikes. There are two main ways this can be added to the model as an adaptive current or a dynamic threshold. The biophysical mechanism of the adaption in p-units is unknown because the cell bodies are not accessible for intra-cellular recordings. Following the results of \cite{benda2003universal} a negative adaptive current was chosen, because the dynamic threshold causes divisive adaption instead of the subtractive adaption of p-units \todo{reference}. This results in an leaky integrate-and-fire model with adaption current (LIFAC) (fig. \ref{fig:model_comparison} LIFAC). The added adaptive current follow the dynamics:
To reproduce this behavior the model needs some form of memory of previous spikes. There are two main ways this can be added to the model as an adaptive current or a dynamic threshold. The biophysical mechanism of the adaption in p-units is unknown because the cell bodies are not accessible for intra-cellular recordings. Following the results of \cite{benda2010linear} a negative adaptive current was chosen, because the dynamic threshold causes divisive adaption instead of the subtractive adaption of p-units \todo{reference}. This results in an leaky integrate-and-fire model with adaption current (LIFAC) (fig. \ref{fig:model_comparison} LIFAC). The added adaptive current follow the dynamics:
\begin{equation}
\tau_A \frac{dI_A}{dt} = -I_A + \Delta_A \sum \delta (t)
@@ -370,10 +363,27 @@ Together this results in the dynamics seen in equation \ref{eq:full_voltage_dyna
This leaves the eight variables to be fitted to the cell. During the fitting and the analysis all models were simulated with at time step of 0.05\,ms.
The stimuli as described in the stimulus protocols section above were recreated for the stimulation of the model during the fitting process. The pure fish EOD was approximated by a simple sin function of the appropriate frequency, but it was decided to keep the amplitude of the sin at one to make the models more comparable. Changes in the amplitude can be compensated in the model by changing the input scaling factor and the time constant of the dendritic low-pass filter, so there is no qualitative difference.
The model during the fitting the baseline stimulus was simulated with 3 times with 30\,s each and the step stimuli were simulated with a 0.5\,s delay, 0.5\,s duration and 0.5\,s recovery time with the same contrasts as used in the cell. The step stimuli were repeated 8 times.
During the fitting the baseline stimulus was simulated 3 times with 30\,s each and the step stimuli were simulated with a 0.5\,s delay, 0.5\,s duration and 0.5\,s recovery time. The contrasts were the same as in the cell recording. The step stimuli were repeated 8 times.
With the simulation data the model characteristics were calculated the same way as for the cells (see above).
The error function was constructed from the absolute differences between the vector strengths, the burstiness and the coefficients of variation, the absolute relative error for the steady-state response slope and mean absolute relative error between their respective $f_{inf}$ detections and $f_0$ detections and finally the root-mean-square error between the ISI-histograms and the first 50\,ms of the frequency response after stimulus onset.
The error function was constructed from both the baseline characteristics: ISI-histogram, VS, CV, SC and burstiness and the fi-curve: the detections of $f_{inf}$ and $f_0$ responses for each contrast, the slope of the linear fit into the $f_{inf}$ and the frequency trace of one step response.
The error of the VS, CV, SC, and burstiness was calculated as the scaled absolute difference:
\begin{equation}
err_i = |x^M_i - x^C_i| * c_i
\end{equation}
with $x^M_i$ the model value for the characteristic $i$, $x^C_i$ the corresponding cell value and $c_i$ a scaling factor that is the same for all cells but different between characteristics. The scaling factor was used to make all errors a comparable size.
The error for the slope of the $f_{inf}$ fit was the scaled relative difference:
\begin{equation}
err_i = |(x^M_i - x^C_i) / x^C_i| * c_i
\end{equation}
For the $f_{inf}$ and $f_0$ responses the average scaled difference off all contrasts was taken and finally the error for the ISI-histogram and the step-response was calculated with a root-mean-square error. For the histogram over all bins but for the step response only the first 50\,ms after stimulus onset as an error for the adaption time constant.
All errors were then weighted and summed up for the full error. The fits were done with the Nelder-Mead algorithm of scipy minimize \citep{gao2012implementing}. All model variables listed above in table \ref{tab:parameter_explanation} before were fit at the same time except for $I_{Bias}$. $I_{Bias}$ was determined before each fitting iteration and set to a value giving the correct baseline frequency.

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\select@language {english}
\babel@toc {english}{}
\contentsline {section}{\numberline {1}Zusammenfassung}{4}{section.1}
\contentsline {section}{\numberline {2}Abstract}{4}{section.2}
\contentsline {section}{\numberline {3}Introduction}{4}{section.3}