first proofread of mat&met v1
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The cell recordings for this master thesis were collected as part of other previous studies (\cite{walz2013Phd}, \citep{walz2014static})\todo{ref other studies} and is described there but will also be repeated below . There were recordings of 457 p-units were inspected. Of those 88 fulfilled the basic necessary requirements of including a measurement of at least 30 seconds of the baseline behavior and containing at least 7 different contrasts with each at least 7 trials for the FI-Curve (see below \todo{ref fi-curve? }). After pre-analysis of those cells an additional 13 cells were excluded because of analysis difficulties.
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The cell recordings for this master thesis were collected as part of other previous studies (\cite{walz2013Phd}, \citep{walz2014static})\todo{ref other studies} and is described there but will also be repeated below. The recordings of 457 p-units were inspected. Of those 88 fulfilled the basic necessary requirements: including a measurement of at least 30 seconds of the baseline behavior and containing at least 7 different contrasts with each at least 7 trials for the FI-Curve (see below \todo{ref fi-curve? }). After pre-analysis of those cells an additional 13 cells were excluded because of analysis difficulties.
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The 75 used cells came from 32 \AptLepto (brown ghost knifefish). The fish were between 11-25\,cm long (15.78 $\pm$ 3.48\,cm) and their electric organ discharge (EOD) frequencies were between 601-928\,Hz (753.1 $\pm$ 82.3\,Hz). The gender of the fish was not determined.
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The in vivo intracellular recordings of P-unit electroreceptors were done in the lateral line nerve . The fish were anesthetized with MS-222 (100-130 mg/l; PharmaQ; Fordingbridge, UK) and the part of the skin covering the lateral line just behind the skull was removed, while the area was anesthetized with Lidocaine (2\%; bela-pharm; Vechta, Germany). The fish were immobilized for the recordings with Tubocurarine (Sigma-Aldrich; Steinheim, Germany, 25–50 $\mu l$ of 5\. mg/ml solution) and placed in the experimental tank (47 $\times$ 42 $\times$ 12\,cm) filled with water from the fish's home tank with a conductivity of about 300$\mu$\,S/cm and the temperature was around 28°C.
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All experimental protocels were approved and complied with national and regional laws (files: no. 55.2-1-54-2531-135-09 and Regierungspräsidium Tübingen no. ZP 1/13 and no. ZP 1/16 \todo{andere antrags nummern so richtig ?})
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For the recordings a standard glass mircoelectrode (borosilicate; 1.5 mm outer diameter; GB150F-8P, Science Products, Hofheim, Germany) was used. They were pulled to a resistance of 50-100\,M$\Omega$ using Model P-97 from Sutter Instrument Co. (No-
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vato, CA, USA) and filled with 1M KCl solution. The electrodes were controlled using microdrives (Luigs-Neumann; Ratingen, Germany) and the potentials recorded with the bridge mode of the SEC-05 amplifier (npi-electronics GmbH, Tamm, Germany) and lowpass filtered at 10 kHz.
|
||||
The in vivo intracellular recordings of P-unit electroreceptors were done in the lateral line nerve . The fish were anesthetized with MS-222 (100-130 mg/l; PharmaQ; Fordingbridge, UK) and the part of the skin covering the lateral line just behind the skull was removed, while the area was anesthetized with Lidocaine (2\%; bela-pharm; Vechta, Germany). The fish were immobilized for the recordings with Tubocurarine (Sigma-Aldrich; Steinheim, Germany, 25–50\,$\mu l$ of 5\,mg/ml solution) and placed in the experimental tank (47 $\times$ 42 $\times$ 12\,cm) filled with water from the fish's home tank with a conductivity of about 300$\mu$\,S/cm and the temperature was around 28°C.
|
||||
All experimental protocols were approved and complied with national and regional laws (files: no. 55.2-1-54-2531-135-09 and Regierungspräsidium Tübingen no. ZP 1/13 and no. ZP 1/16 \todo{andere antrags nummern so richtig ?})
|
||||
For the recordings a standard glass mircoelectrode (borosilicate; 1.5 mm outer diameter; GB150F-8P, Science Products, Hofheim, Germany) was used. They were pulled to a resistance of 50-100\,M$\Omega$ using Model P-97 from Sutter Instrument Co. (Novato, CA, USA) and filled with 1\,M KCl solution. The electrodes were controlled using microdrives (Luigs-Neumann; Ratingen, Germany) and the potentials recorded with the bridge mode of the SEC-05 amplifier (npi-electronics GmbH, Tamm, Germany) and lowpass filtered at 10 kHz.
|
||||
|
||||
During the recording spikes were detected online using the peak detection algorithm from \cite{todd1999identification}. It uses a dynamically adjusted threshold value above the previously detected trough. To detect spikes through changes in amplitude the threshold was set to 50\% of the amplitude of a detected spike while keeping the threshold above a minimum set to be higher than the noise level based on a histogram of all peak amplitudes. Trials with bad spike detection were removed from further analysis.
|
||||
The fish's electric organ discharge (EOD) was recorded using using two vertical carbon rods (11\,cm long, 8\,mm diameter) positioned in front of the head and behind its tail. The signal was amplified 200 to 500 times and band-pass filtered (3 − 1500 Hz passband, DPA2-FX, npi-electronics, Tamm, Germany). The electrodes were placed on iso-potential lines of the stimulus field to reduce the interference of the stimulus in the recording. All signals were digitized using a data acquisition board (PCI-6229; National Instruments, Austin TX, USA) at a sampling rate of 20-100\,kHz (54 cells at 20\,kHz, 20 at 100\,kHz and 1 at 40\,kHz)
|
||||
The fish's EOD was recorded using using two vertical carbon rods (11\,cm long, 8\,mm diameter) positioned in front of the head and behind its tail. The signal was amplified 200 to 500 times and band-pass filtered (3 − 1500 Hz passband, DPA2-FX, npi-electronics, Tamm, Germany). The electrodes were placed on iso-potential lines of the stimulus field to reduce the interference of the stimulus in the recording. All signals were digitized using a data acquisition board (PCI-6229; National Instruments, Austin TX, USA) at a sampling rate of 20-100\,kHz (54 cells at 20\,kHz, 20 at 100\,kHz and 1 at 40\,kHz)
|
||||
|
||||
The recording and stimulation was done using the ephys, efield, and efish plugins of the software RELACS (\href{www.relacs.net}{www.relacs.net}). It allowed the online spike and EOD detection, pre-analysis and visualization and ran on a Debian computer.
|
||||
|
||||
@ -173,7 +174,6 @@ The stimuli used during the recordings were presented from two vertical carbon r
|
||||
For this work two types of recordings were made with all cells: baseline recordings and amplitude step recordings for the frequency-Intensity curve (FI-Curve).
|
||||
The 'stimulus' for the baseline recording is purely the EOD field the fish produces itself with no external stimulus.
|
||||
|
||||
|
||||
The amplitude step stimulus here is a step in EOD amplitude. To be able to cause an amplitude modulation (AM) in the fish's EOD , the EOD was recorded and the multiplied with the modulation (see fig. \ref{fig:stim_examples}). This modified EOD can then be presented at the right phase with the stimulus electrodes, causing constructive interference and adding the used amplitude modulation to the EOD (Fig. \ref{fig:stim_examples}). This stimuli construction as seen in equation \ref{eq:am_generation} works for any AM as long as the EOD of the fish is stable.
|
||||
|
||||
\begin{equation}
|
||||
@ -184,15 +184,15 @@ Stimulus = EOD(t) + AM(t) * EOD(t) \todo{acceptable?}
|
||||
|
||||
\begin{figure}[H]
|
||||
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={left, center}, capbesidewidth=0.45\textwidth}}]{figure}[\FBwidth]
|
||||
{\caption{\label{fig:stim_examples} Example of the Stimuli construction. At the top a recording of the fish's EOD. In the middle a part of the recording multiplied with the AM, a step with a contrast of 130\% between 0 and 50\,ms (marked in \todo{color}). At the bottom the resulting stimulus trace when the AM is added to the EOD. This example stimulus is for visualization purposes 50\,ms short. During the measurements the stimulus was 0.4\,s or 1\,s long. }}
|
||||
{\caption{\label{fig:stim_examples} Example of the stimulus construction. At the top a recording of the fish's EOD. In the middle a part of the recording multiplied with the AM, a step with a contrast of 130\% between 0 and 50\,ms (marked in \todo{color}). At the bottom the resulting stimulus trace when the AM is added to the EOD. This example stimulus is for visualization purposes 50\,ms short. During the measurements the stimulus was 0.4\,s or 1\,s long. }}
|
||||
{\includegraphics[width=0.45\textwidth]{figures/amGeneration.pdf}}
|
||||
\end{figure}
|
||||
|
||||
|
||||
The step stimuli all consisted of a delay of 0.2\,s followed by a 0.4\,s (n=68) or 1\,s (n=7) long step and a 0.8\,s long recovery time. The contrast range measured was for the most cells 80-120\% of EOD amplitude. Some cells were measured in a larger range up to 20-180\%. In this range at least 7 contrasts were measured with at least 7 trials, but again many cells were measured with more contrasts and trials. The additionally measured contrasts were used for the model if they had at least 3 trials
|
||||
The step stimuli all consisted of a delay of 0.2\,s followed by a 0.4\,s (n=68) or 1\,s (n=7) long step and a 0.8\,s long recovery time. The contrast range measured was for the most cells 80-120\% of EOD amplitude. Some cells were measured in a larger range up to 20-180\%. In the range at least 7 contrasts were measured with at least 7 trials, but again many cells were measured with more contrasts and trials. The additionally measured contrasts were used for the model if they had at least 3 trials.
|
||||
|
||||
That means for every cell the FI-Curve was measured at at least 7 Points each with at least 7 trials. If more contrasts were measured during the recording the additional information was used as long as there were at least 3 trials available.
|
||||
All presentations had 0.2\,s delay at the start and then started the stimulus at time 0. The step stimulus was presented for 0.4\,s (7 cells) or 1\,s(68 cells) and followed by 0.8\,s time for the cell to recover back to baseline.
|
||||
%That means for every cell the FI-Curve was measured at at least 7 Points each with at least 7 trials. If more contrasts were measured during the recording the additional information was used as long as there were at least 3 trials available.
|
||||
%All presentations had 0.2\,s delay at the start and then started the stimulus at time 0. The step stimulus was presented for 0.4\,s (7 cells) or 1\,s(68 cells) and followed by 0.8\,s time for the cell to recover back to baseline.
|
||||
|
||||
% Always a 0.2 second delay and 0.8 seconds after stimulus
|
||||
% Stimulus start at time=0
|
||||
@ -222,7 +222,7 @@ All presentations had 0.2\,s delay at the start and then started the stimulus at
|
||||
\subsection{Cell Characteristics}
|
||||
|
||||
The cells were characterized by ten parameters: 6 for the baseline and 4 for the fi-curve.
|
||||
For the baseline the mean frequency was calculated by dividing the number of spikes in the recording by the recording time. Then the set $T$ of all interspike intervals (ISI) of the spikes in the recording further parameter was calculated and the other parameters were calculated from it.
|
||||
For the baseline the mean frequency was calculated by dividing the number of spikes in the recording by the recording time. Then the set of all interspike intervals (ISI) $T$ of the spikes in the recording further parameter was calculated and the other parameters were calculated from it.
|
||||
|
||||
The coefficient of variation (CV) is defined as the standard deviation (STD) of $T$ divided by the mean ISI, see equation \ref{eq:CV} with angled brackets as the averaging operator.
|
||||
|
||||
@ -250,7 +250,7 @@ SC_x = \frac{\langle (T_{i} - \langle T \rangle)(T_{i+x} - \langle T \rangle) \r
|
||||
with the angled brackets again the averaging operator.
|
||||
|
||||
|
||||
Finally the ISI-histogram was calculated within a range of 0-50\,ms and a bin size of 0.1\,ms and the burstiness was calculated as the percentage of ISI smaller than 2.5 EOD periods multiplied by the average ISI. This gives a rough measure of how how often a cell fires in the immediately following EOD periods compared to its average firing frequency. With a cell being more bursty the higher the percentage of small ISI and the lower the firing frequency of the cell.
|
||||
Finally the ISI-histogram was calculated within a range of 0-50\,ms and a bin size of 0.1\,ms and the burstiness was calculated as the percentage of ISI smaller than 2.5 EOD periods multiplied by the average ISI. This gives a rough measure of how how often a cell fires in the immediately following EOD periods compared to its average firing frequency. With a cell being more bursty the higher the percentage of small ISI and the lower the mean firing frequency of the cell.
|
||||
|
||||
|
||||
%burstiness: \todo{how to write as equation, ignore and don't show an equation?}
|
||||
@ -267,7 +267,7 @@ Finally the ISI-histogram was calculated within a range of 0-50\,ms and a bin si
|
||||
\end{figure}
|
||||
|
||||
|
||||
The adaption behavior of the cell was characterized by the fi-curve consisting of the onset ($f_0$) and steady-state ($f_{inf}$) response. First the ISI frequency trace for each stimulus was calculated. The ISI frequency of a time point t is defined as the $1/T_i$ with $T_i$ being the ISI the time point t falls into. This gives a frequency trace starting by the first spike and ending at the last spike. For the further analysis all trials done for a specific contrast were pointwise averaged after cutting them to the same length. This gives an averaged step response for each contrast as seen in figure \ref{fig:f_point_detection} on the left.
|
||||
The adaption behavior of the cell was characterized by the fi-curve consisting of the onset ($f_0$) and steady-state ($f_{inf}$) response. First the ISI frequency trace for each stimulus was calculated. The ISI frequency of a time point t is defined as the $1/T_i$ with $T_i$ the ISI the time point t falls into. This gives a frequency trace starting by the first spike and ending at the last spike. For the further analysis all trials done for a specific contrast were pointwise averaged after cutting them to the same length. This gives an averaged step response for each contrast as seen in figure \ref{fig:f_point_detection} on the left.
|
||||
In this frequency trace the onset $f_0$ and steady-state $f_{inf}$ response were detected (fig. \ref{fig:f_point_detection}). $f_0$ was defined as the farthest deviation from the mean frequency before the stimulus in the range of 25\,ms after stimulus onset. If there was no deviation farther than the variations before the stimulus, then the average frequency in that time window was used. This approximation made the detection of $f_0$ more stable for small contrasts and traces with high variation.
|
||||
The $f_{inf}$ response was defined as the average frequency of the trace in the 0.1\,s time window, 25\,ms before the end of the stimulus.
|
||||
Afterwards a Boltzmann was fitted to the onset response and a rectified line was fitted to the steady-state responses (FI-Curve fig. \ref{fig:f_point_detection}).
|
||||
@ -296,20 +296,17 @@ To reproduce this behavior the model needs some form of memory of previous spike
|
||||
\label{eq:adaption_dynamics}
|
||||
\end{equation}
|
||||
|
||||
It is modeled as an exponential decay with the time constant $\tau_A$ and a strength called $\Delta_A$. $\Delta_A$ is multiplied with the sum of events in the spike train ($\delta (t)$) of the model cell. For the simulation using the Euler integration this results in an increase of $I_A$ by $\frac{\Delta_A}{\tau_A}$ at every time step where a spike is recorded. \todo{image of model simulation with voltage adaption and spikes using the toy model}
|
||||
|
||||
The current of the from equation \ref{eq:basic_voltage_dynamics} can thus be split into three currents for the modeling of the neuron:
|
||||
It is modeled as an exponential decay with the time constant $\tau_A$ and a strength called $\Delta_A$. $\Delta_A$ is multiplied with the sum of events in the spike train ($\delta (t)$) of the model cell. For the simulation using the Euler integration this results in an increase of $I_A$ by $\frac{\Delta_A}{\tau_A}$ at every time step where a spike is recorded. \todo{image of model simulation with voltage adaption and spikes using the toy model} The current of the from equation \ref{eq:basic_voltage_dynamics} can thus be split into three currents for the modeling of the neuron:
|
||||
|
||||
\begin{equation}
|
||||
I = \alpha I_{Input} - I_A + I_{Bias}
|
||||
\label{eq:currents_lifac}
|
||||
\end{equation}
|
||||
|
||||
The stimulus current $I_{Input}$, the bias current $I_{Bias}$ and the already discussed adaption current $I_A$. $I_{Input}$ is the current the stimulus, an amplitude modulated sine wave mimicking the frequency EOD. This stimulus is then rectified to model the receptor synapse and low-pass filtered with a time constant of $\tau_{dend}$ to simulate the low-pass filter properties of the dendrite (fig. \ref{fig:stim_development}). Afterwards it is multiplied with $\alpha$ a cell specific gain factor. This gain factor has the unit of cm because the $I_{Input}$ stimulus represents the EOD with a unit of mV/cm.
|
||||
The stimulus current $I_{Input}$, the bias current $I_{Bias}$ and the already discussed adaption current $I_A$. Note that in this p-unit model all currents are measured in mV because as mentioned above the cell body is not accessible for intra-cellular recordings and as such the membrane resistance $R_m$ is unknown \todo{ref mem res p-units}. $I_{Input}$ is the current of the stimulus, an amplitude modulated sine wave mimicking the frequency EOD. This stimulus is then rectified to model the receptor synapse and low-pass filtered with a time constant of $\tau_{dend}$ to simulate the low-pass filter properties of the dendrite (fig. \ref{fig:stim_development}). Afterwards it is multiplied with $\alpha$ a cell specific gain factor. This gain factor has the unit of cm because the $I_{Input}$ stimulus represents the EOD with a unit of mV/cm. $I_{Bias}$ is the bias current that causes the cells spontaneous spiking.
|
||||
|
||||
$I_{Bias}$ is the bias current that causes the cells spontaneous spiking. Note that in this p-unit model all currents are measured in mV because as mentioned above the cell body is not accessible for intra-cellular recordings and as such the membrane resistance $R_m$ is unknown \todo{ref mem res p-units}.
|
||||
Finally noise and an absolute refractory period were added to the model. The noise $\xi$ is drawn in from a Gaussian noise with values between 0 and 1 and divided by $\sqrt{\Delta t}$ to get a noise which autocorrelation function is independent of the simulation step size $\Delta t$. The implemented form of the absolute refractory period $t_{ref}$ keeps the model voltage at zero for the duration of $t_{ref}$ after a spike.
|
||||
|
||||
Finally noise and an absolute refractory period were added to the model. The noise $\xi$ is drawn in from a Gaussian noise with values between 0 and 1 and divided by $\sqrt{\Delta t}$ to get a noise which autocorrelation function is independent of the simulation step size $\Delta t$. The absolute refractory period $t_{ref}$ is implemented that after a spike the model voltage is kept at zero for the duration of the refractory period.
|
||||
|
||||
\begin{figure}[H]
|
||||
\includegraphics[scale=0.6]{figures/model_comparison.pdf}
|
||||
@ -318,7 +315,7 @@ Finally noise and an absolute refractory period were added to the model. The noi
|
||||
|
||||
\end{figure}
|
||||
|
||||
Together this results in the dynamics seen in equation \ref{eq:full_voltage_dynamics}. Not shown in the equation is the refractory period $t_{ref}$
|
||||
Together this results in the dynamics seen in equation \ref{eq:full_voltage_dynamics}. Not shown in the equation is the refractory period $t_{ref}$ and the $\tau_{dend}$ \todo{extra function describing $I_{input}$? rectify and filtering}
|
||||
|
||||
\begin{equation}
|
||||
\begin{split}
|
||||
@ -373,17 +370,17 @@ The error of the VS, CV, SC, and burstiness was calculated as the scaled absolut
|
||||
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.
|
||||
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 similar 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
|
||||
err_i = |1 - ((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.
|
||||
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 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.
|
||||
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} 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.
|
||||
|
||||
|
||||
|
||||
|
||||
@ -1,11 +1,11 @@
|
||||
\babel@toc {english}{}
|
||||
\select@language {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}
|
||||
\contentsline {section}{\numberline {4}Materials and Methods}{5}{section.4}
|
||||
\contentsline {subsection}{\numberline {4.1}Cell recordings}{5}{subsection.4.1}
|
||||
\contentsline {subsection}{\numberline {4.2}Stimulus Protocols}{6}{subsection.4.2}
|
||||
\contentsline {subsection}{\numberline {4.3}Cell Characteristics}{7}{subsection.4.3}
|
||||
\contentsline {subsection}{\numberline {4.3}Cell Characteristics}{6}{subsection.4.3}
|
||||
\contentsline {subsection}{\numberline {4.4}Leaky Integrate and Fire Model}{8}{subsection.4.4}
|
||||
\contentsline {subsection}{\numberline {4.5}Fitting of the Model}{11}{subsection.4.5}
|
||||
\contentsline {section}{\numberline {5}Results}{12}{section.5}
|
||||
|
||||
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Reference in New Issue
Block a user