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\title{On the role of noise in signal detection}
\author{Dennis Huben}

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\begin{document}

\maketitle

\begin{abstract}

\end{abstract}

\tableofcontents

\newpage


\section{Suprathreshold stochastic resonance}

\subsection{Introduction}

In any biological system, there is a limit to the precision of the components making up that system. This means that even without external input the spike times of each individual neurons will have some variation and will not be perfectly regular. Increasing the precision has a cost in energy requirement \citep{schreiber2002energy} but may not even be desirable.

In populations of neurons, representation of a common stimulus can be improved by population heterogeneity  \citep{ahn2014heterogeneity}. The source of this heterogeneity could for example be a different firing threshold for each neuron. Alternatively, the improvement can be achieved by adding noise to the input of neurons \citep{shapira2016sound}. The effect of adding noise to a sub-threshold signal, a phenomenon known as "stochastic resonance" (SR), has been very well investigated during the last decades \citep{benzi1981mechanism,gammaitoni1998resonance, shimokawa1999stochastic}. The noise added to a signal makes it more likely that the signal reaches the detection threshold so that it triggers a spike in a neuron.
But often in nature the goal is not to simply detect a signal but to discriminate between two different signals as well as possible. For example  in auditory communication it is not sufficient to detect the presence of sound but instead the goal is to encode an auditory stimulus so that an optimal amount of information is gained from the stimulus.
Another example is the electrosensory communication between conspecifics in weakly electric fishes. Those fish need to for example differentiate aggressive and courtship behaviors. 


More recently it has been shown that for populations of neurons the beneficial role of noise can also be true for signals which already are above the threshold\citep{stocks2000suprathreshold, Stocks2000,stocks2001information,stocks2001generic,beiran2018coding}, a phenomenon termed "Suprathreshold Stochastic Resonance" (SSR). Despite the similarity in name, SR and SSR work in very different ways. The idea behind SSR is that in case of no or very weak individual noise the different neurons in the population react to the same features of a common input. Additional noise that affects each cell differently desynchronizes the response of the neurons. The spiking behavior of the neurons becomes more probabilistic than deterministic in nature. However, if the noise is too strong, the noise masks the signal and less information can be coded than would be ideally possible. In the case of infinite noise strength, no information about the signal can be reconstructed from the responses. Because some noise is beneficial and too much noise isn't, there is a noise strength where performance is best. 
This thesis investigates populations of neurons reacting to input signals with cutoff frequencies over a large range. Population sizes range from a single neuron to many thousands of neurons.


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\includegraphics[width=0.5\linewidth]{img/stocks}

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\caption{Array of threshold systems as described by Stocks.}
\label{stocks}
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Here we use the Integrate-and-Fire model to simulate neuronal populations receiving a common dynamic input. We look at linear coding of signals by different sized populations of neurons of a single type, similar to the situation in weakly electric fish. We show that the optimal noise grows with population size and depends on properties of the input. We use input signals of varying frequencies widths and cutoffs, along with changing the strength of the signal.

We also present the results of electrophysiological results in the weakly electric fish \textit{Apteronotus leptorhynchus}. Because it is not obvious how to quantify noisiness in the receptor cells of these fish, we compare different methods and find that using the activation curve of the individual neurons allows for the best estimate of the strength of noise in these cells. Then we show that we can see the effects of SSR in the real world example of \textit{A. leptorhynchus}.

\subsection{Methods}

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\subsection{Simulations with more neurons}

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\subsection{Different frequency ranges}

\subsection{Narrow-/wideband}

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\section{Theory}

\subsection{Firing rates}

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\subsection{Refractory period}

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\section{Electric fish}

\subsection{Introduction}

\subsection{Methods}

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\subsection{How to determine noisiness}

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\subsection{Results}

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\section{Discussion: Combining experiment and simulation}

\section{Literature}

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