Formalizing stuff (WIP).

main.tex

@@ -1,14 +1,28 @@
 \documentclass[a4paper, 12pt]{article}
 
 \usepackage{parskip}
+\usepackage{amsmath}
+\usepackage[
+backend=biber,
+style=authoryear,
+]{biblatex}
+\addbibresource{cite.bib}
 
-\title{Physiologically inspired model of the grasshopper auditory system}
+\title{Emergent intensity invariance in a physiologically inspired model of the grasshopper auditory system}
 \author{Jona Hartling, Jan Benda}
 \date{}
 
 \begin{document}
 \maketitle{}
 
+\newcommand{\bp}{h_{\text{BP}}(t)} % Bandpass filter function
+\newcommand{\lp}{h_{\text{LP}}(t)} % Lowpass filter function
+\newcommand{\hp}{h_{\text{HP}}(t)} % Highpass filter function
+\newcommand{\fc}{f_{\text{cut}}} % Filter cutoff frequency
+\newcommand{\infint}{\int_{-\infty}^{\infty}} % Indefinite integral
+\newcommand{\bi}{b_\theta}
+\newcommand{\feat}{f_\theta}
+
 \section{The sensory world of a grasshopper}
 
 Strong dependence on acoustic signals for ranged communication\\
@@ -53,39 +67,108 @@ How can a human observer conceive a grasshopper's auditory percepts?\\
 - How to integrate the available knowledge on anatomy, physiology, ethology?\\
 $\rightarrow$ Abstract, simplify, formalize $\rightarrow$ Functional model framework
 
-\section{Pre-split pathway: Population pre-processing}
+
+\section{Developing a functional model of\\the grasshopper auditory pathway}
+
+
+\subsection{Population-driven signal pre-processing}
+
+"Pre-split portion" of the auditory pathway:\\
+Tympanal membrane $\rightarrow$ Receptor neurons $\rightarrow$ Local interneurons
+
+Similar response/filter properties within receptor/interneuron populations (\cite{clemens2011})\\
+$\rightarrow$ One population-wide response trace per stage (no "single-cell resolution")
+
+\textbf{Stage-specific processing steps and functional approximations:}
+
+Initial: Continuous acoustic input signal $x(t)$
+
 Filtering of behaviorally relevant frequencies by tympanal membrane\\
-- Bandpass 5-30 kHz
+$\rightarrow$ Bandpass filter 5-30 kHz
+\begin{equation}
+    x(t)\,*\,\bp; \quad\quad \fc\,=\,5\,\text{kHz},\,30\,\text{kHz}
+\end{equation}
 
 Extraction of signal envelope (AM encoding) by receptor population\\
-- Full-wave rectification + lowpass 500 Hz
+$\rightarrow$ Full-wave rectification, then lowpass filter 500 Hz
+\begin{equation}
+    |x(t)|\,*\,\lp; \quad\quad \fc\,=\,500\,\text{Hz}
+\end{equation}
 
 Logarithmically compressed intensity tuning curve of receptors\\
-- Decibel transformation
+$\rightarrow$ Decibel transformation
+\begin{equation}
+    20\,\cdot\,\log_{10} \frac{x(t)}{x_{\text{max}}}
+\end{equation}
 
 Spike-frequency adaptation in receptor and interneuron populations\\
-- Highpass 10 Hz
+$\rightarrow$ Highpass filter 10 Hz
+\begin{equation}
+    x(t)\,*\,\hp; \quad\quad \fc\,=\,10\,\text{Hz}
+\end{equation}
 
-\section{Post-split pathway: Feature extraction}
 
-Template matching by individual ascending neurons\\
-- Separate convolution with each of a set of Gabor kernels\\
-- Pathway splitting: Single population response into several separate branches
-- Expansion into a higher-dimensional sound representation
+\subsection{Feature extraction by individual neurons}
+
+"Post-split portion" of the auditory pathway:\\
+Ascending neurons (AN) $\rightarrow$ Central brain neurons
+
+Diverse response/filter properties within AN population (\cite{clemens2011})\\
+- Pathway splitting into several parallel branches\\
+- Expansion into a decorrelated higher-dimensional sound representation\\
+$\rightarrow$ Individual neuron-specific response traces from this stage onwards
+
+\textbf{Stage-specific processing steps and functional approximations:}
+
+Template matching by individual ANs\\
+- Filter base (STA approximations): Set of Gabor kernels\\
+- Gabor parameters: $\sigma, \phi, f$ $\rightarrow$ Determines kernel sign and lobe number
+\begin{equation}
+    k(t)\,=\,e^{-\frac{t^{2}}{2\sigma^{2}}}\,\cdot\,\sin(2\pi f t\,+\,\phi)
+\end{equation}
+$\rightarrow$ Separate convolution with each member of the kernel set
+\begin{equation}
+    c_i(t)\,=\,x(t)\,*\,k_i(t)
+    = \infint x(\tau)\,\cdot\,k_i(t\,-\,\tau)\,d\tau
+\end{equation}
 
 Thresholding nonlinearity in ascending neurons (or further downstream)\\
-- Step-function (or sigmoid) threshold\\
-- Binarization of response values into "relevant" vs. "irrelevant"
+- Binarization of AN response traces into "relevant" vs. "irrelevant"\\
+$\rightarrow$ Heaviside step-function $H(c\,-\,\theta)$ (or steep sigmoid threshold?)
+\begin{equation}
+    \bi(t)\,=\,\begin{cases}
+        \;1, \quad c(t)\,\geq\,\theta\\
+        \;0, \quad c(t)\,<\,\theta 
+    \end{cases}
+\end{equation}
 
 Temporal averaging by neurons of the central brain\\
-- Lowpass 1 Hz\\
-- Finalized set of slowly changing kernel-specific features\\
-- Different (species-specific) songs are characterized by a distinct combination of feature values
+- Finalized set of slowly changing kernel-specific features (one per AN)\\
+- Different species-specific song patterns are characterized by a distinct combination
+of feature values $\rightarrow$ Clusters in high-dimensional feature space\\
+$\rightarrow$ Lowpass filter 1 Hz
+\begin{equation}
+    \feat(t)\,=\,\bi(t)\,*\,\lp; \quad\quad \fc\,=\,1\,\text{Hz}
+\end{equation}
 
-\section{Pre-split intensity invariance:\\Logarithm-highpass mechanism}
 
-\section{Post-split intensity invariance:\\Threshold-lowpass mechanism}
+\section{Two mechanisms driving the emergence of intensity-invariant song representation}
 
-\section{Conclusion and outlook}
+
+\subsection{Logarithmic scaling \& spike-frequency adaptation}
+
+Song signal $s(t)$ with variable scale $\alpha$ and fixed-scale additive noise $\eta(t)$
+\begin{equation}
+    \alpha\,\cdot\,s(t)\,+\,\eta(t)
+\end{equation}
+
+
+\subsection{Threshold nonlinearity \& temporal averaging}
+
+
+\section{Discriminating species-specific song\\patterns in feature space}
+
+
+\section{Conclusions \& outlook}
 
 \end{document}