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zTvD!eS`%*d4~I@S_)gzDr@>!04$E{hydIrdu(BFMOJ*(uTF6JHqUPY`41*yR2YM$} zXB5IQ-2hMLI^tnN3~9JqOR%-w{_yaMqGt%cnJf2{A^3Te4;$qZ8G?mkl_9tep76Ia z1fyX0;y=QRfBf|2&+m6W-@SSN`Hy#R-oO3&{o{vE_iw(u|Lx6}w_o0U-TwYpCQf|w z^P4~3e)#nB+dm&){C!F<{_E#&-{1c6%g1+jFaF{4pP#\,\thr\\ \;0, \quad c_i(t)\,\leq\,\thr \end{cases} + \label{eq:binary} \end{equation} Temporal averaging by neurons of the central brain\\ @@ -154,6 +174,7 @@ 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} + \label{eq:lowpass} \end{equation} @@ -162,31 +183,103 @@ $\rightarrow$ Lowpass filter 1 Hz \subsection{Logarithmic scaling \& spike-frequency adaptation} -Song signal $s(t)$ with variable scale $\alpha$ and fixed-scale additive noise $\eta(t)$ +Envelope $\env(t)$ $\xrightarrow{\text{dB}}$ Logarithmic $\db(t)$ $\xrightarrow{\hp}$ Adapted $\adapt(t)$ + +Example signal envelope $\env(t)$ ($\env(t)>0$ for all $t\in T$):\\ +- Song signal $s(t)$ with $\sigs=1$\\ +- Variable multiplicative song scale $\alpha\geq0$\\ +- Fixed-scale additive noise $\eta(t)$ with $\sign=1$\\ +- Suitable observed time interval $T$\\ +- Decibel reference intensity $m\,=\,\max[\env(t)]$ \begin{equation} - \alpha\,\cdot\,s(t)\,+\,\eta(t) + \env(t)\,=\,\alpha\,\cdot\,s(t)\,+\,\eta(t),\quad\quad x:T\to(0,\infty) + \label{eq:toy_env} \end{equation} +\textbf{Logarithmic component:}\\ +\begin{equation} + \begin{split} + \db(t)\,&=\,10\,\cdot\,\dec \frac{\alpha\,\cdot\,s(t)\,+\,\eta(t)}{m}\\ + &=\,10\,\cdot\,\big(\dec \frac{\alpha}{m}\,+\,\dec[s(t)\,+\,\frac{\eta(t)}{\alpha}]\big) + \end{split} + \label{eq:toy_log} +\end{equation} + +% \begin{equation} +% \begin{split} +% \db(t)\,&=\,\log{[\alpha\,\cdot\,s(t)\,+\,\eta(t)]}\\ +% &=\,\log{\alpha}\,+\,\log{[s(t)\,+\,\frac{\eta(t)}{\alpha}]} +% \end{split} +% \label{eq:toy_log} +% \end{equation} + +\textbf{Adaptation component:}\\ +\begin{equation} + \begin{split} + \adapt(t)\,\approx\,\db(t)\,-\,\dec \frac{\alpha}{m}\,=\,\dec{[s(t)\,+\,\frac{\eta(t)}{\alpha}]} + \end{split} + \label{eq:toy_highpass} +\end{equation} + +% \textbf{Adaptation component:}\\ +% \begin{equation} +% \begin{split} +% \adapt(t)\,\approx\,\db(t)\,-\,\log{\alpha}\,=\,\log{[s(t)\,+\,\frac{\eta(t)}{\alpha}]} +% \end{split} +% \label{eq:toy_highpass} +% \end{equation} \subsection{Threshold nonlinearity \& temporal averaging} -Convolution output $c_i(t)$ $\xrightarrow{\thr}$ Thresholded response $\bi(t)$ $\rightarrow$ Feature $\feat(t)$ +Convolved $c_i(t)$ $\xrightarrow{\nl}$ Binary $\bi(t)$ $\xrightarrow{\lp}$ Feature $\feat(t)$ -- Convolution output $c_i(t)$ has distribution $\pc$ over time interval $T$\\ -- Within $T$, $c_i(t)$ exceeds the threshold value $\thr$ for time $T_1$ ($T_1+T_0=T$)\\ -$\rightarrow$ Step-function $\nl$ bipartitions distribution $\pc$ around $\thr$ +\textbf{Thresholding component:}\\ +- Within an observed time interval $T$, $c_i(t)$ follows probability density $\pc$\\ +- Within $T$, $c_i(t)$ exceeds threshold value $\thr$ for time $T_1$ ($T_1+T_0=T$)\\ +- Threshold $\nl$ splits $\pc$ around $\thr$ in two complementary parts \begin{equation} \int_{\thr}^{+\infty} p(c_i,T)\,dc_i\,=\,1\,-\,\int_{-\infty}^{\thr} p(c_i,T)\,dc_i\,=\,\frac{T_1}{T} + \label{eq:pdf_split} \end{equation} -- Ratio of time above threshold $T_1$ to total time $T$ because +$\rightarrow$ Semi-definite integral over right-sided portion of split $\pc$ gives ratio +of time $T_1$ where $c_i(t)>\thr$ to total time $T$ due to normalization of $\pc$ \begin{equation} \infint \pc\,dc_i\,=\,1 + \label{eq:pdf} \end{equation} -Approximate lowpass filter as moving average over time interval $\tlp$ +\textbf{Averaging component:}\\ +- Lowpass filter over binary response $\bi(t)$ (Eq.\,\ref{eq:lowpass}) can be +approximated as temporal averaging over a suitable time interval $\tlp$ ($\tlp > \frac{1}{\fc}$)\\ +- Within $\tlp$, $\bi(t)$ takes a value of 1 ($c_i(t)>\thr$) for time $T_1$ ($T_1+T_0=\tlp$) \begin{equation} \feat(t)\,\approx\,\frac{1}{\tlp} \int_{t}^{t\,+\,\tlp} \bi(\tau)\,d\tau\,=\,\frac{T_1}{\tlp} + \label{eq:feat_avg} \end{equation} +$\rightarrow$ Temporal averaging over $\bi(t)\in[0,1]$ (Eq.\ref{eq:binary}) gives +ratio of time $T_1$ where $c_i(t)>\thr$ to total averaging interval $\tlp$\\ +$\rightarrow$ Feature $\feat(t)$ approximately represents supra-threshold fraction of $\tlp$ + +\textbf{Combined result:}\\ +- Feature $\feat(t)$ can be linked to the distribution of $c_i(t)$ using Eqs.\,\ref{eq:pdf_split} \& \ref{eq:feat_avg} +\begin{equation} + \feat(t)\,\approx\,\int_{\thr}^{+\infty} \pclp\,dc_i\,=\,P(c_i\,>\,\thr,\,\tlp) + \label{eq:feat_prop} +\end{equation} +$\rightarrow$ Because the integral over a probability density is a cumulative +probability, the value of feature $\feat(t)$ (temporal compression of $\bi(t)$) +at every time point $t$ signifies the probability that convolution output +$c_i(t)$ exceeds the threshold value $\thr$ during the corresponding averaging +interval $\tlp$ + +\textbf{Implication for intensity invariance:}\\ +- Convolution output $c_i(t)$ = amplitude-based quantity\\ +$\rightarrow$ Values indicate how well template waveform $k_i(t)$ matches signal $x(t)$\\ +- Feature $\feat(t)$ = duty cycle-based quantity\\ +$\rightarrow$ Values indicate how often $c_i(t)$ exceeds threshold value $\thr$ + +- Thresholding of $c_i(t)$ and subsequent temporal averaging of $\bi(t)$ to obtain $\feat(t)$ +constitutes a remapping of an amplitude-based quantity (values indicating the match between) into a duty cycle-based quantity\\ \section{Discriminating species-specific song\\patterns in feature space}