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j-hartling
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces \textbf {Representations of a song of \textit {O. rufipes} during the feature extraction stage.} \textbf {a}:~Kernel-specific filter responses. \textbf {b}:~Binary responses. \textbf {c}:~Finalized features. }}{10}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces \textbf {Representations of a song of \textit {O. rufipes} during the feature extraction stage.} Different colors indicate Gabor kernels with different lobe number $n$ and sign, with lighter colors for higher $n$~($1\,\leq \,n\,\leq \,4$; both $+$ and $-$ per $n$; two kernel widths $\sigma $ of $4\,$ms and $32\,$ms per sign). \textbf {a}:~Kernel-specific filter responses. \textbf {b}:~Binary responses. \textbf {c}:~Finalized features. }}{10}{}\protected@file@percent }
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This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2025.10.28) 20 FEB 2026 14:29
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This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2025.10.28) 23 FEB 2026 16:48
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\OT1/cmr/m/n/12 for con-spe-cific song recog-ni-tion and re-sponse ini-ti-a-tion ([];
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[]
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\OT1/cmr/m/n/12 tym-pa-nal mem-brane acts as a me-chan-i-cal res-o-nance fil-ter for sound-induced vi-bra-tions ([];
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[]
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main.tex
114
main.tex
@@ -86,9 +86,12 @@
|
||||
\newcommand{\thr}{\Theta_i} % Step function threshold value
|
||||
\newcommand{\nl}{H(c_i\,-\,\thr)} % Shifted Heaviside step function
|
||||
|
||||
% Math shorthands - Minor symbols and helpers:
|
||||
\newcommand{\svar}{\sigma_{\text{s}}^{2}} % Song signal variance
|
||||
\newcommand{\nvar}{\sigma_{\eta}^{2}} % Noise signal variance
|
||||
% Math shorthands - Intensity invariance analysis:
|
||||
\newcommand{\soc}{s} % Song component of synthetic mixture
|
||||
\newcommand{\noc}{\eta} % Noise component of synthetic mixture
|
||||
\newcommand{\sca}{\alpha} % Multiplicative scale of song component
|
||||
\newcommand{\svar}{\sigma_{\text{s}}^{2}} % Song component variance
|
||||
\newcommand{\nvar}{\sigma_{\eta}^{2}} % Noise component variance
|
||||
\newcommand{\pc}{p(c_i,\,T)} % Probability density (general interval)
|
||||
\newcommand{\pclp}{p(c_i,\,\tlp)} % Probability density (lowpass interval)
|
||||
|
||||
@@ -387,7 +390,7 @@ sigmoidal response curve over logarithmically compressed intensity
|
||||
levels~(\bcite{suga1960peripheral}; \bcite{gollisch2002energy}). In the model
|
||||
pathway, logarithmic compression is achieved by conversion to decibel scale
|
||||
\begin{equation}
|
||||
\db(t)\,=\,10\,\cdot\,\dec \frac{\env(t)}{\dbref}, \qquad \dbref\,=\,\max[\env(t)]
|
||||
\db(t)\,=\,10\,\cdot\,\dec \frac{\env(t)}{\dbref}, \qquad \dbref\,=\,\max\big[\env(t)\big]
|
||||
\label{eq:log}
|
||||
\end{equation}
|
||||
relative to the maximum intensity $\dbref$ of the signal envelope $\env(t)$.
|
||||
@@ -492,12 +495,12 @@ the left of the two central lobes (odd kernels).
|
||||
\label{tab:gabor_phases}
|
||||
\end{table}
|
||||
\FloatBarrier
|
||||
These four groups of Gabor kernels allow for the extraction of different types
|
||||
of signal features, such as the presence of peaks (even, $+$), troughs (even,
|
||||
$-$), onsets (odd, $+$), and offsets (odd, $-$) at various time scales.
|
||||
These four major groups of Gabor kernels allow for the extraction of different
|
||||
types of signal features, such as the presence of peaks (even, $+$), troughs
|
||||
(even, $-$), onsets (odd, $+$), and offsets (odd, $-$) at various time scales.
|
||||
Following the convolutional template matching, each kernel-specific response
|
||||
$c_i(t)$ is passed through a shifted Heaviside step-function $\nl$ with threshold
|
||||
value $\thr$ to obtain a binary response
|
||||
$c_i(t)$ is passed through a shifted Heaviside step-function $\nl$ with
|
||||
threshold value $\thr$ to obtain a binary response
|
||||
\begin{equation}
|
||||
b_i(t,\,\thr)\,=\,\begin{cases}
|
||||
\;1, \quad c_i(t)\,>\,\thr\\
|
||||
@@ -528,6 +531,10 @@ can be read out by a simple linear classifier.
|
||||
\includegraphics[width=\textwidth]{figures/fig_feat_stages.pdf}
|
||||
\caption{\textbf{Representations of a song of \textit{O. rufipes} during
|
||||
the feature extraction stage.}
|
||||
Different colors indicate Gabor kernels with different
|
||||
lobe number $\kn$ and sign, with lighter colors for higher
|
||||
$\kn$~($1\,\leq\,\kn\,\leq\,4$; both $+$ and $-$ per $\kn$;
|
||||
two kernel widths $\kw$ of $4\,$ms and $32\,$ms per sign).
|
||||
\textbf{a}:~Kernel-specific filter responses.
|
||||
\textbf{b}:~Binary responses.
|
||||
\textbf{c}:~Finalized features.
|
||||
@@ -536,55 +543,62 @@ can be read out by a simple linear classifier.
|
||||
\end{figure}
|
||||
\FloatBarrier
|
||||
|
||||
\section{Two mechanisms driving the emergence of intensity-invariant song representation}
|
||||
\section{Two mechanisms driving the emergence of intensity-invariant song representations}
|
||||
|
||||
\textbf{Definition of invariance (general, systemic):}\\
|
||||
Invariance = Property of a system to maintain a stable output with respect to a
|
||||
set of relevant input parameters (variation to be represented) but irrespective
|
||||
of one or more other parameters (variation to be discarded)
|
||||
$\rightarrow$ Selective input-output decorrelation
|
||||
% Still missing the SNR analysis. Should be able to write around it for now.
|
||||
The robustness of song recognition is tied to the degree of intensity
|
||||
invariance of the finalized feature representation. Ideally, the values of each
|
||||
feature should depend only on the relative amplitude dynamics of the song
|
||||
pattern but not on the overall intensity level of the song. In the grasshopper,
|
||||
the emergence of intensity-invariant representations along the song recognition
|
||||
pathway likely is a distributed process that involves different neuronal
|
||||
populations, which raises the question of what the essential computational
|
||||
mechanisms are that drive this process. Within the model pathway, we identified
|
||||
two key mechanisms that render the song representation more invariant to
|
||||
variations in baseline intensity. The two mechanisms each comprise a nonlinear
|
||||
signal transformation followed by a linear signal transformation but differ in
|
||||
the specific operations and the neural substrate involved, as outlined in the
|
||||
following sections.
|
||||
|
||||
\textbf{Definition of intensity invariance (context of neurons and songs):}\\
|
||||
Intensity invariance = Time scale-selective sensitivity to certain faster
|
||||
amplitude dynamics (song waveform, small-scale AM) and simultaneous
|
||||
insensitivity to slower, more sustained amplitude dynamics (transient baseline,
|
||||
large-scale AM, current overall intensity level)\\
|
||||
$\rightarrow$ Without time scale selectivity, any fully intensity-invariant
|
||||
output will be a flat line
|
||||
\subsection{Logarithmic compression \& spike-frequency adaptation}
|
||||
|
||||
\subsection{Logarithmic scaling \& spike-frequency adaptation}
|
||||
|
||||
Envelope $\env(t)$ $\xrightarrow{\text{dB}}$ Logarithmic $\db(t)$ $\xrightarrow{\hp}$ Adapted $\adapt(t)$
|
||||
|
||||
- Rewrite signal envelope $\env(t)$ (Eq.\,\ref{eq:env}) as a synthetic mixture:\\
|
||||
1) Song signal $s(t)$ ($\svar=1$) with variable multiplicative scale $\alpha\geq0$\\
|
||||
2) Fixed-scale additive noise $\eta(t)$ ($\nvar=1$)
|
||||
%
|
||||
The first emergence of intensity invariance along the model pathway occurs
|
||||
during the preprocessing stage, in the transition from the signal envelope
|
||||
$\env(t)$ to the logarithmically scaled envelope $\db(t)$ and then to the
|
||||
intensity-adapted envelope $\adapt(t)$. In order to disentangle the interplay
|
||||
of logarithmic compression and adaptation, we can rewrite
|
||||
$\env(t)$~(Eq.\,\ref{eq:env}) as synthetic mixture
|
||||
\begin{equation}
|
||||
\env(t)\,=\,\alpha\,\cdot\,s(t)\,+\,\eta(t),\qquad \env(t)\,>\,0\enspace\forall\enspace t\,\in\,\mathbb{R}
|
||||
\env(t)\,=\,\sca\,\cdot\,\soc(t)\,+\,\noc(t), \qquad \env(t)\,>\,0\enspace\forall\enspace t\,\in\,\mathbb{R}
|
||||
\label{eq:toy_env}
|
||||
\end{equation}
|
||||
%
|
||||
- Signal-to-noise ratio (SNR): Ratio of variances of synthetic mixture
|
||||
$\env(t)$ with ($\alpha>0$) and without ($\alpha=0$) song signal $s(t)$, assuming $s(t)\perp\eta(t)$
|
||||
%
|
||||
of a song component $\soc(t)$ with variable multiplicative scale $\sca\geq0$
|
||||
and a fixed-scale noise component $\noc(t)$. Both $\soc(t)$ and $\noc(t)$ are
|
||||
assumed to have unit variance~($\svar=\nvar=1$). If $\soc(t)$ and $\noc(t)$ are
|
||||
uncorrelated~($\soc(t)\perp\noc(t)$), the signal-to-noise ratio (SNR) of the
|
||||
synthetic $\env(t)$ with ($\sca>0$) and without ($\sca=0$) song component
|
||||
$\soc(t)$ is given by
|
||||
\begin{equation}
|
||||
\text{SNR}\,=\,\frac{\sigma_{s+\eta}^{2}}{\nvar}\,=\,\frac{\alpha^{2}\,\cdot\,\svar\,+\,\nvar}{\nvar}\,=\,\alpha^{2}\,+\,1
|
||||
\label{eq:toy_snr}
|
||||
\end{equation}
|
||||
%
|
||||
\textbf{Logarithmic component:}\\
|
||||
- Simplify decibel transformation (Eq.\,\ref{eq:log}) and apply to synthetic $\env(t)$\\
|
||||
- Isolate scale $\alpha$ and reference $\dbref$ using logarithm product/quotient laws
|
||||
%
|
||||
When simplifying the decibel transformation~(Eq.\,\ref{eq:log}), the logarithmically
|
||||
scaled envelope $\db(t)$ can be expressed as a sum of two logarithmic terms
|
||||
\begin{equation}
|
||||
\begin{split}
|
||||
\db(t)\,&=\,\log \frac{\alpha\,\cdot\,s(t)\,+\,\eta(t)}{\dbref}\\
|
||||
&=\,\log \frac{\alpha}{\dbref}\,+\,\log b_ig[s(t)\,+\,\frac{\eta(t)}{\alpha}b_ig]
|
||||
&=\,\log \frac{\alpha}{\dbref}\,+\,\log \left[s(t)\,+\,\frac{\eta(t)}{\alpha}\right]
|
||||
\end{split}
|
||||
\label{eq:toy_log}
|
||||
\end{equation}
|
||||
%
|
||||
|
||||
|
||||
|
||||
|
||||
\textbf{Logarithmic component:}\\
|
||||
- Simplify decibel transformation (Eq.\,\ref{eq:log}) and apply to synthetic $\env(t)$\\
|
||||
- Isolate scale $\alpha$ and reference $\dbref$ using logarithm product/quotient laws
|
||||
|
||||
$\rightarrow$ In log-space, a multiplicative scaling factor becomes additive\\
|
||||
$\rightarrow$ Allows for the separation of song signal $s(t)$ and its scale $\alpha$\\
|
||||
$\rightarrow$ Introduces scaling of noise term $\eta(t)$ by the inverse of $\alpha$\\
|
||||
@@ -597,7 +611,7 @@ interval $\thp$ ($0 \ll \thp < \frac{1}{\fc}$)
|
||||
%
|
||||
\begin{equation}
|
||||
\begin{split}
|
||||
\adapt(t)\,\approx\,\db(t)\,-\,\log \frac{\alpha}{\dbref}\,=\,\log b_ig[s(t)\,+\,\frac{\eta(t)}{\alpha}b_ig]
|
||||
\adapt(t)\,\approx\,\db(t)\,-\,\log \frac{\alpha}{\dbref}\,=\,\log\left[s(t)\,+\,\frac{\eta(t)}{\alpha}\right]
|
||||
\end{split}
|
||||
\label{eq:toy_highpass}
|
||||
\end{equation}
|
||||
@@ -715,6 +729,20 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
|
||||
|
||||
\section{Conclusions \& outlook}
|
||||
|
||||
\textbf{Definition of invariance (general, systemic):}\\
|
||||
Invariance = Property of a system to maintain a stable output with respect to a
|
||||
set of relevant input parameters (variation to be represented) but irrespective
|
||||
of one or more other parameters (variation to be discarded)
|
||||
$\rightarrow$ Selective input-output decorrelation
|
||||
|
||||
\textbf{Definition of intensity invariance (context of neurons and songs):}\\
|
||||
Intensity invariance = Time scale-selective sensitivity to certain faster
|
||||
amplitude dynamics (song waveform, small-scale AM) and simultaneous
|
||||
insensitivity to slower, more sustained amplitude dynamics (transient baseline,
|
||||
large-scale AM, current overall intensity level)\\
|
||||
$\rightarrow$ Without time scale selectivity, any fully intensity-invariant
|
||||
output will be a flat line
|
||||
|
||||
The model pathway includes a rather large number of Gabor kernels compared to
|
||||
the 15 to 20 ascending neurons in the grasshopper auditory
|
||||
system~(\bcite{stumpner1991auditory}).
|
||||
|
||||
@@ -117,32 +117,12 @@ def reorder_traces(handles, signal, zlow=2, zhigh=2.5):
|
||||
return None
|
||||
|
||||
def choose_kernels(kern_specs, features, kern_types, per_type=2, thresh=0.01):
|
||||
embed()
|
||||
mean_feat = features.mean(axis=0)
|
||||
feat_diff = np.abs(mean_feat[:, None] - mean_feat[None, :])
|
||||
feat_diff[features.max(axis=0) < thresh, :] = np.nan
|
||||
feat_diff = np.nanmean(feat_diff, axis=0)
|
||||
|
||||
ranking = np.argsort(feat_diff)
|
||||
|
||||
kern_inds = []
|
||||
for type_id in kern_types:
|
||||
type_inds = np.nonzero(kern_specs[:, 0] == type_id)[0]
|
||||
rank_inds = np.nonzero(np.isin(ranking, type_inds))[0][-per_type:]
|
||||
kern_inds.extend(ranking[rank_inds])
|
||||
return np.array(kern_inds)
|
||||
|
||||
|
||||
mean_feat = features.mean(axis=0)
|
||||
mean_feat -= mean_feat.min()
|
||||
mean_feat /= mean_feat.max()
|
||||
|
||||
feat_diff = np.abs(mean_feat[:, None] - mean_feat[None, :]).mean(axis=0)
|
||||
feat_diff -= feat_diff.min()
|
||||
feat_diff /= feat_diff.max()
|
||||
|
||||
ranking = np.argsort(mean_feat + feat_diff)
|
||||
|
||||
kern_inds = []
|
||||
for type_id in kern_types:
|
||||
type_inds = np.nonzero(kern_specs[:, 0] == type_id)[0]
|
||||
@@ -162,7 +142,7 @@ def letter_subplots(axes, labels='abcd', x=0.02, y=1, ha='left', va='bottom',
|
||||
target = 'Omocestus_rufipes'
|
||||
data_paths = glob.glob(f'../data/processed/{target}*.npz')
|
||||
stages = ['filt', 'env', 'log', 'inv', 'conv', 'bi', 'feat']
|
||||
save_path = None#'../figures/'
|
||||
save_path = '../figures/'
|
||||
|
||||
# PLOT SETTINGS:
|
||||
fig_kwargs = dict(
|
||||
@@ -194,8 +174,8 @@ lw_full = dict(
|
||||
log=0.5,
|
||||
inv=0.5,
|
||||
conv=0.25,
|
||||
bi=0,
|
||||
feat=2
|
||||
bi=0.2,
|
||||
feat=1.5
|
||||
)
|
||||
lw_zoom = dict(
|
||||
filt=0.5,
|
||||
@@ -203,8 +183,8 @@ lw_zoom = dict(
|
||||
log=1,
|
||||
inv=1,
|
||||
conv=1.5,
|
||||
bi=0,
|
||||
feat=2
|
||||
bi=0.2,
|
||||
feat=1.5
|
||||
)
|
||||
loc_full = dict(
|
||||
filt=0.2,
|
||||
@@ -228,27 +208,27 @@ zoom_kwargs = dict(
|
||||
zorder=0,
|
||||
linewidth=0
|
||||
)
|
||||
kernels = np.array([
|
||||
[1, 0.002],
|
||||
[1, 0.016],
|
||||
[-1, 0.002],
|
||||
[-1, 0.016],
|
||||
[2, 0.004],
|
||||
[2, 0.032],
|
||||
[-2, 0.004],
|
||||
[-2, 0.032],
|
||||
[3, 0.004],
|
||||
[3, 0.032],
|
||||
[-3, 0.004],
|
||||
[-3, 0.032],
|
||||
[4, 0.004],
|
||||
[4, 0.032],
|
||||
[-4, 0.004],
|
||||
[-4, 0.032]
|
||||
])
|
||||
# t = [1, 2, 3, 4]
|
||||
# s = [0.001, 0.002, 0.004, 0.008, 0.032]
|
||||
# kernels = np.array([[i, j] for i in t for j in s])
|
||||
# kernels = np.array([
|
||||
# [1, 0.002],
|
||||
# [1, 0.016],
|
||||
# [-1, 0.004],
|
||||
# [-1, 0.032],
|
||||
# [2, 0.004],
|
||||
# [2, 0.016],
|
||||
# [-2, 0.002],
|
||||
# [-2, 0.032],
|
||||
# [3, 0.008],
|
||||
# [3, 0.032],
|
||||
# [-3, 0.008],
|
||||
# [-3, 0.032],
|
||||
# [4, 0.004],
|
||||
# [4, 0.032],
|
||||
# [-4, 0.004],
|
||||
# [-4, 0.032]
|
||||
# ])
|
||||
t = [1, -1, 2, -2, 3, -3, 4, -4]
|
||||
s = [0.004, 0.032]
|
||||
kernels = np.array([[i, j] for i in t for j in s])
|
||||
conv_colors = load_colors('../data/conv_colors.npz')
|
||||
bi_colors = load_colors('../data/bi_colors.npz')
|
||||
feat_colors = load_colors('../data/feat_colors.npz')
|
||||
|
||||
Reference in New Issue
Block a user