Figure 1.
A schematic example of a typical spectro-temporal receptive field, plotted with a reversed abscissa.
This STRF has one excitatory and three inhibitory regions, prefers frequency , and evokes response at a typical latency
. Since the response at time
is
, an input stimulus
exactly as depicted in this plot is most likely to elicit a large response
at time
, or indeed a spike.
Figure 2.
Formulation and components of efficient coding.
(A) A schematic plot of the efficient encoding transform. (B) Signal transformation in the auditory system. The cochlea turns the time-varying waveform into a time-frequency representation
, as the population activities of the auditory nerves, which is the input to the efficient encoding system. Signal and noise pass through a series of brain nuclei such as cochlear nucleus, superior olive, inferior colliculus, etc. The current work proposes that the effective transform STRF of the spectrogram that is collectively realized by these nuclei is, in its linear form, the optimal filter
implied by the efficient coding principle. The output
is the activity of neurons in a higher nucleus. (C) Three steps of signal flow within the linear encoding step
or STRF in (A) and (B). Note that these three steps are merely abstract algorithmic steps, rather than neural implementation processes for the effective transform
or STRF.
Figure 3.
Simulation of the efficient spectral kernel SRF, when the temporal dimension is omitted.
(A) 250 samples of input spectra , each of which is smoothed Gaussian white noise in the frequency domain (equations (11–13),
). (B) Correlation between different frequency channels
. Left: Correlation
; Right: an zoomed-in view, as
vs
. (C) Ten examples of eigenvectors
of the correlation matrix
in B; each is an independent component in
. Smaller indices
are associated with larger eigenvalues. (D) Gain profile (peaking at
), and signal and noise power in decorrelated channels. (E) Four examples (
,
,
, and
) of spectral receptive fields
; each prefers input frequencies around
.
Figure 4.
The effect of signal-to-noise ratio (SNR) on gain and the spectral receptive field (SRF).
Same stimulus ensemble as in Figure 3A except the overall SNR has been scaled by . (A) Gain control (red), signal (blue), and noise power (black) under high, medium and low SNR. (B) The corresponding SRFs of one output neuron (channel #120) in the three SNR cases.
Figure 5.
Adaptation of gain and spectral filter kernel SRF to input correlations under high/low SNR.
Same input ensemble as that in Figure 3A, except that the smoothing parameter, and
, are set for short and long range correlations, respectively. Analogous figure format as in Figure 4, with added illustrations of the adaptation to input correlations. The thick and thin curves correspond to quantities for inputs with large and small correlations respectively, blue/red curves plot signal power
and gain
respectively.
Figure 6.
Simulation of temporal receptive field TRF, when the spectral dimension is omitted.
The same stimulus ensemble is used as in Figure 3A, except the factor in equation (12) to ensure translation invariance of correlation. (A;B) Demonstration of transforming an acausal temporal filter (A) to its causal minimum-phase counterpart (B) at a relatively high input SNR. (C) TRF for a relatively low input SNR.
Figure 7.
The 2D STRFs/MTFs implied by efficient coding and found physiologically.
(A) input power (equation (15),
,
) in decorrelated channels. (B, C) MTF profile
and the corresponding STRFs with two SNRs (scaled by
's). (D)
and STRF as in B;C (when
) except with larger input correlations (
,
in equation (15)). (E;F) Modulation transfer functions (MTFs) and their properties at low and high input sound intensities averaged over 40 IC neurons from Lesica and Grothe [7]. Here,
is the spectral-temporal modulation frequency where the MTF peaks. Modulation frequencies in E and F are normalized by the same value across cells and intensities. Error bars in E indicate standard errors. The magnitude patterns of the MTFs for all neurons are normalized to peak value
. Their average across neurons at each input intensity is then normalized to the same peak value and displayed in F.