Understanding auditory spectro-temporal receptive fields and their changes with input statistics by efficient coding principles

Abstract

Spectro-temporal receptive fields (STRFs) have been widely used as linear approximations to the signal transform from sound spectrograms to neural responses along the auditory pathway. Their dependence on statistical attributes of the stimuli, such as sound intensity, is usually explained by nonlinear mechanisms and models. Here, we apply an efficient coding principle which has been successfully used to understand receptive fields in early stages of visual processing, in order to provide a computational understanding of the STRFs. According to this principle, STRFs result from an optimal tradeoff between maximizing the sensory information the brain receives, and minimizing the cost of the neural activities required to represent and transmit this information. Both terms depend on the statistical properties of the sensory inputs and the noise that corrupts them. The STRFs should therefore depend on the input power spectrum and the signal-to-noise ratio, which is assumed to increase with input intensity. We analytically derive the optimal STRFs when signal and noise are approximated as Gaussians. Under the constraint that they should be spectro-temporally local, the STRFs are predicted to adapt from being band-pass to low-pass filters as the input intensity reduces, or the input correlation becomes longer range in sound frequency or time. These predictions qualitatively match physiological observations. Our prediction as to how the STRFs should be determined by the input power spectrum could readily be tested, since this spectrum depends on the stimulus ensemble. The potentials and limitations of the efficient coding principle are discussed.

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 . 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.