Figure 1.
Optimal sigmoidal stimulus response curves (solid lines) for a stimulus distribution consisting of three peaks (shaded areas) as predicted by two coding hypotheses.
(A) Infomax: the dynamic range of the adapted response curve covers the whole range of input signals. Note that the optimal sigmoidal response curve is shown; generic optimal transmission would be attained by a response curve that has a derivative proportional to the local stimulus distribution. Such a response curve would be steep within peaks of the stimulus distribution and much flatter in between, thus it would be more staircase-like. (B) Selective coding: the response function optimally represents the most intense signal (light gray) whereas other signals (dark gray) are suppressed.
Figure 2.
Typical recording trace from a cricket AN2 neuron (T. oceanicus).
The figure shows the voltage trace during constant stimulation (duration 1 s) with a sinusoidal tone of 16 kHz frequency. The shaded area depicts the spike detection window, bounded by the lower and upper threshold.
Figure 3.
Summary of the experimental protocols.
(A) Adaptation protocol. Amplitude-modulated noise signals (adapting stimuli with 0 dB average relative intensity) of variable duration (from 75 ms to 4800 ms) are followed by a test stimulus (16 kHz sinusoidal tone) with a duration of 1000 ms and a relative intensity ranging from −9 dB to +6 dB (several test stimuli are plotted on top of each other). (B) Recovery protocol. Amplitude-modulated noise signals (adapting stimuli) of 5 second duration are followed by a pause of variable length (from 75 ms to 4800 ms) and a test stimulus as in (A). (C) Adaptation protocol for amplitude-modulated noise stimuli drawn from a bimodal and a trimodal distribution (the corresponding amplitude distributions are shown in the right panel). Relative intensities of the test stimuli range from −6 dB to +6 dB.
Figure 4.
Representative examples of the neural response (AN2 neuron from a T. leo) after adaptation to noise stimuli of duration 75 ms (dotted line), 600 ms (dashed line), and 4800 ms (solid line).
(A–D) Responses (spike rates) during a test stimulus of 1 s duration (cf. protocol of Figure 3A). Relative intensities of the test stimuli range from −3 dB (A) to +6 dB (D); the average relative intensity of the adapting stimulus was 0 dB. Each stimulus was presented 5 times and the recorded spike trains (1 ms resolution) were convolved with a Gaussian kernel (σ = 50 ms). The instantaneous spike rates were estimated by averaging over the 5 repetitions. The increase of the estimated rate during the first 50 ms is an artifact introduced by filtering the neural response with the Gaussian kernel. Note that the onset latency of the AN2 neuron is in the range of 15 to 18 ms. The spike counts during the sample period (shaded) from 100 ms to 300 ms are used to construct neural response curves.
Figure 5.
Representative example of response curves for different adaptation (A) and recovery times (B) (cf. protocols of Figure 3A and 3B).
The average relative intensity of the adapting stimulus was 0 dB. Symbols denote the average spike counts during the sample period (cf. Figure 4) for different test intensities. Solid lines indicate the expected response curve, i.e., the response curve with the set of parameters with the mean value of the posterior distribution (see Methods, Bayesian data analysis). Each stimulus protocol was repeated 5 times (the error bars indicate the standard deviation). The data shown was obtained from a T. leo (the same preparation as used in Figure 4).
Figure 6.
Time course of adaptation and recovery of a T. leo cell (A1,A2) and of a T. oceanicus cell (B1,B2).
The response to the test stimulus is plotted against the duration of the adapting stimulus (A1,B1) and the delay between the adapting and the test stimulus (A2,B2). Displayed are the average spike counts in the 200 ms time window of the test stimulus (cf. Figure 4). The intensity of the test stimulus was equal to the average intensity of the adapting stimulus (0 dB relative intensity). The error bars denote the standard deviation. Solid lines indicate the exponential function with the set of parameters with the highest value of the posterior distribution (see Methods, Bayesian data analysis).
Table 1.
Summary of the adaptation (τa) and recovery (τr) time constants for the T. oceanicus and T. leo AN2 cells.
Figure 7.
Combined posterior distribution (cf. Methods, Bayesian data analysis) of the adaptation time constants τa (A) and the recovery time constants τr (B) for the T. oceanicus (solid line) and T. leo (dotted line) AN2 cells.
Solid (dotted) lines on top of the figures depict the 95% posterior intervals.
Figure 8.
Optimal response curves for the bimodal (circles) and trimodal (squares) stimulus distribution predicted by the infomax principle (A) and the selective coding hypothesis (B).
The figures show the predicted relationship between the response variable (spike rate) and the stimulus intensity. The Gaussian curves depict the probability distributions of stimulus intensity, where the dark shaded areas under the curve denote the bimodal stimulus distribution and the light shaded area under the curve the additional peak of the trimodal stimulus distribution (cf. Figure 1).
Figure 9.
Representative responses of an AN2 cell (T. leo) to the amplitude-modulated noise stimuli of Figure 3C.
(A1,A2) Bimodal stimulus distribution. The envelope of an amplitude-modulated stimulus and the distribution of the stimulus amplitude are shown in (A2), the corresponding instantaneous spike rate is shown in (A1). (B1,B2) Trimodal stimulus distribution. The envelope of an amplitude-modulated stimulus and the distribution of the stimulus amplitude are shown in (B2), the corresponding spike rate is shown in (B1). The stimuli were presented 45 times and the recorded spike trains (1 ms resolution) were convolved with a Gaussian kernel (σ = 5 ms). The instantaneous spike rates were estimated by averaging over the 45 repetitions.
Figure 10.
Typical examples of stimulus response curves after adaptation to the bimodal and to the trimodal stimulus distributions (A1,B1,C1) and posterior densities of the corresponding response curve parameters (A2,B2,C2).
(A1,A2,C1,C2) Results for AN2 cells of T. leo. (B1,B2) Results for an AN2 cell of a T. oceanicus. (A1,B1,C1) Circles and squares denote the mean spike counts in a 200 ms time window of the test stimulus after adaptation to the bimodal and trimodal distributions, measured for 9 different relative intensities of the test stimulus (cf. protocol of Figure 3C). Error bars denote the standard deviation. Solid lines indicate the expected response curve, i.e., the response curve with the set of parameters with the mean value of the posterior distribution (see Methods, Bayesian data analysis). The shaded areas depict the intensity distribution of the stimuli (dark: bimodal stimulus distribution, light: additional peak of the trimodal stimulus distribution). (A2,B2,C2) Marginal posterior densities (cf. Methods, Bayesian data analysis) of the response curve parameters B50 (location) and S50 (slope). The posterior densities after adaptation to the bimodal (solid lines) and trimodal (dotted lines) stimulus distributions are shown in the top panels and the corresponding posterior densities of the changes (ΔB50, ΔS50) between stimulus conditions in the bottom panels. Solid (dotted) lines on top of the figures depict the 95% posterior intervals. Significant changes between stimulus conditions are indicated by a star.
Figure 11.
Summary of adaptation induced changes of the response curve parameter B50 for all 20 AN2 cells.
Distribution of the mean values of the parameters B50 for individual cells (A1) and combined posterior density (see Methods, Bayesian data analysis) over all cells (A2) after adaptation to the bimodal stimulus distribution. (B1,B2) Distribution and combined posterior density of the parameter B50 after adaptation to the trimodal stimulus distribution. (C1,C2) Distribution and combined posterior density of the change of the parameter B50 between the two stimulus distributions. Symbols depict the values predicted by infomax (stars) and the selective coding hypothesis (circles). Triangles denote the median value. The distribution of cells that showed changes in B50 that were significant (Bayesian posterior intervals, see Methods, Bayesian data analysis) is marked black in (A1,B1,C1). Shaded areas depict the two-tailed 95% posterior intervals in (A2,B2) and the right-tailed 95% posterior interval in (C2).
Figure 12.
Summary of adaptation induced changes of the slope S50 of the response curves.
Distribution of the mean values of the parameters S50 for individual cells (A1) and combined posterior density (cf. Methods, Bayesian data analysis) over all cells (A2) after adapting to the bimodal stimulus distribution. (B1,B2) Distribution and combined posterior density of the parameter S50 after adapting to the trimodal stimulus distribution. (C1,C2) Distribution and combined posterior density of the relative change of S50 between the two stimulus distributions. Symbols depict the values predicted by infomax (stars) and the selective coding hypothesis (circles). Triangles denote the median value. The distribution of cells that showed changes in S50 that were significant (Bayesian posterior intervals, Methods, Bayesian data analysis) is marked black in (A1,B1,C1). Shaded areas in (A2,B2,C2) depict the 95% posterior intervals.
Figure 13.
Adaptation induced changes in the mutual information between the stimulus and the neural response.
(A1,A2) Distribution and combined posterior density of changes in the transmitted mutual information when considering the whole stimulus range (relative intensity from −4.5 dB to 4.5 dB) and the trimodal amplitude distribution. For each cell the change of the mutual information is calculated as the difference of the mutual information for the ‘trimodal’ (neural response adapted to the trimodal stimulus) and the ‘bimodal’ (neural response adapted to the bimodal stimulus) response curve. The distribution in (A1) is based on the mean values of changes in mutual information for individual cells. (B1,B2) Distribution and combined posterior density of changes in the transmitted mutual information when considering the stimulus range from −4.5 dB to 1.5 dB (including only the two low-intensity peaks of the trimodal stimulus distribution). (C1,C2) Distribution and combined posterior density of changes in the transmitted mutual information when considering the stimulus range from 1.5 dB to 4.5 dB (including only the high-intensity peak of the trimodal stimulus distribution). Triangles denote the median value. The distribution of cells that showed changes that were significant (Bayesian posterior intervals, Methods, Bayesian data analysis) is marked black in (A1,B1,C1). Shaded areas depict the left-tailed 95% posterior intervals in (A2,B2) and the two-tailed 95% posterior interval in (C2).