Theoretical background

Dynamic range refers to the range of input signal values to which the sensing system responds. If the system’s response changes sharply over a narrow range of signal values, the dynamic range is small. Conversely, if the response changes gradually over a broad range of signal values, the dynamic range is large. In biological systems, the dynamic range of the system dictates many of its dynamic properties such as pathogen response, cellular differentiation and sensing mechanisms40,41. In this work, we use the sigmoidal Hill equation to model neural gain, describing the relationship between a neuron’s firing rate and the input signal (other sigmoidal functions yield similar results; Supplementary Figs. 1–7). The population response is the average of individual neurons’ responses:

$$_}}\left(S\right)=\frac^}^+_^}$$

(1)

where Apop(S) is the averaged population neural gain to an input signal S, n the Hill coefficient and Km the half-activation point (where the response function reaches half its maximum value).

Notably, n dictates the slope of the response function (Supplementary Fig. 8)—higher n values correspond to a sharp population response and therefore to a narrow dynamic range (NDR; Fig. 1, turquoise line), whereas lower n values correspond to a gradual population response and an IDR (Fig. 1, red line). Note that the input S is the rescaled intensity of the input. For example, in a motion coherence task, all dots moving to the right can be considered as S = 0 and all dots moving to the left as S = 1.

Fig. 1: A gradual response function entails an IDR that allows for better discrimination between close-by input signal values away from the middle of the signal range.

The neural gain, the normalized firing rate relative to baseline of a neuron to different levels of input signal for a sharp and a gradual response. For a sharp response (NDR, Hill coefficient n = 16, turquoise line), the dynamic range between 0.1 and 0.9 activation levels is [0.44, 0.57]. Encoding input signals away from the middle of the input signal range (dashed lines B versus A or D versus C) induce similar responses, which are difficult to discriminate (neural gain difference: ΔANDR = 0.0003 (B versus A) and ΔANDR = 0.004 (D versus C)). For a gradual response (IDR, Hill coefficient n = 7, red line), the dynamic range between 0.1 and 0.9 activation levels is [0.36, 0.66], with between one to two orders of magnitude better discrimination between the aforementioned input values (neural gain difference: ΔAIDR = 0.03 (B versus A) and ΔAIDR = 0.05 (D versus C)).

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We proposed that the computational distinction between individuals with ASD and NT individuals resides in the dynamic range of the neuronal population response. NT individuals display an NDR (Fig. 1, turquoise line), whereas individuals with ASD exhibit an IDR (Fig. 1, red line).

IDR better discriminates between close-by input values

Individuals diagnosed with ASD show a heightened ability to discriminate between two close-by signals in perceptual tasks. They show superior search ability for small elements, heightened performance in high-contrast motion discrimination, lower thresholds to detect luminance modulation and oblique orientations, heightened performance in pitch discrimination and higher performance in spatial location modulations compared with NT individuals8,9,10,14,24,25,42 (however, see also ref. 28). This heightened sensitivity arises naturally from the IDR model, comparing between sharp and gradual population responses. A gradual population response (Fig. 1, red line) elicits discernible responses for close-by signals (even far from the half-activation point), thus allowing distinction between the two input signals. This heightened sensitivity is not available for the sharp population response (Fig. 1, turquoise line) because close-by signals away from the half-activation point elicit indistinguishable population responses. Comparing the activation of the neuronal population at close-by input signal levels, we found that the gradual population response outperforms the sharp population response in most of the signal range (for 80% of the signal range, see Supplementary Fig. 9 and two examples in Fig. 1, red versus turquoise lines). Note that, if the discrimination between two close-by signals falls near the center of signal range, an NDR may outperform an IDR.

Findings on orientation discrimination thresholds in ASD support these results. Typically, individuals better discriminate near cardinal orientations than oblique angles, known as the oblique effect43. In ASD, this effect is reduced or abolished42, suggesting better discrimination away from the signal range center and a more uniform allocation of encoding resources. An IDR allows for more accurate signal decoding and representation, explaining the heightened ability to discriminate close-by signals in visual and auditory tasks away from the center of the signal range.

IDR increases the range of elevated neural variability

Enhanced variability in neuronal activity is a salient difference between individuals with ASD and NT individuals7,22,44. We therefore assessed the variability in the neuronal population response in the face of noise (Methods). We tested three different models: a neuronal population with a gradual response (IDR, n = 7), a neuronal population with a sharp response (NDR, n = 16) and a neuronal population with a sharp response and an increased E:I ratio36 (increased E:I model, n = 16, 25% reduction in inhibition; Methods). Comparing the gradual response (IDR) to the sharp response (NDR and E:I) models, we found that the interval of notable variance in the population’s response is wider for the gradual response compared with the sharp response model and the E:I model (Fig. 2 and Supplementary Fig. 10). We note that these heightened variance levels do not change the discriminability results of the previous section, as a signal:noise ratio calculation shows (Supplementary Fig. 11).

Fig. 2: An increased dynamic range increases the range of elevated variance of the neuronal population response.

The variance between population responses of N = 200 neurons with a noisy input signal (\(_}}^=0.1\)). For the sharp response (NDR, n = 16, turquoise curve), the variance is localized and peaked around the half-activation point of the population response (width, measured as the distance between the two variance curve crossings of 1/e of the maximal mean variance: W = 0.26, 99% CI = 0.25, 0.27). For the gradual response (IDR, n = 7, red curve), the variance is spread over an increased range of signal values (W = 0.41, 99% CI = 0.4, 0.41). For the increased E:I model (n = 16, light-blue curve), the variance is localized and peaked around the effective half-activation point of the population response (W = 0.26, 99% CI = 0.25, 0.26; see Methods for more details).

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Importantly, the difference in the width of the variance curves can distinguish between the IDR model and the increased E:I ratio model. By presenting a set of stimuli varying across a single parameter (for example, contrast) and measuring the variance curves of neural and behavioral responses, the IDR model predicts wider variance curves for ASD compared with NT. In contrast, the increased E:I model predicts similar widths for both ASD and NT.

IDR entails slower updating rates to abrupt changes

Individuals diagnosed with ASD show slower updating rates to changes in the environment in auditory and motor tasks15,18,45. We tested whether the increased dynamic range model can account for the slower updating rate when tracking an abrupt signal change with optimal Bayesian inference (Kalman filter). To test this, we simulated an abrupt change of a noisy signal from S = 0.3 to S = 0.7 and computed the population response for a gradual neuronal response and a sharp neuronal response (Methods, Fig. 3 and Supplementary Fig. 12).

Fig. 3: An increased dynamic range entails slower updating in response to abrupt changes.

Optimal Bayesian inference (Kalman filter) is used to estimate the response of a neuronal population to an abrupt signal change with Gaussian noise. a, A single simulation (out of 500) tracking an abrupt change in the mean of a noisy signal, from a level of 0.3 (\(S \sim }(0.3,0.01)\)) to a level of 0.7 (\(S \sim }(0.7,0.01)\)) (the black line is the mean of the signal and the gray line is a noisy realization of the signal). The noisy signal was encoded and then decoded using Bayesian inference of two different populations—one with a gradual population response (IDR, red, n = 7) and one with a sharp population response (NDR, turquoise, n = 16). a, Inset: a histogram of response times to the abrupt change. Response times (RT) are the number of time steps needed to reach 95% of the updated signal level at 0.7 (response time (RT) mean ± s.d.: IDR: 229 ± 15.6, NDR: 5 ± 0.2, two-sided Wilcoxon’s signed-rank test for RTIDR > RTNDR, W = 0, P < 10−5). The variance of RT for IDR was much higher than the variance of RT for NDR (IDR variance RT distribution − NDR variance RT distribution: Δvar = 244, permutation test (10,000 permutations) for variance difference: P < 10−4). b, Histogram of the Hill coefficients fitted to each individual participant data from Vishne et al.18. The Hill coefficients of the ASD group are significantly lower than those of the NT group (mean ± s.e., ASD: 8.5 ± 0.2, NT: 13 ± 0.1, one-sided Mann–Whitney U-test, U = 852, P < 0.0005). c,d, Data and model fit to ASD (c) and NT (d) on the group average tracking dynamics from Vishne et al.18. Different panels show the tracking of the metronome tempo change (dashed black line) for different tempo step sizes. Gray lines are the data from Vishne et al.18, with black error bars for ±1 s.e., colored lines are the fits of the model, mean ± s.e.; nASD = 7.4 ± 0.1 and nNT = 14 ± 0.1, permutation test (1,000 permutations), P < 0.001.

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We found that a gradual response shows slower updating of the estimated signal level (Fig. 3, red (IDR) versus turquoise (NDR) lines, and Supplementary Figs. 13 and 14). Furthermore, a gradual neuronal response function shows higher variability in the response times to the abrupt change (Fig. 3a, inset, and Supplementary Fig. 15). Next, we compared the response of the increased E:I ratio model to an abrupt change in signal levels. We found that the increased E:I ratio model responds quickly to an abrupt change similar to the sharp response model (Supplementary Figs. 16–21). Thus, a sharp neuronal response will react and adapt faster to an abrupt change in the environment compared with a gradual neuronal response.

To test the model’s predictions, we compared it with recent experimental results. In Vishne et al.18, participants performed a tracking task, where they synchronized their finger tapping to a metronome beat. The metronome abruptly changed its tempo during the trial, accelerating or decelerating. The authors found slower updating of the tempo in individuals with ASD compared with NT individuals. We fitted the changing dynamics of each individual in the ASD and NT groups and also fitted the dynamics of the averaged group level in response to acceleration and deceleration of the tempo18 (Methods and Fig. 3b–d). Consistent with our model predictions, at the individual level the fitted curves resulted in lower Hill coefficients for the ASD group (mean ± s.e., ASD: 8.5 ± 0.2, NT: 13 ± 0.1, Mann–Whitney U-test: U = 852, P < 10−3; Fig. 3b and Supplementary Figs. 22–25). Similarly, on the averaged group level, we found lower Hill coefficients for the ASD group (Fig. 3c,d). We further assessed the ASD data fit to the E:I model. We added a fitting parameter of inhibition strength and restricted the Hill coefficients to higher values (between 14 and 20; Methods). This procedure resulted in worse fits (the Bayesian information criterion using average mean squared error over all blocks, IDR model: −5, E:I model: −3.47; Supplementary Fig. 26). Moreover, the fitted inhibition strengths were mostly extremely low, indicating an inhibition reduction of 90% or no inhibition reduction, when the Hill coefficients replicated the Hill coefficients for the IDR model (Supplementary Fig. 26).

The crux of this finding relies on the differences in the neuronal encoding variance of sharp versus gradual population responses. An increased dynamic range incurs an increased variance in the neuronal response over a larger portion of the input signal range (Fig. 2). In turn, the increased variance implies lower confidence in the incoming signal, thus down-weighting the new evidence and eliciting a smaller update of the prior belief. Note that this gradual response can carry benefits: if the abrupt changes stem from noisy transients that quickly disappear, then an IDR with its gradual response will filter them better.

IDR induces slower dynamics in a binocular rivalry task

In binocular rivalry tasks, a different image is presented to each eye simultaneously and perception varies between different states: two ‘pure’ states that are the perception of only one of the two images or a ‘mixed’ state that is the perception of a mixture of both images. Individuals diagnosed with ASD exhibit slower transition rates between the two pure states as well as a decreased duration in the pure states compared with matched NT individuals12,13 (although see also ref. 46).

As the two stimuli are simultaneously presented to each eye on an equal footing and with perceptual noise, we simulated the input as a Gaussian random walk starting from S = 0.5 (clipped at [0, 1]). The signal is then encoded via the response function to produce activation. Activations >0.8 or <0.2 are considered pure states, whereas those in the [0.2, 0.8] range are mixed states (see Supplementary Fig. 27 for threshold sensitivity analysis). As a gradual neuronal response requires more noisy steps to reach a pure state, we expect it to result in slower transition rates and shorter durations in the pure states compared with a sharp neuronal response.

We tested our predictions by simulating input levels for each model (gradual response (IDR), sharp response (NDR) and the E:I model). We found that a sharp neuronal response (NDR, n = 16) resulted in more transitions between the two pure states and that the amount of time spent in the mixed state was overshadowed by the time spent in the two pure states compared with the gradual response (IDR, n = 7). We also found that, in the E:I model, decreasing the inhibition strength by 25% accentuates these differences, further increasing the number of transitions (Fig. 4b and Supplementary Figs. 28–30), as well as increasing the portion of the time spent in the pure states (Fig. 4c and Supplementary Figs. 31–33).

Fig. 4: Increased dynamic range decreases the rate of state transitions and increases the time spent in the mixed state in a binocular rivalry task.

The binocular rivalry task is simulated as the response to a signal generated by a random walk (Gaussian noise, σ2 = 0.03) starting at S = 0.5 and clipped to [0, 1]. a, Example of a single binocular rivalry simulation. The noisy input signal (solid black line), the pure states thresholds at 0.2 and 0.8 signal values (dashed horizontal black lines), the gradual (IDR, n = 7, red), sharp (NDR, n = 16, turquoise) and decreased inhibition (n = 16, 75% inhibition, E:I, light blue) population responses are shown. b, A histogram of the number of state transitions (from one pure state to the other) for sharp (NDR, n = 16, turquoise), gradual (IDR, n = 7, red) and decreased inhibition (n = 16, 75% inhibition strength, E:I, light blue) population responses. The population with the gradual response has a significantly lower number of state transitions than both of the sharp response populations, with the decreased inhibition population having a significantly higher number of transitions than the sharp population response (number of transitions, mean ± s.d.: IDR: 2.6 ± 1.7, 95% CI of mean = 2.5, 2.8; NDR: 9.3 ± 5.0, 95% CI of mean = 8.9, 9.8; E:I: 11.3 ± 6.1, 95% CI of mean = 10.8, 11.9; one-sided Wilcoxon’s signed-rank test IDR < NDR, W = 0, P < 10−5, IDR < E:I, W = 0, P < 10−5; NDR < E:I, W = 976.5, P < 10−5). c, A histogram of the ratios of the number of time steps in a pure state to the number of time steps in the mixed state. The population with the gradual response (IDR, n = 7, red) has a significantly lower \(\frac_}}}_}}}\) ratio than the sharp (NDR and E:I, n = 16, turquoise and light blue, respectively) populations, with the decreased inhibition population having a significantly higher \(\frac_}}}_}}}\) ratio than the sharp population response. (The ratio of the time in the pure state to the time in the mixed state, mean ± s.d.: IDR: 3 ± 3.6, 95% CI of mean: 2.7, 3.3; NDR: 18.4 ± 26.3, 95% CI of mean: 16.2, 20.8; E:I: 24.4 ± 29.4, 95% CI of mean = 21.9, 27.1; one-sided Wilcoxon’s signed-rank test: IDR < NDR: W = 0, P < 10−5; IDR < E:I: W = 0, P < 10−5; NDR < E:I: W = 800, P < 10−5).

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IDR increases detection thresholds in motion coherence tasks

Previous studies of the motion coherence task found that individuals with ASD showed elevated detection thresholds compared with NT individuals11, whereas others found comparable thresholds8,9,11,20,31. In the task, participants viewed randomly moving dots, with a portion moving coherently to create a perception of motion, and were asked to identify the motion direction. The detection threshold is the percentage of coherently moving dots required for participants to reliably report the direction (>80% correct). Importantly, Robertson et al.11 found increased thresholds for individuals with ASD for short stimulus durations and comparable thresholds for longer durations (Fig. 5c).

Fig. 5: Increased dynamic range induces elevated detection thresholds in decision-making for short integration times.

a, Illustration of the LCA model used to simulate the decision-making process47. The stimuli are encoded as a signal I ∈ [0, 0.5], which is passed to two encoding neuronal populations with the Hill equation. The output of the encoding populations (f(I), 1 − f(I)) is the input to the mutually inhibiting, leaky accumulators, which have leak strengths of κ1 and κ2 and inhibition strengths of β1,2 and β2,1. The dynamics are run for T steps, where the first accumulator that passes a predefined activation threshold, θ, generates a decision. b, The LCA model signal detection levels that elicit 80% correct responses for a sharp response (NDR, nNDR = 16) and a gradual response (IDR, nIDR = 8) encoders at different maximal simulated decision times. Simulation parameters were: β1,2 = β2,1 = 0.25, κ1 = κ2 = 1, θ = 0.51, T ∈ [200, 400, 1,500]. c, Adapted from ref. 11 under a Creative Commons license CC BY 4.0. ASD and NT participants’ signal detection levels elicit 80% correct responses at different maximal decision times and error bars indicate ±1 s.e.

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To test our model, we used the leaky competing accumulator (LCA) model47 for decision-making. In the LCA simulations (Fig. 5a), signal strength, I ∈ [0, 0.5], is encoded by a population of neurons with a varying dynamic range (\(f(I)=\frac^}^+^}\)) and then accumulated by one leaky accumulator, whereas the evidence for the other alternative (1 − f(I)) is accumulated by a second leaky accumulator. The two leaky accumulators also inhibit each other. We simulated the LCA model dynamics with an increased dynamic range and a narrow dynamic range (Fig. 5a, second step, IDR in red and NDR in turquoise lines). The simulations yield a percentage of correct responses for a given set of LCA model parameters, signal levels, multiple noise initializations and integration times (Methods). By keeping all other parameters constant and changing only the slope of the encoding function, the model replicated the results of Robertson et al.11. An NDR matched the control group’s detection thresholds, whereas an IDR matched the ASD group’s thresholds across different integration times (Fig. 5a,b).

Using an encoding function with an IDR effectively slows down the LCA dynamics, amplifying noise effects (Methods). This necessitates either higher signal levels or longer stimulus durations for motion detection. With sufficiently long stimulus durations, noise effects average out, resulting in similar performance between gradual and sharp encoding functions.

IDR changes the encoding scheme and reduces total capacity

Fisher information measures how much information an encoding scheme provides about the encoded signal, with higher values indicating more accurate local inference. Thus, it gauges the information capacity of an encoding scheme. In a recent study, Noel et al.27 tested participants’ orientation perception using an orientation reproduction task. Participants briefly saw an oriented Gabor patch followed by a mask and then reproduced the orientation. The authors used the bias and variance in responses to estimate Fisher information via its Cramer–Rao bound27 and observed a reduced total encoding capacity in ASD.

To test whether the IDR model produces similar results, we considered the population response as the mean firing rate of a Poisson neuron and derived a closed-form encoding capacity equation for the population response. A Poisson neuron with mean firing rate \(f\left(S\right)\) has the following Fisher information (Methods):

$$_\left(S\right)=\frac\,f\left(S\right)\right)}^}.$$

(2)

Plugging the Hill equation for: \(f(S)=\frac^}^+_^}\) yields:

$$_\left(S\right)=\frac^_^^}^+_^\right)}^}.$$

(3)

Using equation (3), the total encoding capacity, \(\mathop\nolimits_^_(S)}S\), of a gradual neuronal response (IDR) is reduced compared with the total encoding capacity of a sharp neuronal response (NDR; Fig. 6a), consistent with the findings of Noel et al.27. We note that, although the total encoding capacity is reduced with a gradual neuronal response, it allocated higher encoding capacity for a broader range of the input signal. Importantly, contrary to the IDR model, in the E:I model increasing the E:I ratio by reducing inhibition strength resulted in an increased total encoding capacity (Fig. 6b and Methods).

Fig. 6: Increased dynamic range changes the encoding scheme and reduces the total encoding capacity.

a, The effects of the dynamic range on encoding capacity. A gradual response reduces the encoding capacity at the half-activation point and enhances it away from the middle point of the input signal range. The square root of the encoding capacity curves for response function with different slopes (Hill coefficient values, n) is shown. Inset: total encoding capacity, \(\mathop\nolimits_^_(S)}S\), as a function of the Hill coefficient of the response function. As n decreases and the response function becomes more gradual, the total encoding capacity decreases linearly with n. b, An increased E:I ratio model predicting an increase in the encoding capacity as inhibition is decreased. Encoding via a less inhibited response changes the encoding capacity, shifting the peak of the Fisher information curve and increasing its amplitude. The square root of the encoding capacity curves for each signal level for different inhibitory strength coefficients, ν, is shown. The lower the ν value, the higher the ratio between excitation and inhibition. Inset: total encoding capacity, \(\mathop\nolimits_^_(S)}S\), as a function of the inhibitory strength ν. Total encoding capacity scales as \(\frac\): as ν decreases and the E:I ratio increases, the total encoding capacity increases. c, Distributions of the Hill coefficients that match the total encoding capacity measured in the ‘without feedback’ block in Noel et al.27 for the ASD (red) and NT (turquoise) participants, with \(_ \sim }(15.1,0.35)\), \(_ \sim }(11.3,0.33)\). d, Distributions of total encoding capacities for the ASD and NT participants from Noel et al.27 (ASD participants in dark red, NT participants in dark turquoise) and the model’s fit of the total encoding capacities for the Hill coefficients shown in c.

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To further test our model, we compared it with the data of Noel et al.27. Following the authors’ procedure, we fit the total encoding capacity distributions of the ASD and NT groups. Leveraging the analytical solution for the total Fisher information derived from our model (Methods), we computed the Hill coefficients that yield the 95% highest density interval (HDI) for the behavioral encoding capacity distributions (Fig. 6c). Subsequently, we drew 5,000 samples from a Gaussian distribution with the same 95% HDI, thereby yielding almost indistinguishable total encoding capacity distributions (Fig. 6d).

Heterogeneity in half-activation points may cause IDR

How might an IDR in ASD arise? We propose a biological mechanism based on the heterogeneity of the half-activation point in individual neurons. Consider a population of N neurons with a response to an input signal \(S\in \left[0,1\right]\) following a sigmoidal function, such as the Hill equation:

$$_\left(S\right)=\frac^}^+_^}$$

(4)

where \(_\in \left[0,1\right]\) is the activity of neuron i, Km,i its half-activation point and n the Hill coefficient.

The response of the entire population is given by the average of the individual neuron’s activations:

$$_}}(S)=\frac\mathop\limits_^_\left(S\right).$$

(5)