Autism spectrum disorder variation as a computational trade-off via dynamic range of neuronal population responses

Abstract

Individuals diagnosed with autism spectrum disorder (ASD) show neural and behavioral characteristics differing from the neurotypical population. This may stem from a computational principle that relates inference and computational dynamics to the dynamic range of neuronal population responses, reflecting the signal levels for which the system is responsive. In the present study, we showed that an increased dynamic range (IDR), indicating a gradual response of a neuronal population to changes in input, accounts for neural and behavioral variations in individuals diagnosed with ASD across diverse tasks. We validated the model with data from finger-tapping synchronization, orientation reproduction and global motion coherence tasks. We suggested that increased heterogeneity in the half-activation point of individual neurons may be the biological mechanism underlying the IDR in ASD. Taken together, this model provides a proof of concept for a new computational principle that may account for ASD and generates new testable and distinct predictions regarding its behavioral, neural and biological foundations.

Individuals with autism spectrum disorder show neural patterns different from those of neurotypical individuals. Here the authors show that this variation reflects a computational trade-off between accurate encoding and fast adaptation tuned by the neural population response.

Main

In recent years, research has mapped many differences between people diagnosed with ASD and the neurotypical (NT) population. These differences are reflected in neural activity^1–7, perceptual tasks^8–15, prediction abilities^16,17, adaptation to changes in the environment^15,18 and decision-making^11,19,20, to name a few. Despite great advances in mapping the myriad of behavioral and neural differences, the underlying computational principle that drives them is still unclear.

At the neural level, studies provide insights into differences in brain activity^{1,4,5,7,21–23}. A prominent finding in brain activity studies is the increased variance in neural activity of individuals diagnosed with ASD compared with NT individuals^1,4,5. This increased variability in neural activity has been proposed as a dominant feature of atypicality in ASD^21,22.

On the behavioral level, many studies showed perceptual and cognitive distinctions between individuals with ASD and NT individuals. For example, individuals diagnosed with ASD demonstrate superior performance in tasks involving detailed visual perception, auditory recognition and auditory discrimination^{9,10,14,24,25}. They also exhibit slower integration and adaptation rates^15,18, as well as altered inference processes^16,17,26. In addition, neurophysiological activity and behavioral measurements in binocular rivalry tasks indicate a slower binocular rivalry process in individuals diagnosed with ASD^6,12,13. In tasks that necessitate global integration, such as the motion coherence task, individuals diagnosed with ASD demonstrate elevated detection thresholds^11,13,20. Last, a recent study quantified the information-encoding capacity of individuals with ASD and NT individuals. The authors found reduced total encoding capacity in individuals with ASD compared with individuals with NT²⁷.

Importantly, the ASD literature is complex. Although the findings above were replicated several times, there are other studies showing no difference in perceptual discrimination^28,29, intact priors in ASD³⁰ and unchanged detection thresholds³¹. At the neural level, studies found normal or reduced variance in ASD neural activity²³ (but see ref. ⁷). These heterogeneous findings pose a challenge to computational models that aim to account for ASD variation.

A main goal of autism research is finding the driver for the differences in ASD neural activity and behavior. To this end, several computational models have been suggested: predictive impairment^26,32,33 suggests that ASD generates high prediction errors and this, in turn, hampers performance in dynamic environments and favors stability and repetition. The ‘hypo-priors’^34,35 model stipulates that ASD variation stems from weakly informative priors, making the posterior mainly dependent on new sensory evidence. Aberrant precision weighting^16,17 generalizes this concept to predictive coding. Within the aberrant precision framework, when the precision of incoming sensory evidence is too high or when the precision of the priors is too low, perception will be dominated by the sensory input. Another model proposes that ASD variation stems from slower updating rates of internal representations^15,18. Slow updating creates a mismatch between the priors and the environment, which, in turn, could account for impaired social skills and motor and perceptual atypicalities. Last, the increased excitation:inhibition ratio (E:I) model^3,36–38, inspired by the relatively high co-morbidity of epileptic seizures and ASD (ranging between 6% and 27%)^3,37, suggests that ASD variation stems from a higher ratio of activity of excitatory:inhibitory neurons. The E:I model was also incorporated within a divisive normalization framework to explain ASD variation in perception, local versus global processing, and modality integration^36,39.

Although these computational models provide insights into ASD variation, a unifying computational principle for the wide range of atypical behaviors and neural activity associated with ASD remains elusive. In the present study, we hypothesized that ASD variation can be attributed to a computational principle that connects inference (for example, accuracy, encoding capacity) and dynamic features (for example, adaptation and updating rates) to the dynamic range of neuronal population response during signal encoding. We demonstrate, with numerical simulations and analysis of experimental data, that an increased dynamic range (IDR) of neuronal population responses can drive the diverse differences shown in ASD. Finally, we proposed one plausible biological mechanism that could cause an increase in the dynamic range of neuronal responses in ASD.

Results

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 mechanisms^40,41. 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:

$$A_{\mathrm{pop}}\left(S\right)=\frac{S^n}{S^n+K_m^n}$$ 1

where A_pop(S) is the averaged population neural gain to an input signal S, n the Hill coefficient and K_m 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: ΔA_NDR = 0.0003 (B versus A) and ΔA_NDR = 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: ΔA_IDR = 0.03 (B versus A) and ΔA_IDR = 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 individuals^{8–10,14,24,25,42} (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 individuals^7,22,44. 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 (${\sigma }_{\mathrm{signal}}^2=0.1$). 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 tasks^15,18,45. 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 ($S~\mathcal{N}(0.3,0.01)$) to a level of 0.7 ($S~\mathcal{N}(0.7,0.01)$) (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 RT_IDR > RT_NDR, 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: Δ_var = 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.; n_ASD = 7.4 ± 0.1 and n_NT = 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 individuals^12,13 (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 $\frac{t_{\mathrm{pure}}}{t_{\mathrm{mixed}}}$ 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 $\frac{t_{\mathrm{pure}}}{t_{\mathrm{mixed}}}$ 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 thresholds^8,9,11,20,31. 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 β_1,2 and β_2,1. 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, n_NDR = 16) and a gradual response (IDR, n_IDR = 8) encoders at different maximal simulated decision times. Simulation parameters were: β_1,2 = β_2,1 = 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 ($f\left(I\right)=\frac{{(0.5+I)}^n}{{(0.5+I)}^n+K^n}$) 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):

$$I_F\left(S\right)=\frac{{\left(\frac{\partial }{\partial S}f\left(S\right)\right)}^2}{f\left(S\right)}.$$ 2

Plugging the Hill equation for: $f\left(S\right)=\frac{S^n}{S^n+K_m^n}$ yields:

$$I_F\left(S\right)=\frac{n^2{K}_m^{2n}{S}^{n-2}}{{\left(S^n+K_m^n\right)}^3}.$$ 3

Using equation (3), the total encoding capacity, ${\int }_0^1{I}_F\left(S\right)\mathrm{d}S$, 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, ${\int }_0^1{I}_F\left(S\right)\mathrm{d}S$, 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, ${\int }_0^1{I}_F\left(S\right)\mathrm{d}S$, as a function of the inhibitory strength ν. Total encoding capacity scales as $\frac{1}{\nu }$: 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 $n_{NT}~\mathcal{N}(15.1,0.35)$, $N_{ASD}~\mathcal{N}(11.3,0.33)$. 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:

$$A_i\left(S\right)=\frac{S^n}{S^n+K_{m,i}^n}$$ 4

where $A_i\in \left[0,1\right]$ is the activity of neuron i, K_m,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:

$$A_{\mathrm{pop}}\left(S\right)=\frac{1}{N}\sum _{i=1}^N{A}_i\left(S\right).$$ 5

Importantly, the slope of the population response given in equation (5) depends on the heterogeneity in the half-activation points across the entire neuronal population (Supplementary Fig. 34). Low heterogeneity in the half-activation points of the neuronal population results in a sharp population response and a narrow dynamic range (NDR) (similar to that of a single neuron), whereas high heterogeneity in the half-activation points results in a gradual population response and an IDR (Fig. 7 and Supplementary Fig. 34).

Fig. 7. Heterogeneity in the half-activation point of individual neurons induces an increase in the dynamic range.

The heterogeneity in the half-activation point of single neurons within a population determines whether the average population response is sharp or gradual. a, Low heterogeneity in the half-activation point of individual neurons eliciting a sharp population response (n_eff = 16). Population response (turquoise line) of 250 neurons (gray lines) with individual neurons, with n = 16 and K_m,i = 0.5 + ξ_i, where ξ_i is taken from a uniform distribution ξ ~ U(−σ, σ), σ = 0.01. b, High heterogeneity in the half-activation point of individual neurons eliciting a gradual population response (n_eff = 7). Population response (red line) of 250 neurons (gray lines) with individual neurons, with n = 16 and K_m,i = 0.5 + ξ_i, where ξ_i is taken from a uniform distribution ξ ~ U(−σ, σ), σ = 0.175.

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IDR increases events of spontaneous activation

Individuals diagnosed with ASD exhibit relatively high levels of co-morbidity with epilepsy³. These observations led to the hypothesis that both ASD and epilepsy may have, at least in certain cases, a common neurobiological basis. In the present study, we show that an increased dynamic range can account for a seizure-like hyperactivity. We simulated the activity of a population of neurons (N = 50) with heterogeneity in the half-activation point of individual neurons within a fully connected network. We then initialized the network with a noisy signal (S = 0.4, 2,000 repeats for each noise level) and let the network activity evolve according to the following dynamics:

$$\tau \dot{A_i}=-A_i+g_i\left(\frac{1}{N}\sum _{j=1}^N{\left[A_j+{\xi }_j\right]}_0^1\right)$$ 6

where A_j is the activity of neuron j, τ is the characteristic timescale of the dynamics, ξ_j is a noise term and ${\left[\right]}_0^1$ rectifies the signal to the [0, 1] range. Without heterogeneity in the half-activation points and without signal noise, only input levels >0.5 evolve to full activation of the network. Spontaneous activation of the entire network below that threshold indicates a ‘seizure’—full activity where there should be none.

The simulations showed that, as the heterogeneity in the neuronal half-activation points increases, the probability of spontaneous activation increases as well (Fig. 8). Even at very low signal noise levels (σ < 0.175, including no noise) where a sharp response (n_eff = 16) shows zero probability for full activation, a gradual response (n_eff = 6) has a finite probability for full activation. This indicates that a gradual response lowers the activation thresholds, making the system hyperexcitable (Fig. 8 and Supplementary Fig. 35).

Fig. 8. Heterogeneity in the half-activation point of neurons increases the probability of spontaneous activation of the entire network.

Activation probabilities of 2,000 randomly initialized populations (N = 50 neurons), with bootstrap-calculated (10,000 bootstrap samples) 95% CI in the shaded area around each line. Heterogeneity in neurons K_m is specified by the expected effective Hill coefficient, n_eff, of the population. Full activation of the network is defined as an averaged population activity that goes >0.9 at any timepoint within the simulation. Activation probabilities are the fraction of populations (out of the 2,000) that reached full population activation. The simulation begins with an initial signal level of S = 0.4 and population activity evolves with an all-to-all connectivity (Methods).

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Two suggested drivers for the literature heterogeneity

One main challenge in ASD research is the heterogeneity in findings. Although many differences between ASD and NT are recorded and replicated, some studies show no difference. For example, both altered and intact motion integration have been demonstrated (see refs. ^9,20,31 for reviews). Similarly, several studies show heightened discrimination in ASD^{8–10,14,24,25,42}, whereas others do not^28,29. Previous accounts point to the large heterogeneity in the ASD population to explain these conflicting findings. This is partly the result of conceptual ambiguity ³⁵, hindering efforts to understand the etiology of sensory differences in autism. Others suggested different subtypes within the ASD population as a source for the nonrobust results^48,49. In the present study, we developed a quantitative framework to assess the robustness of ASD to NT experimental results by considering blind sampling of the spectrum of dynamic range values of the neuronal encoding function.

Consider the continuous spectrum of dynamic range values and an arbitrary threshold that separates ASD and NT populations (Extended Data Fig. 1a), transforming a continuous spectrum into a dichotomous view. The sampling procedure would heavily influence the power of any experimental setup. We illustrate this with two hypothetical scenarios. In the first (light-purple boxes), individuals were sampled far from the threshold (n_ASD = 7, n_NT = 13), resulting in a large difference in dynamic range. In the second, the groups were sampled just across the threshold (n_ASD = 9, n_NT = 11). On top of differences in their average values, we also explored scenarios where sampled groups display different levels of variability within each group (equivalent to changing the size of the boxes in Extended Data Fig. 1a).

Extended Data Fig. 1. Subsampling of the ASD spectrum may account for conflicting results in the ASD literature.

(A) The effect of neural within-population variability and dynamic range, illustrating different possible sub-samples of dynamic range levels of individuals in studies. Samples in different studies might probe many combinations of dynamic ranges in the ASD and NT groups, eliciting different effect sizes for any given task. In this illustration we take an arbitrary threshold determining ASD diagnosis - with dynamic ranges below n=10 indicating an ASD diagnosis. We present three possible sampling scenarios - two well separated groups (n_ASD = 7, n_NT = 13, light purple boxes), two closer groups (n_ASD = 8, n_NT = 12, blue boxes), and two groups just across the arbitrary threshold (n_ASD = 9, n_NT = 11, orange boxes). The mean (circle) and the IQR (line) of the heterogeneity leading to the fitted Hill-coefficients obtained in the tapping synchronization task are presented at the bottom of the panel in darker red (ASD) and turquoise lines (NT). See also Supplementary Fig. 47 for similar results with using the medians of each fitted Hill-coefficient. (B) Power analysis for different simulated tasks for the scenario depicted in panel A, light purple boxes (mean Hill-coefficients n_IDR = 7, n_NDR = 13). Power analysis is presented for belief updating, binocular rivalry, motion coherence simulations and the encoding capacity calculation. Presented is the number of participants in each group (IDR/NDR) required to achieve 80% power in the simulated experiment. The dotted gray line indicates the averaged level of variability between fitted Hill-coefficients within each group of the tapping synchronization experiment data. (C) Same as panel B, for the scenario depicted in panel A, for the two groups just across the threshold, orange boxes (mean Hill-coefficients n_IDR = 9, n_NDR = 11).

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For each scenario and different levels of within-group variability, we simulated four experimental tasks—binocular rivalry, belief updating, encoding capacity and motion coherence—to find the group size needed to achieve 80% power (Extended Data Fig. 1b,c). Increased within-group variability led to a nonlinear increase in the required sample sizes and the difference in mean dynamic range between the groups also notably impacted the required group size. We used the within-group variability in the tapping synchronization experiment as a benchmark (Extended Data Fig. 1a, darker-red and turquoise lines). These differences required different group sizes for 80% power: around 40 participants per group in the first scenario (n = 7 versus n = 13), but around 400 in the third scenario (n = 9 versus n = 11, which resembles the tapping task more). Thus, sampling from a spectrum of dynamic ranges with high within-group variability and an arbitrary threshold poses challenges for achieving proper power, indicating that current studies may be underpowered for recognizing behavioral differences between ASD and NT populations.

On top of sampling challenges, previous accounts suggest that differing experimental parameters can yield different conclusions from similar experiments^11,46. Our findings support this: in the motion coherence task, there is a large difference in detection thresholds between IDR (representing ASD) and NDR (representing NT) groups for short stimulus durations, but almost no difference for longer durations. Similarly, our analysis indicated that discrimination ability depends on the signals’ positions within the range. If signals are far from the center, IDR shows higher discrimination probability. If signals are close to the center, NDR shows higher or equal discrimination probability. These findings underscore the crucial role of experimental parameters in detecting differences between ASD and NT.

A comparison with existing computational models of ASD

This section explores the relationship between the increased dynamic range model and existing computational models of ASD. The IDR model extends and complements several previous models. First, it aligns with the aberrant precision account of ASD¹⁷. The IDR model suggests elevated neural variance and increased perceptual uncertainty, which lead to a higher perceived volatility of the sensory environment. Using the hierarchical Gaussian filter (HGF) framework of Lawson and colleagues¹⁷ with increased perceptual uncertainty, we found similar inferred volatility estimates¹⁷ (Extended Data Fig. 2a, Supplementary Fig. 36 and Methods). This increased volatility broadens learned priors, supporting recent Bayesian and predictive coding ASD models like hypo-priors³⁴ and aberrant precision^16,17. It also makes predicting future events harder³². In addition, slower updating^15,18 arises naturally from the IDR model owing to higher neural variance. Although our model differs from the increased E:I model, both could coexist⁵⁰ and their interplay remains to be tested.

Extended Data Fig. 2. Comparison of known models and the increased dynamic range model.

(A) Increased perceptual uncertainty shows elevated volatility estimates. Volatility parameter (ω₃) fits for a Hierarchical Gaussian Filter (HGF) model for low and high perceptual uncertainty (PU). Data was generated using two sets of parameters for an HGF model for binary choices with perceptual uncertainty, one with low perceptual uncertainty (low PU, α = 0.13, akin to a narrow dynamic range) and one with high perceptual uncertainty (high PU, α = 0.6, akin to an increased dynamic range). Fitting the model parameters resulted in increased volatility estimates for high perceptual uncertainty similar to the findings of Lawson et al.¹⁷. Centers represent median values, minima and maxima are Q1, Q3 values and whiskers extend to the farther data point within 1.5 times of the inter-quantile range. (B) Encoding capacity as a function of inhibition strength using the divisive normalization with decreased inhibition model³⁶. Contrary to experimental results by Noel et al.⁴⁴, as inhibition decreases, encoding capacity increases. (C) Decision-making simulations of the Leaky, Competing Accumulator (LCA) model using the divisive normalization with decreased inhibition model³⁶ as the encoding function (see Fig. 5). Decreased inhibition in the encoding function (red solid line) resulted in lower thresholds compared to full inhibition (turquoise solid line), contrary to the experimental findings¹¹ (compare with Fig. 5b,c).

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Rosenberg et al.³⁶ demonstrated how divisive normalization with decreased inhibition explains ASD perception differences. This model introduces a larger dynamic range and counters saturation effects, similar to the IDR model. We therefore simulated their model with belief updating, binocular rivalry, encoding capacity and motion coherence tasks to compare results from both models (Supplementary Figs. 37–42). For belief updating, both models generally agreed (Supplementary Fig. 39). However, the divisive normalization with decreased inhibition model predicted opposite results in three tasks compared with the IDR model and experimental results. First, it showed an increase in encoding capacity with decreased inhibition, contrary to Noel et al.²⁷ (Extended Data Fig. 2b). Second, it predicted lower thresholds for decreased inhibition in the motion coherence task, contradicting Robertson et al.¹¹ (Extended Data Fig. 2c). Third, in binocular rivalry simulations, it resulted in more transitions and less time in the mixed state, conflicting with experimental findings. These discrepancies arise because decreased inhibition models with their higher amplitudes accelerate computation dynamics, unlike the slower dynamics of the IDR model.

Last, previous studies highlighted the role of divisive normalization in neural function and its impact on causal inference³⁹. We explored divisive normalization within the IDR model using two encoded signals (Methods). It is interesting that, aligned with the decreased inhibition hypothesis, the IDR model predicted lower inhibition levels for signals above the half-activation point and higher inhibition levels for signals below it, compared with sharp neural responses (Supplementary Fig. 43). These differences stem from the shape of the response functions: gradual responses rise earlier but saturate later than sharp responses. Therefore, when inhibiting another signal, it will produce stronger inhibitions for low signal levels and weaker inhibitions for high signal levels. This suggests different impacts on causal inference based on whether the signal is below or above the half-activation point³⁹.

Discussion

We demonstrated how a simple computational principle linking inference and computational dynamics to the neuronal population’s dynamic range can explain ASD variation. An increased dynamic range (gradual response) provides accurate, analog-like encoding, whereas a narrow dynamic range (sharp response) offers digital-like encoding with a clear threshold. These encoding strategies span a computational trade-off: gradual responses enhance discrimination and robustness against noise, whereas sharp responses allow fast reaction to changes, higher total encoding capacity and robustness against full activations. We proposed that increased heterogeneity in the half-activation points of individual neurons could modulate the dynamic range of the neuronal population response.

The proposed computational model captured hallmark findings regarding ASD variation in neural and behavioral measures. Although these differences were recorded and replicated across different studies, the ASD literature harbors many conflicting results. Using simulations, we estimated the group size needed to achieve 80% power in four tasks: belief updating, binocular rivalry, motion coherence task and encoding capacity. We found that different sampling scenarios of the spectrum of possible dynamic range levels, divided by an arbitrary dichotomous threshold, can lead to different levels of overlap between groups within an experiment. This intragroup variability requires hundreds of participants per group to conduct high-powered studies. These suggestions complement current theories for the possibility of subtypes of ASD to explain the heterogeneous findings in ASD research^35,48,49. Moreover, our computational simulations and known experimental results suggest that parameters of the experimental setup (for example, integration time, Fig. 5, and signal levels, Fig. 1) could vastly impact the measured effect size in the experiment.

The IDR model extends and complements previous models like slower updating, aberrant precision, hypo-priors and predictive impairment. It aligns with empirical findings that originally motivated the E:I theories on atypical perceptual processing⁵¹, synapse formation⁵² and the increased co-morbidity with epilepsy in ASD³⁸. Our model provides distinct predictions from decreased inhibition models, which cannot explain the decrease in total encoding capacity, elevated decision-making thresholds or slower binocular rivalry dynamics. Analyzing variance in neural responses across the input signal range can further support the IDR model over decreased inhibition models. It is interesting that divisive normalization coupled with an IDR suggests decreased inhibition for signal levels above the half-activation point, which may explain previous findings of reduced γ-aminobutyric acid (GABA)-ergic activity and concentrations ^6,53.

What might be the biological underpinnings of an IDR of the neuronal population? One plausible mechanism is the heterogeneity in the half-activation points of individual neurons. For instance, neurons with deficient CHD8, a genetic risk factor for ASD⁵⁴, exhibit increased transcriptional heterogeneity⁵⁵ and many other ASD-associated genes relate to synapses and synaptic regulation, particularly in the GABA-ergic system⁵³. More generally, an IDR can arise from averaging different sources of neuronal heterogeneity. Deficits in synaptic scaffolding, formation or maintenance as a result of genetic mutations (for example, SHANK3, neuroligins, FMR1) could increase variability in neuronal responses. For example, multiple mouse models of ASD (SHANK3, FMR1, CNTNAP2) exhibit slower updating of priors⁵⁶, a prediction of the IDR model. Changes in receptor density (AMPA/NMDA or GABA_A) can also affect receptor occupancy⁵⁷ and thus the effective half-activation point. Similarly, genetic changes in excitatory and inhibitory neurons^6,57 could also support an effective more gradual population response. For example, averaging over heterogeneity in the E:I ratio for individual neurons has the same effect as heterogeneity in half-point activation, leading to an IDR. Thus, differences in the expression of epigenetic modulators, synapse-related genes or genetic variations in the glutamatergic/GABA-ergic system may result in increased heterogeneity and, consequently, an IDR in neuronal population responses. Other biological mechanisms for an IDR may exist at both the individual neuron and the neuronal population levels. For instance, increased inhibition strength^50,58 can yield an IDR by counteracting the bistable nature of responses from the positive feedback loop of the excitatory neurons. Changes in the electrical properties of neurons, such as membrane potential, cellular conductance and synaptic noise, can also create an IDR at the single neuron level^58,59. These proposed effects have not been the focus of previous research and therefore require further study.

Many of the behavioral and neural differences between individuals with ASD and NT individuals remain unexplained, including global–local bias, responses to cognitive biases, differences in memory encoding and decoding, savant syndrome and social interaction differences. The IDR model in its current form is a simplification that does not account for complex interactions between neuronal populations. Further computational research is needed to explore interactions between neural populations and signal and noise propagation along neural hierarchies. Future work should extend the model from single neurons/populations to hierarchical processing, integrating local signals into global scenarios and examining effects on global versus local processing¹⁴ and causal inferences³⁹. The IDR model should also be tested in animal models and brain activity, with attempts to create new models based on genetic mutations that increase neural dynamic range to see whether they generate ASD-like behavior. Another area of study is how dynamic range evolves with development or feedback and its relationship to ASD symptom severity. Despite its simplified nature, the IDR model captures many ASD-related variations and highlights a simple computational principle as its driver.

In summary, we presented a new computational model of ASD variation that posits ASD as an extreme of a natural computational trade-off between accurate encoding and fast adaptation, based on variations along the spectrum of the dynamic range of neuronal population response. This trade-off highlights the need to understand the benefits and costs of the different computation schemes. Broadly, the increased dynamic range model of ASD offers a new computational principle, opening avenues for investigating the biological, neural, behavioral and computational aspects of ASD.

Methods

All simulations were run in Python v.3.8.10 on a 64-bit PC running Ubuntu 22.04. Simulations were performed using the numpy⁶⁰ and scipy⁶¹ modules. All plots were created using matplotlib.pyplot package⁶². All simulations were run with the same random seed.

IDR better discriminates between close-by input values

We used the Hill equation:

$$A\left(S\right)=\frac{S^n}{S^n+K_m^n}$$ 7

where S is the input signal, n the Hill coefficient, which dictates the slope by which the sigmoid transitions from 0 to 1, and K_m the half-activation point, the point for which the response function is at half its full activation. We generated the sharp neuronal population response (NDR, NT like) with n = 16 and the gradual neuronal population response (IDR, ASD like) with n = 7. Both responses have a mean half-activation point of K_m = 0.5 with heterogeneity created by adding noise in the half-activation points of individual neurons, K_m,i = 0.5 + ξ_i, where ξ is sampled from a uniform distribution ξ ~ U(−σ, σ). We used σ values of, NDR: σ = 0.01 and IDR: σ = 0.175. We then calculated the encoded signal differences as A_pop(0.3) − A_pop(0.2) and A_pop(0.8) − A_pop(0.7), pop ∈ (IDR, NDR).

IDR increases the range of elevated neural variability

For each model (a gradual response (IDR) or a sharp response (NDR) or increased E:I ratio), we simulated 75 realizations of neuronal populations of N = 200 neurons, with n_individual = 16, K_m = 0.5 + U(−σ_pop, σ_pop), σ_IDR = 0.175, σ_NDR = 0.01 and σ_E:I = 0.01. For each realization, we simulated the population responses to 100 different noisy signals (with noise $\xi ~\mathcal{N}(0,0.1)$) for 200 equally spaced input signal values in [0, 1] and calculated the variance of each population activity to the same input signal value. The width of the variance curve was calculated as the difference between the input signal levels in which the mean variance crossed 1/e of the maximal mean variance. Confidence intervals (CIs) were calculated by bootstrapping the populations with a response variance that was considered and calculating the width. The bootstrap procedure was repeated 10,000 times.

We modeled the E:I response by an effective reduction of the inhibition term (ν) in the population response³⁶:

$$A_{E/I}\left(S\right)=\frac{S^n}{\nu S^n+K_m^n}=\frac{1}{\nu }\frac{S^n}{S^n+\left(K_m^n/\nu \right)}.$$ 8

This formulation of the increased E:I ratio model effectively re-scales the amplitude of the entire population response and slightly shifts the half-activation point. We used ν = 0.75 for the simulations to match Rosenberg et al.³⁶.

IDR entails slower updating rates to abrupt changes

Simulation

We simulated a time series of a noisy signal transitioning from a mean level of 0.3 (50 time steps) to a mean level of 0.7 (450 time steps), with noise $\xi ~\mathcal{N}(0,0.01)$. We then simulated the response of two neuronal populations (N = 200 neurons) to the signal (each neuron receiving a different noise realization) and tracked the mean response using optimal Bayesian updating (Kalman filter). Estimation of the parameters of a Gaussian process from a noisy measurement is optimal when the prior parameters (${\mu }_{\mathrm{prior}},{\sigma }_{\mathrm{prior}}^2$) are updated by the noisy measurement (${\mu }_{\mathrm{measure}},{\sigma }_{\mathrm{measure}}^2$) according to the following update rules:

$$\begin{array}{c}{\mu }_{\mathrm{posterior}}=\frac{{\sigma }_{\mathrm{measure}}^2}{{\sigma }_{\mathrm{prior}}^2+{\sigma }_{\mathrm{measure}}^2}{\mu }_{\mathrm{prior}}+\frac{{\sigma }_{\mathrm{prior}}^2}{{\sigma }_{\mathrm{prior}}^2+{\sigma }_{\mathrm{measure}}^2}{\mu }_{\mathrm{measure}}\hfill \\ {\sigma }_{\mathrm{posterior}}^2=\frac{{\sigma }_{\mathrm{prior}}^2{\sigma }_{\mathrm{measure}}^2}{{\sigma }_{\mathrm{prior}}^2+{\sigma }_{\mathrm{measure}}^2}\hfill \end{array}.$$ 9

The higher the variance of the measurement relative to the prior variance, the lower the impact the measurement has on the posterior—reflecting a lower confidence in the accuracy of the measurement owing to its noise. After each measurement, we calculated equation (9) and set ${\mu }_{\mathrm{prior}}={\mu }_{\mathrm{posterior}},{\sigma }_{\mathrm{prior}}^2={\sigma }_{\mathrm{posterior}}^2$. We assumed that there is a finite precision to the input signal, meaning that the variance has a lower bound cutoff that signifies the inherent noise in the estimation. One realization of the variance cutoff comes from assuming that the signal may change its mean value at some point of the tracking process. To account for the possibility of a change in the mean, a non-zero minimum variance cutoff for ${\sigma }_{\mathrm{posterior}}^2$ is employed. A natural way to implement this is to add to the model a small probability p ≪ 1 for re-setting the variance of the posterior to a higher value, ${\sigma }_{\mathrm{new}}^2$. The reset to higher variance renders the prior uncertain, allowing new evidence to affect it¹⁷. Adding this reset probability into the model yields:

$${\widehat{\sigma }}_{\mathrm{prior}}^2=\left(1-p\right)\times {\sigma }_{\mathrm{prior}}^2+p\times {\sigma }_{\mathrm{new}}^2.$$ 10

We set ${\sigma }_{\mathrm{new}}^2$ to $\frac{1}{12}$, which is the variance of a uniform distribution over the range [0, 1] and P = 10⁻⁷ for the simulations. Note that, as the probability of a variance reset p or the value of the new variance ${\sigma }_{\mathrm{new}}^2$ decreases, the sharper the prior distribution is and consequently the update of the noisy signal becomes slower (Supplementary Fig. 12). We calculated μ_measure as the mean response of the neuronal population and ${\sigma }_{\mathrm{measure}}^2$ as the variance between the responses of the neurons in the population, divided by the number of neurons (N = 200). We simulated each neuronal population with different heterogeneity levels, where K_m is uniformly drawn from $0.5\pm {\sigma }_{K_m}$: for the sharp response (NDR): ${\sigma }_{K_m}=0.01$ and for the gradual response (IDR): ${\sigma }_{K_m}=0.175$. Estimated mean values were initialized as the measured population response at time 0 and the estimated noise was initialized at 1. We repeated this procedure for 500 different population realizations and calculated the response time as the number of time steps required after the abrupt change to reach 95% of the new signal level.

Data fitting

The data for the tapping task were acquired from the publicly available repository¹⁸ and imported into a customized Python script. Subsequently, the change dynamics of individuals were computed. For each individual, we identified and averaged all instances of a specific step change in metronome frequency, considering two metronome ticks before the change and seven metronome ticks after the change. Similar to the original study¹⁸, instances involving omitted responses were excluded, as well as dynamics consisting of three or fewer instances. Change dynamics that failed to exhibit tracking were excluded from the fitting process. Tracking change dynamics were defined as successfully passing 500 ms in the correct direction at least one step after the change occurred. Group-level change dynamics were obtained by averaging the individual change dynamics.

To align the signal range in the task ([455, 545]) with the signal range of our model ([0, 1]), we mapped the highest (545 ms) and lowest (455 ms) metronome tempos to 0.8 and 0.2, respectively, with 500 ms being assigned to 0.5. Thus, the transformation of the tempos to signals is given by:

$$f\left(x\right)=0.5+0.3\times \frac{x-500}{45}.$$

We note that changing the mapped range of signal encoding alters the fitted Hill coefficients, albeit preserving the difference between ASD and NT Hill coefficient fit results (Supplementary Fig. 25).

To track the signal, we adopted the Kalman filter formulation described in equation (10). For each possible value of the Hill coefficient n_i, we simulated the tracking of the encoded signal by considering a specific step size and direction (accelerating or decelerating). The tracking process involved a population of 200 neurons, as detailed earlier, with individual Hill coefficients of 20, and the heterogeneity level in the half-activation points corresponding to the average population response with Hill coefficient n_i. To simulate noise in perception, Gaussian noise ($\mathcal{N}(0,0.03)$) was added to the pure signal. Our customized fitting procedure involved exploring potential population-level Hill coefficients within the range [2, 19] in a coarse-to-fine manner for the IDR model and in the [14, 19] range for the increased E:I ratio model. Inhibition level strength was constrained across [0.1, 1]. The fitting aimed to minimize the mean squared error between the change dynamics and the perceived change, starting from one step after the change in metronome tempo occurred.

The resolution of inference points in the simulated tracking was dictated by Akima interpolation with Scipy⁶¹ to interpolate 49 timepoints between the measured steps. This fitting procedure was applied to both the group-level change dynamics and the individual-level change dynamics. Consequently, each individual was associated with a maximum of six Hill coefficients, corresponding to each direction of signal change (accelerating or decelerating) and step size (50, 70 and 90 ms). The fitted Hill coefficient for an individual was determined as the median of these values.

IDR induces slower dynamics in a binocular rivalry task

We simulated 1,000 time steps of a Gaussian random walk starting at 0.5 with σ² = 0.03, clipping the signal to [0, 1]. We then simulated two populations of 200 neurons, one representing a gradual response (IDR) and one a sharp response (NDR), with heterogeneity levels of the half-activation point σ_IDR = 0.175, σ_NDR = 0.01, with K_m,i = 0.5 + U(−σ_pop, σ_pop). Next, we calculated the neuronal population response to the signal over time. We treated responses >0.8 and <0.2 as pure states and intermediate responses as mixed states. Transitions were defined as a switch from one pure state to the other. We repeated this process 500× for different realizations of signal noise and half-activation point values in neuronal populations (see also Supplementary Fig. 27 for the analysis of the effect of the threshold values on the results).

IDR increases detection thresholds in motion coherence tasks

The LCA model follows the following dynamics for two neural populations, y₁, y₂:

$$\dot{y_1}=f\left(0.5+I\right)-{\kappa }_1{y}_1-{\beta }_{1,2}{y}_2+{\xi }_1$$ 11

$$\dot{y_2}=1-f\left(0.5+I\right)-{\kappa }_2{y}_2-{\beta }_{2,1}{y}_1+{\xi }_2$$ 12

with κ_i being the leak parameters, β_1,2 the inhibition of y₂ on y₁, β_2,1 the inhibition of y₁ on y₂ and ${\xi }_i~\mathcal{N}(0,0.32)$ are noise terms, the signal input is I ∈ [0, 0.06] and f(⋅) is the Hill equation (equation (4)). For a specific simulation, y₁ and y₂ are initialized at 0 and the equations evolve for T steps, using the Runge–Kutta approximation. The chosen direction is determined by the first neuronal population that passes the threshold, θ > 0. We note that, for the Hill equation encoding function, with signal encoded as some distance I from 0.5 and when the leak and inhibition parameters are set equal for both accumulators, the dynamics can be re-written as a single equation describing the difference between the two accumulators, Δy = y₁ − y₂:

$$\begin{array}{c}\hfill {\underbrace{\dot{y_1}-\dot{y_2}}}_{\dot{\mathrm{\Delta }y}}=2f\left(0.5+I\right)-1-\kappa {\underbrace{\left(y_1-y_2\right)}}_{\mathrm{\Delta }y}+\beta {\underbrace{\left(y_1-y_2\right)}}_{\mathrm{\Delta }y}+\xi \hfill \\ \hfill \dot{\mathrm{\Delta }y}=2f\left(0.5+I\right)-1+\left(\beta -\kappa \right)\mathrm{\Delta }y+\xi \hfill \end{array}$$

where $\xi ~\mathcal{N}(0,{\sigma }_{{\xi }_1}^2+{\sigma }_{{\xi }_2}^2)$. Using the first-order approximation for $f\left(S\right)=\frac{S^n}{c{S}^n+0.5^n}$ around 0.5, where S is the signal input and c is the inhibition strength parameter, we get:

$$\begin{array}{ccc}\hfill f\left(0.5+I\right)& \hfill =\hfill & f\left(0.5\right)+I{f}^{\prime }\left(0.5\right)\hfill \\ \hfill f^{\prime }\left(S\right)& \hfill =\hfill & n\frac{0.5^n{S}^{n-1}}{{\left(c{S}^n+0.5^n\right)}^2}\hfill \\ \hfill f\left(0.5\right)& \hfill =\hfill & \frac{1}{\left(c+1\right)}\hfill \\ \hfill f^{\prime }\left(0.5\right)& \hfill =\hfill & \frac{2}{{\left(c+1\right)}^2}n\hfill \\ \hfill \Rightarrow f\left(0.5+I\right)& \hfill \approx \hfill & \frac{1}{\left(c+1\right)}+\frac{2}{{\left(c+1\right)}^2}nI.\hfill \end{array}$$

Combining this first-order approximation with the dynamics of the accumulator difference yields:

$$\dot{\mathrm{\Delta }y}=\frac{2}{\left(c+1\right)}-1+\frac{4}{{\left(c+1\right)}^2}nI+\left(\beta -\kappa \right)\mathrm{\Delta }y+\xi$$ 13

and for c = 1 (regular inhibition strength) we get:

$$\dot{\mathrm{\Delta }y}=nI+\left(\beta -\kappa \right)\mathrm{\Delta }y+\xi .$$ 14

Therefore, the velocity by which the difference between accumulators is driven is proportional to the Hill coefficient n (making IDR slower than NDR) and inversely proportional to the inhibition strength c (rendering decreased inhibition scenarios to be faster because c < 1).

Figure 5b was simulated using the following parameters:

$$\begin{array}{c}{n}_{\mathrm{IDR}}=8,n_{\mathrm{NDR}}=16,K_m=0.5,{\beta }_{1,2}={\beta }_{2,1}=0.25,\hfill \\ {\kappa }_1={\kappa }_2=1,\theta =0.51\hfill \end{array}.$$ 15

For each signal value, 200 simulations were performed and the percentage of correct responses was calculated. This was repeated for three different maximum simulation times: 200, 400 and 1,500 steps. To obtain signal detection thresholds, we chose the first signal value that produced a percentage of correct responses >80%. To transform the signal values to percentage of coherently moving dots, we divided by the signal value that is encoded as 98% of the maximal value, which is S_max = 0.158. We added 12% of S_max to the computed threshold levels, because, in the theoretical simulations of the model, no matter how small the signal is, it is always detected, whereas human behavior requires some minimal level of coherence for detection, even at long stimulus durations¹².

IDR changes the encoding scheme and reduces total capacity

To calculate the encoding capacity, we treated the response function as a normalized firing rate of a Poisson neuron. We denoted the mean firing rate of a Poisson neuron by the function $f\left(\theta \right)$ and derived an equation relating the Fisher information to $f\left(\theta \right)$ and its likelihood:

$$P\left(n\mid \theta \right)={\mathrm{e}}^{-f\left(\theta \right)}\frac{{\left[f\left(\theta \right)\right]}^n}{n!}$$ 16

The Fisher Information of this likelihood is

$$\begin{array}{ccc}\hfill -{⟨\frac{{\partial }^2}{{\partial }^2\theta }\log \left(P\left(n\mid \theta \right)\right)⟩}_{P\left(n\mid \theta \right)}& \hfill =\hfill & -{⟨\frac{{\partial }^2}{{\partial }^2\theta }\left(-f\left(\theta \right)+n\log f\left(\theta \right)-\log n!\right)⟩}_{P\left(n\mid \theta \right)}\hfill \\ \hfill & \hfill =\hfill & -{⟨\frac{\partial }{\partial \theta }\left(-f^{\prime }\left(\theta \right)+\frac{n}{f\left(\theta \right)}{f}^{\prime }\left(\theta \right)\right)⟩}_{P\left(n\mid \theta \right)}\hfill \\ \hfill & \hfill =\hfill & -{⟨-f^{″}\left(\theta \right)+\frac{n}{f\left(\theta \right)}{f}^{″}\left(\theta \right)-n\frac{{\left[f^{\prime }\left(\theta \right)\right]}^2}{{\left[f\left(\theta \right)\right]}^2}⟩}_{P\left(n\mid \theta \right)}\hfill \\ \hfill & \hfill =\hfill & f^{″}\left(\theta \right)-\frac{{⟨n⟩}_{P\left(n\mid \theta \right)}}{f\left(\theta \right)}{f}^{″}\left(\theta \right)+{⟨n⟩}_{P\left(n\mid \theta \right)}\frac{{\left[f^{\prime }\left(\theta \right)\right]}^2}{{\left[f\left(\theta \right)\right]}^2}\hfill \\ \hfill & \hfill =\hfill & f^{″}\left(\theta \right)-\frac{f\left(\theta \right)}{f\left(\theta \right)}{f}^{″}\left(\theta \right)+f\left(\theta \right)\frac{{\left[f^{\prime }\left(\theta \right)\right]}^2}{{\left[f\left(\theta \right)\right]}^2}=\frac{{\left[f^{\prime }\left(\theta \right)\right]}^2}{f\left(\theta \right)}.\hfill \end{array}$$ 17

Plugging the Hill equation for $f\left(\theta \right)$, with θ being the input signal level, S, we arrive at a closed-form equation for the Fisher information of a given signal level S, Hill coefficient n and half-activation point K_m. We then plotted the square root of this function for S ∈ [0, 1] with K_m = 0.5 and different Hill coefficients (Fig. 6a,b).

Using Mathematica 12 to calculate the integral of the Fisher information, we get:

$$\frac{1}{2}\left(\frac{1+n+K_m^n\left(1+2n\right)}{{\left(1+K_m^n\right)}^2}+\frac{K_m^{-n}{\left(1+n\right)}_2{F}_1\left[1,\left(-1+n\right)/n,2-1/n,-K_m^{-n}\right]}{-1+n}\right).$$ 18

Within the relevant range of n values, the second term is negligible and we are left with an expression that scales linearly with n.

Utilizing the integral in equation (18), we obtained Hill coefficient values corresponding to the 95% HDI of total encoding capacities in the publicly available dataset provided by Noel et al.²⁷. We then generated an equal number of Hill coefficients within this HDI for both the ASD and NT groups. Subsequently, we calculated the resulting total Fisher information for each Hill coefficient.

Heterogeneity in half-activation points may cause IDR

We simulated the response of a population of 250 neurons while changing the level of heterogeneity of the half-activation points around K_m with U(−σ_IDR/NDR, σ_IDR/NDR), with n_individual = 16, $K_m=0.5\pm U\left(-{\sigma }_{\mathrm{pop}},{\sigma }_{\mathrm{pop}}\right)$, σ_IDR = 0.175 and σ_NDR = 0.01 to signal levels in the range of [0, 1]. The dynamic range was calculated as the difference between the input signals that elicited a population response of 0.1 or 0.9. The ratio R was calculated as the ratio between the two: $\frac{S_{0.9}}{S_{0.1}}$, where S_x is defined by A_pop(S_x) = x.

IDR increases events of spontaneous activation

We simulated a network of N = 50 neurons with an all-to-all connectivity matrix, J, and with self-connections:

$$J=\frac{1}{N}\times \left[\begin{array}{ccc}\hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \end{array}\right]$$

with the following dynamics:

$$\tau \dot{\mathbf{A}}=-\mathbf{A}+g\left({\left[\mathbf{J}\cdot \left(\mathbf{A}+\xi \right)\right]}_0^1\right)$$

where A is the vector of neuronal responses, g a vectorized Hill equation with K_m values uniformly realized around 0.5, n = 16, $\xi ~\mathcal{N}(0,\sigma )$ and τ = 0.2 is the characteristic timescale. We then ran the dynamics for 6 s using the Euler method with Δt = 0.05. We ran this simulation for 2,000 different neuronal populations with different levels of signal noise (σ) and heterogeneity in half-point activation, each with a different noise realization. We considered a population to be activated if, at any point of the simulation, the population activity was >0.9.

Two suggested drivers for the literature heterogeneity

Simulations were performed according to the parameters described in the main text for each simulation. For each pair of mean Hill coefficients corresponding to an NDR and an IDR group, and each level of variability of Hill coefficients within each group, 3,000 heterogeneity levels of K_m values of neurons in a neural populations were sampled, corresponding to 3,000 effective Hill coefficients of different individuals. Each simulation (binocular rivalry, belief updating and motion coherence) was run for each individual and the results were used for the power analysis procedure. For the power analysis procedure, for each group size, 10,000 different samples with replacement were taken from each group. Then, a statistical test was performed and its P value was recorded. The percentage of P < 0.05 was calculated as the power of the hypothetical experiment with the corresponding group size.

A comparison with existing computational models of ASD

Hierarchical Gaussian filter simulations

We simulated a binary hierarchical Gaussian filter⁶³ (HGF) with perceptual uncertainty (using the tapas implementation). NDR parameters were retrieved using the default configuration of the model (‘tapas_hgf_binary_pu_config’) and ‘tapas_bayes_optimal_binary_config’ observation model for an input signal (u) consisting of 160 steps of ‘0’ followed by 160 steps of ‘1’. The ‘NDR’ parameters fitted for this model had a perceptual uncertainty (PU) of α = 0.1341. Responses from a model with an increased PU, akin to the gradual neuronal response (IDR; Fig. 2), were done by increasing the PU to 0.6 and simulating the response of the model to the input u.

Encoding capacity calculation for the divisive normalization with decreased inhibition model

Encoding capacity was calculated as the Fisher information for a Poisson neuron with $f\left(S\right)=\frac{S}{cS+\nu }$ mean firing rate. Following equation (17), the Fisher information for this encoding function is $\frac{{\nu }^2}{S{\left(cS+\nu \right)}^3}$. We used ν = 0.5, S ∈ [0.1, 1] and c ∈ 0.6, 0.7, 0.8, 0.9, 1.0.

Motion coherence with LCA simulations for the divisive normalization with decreased inhibition model

The LCA model was simulated with the same parameters as in equation (15), with an encoding function $f\left(I\right)=\frac{I}{cI+\nu }$ and ν = 0.5, c ∈{ 0.75, 1}, I ∈ [0, 0.4]. Signal levels were normalized to the percentage of coherently moving dots by dividing by 0.5. We note that the change in input signal range was for visualization purposes. According to the dynamics of change in equation (13), we saw that information was integrated faster when inhibition was decreased: for c₁ < c₂, we get $\frac{nI}{c_1}>\frac{nI}{c_2}$. Different simulation parameters change the percentage of correct responses, but keep the trend of faster dynamics for decreased inhibition (Supplementary Figs. 44–46).

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Online content

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41593-024-01800-6.

Author contributions

O.W. developed the model, coded the simulations, produced the graphs, fitted the data and wrote the manuscript. Y.H. conceived the model, developed the model, supervised the work and wrote the manuscript.

Peer review

Peer review information

Nature Neuroscience thanks Jean-Paul Noel, Laurie-Anne Sapey-Triomphe and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Data availability

No new data were collected for this publication and the data used are publicly available from the original authors. The simulations, data fitting and statistical analysis are available on Github. We used two publicly available datasets: tapping synchronization task (ref. ¹⁸) and orientation reproduction (ref. ²⁷). Source data are provided with this paper.

Code availability

Code, figures and supplementary figures can be found on Github:https://github.com/ComDePri/IncreasedDynamicRange.

Competing interests

The authors declare no competing interests.

Extended data

is available for this paper at 10.1038/s41593-024-01800-6.

Supplementary information

The online version contains supplementary material available at 10.1038/s41593-024-01800-6.