Experimental protocol.

Fifteen human subjects with normal hearing listened to sequences of repeating ABA₋ triplets and were instructed to continuously report their ongoing percept by selectively pressing one of two different buttons on a keypad. Subjects began reporting their percept typically 2 s after stimulus onset as integration (I; a single, coherent stream ABA₋ ABA₋) or segregation (S; two distinct streams A₋ A₋ A₋ A₋ and ₋B₋₋₋ B₋). Stimuli were sequences of triplets ABA₋ that consisted of alternating high (A) and low (B) pure tones followed by a 125 ms silent pause “₋” (Fig 1A and 1B). In total, triplets were 500-ms in duration and were repeated 60 times per trial. Tones were separated in frequency by DF semitones chosen from three conditions (DF = 3, 5, 7) with each condition being presented five times per experimental block (nine blocks total). This resulted in group data (from 15 subjects and 45 trials per subject) with 675 30-s trials for each of three DF values.

Behavioral task performance.

For each DF condition the buildup function was constructed by computing the probability of segregation from trial-averaging (Fig 1B and 1C). The buildup functions started at zero and increased over time before stabilizing to certain DF-dependent asymptotic values, similar to reports by [1, 3, 5, 12]. They started at zero due to the latency period (when no percept was identified) and not because the initial percept was I; see Methods, also [4]. While I first percepts were indeed more likely, S first percepts were reported too. The proportion of segregation as initial percept increased with DF from 103 out of 675 trials at DF = 3 to 137 at DF = 5 and 220 at DF = 7. The probability of segregation increased faster and reached higher levels at larger DF, with transient times of approximately 16, 10, 5 s after stimulus onset and with asymptotic values 0.45, 0.6, and 0.65 at DF = 3, 5, 7 respectively.

We computed distributions of normalized phase durations for subsequent durations, separately for each DF, and found them to be gamma-like, consistent with previous results on subsequent percepts [2, 5, 6]. Herein we report that duration distributions of the first percept are also gamma-like (Fig 1C; see also S1 Fig). We used statistical bootstrapping to compute the shape parameter α of each gamma distribution (see Methods), and determined that α ≈ 2 for normalized first durations and α ≈ 2.6 for subsequent durations. The distributions satisfied the scaling property γ₁ ≈ 2CV with skewness γ₁ and coefficient of variation CV ≈ 0.7 and CV ≈ 0.6 respectively, similar to reports by [24]. For integration, first percept durations were found to be longer in the mean than subsequent percept durations (with statistical significance near equidominance; p-value of 0.0003 at DF = 3 and 0.0184 at DF = 5, right-sided Wilcoxon rank-sum test at significance level 5%). Mean durations of first I-percept were 10.9, 5.3 and 3.1 s, decreasing with DF = 3, 5, 7 (p-value of 0.0002 when comparing DF = 3, 5 and 0.0014 for DF = 5, 7). Mean durations of first S-percept were 3.5, 6.6, 8.1 s (comparisons did not produce statistically significant differences, possibly due to fewer instances of first S percepts). For subsequent percepts the means were the following: 5.4, 3.4, 3.1 s for I and 4.9, 5.2, 5.6 s for S at DF = 3, 5, 7, showing a decreasing trend for integration between DF = 3 and DF = 5 or 7.

A basic state-dependent model for evidence accumulation.

Our EVA model describes activity that accumulates and saturates at a target-level, T, just-subthreshold. The activity X_n is updated at the nth triplet according to: (1) where T < 1 (assuming a unitary threshold) and where are independent random variables (Gaussian noise of zero mean and standard deviation σ). The activity increments are state dependent and proportional to the difference T − X, with constant rate r. Accordingly, the activity X drifts towards T stochastically if 0 < r < 1. Accumulation slows with X_n near T and the activity can cross the threshold only due to noise. At each threshold-crossing X_n is reset to a value X_R taken as the initial condition for the subsequent dynamics. The time D between successive threshold crossings represents a percept duration; it equals N_D, the number of triplets between threshold crossings multiplied by the onset time from one ABA₋ to the next (500 ms).

Phenomenologically, Eq (1) accounts for the features of the behavioral data described in section Auditory triplet-streaming: the observed DF-dependent mean durations, the gamma-like shape of the distributions for first and subsequent durations, and the time course of the psychometric buildup. As an example, consider DF = 5 and take r = 0.6, which is the asymptotic, approximate value of the behavioral buildup, near equidominance (Fig 1C; red curve). With initial and resetting conditions X₀ = 0.7 and X_R = 0.6, and with parameter settings T = 0.9, σ = 0.085, we simulated Eq (1) 675 times. The computed distribution of first integration normalized durations and the corresponding trial-averaged buildup function (Fig 2; right panels) are in agreement with Fig 1C and 1D.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. A basic state-dependent model for evidence accumulation yields percept durations that are gamma-like distributed and with mean values similar to those observed in behavioral data.

To demonstrate the robustness of the model results and dependence on parameter values we simulated Eq (1) with various values for target T and noise level σ. Shown for DF = 5 with r = 0.6: (Left) Two-parameter response diagram of the first I-percept with respect to T and σ. There is no switch for very small noise levels (na; gray area). Threshold-crossing activity appears with increased noise and leads to percept durations that are distributed according to normal distributions (region N), gamma-like distributions (region G between the black dashed and black solid curves), or exponential distributions (region E). Parameter values that lie on the sheets of black and gray dots yield numerically generated first integration mean durations within one and two standard deviation(s) of the experimental mean. (Right) Insets are shown for T = 0.9, σ = 0.085 (black diamond in the diagram): computed distribution of first integration normalized durations and the early phase of numerical buildup obtained during one simulation run of Eq (1) are in agreement with behavioral data. For simplicity, the drift rate r was kept constant to 0.6 between all threshold crossings.

https://doi.org/10.1371/journal.pcbi.1008152.g002

To demonstrate the robustness of the model results and dependence on parameter values, we simulated Eq (1) with various values for target T and noise level σ. The simulations took into account the latency period during each trial, and the proportion of first percepts reported as integration and segregation, during the behavioral experiment at DF = 5 (as in Switches and resetting conditions, in Methods). In this way we could assign a “percept”-type identity label to each event between consecutive resets. For any fixed T we found no threshold-crossings when σ was small. Alternations between “percepts” occurred only as σ increased, with dominant durations distributed as follows (e.g. for first I-percept; see Fig 2): normal distributions (region labeled N), gamma-like distributions with shape close to that found experimentally (region G) and exponential distributions (region E), respectively. The values of σ and T for which numerically generated mean percept durations were within one and two standard deviation(s) of the experimental mean were also calculated (Fig 2; sheets of black dots and gray dots). There is, therefore, a region in the parameter space where critical statistical properties of the behavioral data can be reproduced.

Linking neural data with behavioral data in the EVA framework.

As in Micheyl et al. [1] we seek to relate perceptual buildup and bistability reported by human subjects during triplet-streaming to animal neural data. Spiking activity evoked by the B-tone was recorded from tone-A-selective neurons in macaque primary auditory cortex, A1 [1]. At any triplet position in the ABA₋ sequence the mean spike counts, m_DF, decreased with increased DF. The time course of m_DF exhibited fast adaptation and stabilized by the third triplet [1, (Fig.3 in Ref)].

For modeling we assume that individual spike counts are Poisson distributed with means m_DF, and that m₃ > m₅ > m₇ for DF = 3, 5, 7. The responses of A1 neurons will be processed by downstream neurons whose responses at each triplet then feed into the EVA accumulator. Suppose that N_in A1-neurons activate a neuronal unit downstream if the mean input exceeds a threshold C_th (2) where H(⋅) is a Heaviside function. Such a sampler neuron is binary, taking a value u of 0 or 1 with probabilities p and 1 − p, for triplets after a brief transient phase of adaptation in A1. In line with the population separation hypothesis [7], when spike counts are large (so u = 1) we tag the sampler as evidence for integration; likewise when spike counts are small (u = 0), we tag the sampler as evidence against integration. As N_in increases, the averaged spike count variability around the mean m_DF decreases inversely with (Fig 3). As such, probabilities p_DF obtained from m_DF at different DF = 3, 5, 7 vary from values being graded (when N_in is small) to values spread apart (when N_in is large), approaching extreme values of zero or one (when N_in is very large); Fig 3. A suitable variability in the A1-neuronal population can be chosen (more about this later; sections on pages 9 and 16) to ensure that probabilities p_DF at DF = 3, 5, 7 achieve the graded asymptotic values of the behavioral buildup functions (e.g. 0.45, 0.6, and 0.65 as in Fig 1C). This is an important feature of the EVA model; indeed, without adequate variability in the readout of A1 responses we cannot account for graded BUF levels.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Linking neural data with behavioral data in the EVA framework.

Individual spike counts of A1 neurons are assumed to be Poisson with means m_DF such that m₃ > m₅ > m₇ for DF = 3, 5, 7. Averaged spike counts over N_in A1-neurons, , are normal-like distributed with means m_DF and standard deviation decreasing inversely with ; shown in gray (DF = 3), black (DF = 5) and light-gray (DF = 7) for N_in = 10 (upper panel) and N_in = 100 (lower panel). At each triplet, < Spk > activates a sampler unit downstream if it exceeds a threshold C_th (solid black, vertical line). The area under the probability distribution to the left of C_th (white-dots pattern; DF = 5) determines the probability p of the sampler neuron to be inactive (0); the complementary probability 1 − p is for the sampler to be active (1). For each N_in, the threshold C_th was chosen such that p = 0.6 at DF = 5, which is the asymptotic, approximate value of the corresponding behavioral buildup near equidominance (see Fig 1C; red curve). Probabilities p obtained at different DF vary from values being graded when N_in is small (N_in = 10), to values spread apart approaching zero or one when N_in is large (N_in = 100). A suitable variability in the A1-neuronal population is key if aiming to account for graded BUF levels observed in behavioral data (Fig 1C).

https://doi.org/10.1371/journal.pcbi.1008152.g003

A question remains: How can we link the probability of a sampler unit becoming active stimulated by A1-neurons to the neuronal drive of the accumulator, r in Eq (1)? We resolve this problem by including an entire layer of binary units u as above, say N_sl total, and use the percentages p_I, p_S, as cluster sizes, of active and inactive samplers as input-drive to two accumulators: for and against integration, respectively (Fig 4A). In particular, the output p_S of the sampler layer (not binary anymore) is a stochastic process with mean p and variance p(1 − p)/N_sl (for justification, see Statistical properties of SL-activation, in Methods). Noteworthy, under this construction, the output p_S of the sampler layer (the input to the accumulator “against integration”) takes indeed values very close to r, defined as p, in Eq (1) if N_sl is large enough.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Accumulation model as feed-forward auditory network of 3 layers.

A: State of neurons at triplet t in the input layer and sampler layer of the evidence accumulation model. Input layer comprises A1 units with (triplet- and DF-dependent) mean spike counts presented in panel B. Sampler layer has N_sl = 20 binary neuronal units, either in state 1 (blue; favoring I percept) or state 0 (red; favoring S percept). Each unit samples a small number of input units (N_in = 5) and the averaged spike count across the units is compared to C_th (see panel B) to determine the unit’s appropriate perceptual state. B: Mean spike counts (scatter plot) for tone B of tone-A-selective neurons, and exponential fit (solid) of mean spike counts. These values are interpolated for our specific DF = 3,5,7 using data from cortical area A1 of awake macaque extracted from [1]. A Poisson spike count is generated using the mean value at each triplet. Asymptotic values of mean spike count (printed in parenthesis next to corresponding DF values) are used to generate spike counts after the 20-th triplet. Poisson spike counts are averaged across sets of N_in = 5 neuronal units, and the resulting values are subject to a binary neural threshold C_th (black horizontal line). The error bars indicate the standard errors of the mean spike counts. C: Accumulation layer has 2 accumulators drifting over successive triplets towards their own target values T_a and T_f where T_a > T_f. Their activities are governed by input factors from the sampler layer and stochastic factors. The noise level depends on the target (σ_a > σ_f). During a cycle, the suppressed unit accumulates evidence against the current percept. A switch to the other percept occurs when the accumulator of the suppressed unit reaches the switching threshold of 1. A new cycle starts, with accumulators reset to appropriate values, and targets values switched to corresponding perceptual states. Shown for DF = 5. For other DF values, see S2 Fig. For the complete list of parameter values, see Methods—section Parameter values used in model simulations.

https://doi.org/10.1371/journal.pcbi.1008152.g004

The EVA model.

Our proposed EVA model is structured as a three-layer network (Fig 4; see details in Methods). It takes Poisson spike counts from tone-A-selective A1-neurons (the Input Layer, IL) [1, 7, 8] and passes them through binary units in the Sampler Layer, SL (Fig 4A). Only spike counts recorded during tone B are included (Fig 4B). Each SL-unit compares the averaged spike count across a small number N_in of input units to a fixed threshold C_th and places the outcome into either state 0 (for S) or 1 (for I). High activation in IL (above C_th) is assumed to support percept I while low activation facilitates percept S (Fig 4A and 4B). The proportions p_I(t), p_S(t) (p_S = 1 − p_I) of SL-units in states 1 and 0, together with stochastic noise terms ξ_I(t), ξ_S(t), modulate the activity of the Accumulation Layer, ACC (Fig 4C). Two accumulators representing evidence for the percepts drift towards two targets. Their activities x_I, x_S are updated at discrete time steps determined by the position t of each triplet in the ABA₋ sequence. One unit accumulates evidence for the current percept (e.g. x_I during integration) in the presence of additive “neural” noise defined by a Gaussian process of strength σ_I = σ_f, and approaches target T_I = T_f. The other accumulator works against the current percept (x_S during integration). It experiences stronger noise level σ_a, and approaches another target, T_a. Differential noise levels enable the accumulator “against” to be the first to reach the threshold and initiate the switch; meanwhile, the accumulator “for” remains confined to a neighborhood of its target. In the deterministic (noise free) case, alternations between percepts are not possible given that both T_a and T_f are subthreshold targets (T_f < T_a < 1), a distinctive feature of our accumulation model. Instead, the ACC system is bistable with accumulators x_I, x_S reaching either steady state (T_a, T_f) or (T_f, T_a) depending on the initial conditions (Fig 4C, dotted lines in blue and red). In the presence of noise, however, the accumulator against the current percept reaches the decision threshold; a switch to the other percept occurs, the accumulators are reset, the targets are swapped (T_S = T_f, σ_S = σ_f and T_I = T_a, σ_I = σ_a), then another accumulation cycle begins (Fig 4C, traces for x_I, solid blue, and x_S, solid red; the percept’s type is identified by the background color, blue for I, red for S. See also S2 Fig). It is essential that the accumulators are subjected to noise in order for the distribution of threshold crossing events idealizing the percept durations to exist.

EVA model captures DF-dependence of mean durations.

Numerical simulations of the EVA model followed the experimental setup with N_tr = 675 repetitions (trials) per DF. The model-generated mean durations were computed separately for each percept type (I, S; first, subsequent) and DF. They approximated well their counterparts from behavioral data (Figs 5B and 6B). They also captured two important DF-related trends reported by other studies. First, near equidominance (DF = 3, 5) mean durations of the first I percept were found to be longer than those of subsequent I percepts [2, 6]. Secondly, mean durations for I and S showed a “cross-diagram” like behavior [6] with equidominance near DF = 5. Mean durations for I were greater than mean durations for S when DF low (DF <5), and smaller than mean durations for S when DF large (DF >5), results similar to [6]. The model was robust to noise as demonstrated by 100 Monte Carlo runs of each DF simulation that yielded consistent results in terms of average values and 95% CI (Figs 5B and 6B; error bars).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. EVA model yields realistic first percept durations.

A: Mean percept durations (top) and fitted α value (bottom) from gamma distribution of first I (blue) and first S (red) percepts from behavioral experiment for DF = 3,5,7. The error bars indicate 95% CI around the mean and are obtained from statistical bootstrapping (see Methods). The mean durations of I decrease with DF while those of S increase with DF. The shape parameters α from gamma-fit using MLE for DF = 3,5,7 are also presented here. There is no observed trend for α values. B: Mean percept durations (top) and fitted shape parameter α (bottom) from gamma distribution of first I (blue) and first S (red) states from EVA model. The error bars are 95% CI obtained from 100 Monte Carlo runs to show the robustness of the model. The mean values of duration follow the similar trend as those from experiment. Also, the shape parameters show a close resemblance to those from the experiment. Related results are included in S5 Fig.

https://doi.org/10.1371/journal.pcbi.1008152.g005

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. EVA model yields realistic subsequent percept durations and distributions.

A: Distribution of normalized subsequent percept durations (top) and other properties (bottom) from behavioral experiment. (Top) Likelihood ratio test confirms that both subsequent I (blue; N = 1642, p = 0.49) and S (red; N = 1785, p = 0.49) percepts follow a gamma distribution, shown here for DF = 5. The shape parameters α computed using MLE, and the mean durations μ are shown in the graphic. (Bottom) Mean subsequent durations for I (blue) and S (red) for DF = 3,5,7. The error bars indicate 95% CI around the mean, computed using statistical bootstrapping (see Methods). Similar to the first percept, mean durations of subsequent I and S show a “cross-diagram” like behavior [6, (Fig.9B in Ref)] with equidominance near DF = 5; the ratio between mean durations for I and S percepts changes from larger than 1 to smaller than 1 when crossing DF = 5, near equidominance. The shape parameters α from MLE for DF = 3,5,7 are also presented, and no trend for α values is found. B: Distribution of normalized subsequent percept durations (top) and properties (bottom) from EVA model. (Top) Normalized subsequent percepts are gamma distributed for DF = 5 with mean durations μ and shape parameters α shown in the figure; similar results are obtained for other DF values, see S1 Fig. (Bottom) Mean percept durations and fitted shape parameters α for DF = 3,5,7 from EVA model. Mean subsequent durations follow the same trend and the shape parameters have similar values as compared to those from the experiment. The error bars are 95% CI obtained from 100 Monte Carlo runs of the model. The result shows the robustness and consistency of the model. Related results are shown in S5 Fig.

https://doi.org/10.1371/journal.pcbi.1008152.g006

In EVA model, the switch to a new accumulation cycle occurred when the ACC-unit that accumulated evidence against the current percept reached the decision threshold. Target-against T_a, noise level σ_a, and increment rate p_S(t) determined the trajectory of the suppressed unit x_S and the length of the corresponding dominant percept I. Similarly, T_a, σ_a and p_I(t) determined the duration of percept S. We studied the effect of T_a and σ_a on the model-generated mean durations at each DF by varying their values while keeping all other parameters fixed (see Parameter values used in model simulations, in Methods). At a given T_a, EVA model exhibited no alternations if σ_a was small (Fig 7; region in gray). For moderate σ_a values, perceptual switches occurred but yielded percepts of mean durations much longer than those found experimentally (in warm colors); then, for larger σ_a, simulated durations became comparable to (in green) and then much shorter than (in cool colors) the behavioral mean durations. Similar results were obtained for σ_a fixed when varying T_a. As a general rule, the smaller the target-against, the stronger the noise level had to be in order for the accumulator’s trajectory to be pushed above the threshold and to generate acceptable statistical approximations of the data (Fig 7, in green; also black dots; within one standard error to the experimental mean, SEM).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Dependence of mean percept durations and shape of distributions on target T_a and stochastic term σ_a, in the EVA model.

Diagrams show the difference between the mean duration μ derived from numerical simulations of the model and mean μ_exp from the behavioral data, represented as ratio μ/μ_exp; see the color scheme. Results are shown for A: first and B: subsequent integration and segregation percepts at conditions DF = 3,5,7. Given a fixed value for T_a, the dynamics changes from no alternations between percepts at small σ_a (na; in gray); to alternations of mean durations much longer than the experimental mean (region in warm colors); to mean durations that approximate well the corresponding experimental values (within one standard error to μ_exp; in green; black dots depict a discrete selection of values in the green region); then, to mean durations much shorter than μ_exp (region in cool colors). In each diagram, σ_a, T_a that were used to generate model-based results are identified by a red diamond (see Methods, σ_a = 0.085, T_a varies). Besides mean durations, the shapes of the gamma-like distributions that fit normalized percept durations depend on T_a and σ_a as well (α is the shape-parameter in the gamma-fit; see Eq (3) in Methods). There are three main regions that characterize α and they are delineated by the dashed-white and solid-white curves. Low-level of noise σ_a yields normal distributions (region N, to the left of dashed-white line; α ≫ 3) while high-level of noise yields exponential distributions (region E, to the right of solid-white line; α near 1). For intermediate level of noise, the distributions are gamma-like with shape close to that found experimentally (region G, between white contours; α ≈ 2 for first percepts and α ≈ 2.6 for subsequent percepts; α differs from α_exp by relative error up to 20% except for integration at DF = 7 where it is up to 30%). The parameter range where both model-generated mean duration and shape of distribution are good approximations of their corresponding experimental observations is found at the intersection of region G with the sheet of black dots. Related results are shown in S3 Fig.

https://doi.org/10.1371/journal.pcbi.1008152.g007

The decrease in mean durations of I percepts with increasing DF (Figs 5B and 6B, blue) stemmed from the increase in probability of a sampler to support segregation (Fig 8B) which led to an increase of increment rate p_S of accumulator x_S (see Statistical properties of SL-activation, in Methods). The increasing trend of mean durations of S percept, with DF, could also be associated with the decrease of increment rate p_I of accumulator x_I. These DF-dependent properties of p_S, p_I, inherited from A1 spike counts (Fig 8A) enabled the EVA model to capture the correct qualitative trend of the experimental means across all percept-types and DF conditions. Suitable quantitative agreements were then obtained by fine-tuning the value of target-against T_a (Fig 7, red diamond; error between numerical and behavioral results was restricted to 0.1 SEM. See also S3 Fig).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. Parameter fitting for input and sampler layers in the EVA model.

The signal detection algorithm for constructing a neurometric function (the probability of a sampler to support the segregation percept) utilizes spike count time courses as shown in panel A (data extracted from [1] and interpolated for the cases DF = 3, 5, 7); see below for more detail. The behavioral buildup functions (dashed, in panel B) occupy intermediate ranges of probability of S, and show slow initial rise for DF = 3, 5. The simulated functions (solid, in panel B) do not capture the slow-rising phase of behavior buildup, and the spread between the neurometric curves increases unacceptably at larger N_in. For an optimal choice of parameters N_in, C_th, the algorithm yields well-fit asymptotic values of behavioral data. A: Mean spike counts m_t;DF are interpolated at DF = 3, 5, 7 st from data in [1], and then extended for triplets t ≤ 60; see Methods. (Note: In [1, (Fig.3 in Ref)] mean spike count data were shown for A-tone selective neurons in A1 during triplet tones at DF = 1, 3, 6, 9. They decreased exponentially and stabilized within a few seconds. Mean spike counts changed with DF only during B-tone.) Herein, spike counts during B-tone are generated using Poisson processes of means m_t;DF (DF = 3, 5, 7) and then averaged over N_in neuronal units of the input layer IL (e.g. N_in = 1, 5, 30, 100). The average values of the mean spike counts and the standard error to the mean (SEM) are computed over 675 trials. Averages, including asymptotic values (written in parenthesis, at each DF), do not change with N_in but SEM decreases with a factor of . B: The signal detection algorithm [1] generates neurometric functions using numerical data from IL-pools of N_in neuronal units; parameter C_th is chosen to yield the least-squares error of the experimental buildups and the computer-simulated neurometric functions for DF = 3,5,7. If N_in is small the neurometric curves tend to bunch together due to overlapping and large SEM regions across conditions. As N_in gets bigger, the neurometric curves are pushed apart. The best approximation to the set of psychophysical buildups is obtained for N_in = 5, C_th = 4.21.

https://doi.org/10.1371/journal.pcbi.1008152.g008

EVA-modeled first and subsequent percept durations match observations.

The model reproduces an important statistical feature of the behavioral data, the distributions of normalized durations for all first and subsequent I, S percepts at DF = 3, 5, 7. Histograms were drawn and fitted by gamma probability density functions of shape parameters α (see Eq (3) in Methods) whose values agreed with those from the behavioral experiment. The shape of distributions was tested and confirmed statistically by 100 Monte Carlo runs of the model for each DF condition separately (Figs 5B and 6B; error bars indicate 95% CI around α-mean values). Exemplar distributions for first and subsequent durations are shown in Figs 1D and 6B at DF = 5. For other DF values, see S1 Fig.

Since alternations were caused in the model by the accumulator that gathered evidence against the current percept, the distribution of threshold crossing event times depended on T_a and σ_a. In particular, for a given T_a, EVA model generated percept durations that were normally distributed for σ_a small (Fig 7, region N; α was much bigger than 3) and exponentially distributed for σ_a large (Fig 7, region E; α was close to 1). For intermediate σ_a, the distributions were gamma-like matching those fit to the observed data (Fig 7, region G; model-generated α values were similar to those determined from experiments, α_exp, at relative error up to 20-30%). The closer T_a was to the decision threshold 1, the easier it was to find σ_a that yielded gamma-like distributions. With decreasing T_a, the transition to a narrower region G was either sharp-edged (e.g. DF = 7, first I) or rather smooth (DF = 7, first S). Percept durations that approximated well both the distribution shape and the mean duration of the experimental data were obtained by using parameters from region G that overlapped with the black dotted sheet.

EVA model captures DF-dependence of stream segregation buildup.

The model-generated buildup functions captured both the rising and the asymptotic phases of the behavioral buildup for each DF = 3, 5, 7 (Fig 1D). These trends were a consequence of already having simulated percept durations and percept means well-fit to behavioral data, in accordance with previous works describing the buildup of stream segregation as a byproduct of an alternating renewal process [17].

Computational advantages of the EVA model.

The model is pseudo-neuromechanistic; it takes A1 responses as input, it allows for attractor-states, and it includes accumulators that are saturating akin to synaptic currents. The spike counts are in accordance with neurophysiological data from A1 [1] and provide input to the computation of perception dominance downstream, as in the conceptual population-separation model of [7] and in competition-based model of [6]. The model incorporates fast habituation (after one triplet or so, Fig 4B) as in [1, (Fig.3 in Ref)] and it accounts for the decrease in response amplitudes and in spatial activity patterns evoked by tone B at tone A tonotopic locations observed as DF increases [1, 7]. Indeed, if most of tone-A-selective A1-neurons are active, the model predicts a large proportion of samplers in SL to be active (p_I large) and thus favors I percept. If the opposite happens and A1 is mostly inactive, a large proportion of samplers are inactive (p_S is large) and the model favors S percept. Activation in IL decreases with larger DF (fewer IL-units have mean spike counts above threshold C_th) and so does p_I(t); Fig 4A and 4B, compare DF = 3, 5, 7; see also see also [1, (Fig.3 in Ref)] and [7, (Fig.11 in Ref)]. This affects the dynamics of the accumulators since x_I(t), x_S(t) gather evidence about percepts with increment rates proportional to p_I(t), p_S(t) respectively, while also being modulated by a certain level of noise (Fig 4C, Eqs).

The accumulators resemble discrete time versions of the leaky integrate-and-fire neuron model with conductance-based synaptic input [25], . The voltage-like variable (here, normalized, by the threshold value for switching) has a maximum amplitude of one. The reversal potential V_R is set to either target T_a or T_f depending on the type of the dominant percept and the type of the accumulator. The synaptic drive consists of feedforward input from SL and is analogous to the reciprocal of the time constant. Finally, Gaussian white noise represents input from other brain sources or internal to ACC. Its strength σ needs to reach an appropriate level for the statistics of percept durations generated by the EVA model to match the behavioral data (see Methods).

Our EVA model is data-driven. Initial conditions are set using the latency periods and the proportion of first S-percepts from the experimental data. Poisson spike counts of IL neuronal units at each triplet t and semitone difference DF are generated using mean values m_t;DF derived from macaque A1 multi-unit spiking neural data [1] (Fig 8A). Parameters N_in and C_th are obtained by least-squares fit between the probability of a sampler to support an S percept and the behavioral buildup at all three DF values (Fig 8B; see also Methods). Neuronal granularity as a suitable substrate for perceptual representations [24] is implemented through SL. The number of samplers is chosen flexibly from a wide range of values (N_sl ≥ 1; here N_sl = 20). There are few other free parameters, T_a, σ_a, T_f, σ_f, b (baseline), but only the former two are major players in fitting the model to data (as shown in sections on pages 9 and 11).

Modeling the buildup with Micheyl’s model for auditory streaming.

The signal detection algorithm of Micheyl et al. [1] (see also [26, 27]) has been extensively cited in the auditory streaming research [3–6, 12, 21] in regard to computing time-varying probabilities of stream segregation from neuronal responses in A1. The model was based on choosing a threshold number, C_th, for mean spike count (first, trial-averaged; then averaged across sampled neurons) to classify each triplet as I or S; doing this for each ABA₋ in the sound sequence generated a time course, a “neurometric” function. Briefly, for a given triplet position, a probability distribution was constructed for the B-tone responses measured at and convolved over A1 neurons whose best frequency was that of the A-tones. The area under the probability distribution to the right of C_th determined the probability that tone B was detected and, consequently, that the triplet belonged to I percept; the complementary probability was associated with S percept. The algorithm classified each triplet independently and assumed no memory among nearby triplets.

A simplified view of this procedure is to consider the distribution of trial-averaged counts for the neurons as straddling the mean for each triplet in the time course of an ABA₋ sequence [1, (Fig.3 in Ref)]. Conceptually, one chooses a level C_th that will cut across the distributions, say for DF = 3, and correspond to low probability of S for early time and correspond to the asymptotic level (from behavior) for late time (e.g. C_th, thin horizontal line, in S4 Fig). This classification could likely provide a decent fit for DF = 3 but for DF = 6 the spike counts will fall below the threshold, leading to an overestimate of probability of S. The remaining cases will yield extreme classifications: for DF = 1, spike counts for each triplet in the sequence will be above C_th and probability of S will be estimated as near zero; for DF = 9, all spike counts will be below C_th (except maybe for the initial triplet) and therefore probability of S will be near one for the time course. The spread of the behavioral time courses in [1, (Fig.4 in Ref)], one lying intermediate for DF = 3 and the others at very low or quite high levels, provided an opportunity for reasonable fitting with a single C_th level [1]. For details on numerical fitting with such signal detection algorithm, see S4 Fig.

In the case of our data, the conditions were DF = 3, 5, 7 and the behavioral buildup functions lay in more intermediate levels and clustered around the asymptotic value of 0.5 (Fig 1C; also Fig 8B, dotted lines). It thus became challenging to fit the buildup functions (especially the early, slower rising portions for multiple DF values, 3 and 5 semitones) using Micheyl’s model with a unique C_th.

We attempted to meet this challenge by applying the signal detection algorithm to our behavioral data while using interpolated and Poisson distributed spike counts based on the neural data from [1]. The approach was equivalent to the computation of the probability of a sampler to support segregation from the EVA model, observing only the input layer, IL, and passing its output through one single sampler, N_sl = 1. While the mean spike counts over a pool of N_in A1-neurons did not change significantly with N_in, the standard error to the mean did (Fig 8A). The decrease in the spike count error to the mean made the horizontal line C_th intersect fewer local distributions and biased the data-fitting towards the behavioral curve for a certain DF, at the expense of others. Choosing more A1 neuronal units (larger values of N_in; Fig 8A) led to larger spread in the simulated neurometric functions and poorer fitting (Fig 8B; at DF = 5, 7 the neurometric functions (solid lines) plateaued at probability approximately 1 after triplet-sequence onset, as N_in increased; e.g. case N_in = 100). The best approximation of the asymptotic levels of the behavioral buildup for all DF conditions was found at a relatively low N_in however the rising transients of the neurometric functions were still much faster than in the experiment.

Modeling percept durations with Micheyl’s model.

We extended the work from [1] by using the signal detection algorithm to generate “percepts” and characterize their distributions. For each DF, adjacent triplets of the same type (I or S) were grouped together to create percept phase durations and construct frequency graphs (Fig 9). Theoretical calculations showed that subsequent percept durations generated by Micheyl’s model were exponentially distributed, as opposed to gamma-like. During subsequent durations we could assume that the buildup functions of stream segregation had reached an asymptotic level p (Fig 9A and 9B, upper panel) and that the activity in the A1 pool was independent at each triplet. Then the probability that percept S consisted of n-triplets could be calculated as Prob(D_S) = pⁿ(1 − p) = (1 − p)e^{n ln p}, depending exponentially on n. Likewise the probability that I consisted of n-triplets was Prob(D_I) = p(1 − p)ⁿ = pe^{n ln(1 − p)}. Similar results were obtained from numerical simulations of Micheyl’s model. The probability density functions were found to be discrete versions of exponential curves and the mean durations were small at about 1 s (Fig 9A and 9B, middle and lower panels), suggesting that the signal detection algorithm is not appropriate to describe perceptual alternations and percept durations, key aspects of bistable stream segregation. (Compare to Fig 1C; also S1 Fig and [2, 5, 6]).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. Signal detection algorithm adapted from [1] yields exponential distributions and unrealistic mean durations of percepts.

(Top) Binary threshold C_th is chosen to yield the least-squares error between neurometric buildups (solid) and behavioral buildups (dashed) at DF = 3,5,7. Poisson spike counts are averaged across a sample of N_in = 5 neuronal units and compared to C_th to classify a triplet either as I or S. Trial-averaging the S-tagged responses produces the neurometric functions. The threshold value is determined by least-squares fit for A: the first 15 seconds of the stimulus to match the transients, or for B: the last 15 seconds of the stimulus to match the asymptotic level of the behavioral buildup; See also Fig 8. (Bottom) Trial-by-trial applications of the signal detection algorithm from [1] with A: C_th = 4.01 and B: C_th = 4.21 yield exponentially distributed subsequent percept durations for I (blue) and S (red). Their mean values μ are significantly smaller than those reported in the experiment. Note: Same parameter values N_in = 5, C_th = 4.21 were used in the EVA model for activation of the sampler layer SL (Fig 4A and 4B) and obtain gamma-like distributions of percepts (Figs 1D and 6B).

https://doi.org/10.1371/journal.pcbi.1008152.g009

Participants.

Fifteen human subjects with normal hearing (8 female and 7 male; ages 18-45 yrs.; median 22 yrs.) were included in the behavioral study. They listened to sequences of repeating ABA₋ triplets and were instructed to continuously report their ongoing percept by selectively pressing one of two different buttons on a keypad. Subjects began reporting their percept typically 2 s after stimulus onset as integration (I; a single, coherent stream, the galloping pattern ABA₋ ABA₋) or segregation (S; two simultaneous distinct streams A₋ A₋ A₋ A₋ and ₋B₋₋₋ B₋); Fig 1A and 1B.

Stimuli.

Stimuli were 30-s long sequences of triplets ABA₋ that consisted of alternating high (A) and low (B) pure tones gated with 10 ms raised cosine ramps and followed by a 125 ms silent pause “₋”; Fig 1A. In total, triplets were 500-ms in duration and were repeated 60 times per trial. Tones were separated in frequency by DF semitones chosen from three conditions (DF = 3, 5, 7) with each condition being presented five times per experimental block. To prevent habituation to a certain frequency, for each DF the tones were generated by roving through variants of frequencies taken 0, ±1 or ±2 semitones apart from their geometric mean (middle pair in the list below); see also [4]. Frequencies (f_A, f_B), in Hz, were chosen as: (494, 415), (523, 440), (554, 466), (587, 494) or (622, 523) Hz at DF = 3; (523, 392), (554, 415), (587, 440), (622, 466), (659, 494) at DF = 5; and (554, 370), (587, 392), (622, 415), (659, 440), (698, 466) at DF = 7. Stimuli were digitally generated in Matlab at 48 kHz sampling rate and were delivered through earphones in a soundproof isolated room. Subjects had the sound volume adjusted to their comfortable hearing level.

Experimental protocol.

Each subject performed 9 experimental blocks. Each DF condition was randomly presented 5 times per experimental block, using different combinations of frequencies for tones A and B, without repetition (see Stimuli). A Latin square design was used to determine the order of presentation of each condition in each block. This resulted in group data with 675 30-s trials for each of three DF values. The frequency separation values (DF = 3, 5, 7) were chosen to fall within the range of ambiguity of the van Noorden diagram in which listeners can perceive both integration and segregation [10, 40]. All subjects underwent a training session in which they were given verbal explanations and auditory illustrations of the two possible percepts, and they practiced distinguishing between them. Then, during the recording session, listeners were instructed to press and hold one key on a keypad when they perceived stimuli as I, and to release it while pressing another key when they perceived stimuli as S, and so on. They were encouraged to respond as soon as they heard the change in percept. The key-response data were converted to binary vectors with value 0 assigned to I (and to the latency period defined as the time before identification of either percept) and 1 to S, for further analysis. Experiments were performed in a dedicated soundproof booth in the Human Brain Research Laboratory, Neurosurgery Department at The University of Iowa. Written informed consent was obtained from all subjects. Research protocols were approved by the University of Iowa Institutional Review Board.

Build-up functions and the latency period.

The time course of S percept after stimulus onset was computed from key-pressed data, 0 for I and 1 for S. Those were sampled at 1 ms to create vectors of binary values corresponding to appropriate percept type at a particular time instance. At each DF condition, binary vectors were averaged across 675 trials to obtain the build-up function of S; bootstrapping was also used to compute the 95% confidence interval (CI) around the mean (Fig 1B and 1C). The time course of I was computed with the same procedure but over key-pressed data labeled as 1 for I and 0 for S. At a particular time t, the proportion of trials classified as I, S or neither (during the latency period/the first few seconds after stimulus onset) were p_int(t), p_seg(t) and p_Lat(t), and summed up to p_int(t) + p_seg(t) + p_Lat(t) = 1. The first percepts were typically I. However, for larger values of tone frequency separation, subjects tended to report S as first percept more often which led to an increase in p_seg(t).

First durations and subsequent durations.

All complete percept durations across trials and conditions were included in the behavioral analysis. Unfinished percepts and button presses recorded after the end of stimulus presentation were discarded. For each DF = 3, 5, 7, the statistics was evaluated over four subsets of data, separately: first I, first S, subsequent I and subsequent S. For each DF and each of these four percept types, the mean dominance duration was computed in two steps: first, it was computed per subject, say μ_i for subject i = 1, …, 15; then the mean duration μ of the group data was defined (and reported) as the unweighted average across all subjects, μ = (μ₁+ μ₂ + ⋯ + μ₁₅)/15 (e.g. Fig 1C; mean μ is shown for first duration distributions at DF = 5). By this approach, any potential bias of the calculation towards fast switchers who might contribute more durations to the pool and concurrently spend less time in a particular percept, was mitigated. Error bars at 95% CI of the mean were also determined; error bars corresponded to 1.96 SE; standard error SE was computed from the standard deviation std_exp over the group of means μ_i. For analysis of grouped data we used subject-specific normalized (by individual subject mean) percept durations as follows: at each DF condition and each percept type (first/subsequent, I/S), any raw percept duration D of subject i was normalized by the corresponding mean μ_i to . Histograms of normalized phase durations for each DF condition and percept type were computed and fit by gamma distributions with density functions (3) Mean was well-fit to 1 due to normalization. Then the coefficient of variation and the skewness depended exclusively on the shape parameter α. If α was large then Eq (3) was equivalent to a normal distribution. If α ≈ 1 then Eq (3) was equivalent to an exponential distribution. On the other hand, for α ≈ 2 (as observed for first durations in behavioral data) and α ≈ 2.6 (as observed for subsequent durations), the distributions satisfied the scaling property γ₁ = 2CV with CV ≈ 0.7 and CV ≈ 0.6 respectively. The latter case was similar to the findings of [24] that described the statistics of percept durations for other examples of perceptual bistability.

Distribution testing of phase durations.

The fitting of the experimental (and numerical) data was obtained by calculating the values of α and with the Maximum Likelihood Estimation (MLE) algorithm. The goal was to determine α and that yielded the maximum product ∏_k y_k of all y_k gamma-likelihood values of the normalized percepts counted by index k for each run of the experiment. This was equivalent to maximizing the log-likelihood based on Eq (3). The optimization of LL was implemented numerically with MATLAB function fminsearch. Distribution testing on normalized durations was done by statistical bootstrapping. We generated 10000 bootstrapping sets of gamma distributions with fitted parameters α and and constructed the distribution of maximum log likelihood values for those sets. The test statistics LL was compared to this distribution to obtain the probability of log likelihood to be less than LL (the p-value). The normalized durations were well fit by a gamma distribution with the optimal values α and (as null hypothesis) at significance level 0.05 if p-value ≥ 0.05.

Statistical analysis of model-generated data.

The histograms of first and subsequent durations I and S in trials generated by the model (see below), and their fitting by gamma distributions, were computed in a similar manner as for the experimental data. Likewise, build-up functions for the model were constructed as those for the behavioral data.

Input layer (IL).

The IL-units were assumed to be tone-A selective neurons from primary auditory cortex (A1) as described in [1]. The averaged (over trials) spike counts m_t,DF of the IL-units were derived from data (see section Data-driven parameters for EVA model) and depended on the position t of the triplet in the ABA₋ sequence (t = 1, …, 60 for a 60-triplet long stimulus; 30 s in duration) and on the semitone difference DF. The model was simplified by focusing only on the spike counts during the B-tone presentation at A-tone selective neurons in A1. As reported by multi-unit recordings in monkeys, such A1-neurons adapted strongly and rapidly during presentation of triplet-repeating auditory sequences [1, total of 91 neurons]. Temporal correlations between the means of an A1-neuron from triplet to triplet were captured in the model by the trend of m_t,DF that decreased exponentially with t (Fig 4B). Trial-to-trial variability of the dynamics of IL units as well as unit variability in IL during a single trial were implemented using Poisson point processes. (We used this approach because we could extract mean spike counts from published data of [1] but did not have access to the original spike times.) For an A1-neuron with mean spike count λ = m_t,DF we supposed that its instantaneous spike count k (k = 0, 1, 2, …) at triplet t and condition DF, was randomly generated from a Poisson distribution with probability P(X = k) = λ^k e^−λ/k!. The Poisson spike counts are generated independently for each neuronal unit in IL, each triplet and each semitone difference condition. Note that the model could be generalized by assuming neuronal heterogeneity, with averaged spike counts at neuron j chosen from a normal distribution of mean m_t,DF and standard deviation s_t,DF derived from [1]. However, the impact of heterogeneity on the model’s outcome would be negligible given that EVA model was formulated to use mean spike counts over IL neuronal pools as input rather than mean spike counts of individual neurons (see below).

Sampler layer (SL).

The SL-units were tasked with summating and classifying spike counts from subsets of N_in IL units (A1-neurons). Consider that a trial of length N_t (N_t = 60 triplets) during a certain DF condition was simulated by EVA model: each sampler summed the input of a pool of N_in neuronal units from IL; weighted by N_in, this gave the mean spike count for B-tones of the corresponding pool of A-tone selective A1-neurons, for triplet t; then was compared to a DF-independent, fixed, neuronal threshold C_th. If the averaged spike count was large () then the subset activity was high and the pool was assumed to support, for this triplet, the integration percept I. The sampler’s response at triplet t was classified as “1”. If the averaged spike count was small (), the subset activity was low and the IL-pool was said to support the segregation percept S. The sampler’s response was classified as “0” (Fig 4A). Therefore, at each triplet t, each sampler behaved like a biased coin being flipped with probabilities p_0;t,DF and p_1;t,DF over the binary probability space of 0 and 1. Since neuronal units in each IL-pool followed independent Poisson distributions of parameters m_t,DF, the pool itself was also a Poisson process defined by the product N_in m_t,DF. Then, the samplers were binary signal detectors with probabilities (4) and p_1;t,DF = 1 − p_0;t,DF calculated over 675 repetitions of the model in order to maintain similarities to the behavioral experiment, and with expected value and variance and .

Three DF-independent parameters were associated with SL: N_in, the number of A1 inputs to a sampler unit; C_th, the neuronal counting threshold that categorizes ensemble activity in A1 as high (class 1) or low (class 0); and N_sl, the number of neuronal units in SL. The values of N_in and C_th were obtained by least-squares fit between probabilities p_0;t,DF and the “asymptotic” levels 0.45, 0.6, 0.65 of the psychometric buildup functions (last 15 seconds of trial duration) for all DF = 3, 5, 7 (Fig 8B; also section Data-driven parameters for EVA model). The psychometric buildup represented the fraction (over the trials) of the segregation percept S reported by all subjects at each time point and DF during the 30-s long trial. Through this optimization procedure (Fig 8B; optimal values obtained for N_in = 5, C_th = 4.21) probabilities p_0;t,DF of a sampler to be in a state that supported percept S were estimated—at least for triplets several seconds from the stimulus onset—as (5) The inclusion of the sampler layer in the model (N_sl > 1) ensured neuronal granularity that was found by other studies to be a suitable substrate for perceptual representations [24]. In particular, at each triplet t, some of the N_sl samplers were in class 0 showing low A1 spiking and thereby associated with segregation [1, 7] while others were in class 1 supporting integration. The percentages p_S(t) and p_I(t) = 1 − p_S(t) of such samplers were taken herein as stochastic (over trials) output of SL (Fig 4A).

Accumulation layer (ACC).

The accumulation layer consisted of two units whose dynamic states x_I and x_S changed from triplet to triplet according to Eqs (6) The accumulator for x_I gathered evidence that favored integration (through input p_I(t) from SL) while the accumulator for x_S gathered evidence that favored segregation (through input p_S(t) from SL). Importantly, their states were influenced by the perceptual context as well (Fig 4C, solid lines in blue and red illustrated traces for x_I and x_S; the background color showed the percept’s type, blue for I and red for S). In particular, if segregation was the current dominant percept then x_I accumulated evidence against segregation and aimed to reach target T_I = T_a; simultaneously, x_S accumulated evidence for the current percept and it drifted instead towards target T_S = T_f. Discrete-time Gaussian white noise processes σ_Iξ_I(t), σ_Sξ_S(t) with zero mean and standard deviations σ_I = σ_a and σ_S = σ_f, interacted with the stochastic inputs from SL to produce certain levels of fluctuations. The additive stochastic terms in ACC were target-dependent with σ_a > σ_f for T_a > T_f (Fig 4C). On the contrary, if the current percept was integration then x_I accumulated evidence for I and approached T_I = T_f while x_S accumulated evidence against I and approached T_S = T_a. The level of local noise was adjusted accordingly to values σ_I = σ_f and σ_S = σ_a.

Switches and resetting conditions.

In the experiment, subjects identified the dominant percept by pressing a certain button on the keypad. Equivalently, in EVA model, the switch from one dominant percept to the next occurred when either x_I(t) or x_S(t) crossed the ACC threshold set to 1. Herein, the alternations were initiated by the accumulator that observed how many samplers in SL opposed the current percept at each triplet t. Its trace was attracted to target T_a that lay near the threshold, then was pushed across the threshold by the noise of strength σ_a. In the meantime, the accumulator in favor of the current percept hovered around T_f with fluctuations set by σ_f. For example during percept I, accumulator x_S was the first to reach threshold 1 producing a switch to percept S; then, during S, x_I reached threshold 1 leading to another switch to subsequent percept I, and so on (Fig 4C). At every change in percept, the accumulators were reset to new levels , . These were defined as where was the value that the evidence-for accumulator x_* (* = I, S during current percept I, S respectively) reached just before the switch.

The simulations took into account the proportion of first percepts reported as segregation at each DF during the behavioral experiment as well as the latency period during each trial. The initial conditions of the accumulators were set to a DF-independent baseline value b and kept constant during the entire latency period (calculated in length of T_Lat triplets) of any given trial. We defined x_I(t) = x_S(t) = b for all triplets t between 1 and T_Lat; then at t = T_Lat the type of the current first percept, I or S, was imported from the behavioral data; the dynamics of the accumulators for t ≥ T_Lat were then determined according to Eq (6) and the associated reset conditions.

The choice of parameters (σ_f small) and of reset conditions ( where x_a, x_f are accumulators “against” and “for” the dominant percept) ensured that the switch was triggered by the dynamics of x_a. Rare events when x_f might have crossed the threshold ahead of x_a were disregarded. Another possible implementation of resetting would allow for correlations between consecutive percepts; it could depend on each accumulator state just before a switch, a simple interchange of roles such that ACC variables remained continuous, , .

Parameter values used in model simulations.

All figures for the full EVA model (Figs 1, 4—8, and S1—S3 and S5 Figs), were generated with the following parameter values:

1. N_in = 5, C_th = 4.21, N_sl = 20, b = 0.7, T_f = 0.6, σ_a = 0.085, σ_f = 0.03

and decision threshold θ = 1, unless otherwise stated in their caption. (For parameter values m_t,DF associated with IL, see Data-driven parameters for EVA model.) Target T_a was initially chosen equal to 0.9 and then was fine-tuned to best fit the mean dominance durations of the first and subsequent integration and segregation percepts from the behavioral data—within ±10% standard error (SE) of the experimental mean values (Figs 5 and 6; also S5 Fig). Its values changed with DF, and with classification (first or subsequent) and type (I or S) of the percept. We used the notation T_aI1 for “target against integration, first percept”, T_aS1 for “target against segregation, first percept”, T_aI2 for “target against integration, subsequent percepts”, and T_aS2 for “target against segregation, subsequent percepts”, respectively. Therefore, in the model, whenever acting as target-against, T_S = T_aI1 or T_aI2 while T_I = T_aS1 or T_aS2. In simulations we used the following values (see red diamonds in Fig 7):

1. At DF = 3: T_aI1 = 0.8273, T_aS1 = 0.9273, T_aI2 = 0.8924, T_aS2 = 0.8924;
2. At DF = 5: T_aI1 = 0.9000, T_aS1 = 0.8909, T_aI2 = 0.9288, T_aS2 = 0.9106;
3. At DF = 7: T_aI1 = 0.9348, T_aS1 = 0.8773, T_aI2 = 0.9242, T_aS2 = 0.9318.

Selection of target-against T_a values.

The mean duration μ obtained by numerical simulations of EVA model was compared to its behavioral counterpart μ_exp for first and subsequent I, S percepts, and each DF = 3, 5, 7. Behavioral results from 15 subjects were characterized by group mean data μ_exp and 95% CI with CI corresponding to 1.96 SE (Figs 5A and 6A, lower panel). Then the model was considered to provide a good approximation of the experimental data if μ belonged to a narrow band within 1 SE from μ_exp (Fig 7, green region and black dots). This was equivalent to the relative error where CV = std_exp/μ_exp was the coefficient of variation computed over the group of subjects. Parameters T_a, σ_a used for the numerical simulations of EVA model (Fig 7, red diamond) were chosen as follows: σ_a = 0.085 was kept fixed while values of T_a were determined by restricting the error magnitude to only 10% SE (i.e. |μ − μ_exp| ≤ 0.1 SE); then, among the latter set we selected the value T_a that yielded the least error in shape of the gamma-fit distributions (see Eq (3)).

Statistical properties of SL-activation.

At any fixed triplet position t in the ABA₋ sequence presentation, each of the N_sl samplers was equivalent to an independent Bernoulli process (during repeated trials) with probability of success p_1;t,DF and probability of failure p_0;t,DF = 1 − p_1;t,DF. Likewise, the state of SL described a binomial process equivalent to the random variable, over trials, N_sl p_I(t) where p_I(t) represented the percentage of samplers in class 1 that favored integration at triplet t. The stochastic process had mean N_sl p_1;t,DF and variance N_sl(1 − p_1;t,DF)p_1;t,DF. As a result, the first two moments of the output p_S(t) and p_I(t) of SL were well-approximated, for sufficiently large triplet-indexes (t ≥ 30; see Eq (5) and Fig 8B, 2nd panel), by (7) In particular, for large N_sl the variance of p_S(t) and p_I(t) became negligible while their means remained unchanged.

Selection of noise level σ_a.

Our EVA model features accumulation that could saturate, given that target-against T_a was assumed to be subthreshold (T_a < 1). Hence the noise level σ_a has to be sufficiently large in order for the trajectory of the accumulator drifting towards T_a to reach the ACC-threshold. A theoretical lower-bound estimate for σ_a was obtained by assuming N_sl large and focusing only on the properties of the subsequent percept durations. Under such assumptions, p_S(t) and p_I(t) were approximately constant as demonstrated by Eqs (5) and (7). Then, after each switch, both accumulators satisfied an equation of the form X_n+1 = X_n + (T − X_n)p + ε_n+1 with X₀ ≈ T_f; T, σ taken as either T_a, σ_a or T_f, σ_f; and independent random variables ; see also Eq (1) in section A basic state-dependent model. Such an equation describes a stationary first order autoregressive model with parameter λ = 1-p [41]. Therefore, at the nth triplet during a percept immediately following a switch, the states of the accumulators followed a normal distribution with mean E[X_n] = T-(T-T_f)λⁿ and variance Var[X_n] = σ²(1 − λ²ⁿ)/(1 − λ²). In particular, at the nth triplet during integration, the mean and variance of x_S (x_I) that accumulated evidence against (for) the percept were E[x_S] = E_a, Var[x_S] = V_a, E[x_I] = E_f, Var[x_I] = V_f where with p = 1 − p_seg;DF. Then at the nth triplet during segregation they were E[x_I] = E_a, Var[x_I] = V_a, E[x_S] = E_f, Var[x_S] = V_f with E_a, V_a, E_f, V_f defined as above but computed with p = p_seg;DF. Given that 3 times the standard deviation from the mean accounts for 99.73% of values in a normal distribution, if σ_a was too small the accumulators could cross the threshold 1 only with very small probabillity. From the calculation above, a lower bound for σ_a in the model was estimated at σ_a > σ_a,min with , where p_M was the maximum of p_seg;DF and 1 − p_seg;DF for all DF = 3, 5, 7. For example, if T_a = 0.9 then a necessary condition for switching was σ_a > 0.022.

Statistical properties of ACC-activation.

As explained in the previous section, for small σ_a the accumulators in the EVA model could not cross threshold 1 (Fig 7; na, gray region). Numerical simulations showed that when σ_a increased to the right of curve σ_a = σ_a,min in the (σ_a, T_a)-plane, alternations between percepts occurred and the dominant durations were distributed according to: normal distributions at σ_a small (the parameter α in Eq (3) was very large) or to exponential distributions at σ_a large (α in (3) was near 1); see Fig 7, regions labeled “N” and “E”, respectively. At intermediate values σ_a, the distributions were gamma-like with shape close to that found experimentally (Fig 7, region labeled “G” between the two white curves). In the latter case, parameter α in Eq (3) was either near 2 (for first percept durations) or near 2.6 (for subsequent durations), and it differed from α_exp by relative error up to 20%, |α/α_exp − 1| ≤ 0.2 (except for first and subsequent I at DF = 7 at which the range for α was extremely narrow and we allowed for a larger error, up to 30% instead). The range for σ_a that led to gamma-like distributions varied slightly with N_sl with the biggest difference being identified at N_sl = 1; see S3 Fig.

Data-driven parameters for EVA model.

Parameter mean values m_t,DF were used to generate Poisson spike counts of IL-units at each triplet t (t = 1, 2, …, 60) in the ABA₋ sequence and for each semitone difference DF = 3, 5, 7; see section Input Layer and Eq (4). They were derived from multi-unit spiking neural data recorded from macaque monkey primary auditory cortex A1 by [1], using a procedure that combined exponential fitting with numerical interpolation. First, mean spike counts m_t,DF at A-tone selective neurons in A1 during presentation of tones A, B and A in ABA₋ were extracted from [1] for each triplet t ≤ 20 in the sequence and for each DF = 1, 3, 6, 9; See scatter points in [1, (Fig.3 in Ref)]; also S4 Fig. The mean spike counts at each tone decreased from value m_1,DF measured at the first triplet to some level at which they stabilized after a few seconds since stimulus onset. They were fitted by functions (8) with parameters chosen to minimize the least-squares error between the extracted mean spike counts m_t,DF and the corresponding exponential curve (S4 Fig, solid curves). In particular, significant differences in mean spike counts at different DF values were observed only during tone-B presentation with fitting in Eq (8) achieved for parameter values m_1,1 = 7.25, m_1,3 = 6.25, m_1,6 = 6, m_1,9 = 5.25 (according to data from [1]) and , , , respectively. Secondly, simulations of EVA model were performed for DF = 3, 5, 7 instead of 1, 3, 6, 9, and for a total of 60 rather than 20 triplets. We implemented these constraints in two steps: for DF = 3 we chose the mean spike counts m_t,DF as in [1] for t ≤ 20 and as for t > 20. Then for DF = 5 and DF = 7 and each triplet t we defined the mean spike counts by interpolation using the power function whose coefficients a_t, b_t satisfied the least-squares fit between this curve and the points (DF, m_t,DF) defined by Eq (8) for all DF = 1, 3, 6, 9 at any fixed t.

The mean spike counts for B-tones of any pool of A-tone selective IL neuronal units, as well as the standard error to the mean, were computed from simulation of Poisson processes with parameter m_t,DF while varying N_in (Fig 8A). Then a threshold value C_th was chosen to minimize the squared differences error between the model-based probabilities from Eq (4) of a sampler to support the segregation percept for all DF = 3, 5, 7 and the behavioral buildup functions, applied to the last 30 triplets (15 seconds) of the stimulus (Fig 8B). The pair of parameter values N_in = 5, C_th = 4.21 that generated the minimum error was then used in numerical simulations of the EVA model.