Figure 1.
Model overview.
Figure 2.
A) A linear chain describing event A, followed by 100 ms of silence, event B, which is then followed by 50 ms of silence; and a looping chain which alternates between events A and B every 100 ms. The red “phase” triangle indicates that the looping chain is currently passing through event B. B) The building of chains in response to the input sequence ABAC. Each input event adds a new singleton chain and causes existing chains to split into two new versions (marked by wide arrows), one adding the event and the other omitting it. When A is input for a second time, two potential loops are noticed: AB← and A−←. When event C arrives, the first of these fails to predict B correctly and is therefore removed.
Figure 3.
A) Profile showing the probability that an event is added, given the parameters listed in Table 1 and assuming . The preceding event is denoted using a heavy black dot, the event to be potentially added is characterised by its temporal distance,
, and feature distance,
, from the previous event. B) Profile showing the probability that an event is omitted, given the parameters listed in Table 1 and assuming
. C) The role of
when segregation is discovered first. Top: the input sequence. Below: Once the A−← chain is discovered, A is predicted by one chain and B is not predicted, so
in the exclusion formula for excluding A from a chain currently ended by B. This difference thus facilitates the construction of −B−−←, because the A events are easy to exclude. D) The role of
when integration is discovered first. Top: the input sequence. Middle: When the ABA−← chain is discovered, both A and B are predicted by one chain, so
. Bottom: However, once A−← is also discovered, A is predicted by two chains and B is predicted by one chain. Thus,
, again facilitating the construction of −B−−←.
Table 1.
Chain building parameters.
Figure 4.
Dynamics associated with a single chain.
Each chain, , is associated with an excitatory and inhibitory population of abstract neurons (discs). The termination of each arc onto a population denotes an additive term, which affects the activity in a population (see equations 1 and 2). A red terminal indicates that the influence is always positive. A blue terminal indicates that the influence is always negative. A green terminal may be positive or negative in its influence. The source of each arc in the diagram denotes an additive term in equation (1) or (2), the expression attached to each arc is the coefficient which scales that effect. Note that this diagram shows the dynamics associated with chain
, and all other chains are referred to as
.
Table 2.
Dynamical state variables.
Table 3.
Dynamical system parameters.
Figure 5.
Dynamical system formed in response to an ABA− sequence.
A and B) Collision and success rate effects shown on the excitation/inhibition (dynamics, top; see Figure 4) and the sound-group depiction (chains, bottom; see Figure 2) of the chains formed in response to a repeating ABA− sequence. Columns represent the three most stable chains formed: ABA−←, A−←, and −B−−←, from left to right. DYNAMICS (top panels): The inhibitory neuronal population is shown at the top, the excitatory one at the bottom of the panel. The strength of each population is marked by the filling of the circles (empty circle = weak, filled circle = strong). The size of the suppressing effect of the inhibitory population on the excitatory one is marked by the width of the blue line connecting them. The inhibitory population of a chain is strengthened by collisions with other chains (see section “Successes (S) and Collisions (C)”); the number of collisions and the amount of strengthening they provide to the inhibitory population of the given chain are noted over the inhibitory population. The effects of collisions are marked by red arcs connecting the excitatory population of each chain with the inhibitory population of those chains with which it collides (A−← and −B−−← don't collide, all other pairs do). The size of the strengthening effect to the inhibitory neuronal population (dependent on the strength of the excitatory population of the other colliding chain) is marked by the width of the arc. Excitation is strengthened by the rate of successful predictions made by the given chain; the number of successful predictions is noted below the excitatory population. For simplicity, the rediscovery, noise and self-excitation terms are not depicted here. Chains (bottom panels): Blue shading marks the currently dominant chain (i.e., the chain(s) whose excitatory population is stronger than that of the other chains). A) Integrated organisation dominant. Whilst the ABA−← chain dominates, the excitatory activity associated with the A−← and −B−−← chains is low. B) Segregated organisation dominant. Whilst the A−← and −B−−← chains dominate, the excitatory activity associated with the ABA−← chain is low. The events in the A−← and −B−−← chains do not collide with each other, so they have no inhibitory effect on each other. C) System state showing the various trajectories that the variables associated with the three chains (represented by and marked on the three axes) take, given 20 randomly-chosen initial values (green dots). In the absence of noise, the system permanently settles into one of the two organisations associated with diagrams in (A) and (B) (red dots), moving along a deterministic trajectory (blue lines). That is, some time after the start of the sequence either ABA−← becomes highly excited with A−← and −B−−← becoming weak (lower left red dot) or vice versa (upper right red dot) and the excitation and inhibition values of the three chains do not change anymore (i.e., the model without a noise effect would predict stable perception).
Figure 6.
The left panels show the excitation and other dynamical state variables of the chains that arise in response to a four-minute long ABA− sequence with ,
. The excitation variables (
) alternate at random intervals between two stable organisations once they are both discovered (at around 40 seconds): “integrated” (blue only) and “segregated” (red [“B”] and green [“A”] together). The percepts that would correspond to the chain with maximum momentary excitation are plotted above, calculated from low-pass filtered excitation time-courses (to avoid bouncing). Segregation dominates 74% of the time; the mean phase duration is 23.7 s. The right panels plot the changes in the state variables during a perceptual switch at 110 seconds on a magnified time-scale. The corresponding time period in the left panels is highlighted in bright yellow. Chain excitations are modulated by the noise variables
(not shown). The inhibitory populations (with activities
) serve to achieve exclusivity of the stable organisations by suppressing chains colliding with the dominant one. The adaptation and self-excitation state variable (
) renders switches in close succession unlikely (self-excitation) while increasing the probability of a switch as the duration of the perceptual phase grows (adaptation). The probabilistic rediscovery of a chain supports its excitation through the rediscovery rate (
).
Figure 7.
A) Distribution of the perceptual phase durations obtained from the perceptual experiment data with and
(110 phases from 15 participants). B) Distribution of the “perceptual” phase durations obtained from the model for the same
and
parameters as in panel A) (53 phases from 15 simulations). Note that a small number of outliers are not visible (a 213 and a 223 seconds-long perceptual phase on panel A and a single 179 seconds-long phase on panel B). *Empirical phases exclude “both” and “neither” responses.
Figure 8.
Proportion of time spent in the segregated organisation.
A) An image displaying the proportion of experimental subjects () that reported hearing segregation first for the
combinations of the stimulation parameters. B) An image displaying the proportion of time spent perceiving the segregated percept after the first perceptual phase has ended. C), D) The results from the Chains model (15 simulations) depicted in a
grid of the same parameter space corresponding to those presented in (A) and (B), respectively. Colour calibration of proportions (in %) is shown at the upper right corner.
Figure 9.
Durations of all perceptual phases.
A) An image displaying the group-average () durations of the first perceptual phases, as reported by experimental subjects
for the
combinations of the stimulation parameters. “Integrated” and “segregated” phases were analysed together. B) An image displaying the mean durations of the perceptual phases subsequent to the first phase. C), D) The results from the Chains model (15 simulations) depicted in a
grid of the same parameter space corresponding to those presented in (A) and (B), respectively. Colour calibration of phase durations (in s) is shown at the upper right corner.
Figure 10.
Durations of integrated and segregated phases.
A) Surfaces showing the group-average () mean perceptual phase durations for integrated and segregated phases (cells outlined in black and white, respectively) as reported by listeners
. The first and subsequent phases were analysed together. B) The corresponding results obtained from the Chains model (15 simulations). Both surfaces were based on a
grid of the stimulation parameters (
and
axes). Phase durations are calibrated in seconds on the
axis.
Figure 11.
The time course of the probability of segregation.
A) Four curves showing the group-average probability by which listeners () reported hearing the segregated percept at various times during a trial. The parameter combinations for each coloured curve are shown on the side map. B) The corresponding results from the Chains model (15 simulations). The probability of the streaming percept is always zero at the onset of the stimulus train as there is a delay to the first reported/modelled percept. This does not mean that listeners necessarily report (or that the model would find) the integrated percept before the segregated one (i.e., the probability of the integrated percept is also zero at the onset of the stimulus train).
Figure 12.
Illustration of the role of some model parameters.
The left columns show the proportion of time spent in the segregated organisation separately for the first and subsequent phases, while the right columns display the durations of all perceptual phases (again, in separate columns for the first and subsequent phases). A) Results obtained with the original parameter set, specified in Tables 1 and 3. These charts are identical to panels C) and D) of Figure 8 and the same panels of Figure 9, respectively. B) Chain building parameter is changed from 0.00015 to 0.00075. Increasing the effect of rate-of-change on the inclusion probability renders it more difficult to form the ABA chain and thus the segregated percept is more prominent (especially with small
and large
in the first phase and small
and
in subsequent phases). C) Chain building parameter
is changed from 0.0055 to 0.0035. Decreasing the probability of skipping over auditory events promotes the chains of the integrated organization, especially when rate of change is small. D) The weighting coefficient of success rate
is changed from 3.8 to 3.9. As the number of successful predictions a chain makes in unit time have a larger effect on its excitation, the integrated percept (with the highest success rate) is more dominant in subsequent phases than with the original parameter value. E) The weighting coefficient of the inhibitory signals towards the excitatory populations
is changed from 8.1 to 8.2. The resulting increase in the effectiveness of collisions in lowering chain excitation is manifested by a small bias towards the segregated percept (whose corresponding chains incur fewer collisions) in subsequent phases. Further, switches are less probable, i.e. phase durations are higher, when inhibition is more efficient in suppressing momentarily non-dominant chains. F) The weighting coefficient of noise
is changed from 3.4 to 3.0. As noise is responsible for the perceptual switches, decreasing its contribution to the excitation of the chains lengthens subsequent phases (especially when
is small). Note that adjusting the weighting coefficients of the dynamical state variables in panels D), E), and F) has no influence on the first phases (that is governed exclusively by chain discovery). Colour calibration is shown on top.