From Distributed Resources to Limited Slots in Multiple-Item Working Memory: A Spiking Network Model with Normalization

Population coding gives rise to the discrete-slot model in a continuous attractor network with normalization

Using a continuous network model of spiking neurons selective for an angle θ representing an analog feature such as the position on a color wheel (Fig. 1A), we investigated WM capacity by examining how the system responded to the presentation of an array of directions (Fig. 1B–D). Figure 2A shows the spatiotemporal spiking neural activity pattern of a network with wide connectivity (J⁺ = 3.62, σ = 11.25°), with firing rates plotted as a color-coded map, for a uniform array of 2, 3, 4 or 6 directions. Several characteristics are worth noting. First, pyramidal cells spontaneously discharge at a low rate (<3 Hz) without tuning to any specific directions before the onset of the cue array. Second, when the cue array is presented, the pyramidal cells, whose preferred directions are close to the stimuli in the cue array, increase their firing rates and form distinct bell-shaped activity profiles (bumps) that encode the directions of the corresponding stimuli. Third, these activity bumps continuously develop after the cue array withdrawn. When the set size is small (Fig. 2A, top), the WM load is low, and all the activity bumps can persist throughout the WM delay with slight drifts. For instance, Figure 2A (upper left) shows that two activity bumps are elicited in the cueing stage and persist during a 1 s delay with almost the identical bump width. Therefore, the representations of directions are actively maintained in WM and can be read out accurately after the delay. On the other hand, when the set size is large, the WM load is high; some activity bumps may fade out or merge in the WM delay. For instance, in a sample trial with 6 directions (Fig. 2A, lower right), one activity bump persists throughout the delay, three bumps fade out and two bumps merge in the early phase of the delay. Hence, after a short delay, i.e., 1 s, the information of three fade-out cues is lost; that of the original directions of two merging cues is blurred.

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Figure 1.

Network model structure and simulation protocol. A, Model scheme. The network is composed of spiking excitatory pyramidal cells (Exc) and inhibitory interneurons (Inh). Pyramidal cells are uniformly placed on a ring, labeled by their preferred directions (shown by arrows). The connections between pyramidal cells are structured as a Gaussian function of the difference in the preferred directions (top), and the connections onto and from the interneurons are uniform. B, Simulation protocol. A cue array is presented to the network from 0.25 s to 0.5 s, followed by a delay period up to 9 s. C, D, Sample cue arrays of 6 uniformly and randomly distributed directions, respectively.

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Figure 2.

Neural spiking activity, WM performance and normalization. The network has a wide E-E connectivity. A, Spatiotemporal neural activity pattern of pyramidal cells in response to an array of 2, 3, 4, or 6 directions. Pyramidal cells are labeled along the y-axis according to the preferred directions. The x-axis represents time. Firing rate is color coded. After being briefly presented during the cue period (marked as C on x-axis), each stimulus evokes a bell-shaped activity pattern of localized pyramidal cells (bump). In the lower right panel, some activity bumps fade out, some merge with each other, while all the elicited activity bumps in the other three panels persist in a 1 s delay period. Notably, the width of persistent activity bumps decreases with the set size. B–D, The performance of the network. B, The product of the set size and its correct rate of the reports exhibits a maximum. The corresponding set size defines WM capacity. WM capacity is smaller with longer delay duration. C, The correct rate is ∼100% for a set size below WM capacity, but declines sharply to ∼20% for a set size above it. D, SD is ∼2° for a set size below WM capacity, and sharply increases to ∼18° for a set size above it. E, F, A constant memory resource by network normalization. The total width of mnemonic activity bumps (E) and average firing rate of pyramidal cells (F) are almost invariant to set sizes, and roughly the same for 1 s and 9 s delays.

We assessed the network performance based on the readout from neural population activity in the last 0.25 s of the delay using a population decoding algorithm (Materials and Methods). Consistent with the observations from human visual experiments (Luck and Vogel, 1997; Zhang and Luck, 2008), the network model shows high (poor) performance at small (large, respectively) set sizes. With a 1 s delay (black, Fig. 2B–D), both the correct rate of reports (P_c ≈ 100%) and memory resolution (SD ≈ 2°) are high when the set size is smaller than a critical number (∼4), while P_c sharply decreases to a low level (∼20%) and SD drastically increases to a plateau (∼18°), once the set size exceeds this critical number (Fig. 2C,D). We found that this critical set size not only defines the WM capacity, which maximizes the product of the correct rate of reports and the set size (Fig. 2B), but also sets an upper bound of the number of the distinct activity bumps. In our model, since recurrent network dynamics continue to unfold over time, merging or fade-out could occur later in the delay period, therefore WM capacity depends on the delay duration, as it is shown by comparison of performance with 1 s versus 9 s delay (black vs gray, Fig. 2B–D). We will return to this model prediction later. The plateau of response precision implies that the network represents the memorized directions with an almost constant accuracy, and the low correct response rate indicates that the network forgets some of them if the set size outnumbers WM capacity. Therefore, even though our model is a continuous network and cue directions are encoded by a distributed neural population, it reproduces the defining behavior of a discrete-slot model.

Interestingly, in our model, the neural population activity is normalized, in the sense that the total width of activity bumps and the average firing rate of pyramidal cells are almost independent of the set size. Although the width of a single activity bump in the upper left panel of Figure 2A is obviously wider than that in the upper right panel, the total width of activity bumps are the same. Even the activity bump fades out in Figure 2A lower right, the total width of activity bumps remains essentially constant, as the merging bumps expand while the fade-out bumps shrink. To quantify this intuitive observation, we calculated the total width of activity bumps and the average firing rate of all pyramidal cells in 0.25 s preceding the end of the delay period, and found that both are almost constant despite the set size and the delay durations with or without fade-out and merging (Fig. 2E,F). If we define the WM resources as the activity of pyramidal cells, network normalization indicates that the system recruits a roughly constant amount of memory resource (Bays and Husain, 2008; Buschman et al., 2011), which is dynamically shared by the memorized items (Fig. 2A). With a set size smaller than WM capacity, each activity bump has the same width; the neural representations of items in WM thus equally share the resources. With a set size above WM capacity, an activity bump may merge with another activity bump or fade out, and the WM resources are dynamically shifted from a fade-out activity bump to another activity bump or absorbed by merging activity bumps, which agrees with the defining behavior of shared-resource model (Bays and Husain, 2008). Moreover, to maintain the persistent activity, a distinct activity bump must recruit a minimum number of pyramidal cells to make the local excitation strong enough. Therefore, the normalization, which implies a fixed total bump width, gives rise to a maximum number of distinct activity bumps.

Working memory capacity depends on the strength and width of recurrent excitatory connections

The model has structured excitatory recurrent connections and unstructured inhibitory connections, allowing us to focus on the effects of excitatory-to-excitatory (E-E) connections on WM capacity. Specifically, we gradually varied the E-E connection strength J⁺ and spatial footprint σ to examine how WM capacity depends on the local recurrent excitation (Fig. 3A). With weak or narrow connections (navy blue), recurrent excitation is insufficient to support any persistent activity bump. Otherwise, WM capacity varies from 2 to 7, which is consistent with the human reports of the single-feature WM capacity (Xu and Chun, 2006). The E-E connectivity thus plays an important role in modulating WM capacity. We found that WM capacity monotonically increases with J⁺ given a fixed σ (horizontal white line, Fig. 3A). The increase of J⁺ enhances iso-directional and weakens cross-directional E-E connections. Consequently, neurons within an activity bump receive stronger mutual excitation among themselves, but neurons in different bumps excite each other less effectively, hence self-maintenance of distinct activity bumps is favored and WM capacity is larger. In contrast, with increasing σ given a fixed J⁺, WM capacity increases at first, then decreases (vertical white line, Fig. 3A). A narrower connectivity (smaller σ) results in less pyramidal cells recruited to represent each item, as well as decreased mutual excitation; it thus is detrimental to the maintenance of an activity bump and leads to a smaller WM capacity. On the contrary, if σ is large, there would be an excessive number of pyramidal cells for the representation of each item (bumps are wide). These wide bumps merge with a high probability, and WM capacity is thus small. Therefore, a large WM capacity requires strong recurrent excitation with an optimal spatial footprint.

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Figure 3.

WM capacity depends on E-E connections. A, Dependence of WM capacity on E-E connectivity. The WM capacity is color coded and shown on the plane of two parameters characterizing E-E connections: the strength J⁺ and spatial width σ. When J⁺ is too weak or σ is too small (navy), the persistent activity is absent. Outside of that region, WM capacity ranges from 2 to 7; it is larger with more narrowly structured local synaptic excitation (smaller σ), which needs to be compensated by larger connection strength J⁺ to ensure sufficient recurrent excitation for WM maintenance. For a fixed J⁺ = 4.02, WM capacity increases at first and then decreases with the increasing connection width, peaking at 7 (vertical white line). For a fixed σ = 5°, WM capacity increases with the increasing connection strength (horizontal white line). B, C, Performance of a narrow connectivity network (×a in A) with uniform and random cue arrays. Correct rate and SD show a step-like transition as set size increases for uniform cue arrays (black), while there is a smooth function for random cue arrays (red). All the fitting curves are sigmoid functions. D, The relative precision for random cue arrays exhibits power-law dependence on set size. The total width of activity bumps (E) and average firing rate of pyramidal cells (F) are normalized for both uniformly and randomly distributed arrays of directional cues.

Although WM capacity depends on E-E connections, the typical characteristics of the networks with different E-E connections are similar. Given the uniform cue arrays, the performance curves of the network with narrow connectivity (×a, Fig. 3A) are similar to those of the network with wide connectivity (×b, Fig. 3A). Given uniform cue arrays, P_c drastically decreases to a low level and SD sharply increases to a plateau when the set size exceeds WM capacity (∼3 in Fig. 2 and ∼6 in Fig. 3B,C for a 9 s delay), which have a step-like shape. However, given the random cue arrays in which the minimum distance between items is ≥24° (Zhang and Luck, 2009), P_c continuously decreases and SD continuously increases with the set size until reaching the same plateaus for the uniform cue arrays (Fig. 3B,C). Of note, due to the attractor dynamics in our model, the memorized items are stored separately in different discrete slots during retention process (discrete-slot feature), wherefore SD (Fig. 3C) for random cue arrays resembles the human experimental data from the discrete-slot model. Using the global inhibition, the roughly constant memory resource is dynamically allocated to the memorized items (shared-resource feature), wherefore the relative precision, which is normalized by the maximum P_s over all trials (Materials and Methods), follows a power-law decay function of set size, which resembles that observed in the shared-resource model (compare Bays and Husain, 2008, their Fig. 3B, with our Fig. 3D).

Surprisingly, although the persistent activity pattern and performance are considerably different with random versus uniform array of cues, network's normalization is remarkably similar for the two types of cue arrays (Fig. 3E,F). Therefore, the total WM resources are independent of the details of external inputs, but are determined by the E-E connectivity profile (comparing Fig. 2E,F with 3E,F).

Working memory capacity depends on delay duration

Figure 4A shows the same sample spatiotemporal patterns as those in Figure 2A, except for a longer delay of 9 s. Notably, activity bumps may fade out or merge at different times in the delay. This explains why the performance is different for a delay of 1 s versus 9 s (Fig. 2B), when the set size is in an intermediate range. For a small set size, none of the activity bumps will fade out or merge in a prolonged delay, whereas for a large set size, bump fade-out or merging takes place early. In both cases, WM performance is insensitive to the delay duration. On the other hand, for intermediate “critical” set sizes, bump merging and fade-out exhibit slow stochastic dynamics during the delay. Consequently, WM capacity exhibits a dependence on delay duration (Fig. 4B, top). SD also depends on delay duration in a trend mirroring that of correct rate (Fig. 4B, bottom), and the set size at which SD starts to saturate is roughly a linear function of WM capacity (Fig. 4C).

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Figure 4.

The effects of delay duration on performance. A, Same sample trials as in Figure 2A, except shown for a 9 s delay. For a set size, 4, near WM capacity (lower left), bump fade-out or merging may occur late in the delay. B, Top, WM capacity depends on the delay duration and the E-E connections (left for wide; right for narrow connectivity). Bottom, SD increases as a function of set size for different delay durations, with wide (left) or narrow (right) connectivity. The network performance is essentially independent of the delay duration for small or large set size. However, for an intermediate set size, the performance of the network deteriorates with a prolonged delay period, as found in the human experiment (Zhang and Luck, 2009). C, The set size at which SD reaches a threshold level (10°) is linear with WM capacity for all the conditions considered, different delays, and narrow and wide connectivities.

Comparing capacity estimation using different measurements

The behavior of WM performance “near a critical set size” has been examined in a human study as a probe to the forgetting mechanism of WM (Zhang and Luck, 2009). It was assumed that a subject would report a random value in the case he or she forgets the memorized item, or report a value around the original cue when he or she remembers it (Zhang and Luck, 2008). This discrete-slot model was formulated using a 2-parameter von Mises fit. To compare with the psychophysical data from the discrete-slot model, we performed this fit to the distribution of the response offset, θ_out − θ_in (unbinned data), for random cue arrays. With a 1 s delay, we found that the discrete-slot fit of simulated data (Fig. 5A) is comparable with that of Zhang and Luck (2008), their Figure 1c. We compared quantities, P_m and s.d. of discrete-slot model with P_c and SD of our model using the same data and plotted as functions of set size (Fig. 5B,C) and delay duration (Fig. 5D,E). These two quantification methods display the same trend of performance: P_m and P_c decreases (s.d. and SD increases) as a function of set size and delay duration. Of note, (1) P_c with high threshold displays a smooth decrease, which resembles P_m (Fig. 5B,D); (2) although SD from our model exhibits a continuous and smooth increasing against set size (Bays and Husain, 2008; Bays et al., 2009) and delay duration, its alternative fit, s.d., reaches a plateau when set size is >4 (capacity), which is consistent with the behavioral data in recent studies using brief delays (Zhang and Luck, 2008; Anderson et al., 2011), and the prediction by Fukuda et al. (2010). Notably, s.d. is nearly constant against the delay duration (Fig. 5E), indicating that a declined performance with longer delays mostly results from the sudden death of the mnemonic items (fade-out; Zhang and Luck, 2009).

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Figure 5.

Performance fit using different models. The simulation uses the wide-connectivity network and random cue arrays. A, Typical response offset histograms for set size 3 and 6 (left and right, respectively) with 1 s delay, fitted by discrete-slot model (P_m and s.d., red line) and our model (S.D., green line). B, P_m and P_c decrease as a sigmoid function of set size (delay duration is 1 s). P_c at high threshold (blue line) decreases more smoothly than that at low threshold (black line), which resembles P_m. C, s.d. and S.D. increase as a sigmoid function of set size (delay duration is 1 s). s.d. reaches a plateau as set size is >4 (capacity). D, P_m and P_c decrease as a function of delay duration (set size is 4 at the capacity); P_c at low threshold is nearly constant against the time. E, s.d. and S.D. increase as a function of delay duration (set size is 4 at the capacity); s.d. is nearly constant against the time.

Network mechanism underlying fade-out and merging of activity bumps

We have identified the fade-out and merging as the main dynamic effects limiting WM capacity. Indeed, the probability of a fade-out or a merging bump sharply increases from a low level to a high plateau when the set size increases above the WM capacity, as shown for both wide (Fig. 6A) and narrow (Fig. 6B) network connectivities. The sum of the fade-out and merging probabilities approaches 100% when set size is much larger than WM capacity.

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Figure 6.

Dependence of fade-out and merging of mnemonic activity bumps on set size. The fraction of mnemonic activity bumps that fade out (black) or merge with each other (gray) during a 9 s delay is plotted as a function of set size. When set size is below WM capacity (3 in A with wide connectivity, 6 in B with narrow connectivity), activity bumps seldom fade out or merge. For a set size above WM capacity, the probabilities for merging and fade-out increase sharply. With a sufficiently large set size, a plateau is reached where the sum of the fade-out and merging probabilities is ∼100%, hence an activity bump either fades out or merges with another bump.

With the same parameters as in Figure 6B, and a set size of 6 (capacity), all the activity bumps persists throughout a delay of 9 s (Fig. 7A). However, with a set size of 8 (above capacity), only four activity bumps persist until the end of the delay, whereas two activity bumps fade out (at 67.5° and 337.5°) and two others merge (at 247.5° and 292.5°) into one (Fig. 7B). A persistent activity bump has a bell-shaped spatial distribution of neural activity, while the merging activity bump displays a wide plateau in the spatial profile of neural activity, as shown by the population activity for the last 1 s of the delay (Fig. 7A,B, right).

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Figure 7.

Dynamics of persistent, fade-out, and merging activity bumps in WM delay. The network has a narrow connectivity. A, Spatiotemporal activity in response to a brief stimulus of uniform cue arrays with 6 items (capacity). All activity bumps persist throughout the delay. White lines are memory traces. The spatial distribution of pyramidal cells' firing rate (last 1 s in the delay) shows a bell-shaped profile (activity bump). B, Same as in A except for a set size above WM capacity. Two bumps (67.5° and 337.5°) fade out, and two bumps (247.5° and 292.5°) merge into one wide bump. The other bumps persist throughout the delay. The activity profile shows a wide plateau for the merging bumps and comparatively sharp peaks for the persistent bumps. C, Firing activity of single neurons marked as a, b, and c in B. Left, Neurons near the peak of a persistent bump (22.5°) spike at a high rate in the cueing stage and show persistent activity at ∼50 Hz during the delay. Middle, Neurons in a fade-out bump (67.5°) abruptly drop their activities in the middle of the delay, implying a sudden death of the corresponding item. Right, Firing activity of neurons (from 247.5° to 292.5°) within and between two bumps that eventually merge into one. Neurons within bumps (e.g., red) behave in a manner similar to those in the left panel. Neurons at the edge of bumps (e.g., yellow) are not boosted by the cue stimulus, but their firing rates gradually ramp up to a stable level during the delay. Neurons in the middle of two bumps (e.g., dark green, ∼270°) spike at a low rate in the early phase and suddenly jump to persistent activity in the late delay. D, E, The feedback dynamics between neural firing and recurrent excitatory drive (data from B). D, Instantaneous average firing rate, R(t), of each bump as a function of time. M, P, and F denote merging, persistent, and fade-out bumps, respectively. R(t) values of fade-out bumps suddenly drop to 0 Hz at a random time in WM delay as an all-or-none process. R(t) values of persistent bumps stay at ∼25 Hz with small fluctuations. R(t) values of merging bumps gradually increase (black) or jump (yellow) to ∼45 Hz after merging. E, Instantaneous average excitatory synaptic conductance, G(t), of each activity bump as a function of time. G(t) values of fade-out bumps quickly decay to ∼28 nS preceding the sudden decreases of R(t); G(t) values of merging bumps increase above 35 nS (larger than the maximum value of G(t) of persistent bumps) preceding the merging process. F, The average firing rate R̄ plotted against the average excitatory synaptic conductance Ḡ for different activity bumps (from 100 simulations using the same network and cue arrays as those in B). Three activity groups can be clearly discerned: merging bumps (blue) have high R̄ and Ḡ, while fade-out bumps (black) have low R̄ and Ḡ.

The single neurons inside these three types of bumps display the distinct firing activities during the delay (Fig. 7C). First, neurons around the peak of the persistent bump at 22.5° (a in Fig. 7B,C), spike at a high frequency in the cueing stage and at a moderate frequency ∼50 Hz with small fluctuations during the delay, which maintain the memory trace of the cue direction. Second, neurons in the bump at 67.5° (b in Fig. 7B,C) that eventually fades out, increase their firing rates upon cue presentation, exhibit persistent activity (∼50 Hz) in the early phase of the delay, but abruptly cease firing at ∼3 s in the delay. This sudden disappearance of mnemonic activity reveals an “all-or-none” mechanism for losing the memorized information. Third, for neurons within and between two bumps at 247.5° and 292.5° (c in Fig. 7B,C), the firing rates are initially quite different. Over time, however, they all converge to a similar activity level of ∼50 Hz late in the delay, when the two bumps eventually merge with each other. Neurons near the centers of the two activity bumps behave similarly as those in a persistent bump, whereas firing rates of neurons located at the edges of the original two distinct bumps slowly ramp up; neurons in the midpoint between the two distinct bumps are essentially silent in the early phase of the delay period, but display a sharp jump of activity to ∼50 Hz in the late phase of delay. Therefore a gradual ramping and a sudden transition from spontaneous activity to persistent state in the delay may be manifestations of a merging phenomenon that is observable at the single cell level.

E-E interactions support the persistent activity (Fig. 3A), which might be also a key factor determining whether a bump fades out or merges with another. To test this, we calculated two quantities for each activity bump in Figure 7B: the instantaneous average recurrent excitatory synaptic conductance, G(t), and the instantaneous average firing rate of pyramidal cells R(t) throughout the delay, and classified them into three groups: persistent bumps (P), fade-out bumps (F), and merging bumps (M) (Fig. 7D,E). In a fade-out bump, G(t) exhibits a sharp decrease at an unpredictable time in the delay, to a small but nonzero level. R(t) decreases to zero Hz, implying that the excitatory drive they receive is below firing threshold. Note that the sudden drop of G(t) precedes that of R(t), as expected for a fade-out process: the decrease of excitatory currents leads to less spikes in a localized activity bump, which in turn results in further weaker recurrent excitation; the cycle continues until the overall excitatory drive becomes too small and the bump fades out. For the bumps that eventually merge, G(t) and R(t) increase during the merging process and reach a high level afterward. The increase of G(t) preceding that of R(t) displays the process opposite to the observation of fade-out: stronger excitatory currents lead to more spikes in the localized activity bump, which results in even more recurrent excitation; when this positive feedback exceeds a certain level, neurons between the two activity bumps receive enough excitation to switch to a high activity state (Fig. 7C, right), and the two activity bumps merge with each other. For persistent bumps, G(t) and R(t) fluctuate but remain roughly constant over time. Their values are smaller than those of merging bumps and larger than those of fade-out bumps. Therefore, insufficient excitation leads to fade-out, while excessive excitation results in merging.

To better examine the correlation between recurrent excitation and neural activity, we calculated the average firing rates R̄ and the average excitatory synaptic conductance of each activity bump Ḡ. Figure 7F shows R̄ plotted against Ḡ for different activity bumps using a uniform cue array of set size 8. Three groups can be clearly discerned: the values of R̄ and Ḡ for merging bumps are larger than those for persistent bumps, which are larger than those for fade-out activity bumps. Specifically, insufficient local excitation (in nS), Ḡ < 32, leads to fade-out; strong recurrent excitation, 32 < Ḡ < 35, ensures a persistent bump; and excessive recurrent excitation, Ḡ > 35, results in merging of activity bumps.

Working memory capacity estimation using change-detection tasks

In addition to the DRTs, the CDT is an alternative experimental scheme widely used to assess WM capacity. However, it is not trivial to base the change-detection performance on that of DRTs, because a CDT includes three stages: (1) sample stage, for encoding the visual inputs; (2) retention stage, for working memory; (3) retrieval stage, for a decision upon the memory, and any of these stages can independently influence the post hoc performance in the detection task. The previous research exhibits either that the sample stage plays a bottleneck role of limiting the number of the encoded items in memory in a bottom-up manner via attention, and thus the working memory capacity (Zhang and Luck, 2008; Buschman et al., 2011), or that retrieval process affects the detection accuracy through an inhibitory reciprocal network (Johnson et al., 2009), nevertheless, little has been unfolded from WM retention process per se. The previous behavioral observation (Basile and Hampton, 2011) demonstrates that the psychometric curves from CDTs mimic that from DRTs; one can therefore predict that change-detection performance would decrease with increasing the set size (Luck and Vogel, 1997; Vogel et al., 2001; Wilken and Ma, 2004; Basile and Hampton, 2011; Elmore et al., 2011). To test this hypothesis, we performed a change-detection task with different set size (Fig. 8).

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Figure 8.

Performance as a function of size set in a change-detection task. A, Experimental scheme of a change-detection task. In each trial, network views a cue array (with 2, 4, 6 or 8 colors) and a test array (with the same set size as the cue), separated by a 1 s delay, and identifies whether they are the same. In half of the trials, the test arrays are identical to the cue arrays, namely same trials, where the amplitude of change is 0°, while in the other half of the trials, one color in the cue array is changed to a color with an amplitude from 10° to 90° away from its value, namely diff trials. B, A downstream match-nonmatch neural circuit underlies the probability of responding to same. C, Hit rate decreases and false-alarm rate increases as a function of set size. D, Psychometric curves show the probability to respond to different as a function of the amplitude of change, |θ_in − θ_test|, for different set sizes. E, Performance curve from CDT mimics that from DRT, showing that the change-detection performance (the probability of correct response) would decrease when the set size increases.

In this CDT, the set size of the cue array is the same as that of the test array. The network reports that the test array is the same (match) as or different (nonmatch) from the cue array. In half of the trials, we used the test arrays which are identical to the cue arrays, namely same trials (the amplitude of change is 0°; Fig. 8A, top), while in the other half of the trials, one color in the cue array is changed to a color with an amplitude from 10° to 90° away from its original value, namely diff trials (nonmatch; Fig. 8A, bottom). The probability to report match is obtained from a downstream match-nonmatch decision neural circuit (Fig. 8B; Engel and Wang, 2011). In simulations, we found that hit rate decreases and false-alarm rate increases as a function of set size (Fig. 8C), which is consistent with the behavioral observations (Wilken and Ma, 2004, their Fig. 4). Psychometric curves shows the probability to respond to diff as a function of the amplitude of change, |θ_in − θ_test|, for different set sizes (Fig. 8D). Of note, when the amplitude of change is small, e.g., |θ_in − θ_test| = 10°, the change-detection performance is improved as the set size increases, implying that “similarity” could improve the change-detection performance in some parameter regime. When the amplitude is large, e.g., |θ_in − θ_test| > 50°, the psychometric curves are saturated, and the performance curve from CDT mimics that from DRT (Fig. 8E) (Wilken and Ma, 2004; Basile and Hampton, 2011; see also the comparison between the change-detection tasks with different amplitudes of change by Fougnie et al., 2010). Overall, our simulations demonstrate that the change-detection performance would decrease when the set size increases, which agrees with the predicted performance in DRTs in Figure 5, B and C.

Similarity effect on working memory performance

Our model exhibits two distinct mechanisms underlying the decrease of memory precision with the increase of WM loads or delay duration in DRT: fade-out (complete loss of stored information), and merging (more quantitative blurring of stored information). However, a misreporting error from merging can easily be overlooked in analysis of a DRT with a minimum distance ≥24°, using binned data (Bays et al., 2010). We thus proposed two testable tasks to investigate merging or similarity effect on the WM performance.

First, we investigated the merging process using 2-item cues with different similarity and found that (1) merging can take place given a long delay (∼8 s) when the items are of the weak similarity (100°; Fig. 9A); (2) merging leads the memory traces bias to the convergent side (Fig. 9B), where similarity of items gets enhanced (still 2 items), and network could thus confuse one another in a CDT using test arrays which are similar to the cue arrays (Fig. 8), e.g., purple to blue (Elmore et al., 2011).

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Figure 9.

Behavioral manifestation of merging in the change-detection task and free-recall task. A, Merging happens even when two cues are sufficiently separated from each other: θ_in,1 = 130° and θ_in,2 = 230° (white arrows) in DRTs. Two stimuli evoke two activity bumps, which eventually merge into a single wide bump. Therefore, the reports bias to the center of θ_in,1 and θ_in,2. B, The distribution of the difference between the report and the original cue, θ_in,2 − θ_out,2 or θ_out,1 − θ_in,1 (θ_in,1 < θ_in,2), across 100 trials. A positive (respectively, negative) distance from the original cue implies convergence (respectively, divergence) of two activity bumps. The distribution skews significantly to the positive side, indicating that merging happens in a large amount of trials. Such a skewed distribution of reports can be tested in behavioral experiments. C, Experimental scheme of a change-detection task to test merging. In each trial, network views a cue array (3 colors) and a test array, separated by a 1 s delay, and identifies whether they are the same. Three types of cues, i.e., far (low-similarity), close (high-similarity), and far+close, and three types of tests, i.e., same, diff1 (divergent side), and diff2 (convergent side; changed colors in diff1 and diff2 are circled) are applied in the task; the Lin and Luck (2009) task is a mixture of same (50%) and diff1 (50%) tests. D1, D2, A sample from far+close trials (2 greens + 1 blue) exhibits a merging process between 2 greens (a, b), the memory traces of which converge to an intermediate level (still greens); the memory trace of the blue (c) only drifts around its initial cue. D3, Distributions of the response offset in different trials. That with low similarity (black bars) centers at 0°; that with high similarity (red bars) shows a strong bias to the convergent side (>0°). E, Performance for each test. Low- and high-similarity trials show similar performance in the same test (upper left). High-similarity trials show a better performance in the diff1 test (upper right), indicating that similarity of the cue array improves the change-detection performance in the Lin and Luck (2009) task (lower left), whereas it also shows that the similarity could deteriorate the performance in the diff2 test (lower right).

We then conducted a Lin and Luck (2009) CDT (Materials and Methods), using three types of the cue arrays, namely far (low-similarity), close (high-similarity), far+close, and three types of the test arrays, namely same, diff1 (nonconvergent side), diff2 (convergent side; Fig. 9C). In simulations, merging occurs only between the high-similarity items. Consistent with 2-item-cue result, the memory traces of high-similarity items converge to an intermediate level, while that of a low-similarity item drifts around the cue (Fig. 9D). As a result, the distribution of response offset of the high-similarity items biases to the convergent side, implying an increase (decrease) of the distance between the cue and test arrays for diff1 (diff2, respectively) trials, while that of the low-similarity items is centered at zero. One can thus argue that the similarity would show a great effect on the tests of diff1 and diff2, but little on that of same. To test this, we assessed the probability of choosing same for each trial using a downstream match-non-match decision circuit (Fig. 8B). Figure 9E demonstrates that all three types of trials exhibit similar performance in the same test; trials with high similarity show a better performance in the diff1 test, which resembles the behavioral observation that similarity improves the performance in a Lin and Luck (2009) task, whereas the similarity deteriorates the performance in the diff2 test.

To conclude, we proposed testable tasks to detect the merging in WM delay, and showed that similarity in cue arrays can either improve (Johnson et al., 2009; Lin and Luck, 2009) or impair (Elmore et al., 2011) the detection accuracy, mainly relying on the post-WM comparison process (compare also performance curves in DRT and CDT in Fig. 8E; Hollingworth, 2003; Mitroff et al., 2004).