OP dynamics.

Depending on the timing and strength of the stimulus given to each population, a selected population may either oscillate out-of-phase with currently active populations in the network cycle, or it may compete with one or more populations, quenching their activity while remaining active itself. More concretely, consider the case of a 5-population network (N = 5) with a network capacity of 3 active OP populations. If no input is given, all of the populations fire at low, but non-zero, rates. Selectively stimulating first one of the populations, and at a later time a second population, allows for these two populations to be simultaneously active, exhibiting OP oscillations as shown in Fig 2A. The result of subsequently selectively stimulating a third population depends on the strength, width, and onset time of the stimulus. Stimuli of sufficient width and amplitude can result in a winner-take-all (WTA) scenario (Fig 2F), where in the 3-population-capacity network the activation of the third population suppresses the first two populations sufficiently to become the only active population. Weaker stimuli allow for a winner-take-some (WTS) scenario (Fig 2D and 2E); in this case, the third population quenches one population (whose activity returns to baseline level) and becomes persistently active, leaving two OP populations. Both of these cases could represent selective forgetting due to interference. For example, they could correspond to situations in which attention is shifted to one item at the expense of other attended memoranda, effectively resulting in the forgetting of those items that had been held active in working memory.

If the stimulus is of moderate strength (that is, of sufficient amplitude and duration to activate the population, but not sufficient to quench another active population), then, depending on the onset time of the stimulus, the selected population may become interleaved with the other active populations in the current cycle (Fig 2B and 2C). In this case, all populations may fire OP, with the ordering of the newly activated population with respect to the already-active populations within the cycle determined by the stimulus onset time (e.g., Fig 2B vs 2C).

OP oscillations and working memory capacity.

As mentioned above, for any set of nontrivial parameters (e.g., c_ei ≥ ϵ > 0) there is a maximum number of OP populations that may be active, resulting in a finite network capacity.

We note that a working memory capacity is entailed either by nonzero coupling (i.e., one of c_e, c_ei, c_ie is positive), or by the temporal resolution of the network. That is, in a biological network, we would not expect perfect synchronization to occur with any regularity; rather, the detection of synchrony by the network would need to operate within certain temporal bounds, so that oscillations that are within some threshold time measure of being synchronous are in fact read out as synchronous. Indeed, the same is true numerically in our model simulations, if for very narrow temporal bounds. In either case, there is a minimum distance, say η > 0, that the excitatory peaks, e.g., may be from each other (in the first case, the minimum distance may arise due to the attracting or repelling basins of the synchronous state that result from nonzero coupling strengths, for example). Thus, if T is the period of the network oscillation, the maximum working capacity would be , which is finite for any fixed set of parameters.

Both the mutual inhibition, c_ei, and the timescales, τ_i and τ_n, are critical in determining this capacity. We expect the value of c_ei to play a strong role in determining the network capacity since the OP dynamics fundamentally arise from mutual inhibition between the populations in the network (see Methods and Two populations: Effects of coupling strengths). The effect the timescales have on the capacity is somewhat more nuanced. The timescale of inhibition works differentially on the excitatory populations, depending on their phases. Briefly, we find that, in particular, larger values of τ_i increase the time the excitatory populations spend near zero more than the time they spend away from zero, providing a greater window of opportunity for other populations to be active. We now look at this in greater detail.

We observe that there appear to be two qualitatively different features to the excitatory oscillation: (1) a pulse-like portion, where the activity rapidly increases and rapidly decreases, which we will call the “active phase”; (2) a portion where the firing rate stays very close to zero, which we will call the “quiescent phase”. Although the division between these phases is necessarily arbitrary, a natural threshold choice is the low fixed point, u_* (the baseline firing rate). Thus, for some fixed j, the active phase of a given oscillation, T_a, is defined as the interval of time such that u_j > u_*, while the quiescent phase of an oscillation, T_q, is the interval of time such that u_j ≤ u_*. Thus, T_a is a measure of the width of the pulse, while T_q is a measure of the time it spends near zero. Note that T_a + T_q = T, the period of the oscillation. T_a is mostly determined by the rise time of the inhibition, which is strongly affected by the excitatory population. In contrast, T_q is mostly determined by the decay time of the inhibition. See S2 Text for more detail. Thus, while both T_a and T_q increase as τ_i increases, we expect to increase as well. Indeed we find this to be the case; for example, with our original parameters (including τ_i = 12) and one active population, T ≈ 50ms, and . If we increase the timescale of inhibition up to τ_i = 20, T ≈ 76ms, and . Thus, the ratio has increased by a factor of about 1.3. We therefore might intuit that more populations can be active OP as τ_i increases, since more pulses from other populations may “fit” between the pulses of an already-active population. That is, since there is a proportionally larger interval of time the population spends near zero than the interval of time it is active, there is greater opportunity for other populations to be active during this quiescent phase. We also note that while it is important for τ_n to be large enough so that the NMDA population may reactivate the excitatory population (see S2 Text), it otherwise does not affect the period of the oscillation very much. For example, with our original parameters (including τ_n = 144) and one active population, recall that T ≈ 50ms; increasing τ_n to 240 slightly decreases the period to T ≈ 49ms. (We note that it is not surprising for τ_n to have an inverse relationship with the period, since τ_n mostly controls the decay time of the NMDA; thus, n stays slightly higher for larger τ_n, resulting in slightly faster activation times for u and v. Since τ_n has little effect on the decay time of inhibition, which is mostly determined by the time constant τ_i, we see that increasing τ_n may in fact decrease the period).

We may follow the oscillatory solutions in AUTO to determine exactly how the network capacity depends on the mutual inhibition and the system’s timescales. The OP states are lost as folds of limit cycles or destabilize via torus or period-doubling bifurcations as τ_i increases or decreases with τ_n fixed (Fig 3A illustrates this for the 3-OP state). In particular, we found that for N = 5 or 10, the solutions were lost as folds of limit cycles as in Fig 3A, while for N = 20 the oscillations sometimes first lost stability via the torus or period-doubling bifurcations (see Fig 3D). For N = 5 or 10, the 1-OP state (i.e., just one active oscillating population) is also lost as a fold of limit cycles (in τ_i), suggesting we may gain intuition into why the M-OP states exist for certain τ_i and τ_n values for M > 1 by examining the 1-OP case in more detail, which we do at the end of S2 Text by examining how the separation of timescales allows for the existence of these solutions (i.e., why the ratio cannot be too large or too small). While there are certain trends in the behaviors of the solutions, we did not observe any significant changes in the dynamics near bifurcation. In the case of the inhibition, as τ_i increases, the period increases (as discussed at the beginning of this section), more than doubling (from 50 to 117 ms), and the maxima of the excitatory populations increase by about 25%, while the minima of the NMDA populations decrease by ≈33% (note that the NMDA generally peaks at or near saturation at 1, so this does not tend to change as we adjust parameters) and the maxima of the inhibitory populations decrease only very slightly, staying virtually constant (a change of ≈ 5%). We note that not all of these behaviors (or those described below for the case of varying τ_n) are strictly monotonic, but rather they indicate the trends. For example, the amplitude of the inhibition increases very slightly with τ_i before it decreases; however, up to two significant figures it remains at 6.8. If τ_i increases beyond the high fold, the solutions tend to change from M-OP to (M − 1)-OP; e.g., 3-OP solutions are lost to 2-OP solutions. If τ_i decreases beyond the low fold, we get slightly different behavior. In this case, the solutions tend to change from M-OP to M-MP; i.e., one of the populations will synchronize with another one. In both cases, we see that the number of mutually OP groups decreases from M to M − 1.

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Fig 3. Stability of OP oscillations and working memory capacity.

For fixed τ_n, there is a window of τ_i values for which oscillations exist. We examine the existence and stability of the active out-of-phase populations, corresponding to distinct, single-featured memoranda. (A) Here, we look at the 3-OP state for a network of N = 5 populations, with τ_n fixed at 144 and the mutual inhibition c_ei fixed at 0.03. For τ_i too large or too small, the oscillations are lost as folds of limit cycles (for larger N, they may also be lost through torus or period-doubling bifurcations). (B) By following the limit points in (A) and keeping N fixed at 5, we may examine the dependence of the oscillatory states on both timescales, τ_n and τ_i, for different OP states. Thus, we see how the capacity of the system depends on the timescales. Each curve is a curve of the limit points as shown in (A). Thus, the OP state with 2 active populations exists stably above the blue curve, and the 3-OP state exists stably above the black curve. (C) We may further examine how the capacity is affected by the strength of the mutual inhibition, c_ei. Here, the 3-OP state exists above each curve for different c_ei values as indicated. As the mutual inhibition increases, the minimum τ_n value that supports the 3-OP state increases. Thus, we would like to keep the mutual inhibition low enough to support the 3-OP state within physiologically realistic synaptic timescales, but high enough to allow for the WTA state. (D) If we fix c_ei = 0.03, we may further explore how the network size N affects the 3-OP state. Overall, as N increases, the set of timescales that supports the 3-OP state does not change very much, generally increasing slightly. The bifurcation structure for N = 20 changes somewhat as well, so that the 3-OP state may destabilize through a torus or period-doubling bifurcation for lower τ_i values as well.

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For NMDA, as τ_n increases, the period decreases (as discussed at the beginning of this section) from 77 to 63 ms, the maxima of the excitation and inhibition stay virtually constant (changing by less than 1%) and the minima of the NMDA populations increase by ≈50% (since, of course, τ_n increases). As we can see from Fig 3D, there is a fold of limit cycles (for N = 5 and N = 10) for low τ_n; increasing τ_n does not appear to cause the loss of existence or stability of the solutions. This is as expected, since in the limit τ_n → ∞, NMDA will simply stay high (once activated) as it acts essentially as a parameter. Thus, as we explore in more detail in S2 Text, the NMDA will always outlast the downstroke of the inhibition, allowing the excitation to reactivate. If we decrease τ_n to below the fold, we again can lose the M-OP state to the M-MP state, as one of the active populations will tend to synchronize with another one.

By following the above bifurcations, we may partition τ_i − τ_n space based on how many OP populations can be active for fixed c_ei (Fig 3B), or on the maximum number of OP populations that may be active for different values of c_ei (Fig 3C). In Fig 3 we see that we have open sets of τ_i and τ_n values that lie within physiological ranges where the network capacity is 3, matching experimental ranges of a capacity of 3–5 items in working memory. As we increase the number of populations, the capacity of the system remains more or less the same (Fig 3D), suggesting the capacity is only weakly dependent on the network size.

S dynamics.

In Fig 4, we see three such possible paths based on the pattern of inputs. Unsurprisingly, if two inactive populations, say populations 1 and 2 without loss of generality, are stimulated with similar stimulus parameters (onset time, amplitude, and duration), they will oscillate S (Fig 4A). If two populations are active OP, we may apply a stimulus to either one so that they synchronize (Fig 4B). Finally, if population 1, for example, is oscillating in isolation, population 2 may also synchronize with population 1 if given a small stimulus within a certain time window of the phase of population 1 (in particular, close to when the excitatory component has an upstroke) (Fig 4C). The latter two cases illustrate the attractive nature of S oscillations in the model, which we further confirm through numerical continuation. We note there are other ways in which populations may pairwise oscillate S in the network, some of which are explored in Mixed-phase oscillations: Synchronous and out-of-phase and Biological considerations and applications.

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Fig 4. S dynamics and capacity.

(A) Simultaneous stimuli can cause the selected populations to exhibit S oscillations. (B) If two populations are pairwise OP, selectively stimulating one can alter its relationship so that the two populations subsequently oscillate pairwise S. (C) Sequential stimuli of the right timing may also cause the selected populations to dynamically bind and oscillate S. (D) Synchronous capacity as a function of τ_i and c_ei. For M = 1, …, N (here N = 5), the M-S solution is stable for all τ_i values between each respective pair of (same-colored) points. For example, the 5-S solution with c_ei = 0.03 is stable for 3.7 < τ_i < 36.4. The synchronous solutions are not stable outside of these intervals and are lost to different solutions, as indicated in the text.

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Maximum S populations.

We first observe from Eq (1) that if all N populations are active S in a given network, they oscillate as a single population with inhibition given by a_ei. Thus, we might expect the maximum number of populations that can oscillate synchronously to depend more on the intrapopulation parameters, such as a_ei. Indeed, this solution is eventually lost as a fold of limit cycles as a_ei increases. However, the interpopulation coupling values, such as c_ei, do affect the stability of this oscillating solution. We have observed these solutions to be lost as branch points (with no stable alternate branch) or period doubling bifurcations. Although the types of bifurcation may change, the types of changes the parameters cause match our expectations: As c_ei increases, the a_ei value at which the solution destabilizes decreases. Increasing the c_e value correspondingly increases the a_ei value at which the bifurcation occurs. Furthermore, decreasing the number of active populations increases the a_ei values at which the stable oscillations are lost, so that, for example, with the same intrapopulation and interpopulation coupling strengths, 4 S populations remain stable for higher a_ei values than 5 S populations do.

For other parameters, the behavior may be somewhat more complicated. For example, consider the range of τ_i values that allows for the stable existence of M-S solutions for M ∈ {1, … N} (of course the 1-S oscillation is just a single oscillator, SO). As we see in Fig 4D, as M increases from 1 to 5 with c_ei fixed at 0.03 or 0.07, the interval of τ_i that allows for the stable existence of the M-S oscillations narrows monotonically, resulting in a nest of intervals, each of which is strictly contained in the interval below it. However, the result of increasing c_ei from 0.03 to 0.07 varies somewhat on the number of synchronous oscillating populations. The right boundary, corresponding to the largest τ_i that supports the synchronous solution, increases with increasing c_ei, especially for fewer oscillating populations. The left boundary, corresponding to the smallest τ_i that supports the synchronous solution, increases for larger M (M = 4, 5) and decreases for smaller M (M = 1, 2, 3). As mentioned above, when M = N, the whole network oscillates as a single group, as can be seen with the appropriate substitutions in Eq (1); thus, while varying c_ei does change the stability of this oscillation, as can be seen for M = 5 in Fig 4D, the other characteristics, such as the amplitudes and period of oscillation, do not change at all. However, these can and do change for M < N; in particular, increasing c_ei tends to increase the amplitudes of the excitatory and inhibitory populations. We explore these changes in dynamics and explore the changes we see in Fig 4D in more detail in S4 Text.

MP dynamics.

When a quiescent or active population is stimulated, it may oscillate pairwise OP or S with currently active populations. Here we see three ways whereby we can obtain MP oscillations with a selective stimulus, so that a feature is added to an item in memory.

In Fig 5A–5C, there are initially 3 active OP populations (say populations 1–3). If population 3, for example, receives a stimulus, then its timing may be readjusted to synchronize with population 2. Alternatively, a quiescent population (say population 4) may be stimulated. Since the capacity of the network is reached with 3 OP populations (or really, 3 groups of mutually OP populations; that is, populations within each group oscillate S and populations in different groups oscillate mutually OP), in order for population 4 to remain active, either at least one of populations 1–3 must be quenched or at least two of populations 1–4 must synchronize. In Fig 5A, activating population 4 with a somewhat weaker stimulus quenches the activity of population 1 as might occur in forgetting, suggesting independently of the numerical analysis in OP oscillations and working memory capacity that the capacity of the network is indeed 3. Further stimulating population 4 causes it to synchronize with population 2, in a similar way that occurs in Fig 4C when sequential stimuli cause two populations to oscillate S. However, since all of the populations are coupled to one another, more complicated effects may occur as well. For example, in Fig 5C, once population 4 is stimulated, it also synchronizes with population 2 as in Fig 5B; however, populations 1 and 3 are perturbed enough to synchronize as well. We note that the only differences between Fig 5B and 5C are the onset times of the stimulus to population 4. More generally, the stimulus parameters determine which populations oscillate S and which do not. We also see that in both Fig 5B and 5C four populations are active following the stimulus. Thus, the number of active populations depends in part on how many OP populations may be active (see OP oscillations and working memory capacity) and in part on how many S populations may be active (see Maximum S populations). We consider additional exmaples of MP states in Feature binding and Variable binding.

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Fig 5. MP dynamics.

(A) Stimulating an inactive population when the network is already at capacity (3 OP populations here) may cause one of the active populations to become quiescent, even without a strong stimulus. A subsequent stimulus produces similar behavior as in Fig 4B, so that stimulating an active population may allow it to change its relationship from pairwise OP with both other active populations to oscillate S with one of the two other active populations, and OP with the second. (B–C) Stimulating a quiescent population can cause various dynamic bindings and MP dynamics. For example, changing only the timing of the stimulus can alter the resulting patterns of synchronization. In (B), the stimulated population synchronizes with one of the three OP populations. (C) Adjusting the onset time of the stimulus may result in additional interactions, so that the network transitions from 3 OP populations to 2 OP pairs, where both populations in each pair oscillate S.

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Two populations: Effects of coupling strengths.

Having seen that our model supports both OP and S dynamics that naturally combine to allow for richer MP dynamics, we look at the reduced case of two populations to distinguish how the interpopulation coupling affects the existence of the OP and S states, as well as the case where only a single oscillator (SO) is active. We note that the stable existence of the SO state is necessary for WTA dynamics; however, the SO state may also stably exist with lower coupling values that do not allow for WTA, and so it is not sufficient for WTA dynamics. The coupling strengths are crucial in being able to maintain any of these oscillatory dynamics. We see that in the coupling-strength space in Fig 6D, only the region * supports all three oscillatory states. Both SO (Fig 6B) and OP (Fig 6C) states exist stably in bounded sets, so that if either c_e or c_ei values become too large, the states will be lost. In contrast, there is no indication that the S state only exists in a bounded region (Fig 6A). If c_ei values are large enough with c_e small, the S state may lose stability (to the right of (i) in Fig 6A). However, we see from Eq (2) that as c_e → ∞, each excitatory population only gets excitation from the complementary population, so we expect synchrony to continue to stably exist for arbitrarily large c_e values. All three oscillatory states appear to exist stably when either c_e or c_ei is zero and as the other value approaches zero (i.e., along one of the axes near the origin), suggesting that weak coupling is sufficient for these states to exist. In S3 Text, we explore the attracting states in the weak coupling limit.

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Fig 6. Stability as a function of coupling strengths for N = 2.

The dashed line at c_ei = 1 indicates the interpopulation and intrapopulation inhibition are the same. All of the dynamics of interest exist stably when the interpopulation coupling is less than the intrapopulation coupling (to the left of the dashed line), as desired. (A–C) Colored regions indicate areas that the oscillatory states shown stably exist. Roman numerals refer to the boundary curves of the regions as indicated. (A) The S state exists to the left of (i), which is a curve of branch points of limit cycles (BPLCs). (B) The SO state exists to the left of (ii), a curve of folds of limit cycles (FLCs) (these folds are the only ones in this diagram that arise from a subcritical Hopf bifurcation) and to the right of (iii), another curve of FLCs. (C) The OP state exists below (iv), a curve of FLCs, and (v), a curve of torus bifurcations, and to the left of (vi), a curve of BPLCs. The line style for each curve is shown to the right of each roman numeral for clarity. (D) All of the regions in (A–C) superimposed. (*) indicates the region of interest, where all three oscillating states (S, SO, and OP) exist stably.

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Although the general features described above for Fig 6 match our intuitions, some of the particular features are unforeseen and may warrant further scrutiny. We now look more carefully at the regions that the S, SO, and OP states stably exist in coupling-strength space in turn, describing the curves of bifurcations and the behavior of the oscillations with changing coupling strengths.

The S state (Fig 6A) exists for all relevant coupling strengths. Neither the period nor the amplitude of the oscillations change as c_e and c_ei change. However, if c_ei increases beyond (i), a curve of branch points of limit cycles (BPLCs), only the low and high steady states are stable.

The SO state (Fig 6B) lies to the right of (iii), a curve of folds of limit cycles (FLCs). As we can see in Fig 6D, the oscillatory states that are available for lower c_ei, to the left of (iii), are either S or, for sufficiently small c_e values, OP. The SO state is bounded on the left by (ii). Curve (ii) is also a curve of FLCs; however, unlike the other such curves in Fig 6, these folds arise from a subcritical Hopf bifurcation.

For fixed c_e and increasing c_ei, the SO solutions show increases in the period (from 48 to 74 ms) and the maxima of both the excitatory (increasing by ≈ 20%) and inhibitory (≈ 40%) oscillations. These changes continue until the SO state is lost to a high fixed point. As we might expect from a subcritical Hopf (in particular, one that involves a fixed point away from zero), this change happens rapidly, with no changes before bifurcation suggestive of the loss of stability of the solution. For fixed c_ei and increasing c_e, the SO solutions show slight decreases in the period (from 53 to 51 ms) and the maxima of the excitatory and inhibitory populations (both decrease by ≈ 12% or less).

We note that once c_ei increases beyond the subcritical Hopf bifurcation (curve (ii)), the system displays steady state bistability between a down state and an up state with the same inhibitory timescale as for the case of oscillations. However, a much larger input is required to switch between active populations in this case. For example, suppose c_e = 0.001 and population 1 is active. If c_ei = 0.7 (“after” the Hopf, so that population 1 has a large-amplitude oscillation), stimulating population 2 with a width of 50ms and an amplitude of 3 (arbitrary units) is sufficient to allow population 2 to become active and quench the activity of population 1. However, if c_ei = 0.8 (“before” the Hopf, so that population 1 is in the up state), a stimulus to population 2 with the same width requires an amplitude of greater than 36 in order for population 2 to activate, sending population 1 back to the down state. This up state is only stable for smaller c_e values; for larger c_e the only stable state of the system is S activity. We see in Fig 6D that the S state exists to the right of (ii) for low c_e values as well.

The region of definition of the OP state is more complicated. It is defined by (iv), a curve of FLCs, (v), a curve of torus bifurcations, and (vi), a curve of BPLCs. These curves do not decrease in c_e monotonically as c_ei increases. We see that that there is a small interval of c_e values around 0.06 where the OP state exists stably for small c_ei values, but is lost for higher values (e.g., c_ei not quite 0.2). Surprisingly, once the OP state is lost here, we initially can only get the S state. As c_ei increases further, we may also obtain the SO state. Perhaps the most unanticipated feature of the OP region occurs near the local minimum of (iv). There is a very small c_e interval, near c_e = 0.05, where the OP state is lost (to a region where S and, for larger c_ei, SO both exist, as seen in Fig 6D) and then regained as c_ei increases, before finally being lost again for large c_ei. We explore the behaviors of the OP state further in S4 Text, including changes in the dynamics that occur with increasing c_e or c_ei that may lead to the bifurcations we observe.

Weak coupling.

Surprisingly, both S and OP oscillations are stable as the coupling strengths c_e and c_ei approach 0 with N = 2, as we saw above in Fig 6. Although we have chosen coupling strengths that allow for stronger interactions between populations (allowing for, e.g., WTA and WTS dynamics), we may gain some insight into our system by examining the limit of weak coupling. In this case we first assume that the populations are oscillating.

In S3 Text, we outline how we may use weak coupling theory to reduce our system of N coupled populations from a (3 ⋅ N)-dimensional system to an (N − 1)-dimensional system involving the phase differences. Briefly, we restrict our attention to weakly-coupled oscillatory populations. Since each 3-dimensional population (involving u, v, and n variables) is assumed to be on a limit cycle, its state can be referenced by its phase on that limit cycle, θ_j, where j is the population index. We can further reduce the number of dimensions by one by referencing each population’s phase relative to θ₁, the first population’s phase. Thus, we are left with phase differences ψ_j. Since ψ₁ ≡ 0, we can examine the attractor structure of three weakly-coupled populations in the plane.

With just EI or IE coupling, we can obtain all three relevant states of interest (OP, S, and MP). In Fig 7, we see the basins of attraction with only c_ei coupling (cf. Fig 2 in Horn and Opher 1996 [39]). Adding EE to EI coupling (c_e = 0.1c_ei) predictably increases the basin of attraction of the synchronous state. Based on the geometry we see in Fig 7, we might also expect transitions between regions that share boundary curves to be easy to realize (e.g., the S and MP regions), and transitions between regions that only share isolated boundary points (the S and OP regions) to be more difficult to realize, if even possible. In fact, we found just such a correspondence when we more thoroughly examined accessible states (see Accessible operations).

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Fig 7. Basins of attraction for the case of weak EI coupling with 3 active populations.

When three populations are active (note, ψ_i is the phase of population i relative to θ₁), OP, S, and MP states all have open sets as basins of attraction, as we found in the full model. These basins are defined by the stable manifolds of saddle points, as shown.

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Feature binding.

Dynamic binding is required in working memory in order to quickly associate and dissociate different elements together [40], such as different features of an item in the environment. In the context of our model, this means that items must be able to be both synchronized and desynchronized in response to different stimuli. We have seen above in Fig 4 how synchronization may occur via either simultaneous (Fig 4A) or sequential (Fig 4B and 4C) selective stimuli. Such synchronization allows different features to bind together into one item in memory. In Fig 8, we see two situations in which the patterns of synchrony change as the associations between different features change. For example, if a red traffic light turns green (Fig 8A), red light can unbind from stoplight (and indeed may become quiescent, as in this example), while green light then binds to stoplight as the two associated neuronal populations synchronize. Alternatively, an observer may perceive the static environment in a unified manner, so that the car at the red light and the street, for example, are bound together (Fig 8B). Once the light turns green, the car begins to move, so that car and street, for example, unbind and are represented as distinct items in memory. Thus, attentional mechanisms or environmental changes may act to change the patterns of synchronization, and thus of the binding of different items in working memory, by stimulating either quiescent or already-active populations.

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Fig 8. Feature binding examples.

For simplicity, memoranda that may require several different bound features are represented as a single oscillating population (e.g., a street). (A) Feature binding can be achieved through the synchronous firing of the individual features as in the left half of (A). A new input causes an unbinding and a new binding of an object with a modified feature. For example, the initial binding might represent a red stoplight. The light changes to green, causing the stoplight to bind with green light. (B) Feature binding in which a new input (e.g., focus) results in the decoupling of a single memorandum (e.g., some background such as a street and stationary car). For example, the car is observed to begin moving, and so is now perceived as a separate object from the street.

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Variable binding.

The combinatorial dynamics emerging within the model are rich enough to produce variable binding as is characteristic, for example, of language and abstract reasoning within the context of working memory. Below, we illustrate some simple examples from the model consistent with the most expected and widespread approaches to variable binding and language (e.g., the SHRUTI inference network [41]). These utilize temporal phase binding and represent simple predicate calculus rules. Consider the predicate calculus rule where x and y represent variables in the above rule, and a Boolean query (for example, as presented in Feldman [12]), such as “Does Tom (x) own a book (y)”. In most common inference network models, for example, SHRUTI [42], separate clock phases are assigned to the pairings of Tom with owns and book with owns. Tom here represents a variable agent who could come to own something (a book, for example) through purchases. There are other possibilities, of course, and they could all be linked or synchronized with the owns relation. We note that while a mediator circuit is implicated in the SHRUTI inference network, here the relevant association may occur in a single step as a result of synchronous firing due to temporal associations and the pattern of input, as shown in Fig 9. The active populations in working memory could represent an owns node (e.g., the active population in line 1 of Fig 9A), and Tom (as an agent node; e.g., the active populations in lines 2 and 3). The active populations of Tom and book then become associated and paired as a result of the activation of a node or population representing purchases in working memory (active population in line 4 of Fig 9A). Then, over several clock cycles “Tom owns a book” becomes established, or possibly instantly as here with the activation of the purchases node (i.e., population). Additional populations could become activated and linked via synchronization, such as “Tom purchases The Awakening” where The Awakening would become synchronized with a book node. We note that, whereas Tom requires two populations to be active in this example in order to form distinct bindings with both purchases and owns, different, n: m locking ratios or aperiodic oscillatory dynamics could allow for such unambiguous bindings while just one population is maintained active for Tom, as we touch on at the end of Discussion.

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Fig 9. Variable binding, and sentence construction examples.

Specific sequential inputs result in a cascade of bindings that emerge, following one another within a single clock cycle. (A) A fragment of the working memory system demonstrating how an inference emerges and is established in working memory. Here, the predicate calculus rule examined is purchases(x, y) ⇒ owns(x, y) (from an example in Feldman [12]). After a query is made (“Does Tom own The Awakening?”), a statement is provided, “Tom purchased The Awakening”. This statement first activates a second instantiation of Tom, and then the verb purchases(y), causing owns(y) and Tom to synchronize so that the inference is made. (B) An illustration of that same combinatorial structure is shown as it could apply towards a mechanistic realization of a phrase structure grammar. This could be based upon an already established or innate structure in the cortex. Upon reading the sentence, the words are sequentially input and the appropriate nodes are activated and bound in working memory, forming a determinate noun phrase (DNP) and a verb phrase (VP). For clarity, bindings between nodes and variables (i.e., words here) are not explicitly illustrated. (C) A final binding occurs, as the components of the DNP receive selective and equal simultaneous stimuli, binding all of the components together to form a sentence (S).

https://doi.org/10.1371/journal.pcbi.1006517.g009

We can also consider how the combinatorics allowed through the present model’s dynamics may apply in language by illustrating how they may facilitate the presence of grammars. We note that the predicate-argument structure used above (e.g., owns(x, y)) fits well within some dependency grammars. However, the model is agnostic to any particular type of grammar, and we now consider an example as might be implemented within a phrase structure grammar, illustrated in Fig 9B and 9C. The words are represented by the activation of populations (presumably stored in the structure of networks in long-term memory) and become synchronized with the appropriate activated nodes. For clarity, we have simplified the example so that the nodes, and therefore the bindings of words to nodes, are not displayed. For example, a determiner node is first activated and then bound to the, but in Fig 9B we only show the activity of the population corresponding to the. A noun node is then activated, which is bound to the variable woman. A verb node is activated next, which is bound to writes. Finally a second noun node is activated and then bound to music. This results in the binding of the determiner node and noun node of the subject into a determinate noun phrase (displaying the phrase structure rule D + N → DNP), and within a single clock cycle the dynamic binding of the verb with the direct object to form a verb phrase (V + N → VP). These states are activated sequentially within a single clock cycle, and thus could be recognized as a grammatical sentence maintained in working memory based on the phrase structure rule that a DNP followed by a VP produces a grammatical sentence S. Alternatively, the sequential DNP and VP representations could be input with a subsequent final synchronization of the DNP and VP taking place to produce S, and recognized as a grammatical sentence that is maintained in working memory, as illustrated in Fig 9C.

Ungrammatical sentences could be recognized when bindings occur that do not correspond to valid production rules. We do not consider here the specific mechanism or details by which the particular variables become bound, but rather show how they can dynamically emerge within working memory representations. In principle this could arise via some mapping and closeness of the associations in that cortical map, “hardwired” architectural associations in long-term memory, or some combination.

Accessible operations.

We have shown that the dynamics of our model map onto a number of working memory and binding examples. Indeed, it appears that the relevant attractor patterns are exactly the combinatorial possibilities, limited by the network size and the number of MP patterns available. Beyond ascertaining what states are available to the network, we are further interested in how the network can transition from one pattern to another. For example, in Fig 2B we see that if a third, inactive population is stimulated, the network can transition from 2 OP populations to 3 OP populations. We will consider this to be an operation available to the network; i.e., the operation of transitioning from 2 OP populations to 3 OP populations by providing a stimulus of the right strength and timing (see Methods for further protocol detail). Different patterns may require a different number of operations to attain from a given starting pattern. We refer to a transition that requires n selective, sequential stimuli to instantiate as an nth-order operation. We may assess the capabilities, and thus to some degree the plausibility, of the model more systematically by delimiting the available operations. Determining whether a pattern is accessible through a first-order or a higher-order operation may also allow for predictions to be made from the model, such as the timings involved in certain cognitive processes. We restrict the number of active populations to either two (diads) or three (triads). Our parameter choices are as above, so that WTA is always an accessible state. Thus, if we begin with a diad or triad, the only resultant patterns of interest are also diads and triads.

Fig 10 shows what operations related to diads and triads are available to the network, under the limitations described above and in Methods. In each gray circle, 1st-order operations are displayed; we may determine available higher-order operations by stringing together sequences of 1st-order operations. For example, we may transition from an OP diad to an S triad as a 2nd-order operation by stimulating an inactive population so that the network first transitions to an OP triad, and then stimulating an active population so that the network then transitions to an S triad. In particular, examining Fig 10B and 10C reveals that the network may transition from any diad or triad to any other (through transitions involving only other diads and triads) using no higher than a 3rd-order operation (i.e., no more than 3 selective, sequential excitatory stimuli are required).

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Fig 10. Accessible diad and triad operations.

(A) Using N = 5, a stimulus is given to each of the activity patterns shown. For (B) and (C), the starting activity pattern is indicated in the center of the gray circles, while the subscripts indicate which population was stimulated. The observed resultant activity patterns are indicated in smaller text around the circumference of the gray circles. For (B), one of the already-active populations, population 1, was stimulated, whereas an inactive population, population 3, was stimulated for (C).

https://doi.org/10.1371/journal.pcbi.1006517.g010

Since the stimulus is excitatory, the stimulated population will remain active. Thus, every beginning pattern will have at least two resultant patterns that will not be accessible for a given selected population that receives a stimulus (e.g., beginning with O and stimulating population 1 will ensure that s₂₃ and o₂₃ are not accessible—see Fig 10A for a notation key). By changing the population that receives the stimulus, other patterns become available in a way that is obvious by inspecting Fig 10. For example, we see in Fig 10B that an MP triad and OP diads and triads can transition to most of the remaining diads and triads: Discounting order, there are 5 possible triads and 6 possible diads; disregarding which population receives the stimulus, we see that both an MP and an OP triad can directly transition to 6 of the 10 patterns that are different from themselves. By contrast, the S populations have fewer populations accessible by 1st-order operations. This is in part an outcome of maintaining a uniform connectivity. In order to differentially affect two S populations, one of them must receive a stimulus, as any other population will affect the two S populations symmetrically. So, for example, it is as expected that the only accessible patterns for 3 S populations are the ones consisting of 3 MP populations where the selectively stimulated population oscillates out-of-phase with the remaining two. The results for s₁₂ as shown are likewise entirely expected.

One of the patterns that is difficult to directly obtain from a diad or triad is an S diad (e.g., s₁₂). We were only able to obtain this pattern by stimulating one of the active populations in an OP diad (e.g., o₁₂). Thus, going from any other pattern to an S diad requires at least a second order operation. For example, transitioning from an S triad to an S diad is at least a third order operation: With the first stimulus the network may transition to an MP triad, with the second the network may transition to an OP diad, and with the third the network may finally transition to an S diad. We note that there may be multiple paths that allow one pattern to evolve to another pattern. For example, an alternative route from the S triad to diad would be through a WTA scenario: A strong first stimulus can cause the selected population to quench the activity of the other populations, and a second stimulus to a nonactive population could cause the network to transition to an S diad.

These results show that there are a number of options to get from one pattern to another for two and three active populations, even with strong limitations on the network architecture and the stimulus protocol. For the operations that take triads to triads (as shown in Fig 10B, top), we note that the accessible operations are almost exactly what we would have predicted from the weak coupling analysis. That is, as we mention in Weak coupling, we expect transitions between basins of attraction with boundaries given as curves in Fig 7 (i.e., between OP and MP basins and between MP and S basins) to be easier to realize, and transitions between basins of attraction with boundaries given as points (as in the unstable fixed points separating OP and S basins) to be more difficult to realize. Indeed, we found it easy to transition between OP and MP states and between S and MP states, and found it more difficult to transition between OP and S states (note that we were unable to transition from S to O within the constraints given in Methods, while there is only a very narrow 7ms window that allows us to transition from O to S, just within our stated protocol). It may be of interest in future work to loosen some of the above restrictions (for example, examining heterogeneous networks or having the stimulus also drive the inhibitory components), and to further quantify the level of difficulty for a particular transition (e.g., some transitions are much less dependent on the stimulus parameters than others).

Frequencies.

In constructing our model network, we chose parameters that were biologically plausible and further found constraints on these parameters that allowed OP, S, and MP modes and transitions between them that are relevant to working memory situations. While we did not tune parameters in order to fit certain frequency bands that have been found to be of potential relevance to working memory and binding, we indeed observed some correspondence between these and the frequency relationships in the model network. For example, if we have populations that oscillate OP, then as each population becomes active, the oscillation frequency of the network as a whole (or, equivalently, of each individual population) decreases, as illustrated in Fig 11. If the number of active OP populations remains constant, the frequency of the network oscillations also decreases with increasing network size. Thus, for example, if there are three active populations with OP dynamics, each population oscillates at just over 15 Hz for a network with N = 3, and at approximately 13 Hz for a network with N = 20. We see in Fig 11 that if either N or the number of active populations corresponding to distinct memoranda increases, the frequency of oscillations for any individual population trends downward towards the alpha band.

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Fig 11. Frequency of population oscillation for different network sizes N.

We examine the effect of increasing both system size and number of active OP populations on the oscillation frequency. In both cases, the frequency decreases monotonically. The frequency of interest may also be that based on the time between peaks of active populations (e.g., the time between the peak firing rates of population 1 and population 2 in a given network). This “interpopulation frequency” may be obtained by multiplying the given population by the number of active populations. For example, the interpopulation frequency for a network size of 20 when 2 populations are active would be approximately 30 Hz.

https://doi.org/10.1371/journal.pcbi.1006517.g011

Since we are considering OP dynamics, we also note that the period between successive peaks in the case when, for example, three populations are active, is one third the overall period of the network. We may refer to the associated measure as the “interpopulation period/frequency”. As an example, let us consider a network size of N = 20 with three active OP populations. As mentioned above, the network oscillation, and thus the oscillation of any particular population, is 13 Hz. However, since three populations are active OP, the interpopulation frequency is 39 Hz. Depending on the distances between these active populations of neurons and the spatial resolution of the measurement in an experimental setting, then, increased activity near the alpha and/or gamma band may be detected. Greater numbers of populations that oscillate S may cause increased power near the alpha spectrum, since the interpopulation frequency would not increase. Increases in power in gamma frequencies and in frequencies near alpha are both consistent with neurophysiological experiments that have reported increases in gamma or alpha contributions with working memory tasks and increasing working memory load [6, 23–25, 43–47]. In addition, increased alpha-band oscillations have been reported to play an important role in mental processes related to attention and memory [48–53].