Stability as a function of coupling strengths for <i>N</i> = 2.

The dashed line at cei = 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.</p

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

When three populations are active (note, ψi 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.</p

OP dynamics for up to 3 active populations.

For a network size of 5 populations (N = 5), all combinatorial possibilities for up to 3 active OP populations may be realized. Square waves in these and similar figures indicate the stimuli given to the population just above the given wave, and the vertical dashed lines in each plot allow for phase comparisons across different active populations. The first bar below each group of traces shows the interval of starting times in which the same stimulus (fixed amplitude and width) will produce the same result; the second bar shows the length of the period of the oscillation that is active before the stimulus is applied (note: the period may change after the stimulus is applied); the third bar in (B) is explained below. (A) The network starts at a nonzero, nonactive baseline firing rate. The first stimulus selectively activates the first population, while the other populations remain inactive with low firing rates. A second population is then activated; for these parameters and stimulus strength, almost any stimulus onset time will induce the OP state with 2 active populations, as the bars show. (B, C) Stimulating a third population with a short stimulus induces the OP state with 3 active populations. Either activation ordering may occur, depending on the phase the stimulus is presented; (B) and (C) show the two different orderings. The third, dashed, bar below the stimulus trace in (B) shows the interval of onset times that induce the OP state with three active populations; the first interval shows the onset times that produce (B) while the second interval shows the onset times that produce (C). (D, E) Larger and wider stimuli may deactivate either of the active populations, so that the network remains in the OP state but with different active populations (WTS scenario). (F) Maintaining the amplitude of the stimulus from the WTS case but increasing the stimulus width allows the third selected population to deactivate both active populations and become the only active population (WTA scenario).</p

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).</p

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.</p

Model connectivity.

Left: One population as described in Eq (1). u is the fast excitatory AMPA synaptic activity, v the inhibitory GABA activity, and n the slow excitatory NMDA activity. Feedforward excitation to the AMPA synapses (u) triggers activity in the system. Middle: An example network with N = 7 populations. The elements are coupled in an all-to-all fashion as shown on the right. Right: Connectivity between two populations, each with three components as shown in the left panel. The populations are connected with excitatory (ce) and inhibitory (cei) coupling as described in Eq (2).</p

Stability of OP oscillations and working memory capacity.

For fixed τn, there is a window of τi 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 τn fixed at 144 and the mutual inhibition cei fixed at 0.03. For τi 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, τn and τi, 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, cei. Here, the 3-OP state exists above each curve for different cei values as indicated. As the mutual inhibition increases, the minimum τn 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 cei = 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 τi values as well.</p

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).</p

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 τi and cei. For M = 1, …, N (here N = 5), the M-S solution is stable for all τi values between each respective pair of (same-colored) points. For example, the 5-S solution with cei = 0.03 is stable for 3.7 τi < 36.4. The synchronous solutions are not stable outside of these intervals and are lost to different solutions, as indicated in the text.</p

Frequency of population oscillation for different network sizes <i>N</i>.

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.</p