Oscillations in working memory and neural binding: A mechanism for multiple memories and their interactions

Neural oscillations have been recorded and implicated in many different basic brain and cognitive processes. For example, oscillatory neural activity has been suggested to play a role in binding and in the maintenance of information in working memory. With respect to the latter, the majority of work has focused primarily on oscillations in terms of providing a “code” in working memory. However, oscillations may additionally play a fundamental role by enabling or facilitating essential properties and behaviors that neuronal networks must exhibit in order to produce functional working memory and the processes it supports, such as combining items in memory into bound objects or separating bound objects into distinct items. In the present work, we present a biologically plausible working memory model and demonstrate that specific types of stable oscillatory dynamics that arise may play critical roles in providing mechanisms for working memory and the cognitive functions that it supports. Specifically, these roles include (1) enabling a range of different types of binding, (2) both enabling and limiting capacities of bound and distinct items held active in working memory, and (3) facilitating transitions between active working memory states as required in cognitive function. Several key results arise within the examinations, such as the occurrence of different network capacities for working memory and binding, differences in processing times for transitions in working memory states, and the emergence of a combinatorially rich and complex range of oscillatory states that are sufficient to map onto a wide range of cognitive operations supported by working memory, such as variable binding, reasoning, and language. In particular, we show that these oscillatory states and their transitions can provide a specific instantiation of current established connectionist models in representing these functions. Finally, we further characterize the dependence of the relevant oscillatory solutions on certain critical parameters, including mutual inhibition and synaptic timescales.

As c e and c ei increase, the small peak becomes more pronounced. (A) c e = 0, c ei = 0.01. Here no small peak occurs. (B) c e = 0, c ei = 0.54. The primary population (top plot at the beginning of the timecourse) begins to rise again after its first large excursion from its baseline value, but the secondary (bottom plot at the start of the trace) population comes on and is seen to stop the first population in its tracks, so that only a small, secondary excursion occurs. (C) c e = 0.055, c ei = 0.0855. The presence of the excitatory coupling from nonzero c e pushes the small peak in such a way as to be more coincident with the large peak of the alternate population; however, the inhibition due to c ei still suppresses it, again only allowing for a small excursion from baseline. components, perhaps the most important change is the increase in the amplitude of the 18 small peak. Once the the small peak increases too much, the secondary population may 19 no longer be suppressed, becoming active and perhaps synchronizing with the primary 20 population. This is manifested as a bifurcation (see Fig 6C in  Fixing c ei and varying c e results in (1) slightly decreased period; (2) decreased large 26 peak; (3) increased small peak (Table 1). Suppose again that P1 is the primary 27 population and P2 is the secondary population, so that u 1 > u 2 in the interval of 28 interest. We first observe that as c e increases, u 2 receives significantly more excitation, 29 while u 1 receives less excitation. This is a direct result of the normalization we have 30 used. In particular, from Eq (2) in the main text we see that for two populations, 31 u 1 = u 1 + c e u 2 1 + c e . An example of the main changes to the OP state for N = 2 that occur as c ei increases while c e remains fixed 0. Note that the value for the small u peak indicates "not applicable" since the peak in the secondary excitatory component only occurs for large enough c ei and c e .

PLOS
Differentiating u 1 with respect to c e shows that it monotonically decreases as c e 32 increases if u 1 > u 2 , as we have assumed. As a result, u 1 's maximum decreases, so that 33 the width of the pulse of u 1 decreases, whereas the length of its quiescent phase 34 experiences almost no change. Thus, the small decrease in period is mostly due to the 35 decrease in the amplitude of the large peak. For nonzero c ei , the increase in the 36 maximum of u 2 in turn further excites v 2 , so that u 1 experiences greater inhibition.

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Thus, in addition to the effect just now described, increasing c e results in a lower 38 maximum for u 1 since v 2 inhibits u 1 more when c ei is nonzero. In either case, once the 39 maximum of the secondary u becomes too large relative to that of the primary u, the 40 splay state is lost to synchrony.

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Fixed c e , varying c ei

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As c ei increases, we observe the following changes: (1) the peak of the large 43 amplitude changes, increasing monotonically until just before bifurcation; (2) the small 44 peak changes, generally increasing monotonically; (3) the period of the oscillation 45 changes, increasing monotonically until just before bifurcation; (4) the inhibition 46 generally peaks at larger values and decays to smaller values; (5) the NMDA generally decays to lower levels ( Table 2). There are some subtle differences when c e is low or 48 high, and we discuss each in turn.
suppose P1 is the primary population and P2 is the secondary population. For 51 simplicity we will focus on c e = 0. The excitatory component of the primary population, 52 u 1 , cannot begin its larger upstroke until v 1 is sufficiently low. Since v 1 is very low 53 before u 1 begins its large upstroke, v 2 provides most of the inhibition that keeps u 1 low 54 before its large upstroke. However, we again look at the coupling term: Differentiating with respect to c ei , we see that v 1 increases (for fixed v 1 and v 2 ) with 56 increasing c ei when v 2 > v 1 . Thus, as c ei increases, both v 1 and v 2 must decay to lower 57 values before releasing u 1 .

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For identical reasons, u 2 receives less inhibition at the beginning of the large 59 upstroke of u 1 as c ei increases. That is, v 2 decreases (for fixed v 1 and v 2 ) as c ei 60 increases when v 2 > v 1 . This leads to lower inhibition for u 2 as it begins its small 61 upstroke, allowing it to peak at a higher value (we note that n 2 has also decayed to a 62 lower value, leading to less excitation for u 2 ; however, since a en is much smaller than 63 a ei , this effect is much smaller).

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In summary, larger c ei requires lower inhibitory component values in order for the directly to the nonmonotonic behavior of curve (iv) in Fig 6C in  for some c e values, the OP state is lost, regained, and lost again as folds of limit cycles 79 with increasing c ei . This appears to be due directly to the nonmonotonic behavior of the 80 small peak for the reasons we described above. In particular, the c ei values at which the 81 amplitude of the small peak begins to decrease converge to the minimum of curve (iv) in 82 Fig 6C in the main text. We note that this competition that leads to the nonmonotonic 83 curve depends on the particular parameters. We have explored other parameter sets, 84 e.g., with which this curve simply decreases monotonically as c ei increases. we provide some heuristic reasoning for why this may be the case. 89 We first note that when M = N , as we described at the beginning of Maximum S as for M = 2 or 3 and nearly in the same way as for M = 4, we will focus on the simplest case of M = 1.
As τ i increases, the period lengthens and the maxima of the excitatory and inhibitory 108 solutions increase. As we explain in S2 Text, if τ i is too large relative to τ n , the NMDA 109 will not outlast the inhibition and the oscillations will cease. If τ i is too small (and we 110 are not in a parameter regime that allows for a stable high steady state; see S2 Text) the 111 inhibition activates very rapidly, quenching the excitatory activity before it increases this might occur, consider M = 1, suppose that P1 is the active population, and note 119 that the populations that are inactive are not only at similarly low levels, but in fact are 120 themselves oscillating synchronously at low values. Therefore, the inhibition that u 1 121 receives is given by Eq (2) in the main text, which we specify for the case of v 1 here: where j can be anything in {2, ..., N } since, as we mentioned, {P2, ..., PN } are 124 synchronous. This, of course, is identical to Eq (1), and so the same analysis can be 125 applied as was done in that case. In particular, we note that since P1 is the only active 126 population, u 1 is always (or nearly so) larger than u j , for j ∈ {2, ..., N }. Thus,

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following the above analysis (where P1 is essentially always the primary population), v 1 128 decreases with increasing c ei , so that u 1 receives less inhibition. This exactly explains 129 the increase in the maxima of u 1 with increasing c ei (and, since v j is excited by u j , the 130 increase in the maximum of v 1 as well). More relevantly, u 1 needs less excitation from 131 n 1 to maintain the large oscillations for larger c ei values, allowing the oscillations to 132 remain stable for larger τ i values, as we see in Fig 4D in the main text.

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The same explanation holds for low τ i values. As c ei increases, u 1 receives less 134 inhibition, so that u 1 can maintain its activity for smaller τ i . For example, suppose 135 c ei = 0.03 and τ i = 2.9, the lower point for M = 1 in Fig 4D in the main text. As c ei 136 increases, u 1 receives less inhibition, so that v 1 needs to activate even faster to prevent 137 u 1 from its large excursion from baseline. Thus, τ i must be lowered still further for 138 c ei = 0.07 before P1 will be unable to remain active (down to τ i = 2 in this example).