Two different time scales of memory

Consider, for instance, a model neural circuit (see Materials and Methods) of locally connected rate-coded units. Each unit is described by a leaky membrane potential and a firing rate or activity which depends in a non-linear way (here sigmoidal) on the unit's actual membrane potential (). This formulation allows for a general interpretation of each unit as either a rate-coded neuron [40], [41] or a population of neurons [42]–[44]. Thus, the here presented results are independent of the spatial scale of the neural circuit. In the following, we will use the terms ‘unit’ and ‘neuron’ synonymously.

In the basic state every neuron receives a small noisy background input of about . For a certain period of time (here about two hours), only a local patch of neurons receives an external input of stronger intensity (see green striped area in Figure 1 A and green pulse ‘L = Learning’ in Figure 1 B,C; all inputs are noisy) while others do not and serve as control. This input mimics localized rate-coded signals from the environment or other brain areas delivered to the circuit. In the circuit plastic excitatory synapses to the nearest neighbor neurons exist (see, for instance, in Figure 1 A the purple area regarding the blue unit), as well as, short- and long-range lateral inhibition with unchanging synaptic strengths (purple and bluish gray area). For simplicity, we assume that each unit provides excitatory and inhibitory synapses. The dynamics of the excitatory synapses between neuron and is governed by the combination of synaptic plasticity and scaling defined as [31]:(1)where is the activity, a time constant of synaptic changes, the ratio of plasticity and scaling time constants, and the desired homeostatic level of activity. As shown in previous work [31], [32], the quadratic weight-dependency of the synaptic scaling term guarantees convergent synaptic weights without the need of additional constraints [45]–[49]. The synaptic plasticity part consists only of a correlation-based LTP-term. Analytical and numerical results demonstrate (see below and Text S1) that a synaptic plasticity rule consisting of a combination of LTP and LTD does not alter the general dynamics, we will discuss in the following.

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Figure 1. Increasing the input frequency yields synapses that recover their weight by global, consolidation-like stimulation.

(A) The network consists of a square grid of units with periodic boundary conditions in both directions. Each unit connects excitatorily with its nearest neighbours (see purple area regarding blue neuron) and inhibitorily with the nearest and next-nearest neighbours (purple and bluish gray area). Each unit receives an external projection (only a subset is shown). Two different input types are delivered: (i) a local learning stimulus (‘L’, green area) and (ii) a global input to all neurons (‘C’, yellow). (B,C) Different input intensities induce different activities (middle row) and weights (bottom row) of the input-target neurons (red). Pulses for local learning L are 5–10 times longer than for global consolidation C stimuli (see panels D–F for accurate stimulation-response details). Before learning short activation of all neurons (‘contr’) has no significant effect on the weights. (B) Learning signal L with . Synaptic weights of the red neurons grow but not the control weights (gray). After learning all activities relax back to background () and weights decay. Subsequent consolidation stimuli (C1,C2; ) change weights minimally. (C) Stronger learning signal L () induces stronger weight growth (red curve) than in B. Now consolidation pulses (C1,C2; as before) yield weight recovery. This happens for all stimuli that drive weights across the bifurcation level of weight decay versus recovery (dashed horizontal line). (D) Stimulation protocol during learning. (E) Mean synaptic weight shows for increasing inputs an abrupt transition ( and ). (F1,F2) Different combinations of input interval and duration robustly lead to the same weights (red neurons) for different input intensities (, ). B–F: Background input has an intensity of and all inputs are noisy (see Materials and Methods).

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Depending on the intensity of the external input, differently strong synaptic weights between the stimulated units are induced by the combined rule of plasticity and scaling (bottom panels in Figure 1 B,C). Thus, the units of the stimulated patch form a local cell assembly similar to those found in recent experiments [50]–[52] and represent a memorized version of the local external input. Small differences in input intensity ( vs. ) induce large differences in weights (bottom panels, red curves). The gray curves represent the controls from neurons that do not receive the strong external input. As we show below, these strong weights differences (red curves) arise from a generic nonlinear property of the network, where weight-formation follows a saddle-node bifurcation. This nonlinearity exhibits an intriguing phenomenon: When all units in the circuit (within and outside the cell assembly) receive a strong () but brief input (here about 15 minutes; yellow needles, ‘C1,C2 = consolidation’, in panels B,C) only the strong synapses will recover (panels C), while the weak ones continue to decay (panels B). Here this brief and global input takes the role of the coherent, but unspecific neural activation during slow-wave-sleep, which is commonly considered as a potential basis of synaptic consolidation [5], [7]. This observation is the first indication that the combination of plasticity and scaling in a simple dynamic model allows differentiating between synapses for short-term storage, which decay, from those for long-term storage, which can be recovered (or rather consolidated).

Furthermore, we note that the network has only increased activity during external stimulation. Such a stimulation yields an imbalance in neuronal circuit activity depending on the recurrent synaptic weights. Thus, the learnt cell assemblies are stronger activated than controls and the memory contents stored in the network are read-out (see below). As soon as the external input is not present any more and only background input remains, all activities relax back to background firing rate () although recurrent weights are still high (Figure 1 B,C). This is an important difference to attractor memory models [53]–[55], which will continue to be active after stimulus withdrawal for (theoretically) infinitely long time. This persistent activity is important for explaining the dynamics of working memory (seconds) but contradicts the idea of long-term memories which are not permanently active. Here, the memory content is transferred from the input to the synaptic weights [14]. The activities can relax back to background state.

We remark that the emergence of the here shown phenomena does not rely on saturation effects and fine tuned topology (see Text S1). A detailed quantification is provided below. First, we show the impact of a memory recall on the spatial structure of the LTS-synapses.

Learning and recall

During recall the spatial distribution of weights and activities reveals an interesting competitive effect (Figure 2), that is important for the formation of different memory cell assemblies and also leads to the paradox of memory loss during recall ([36], see below). Initially, during learning only the a local patch of units is stimulated and the synapses of their target neurons all grow (purple square in Figure 2 A; L-phase in Figure 1 C), where we have used a strong and local stimulus to drive all synapses into the LTS-regime. Consolidation stimulates the complete network and all synapses within the assembly recover or exceed their initial strengths (Figure 2 B; C1,C2-phase in Figure 1 C). The process of remembering (recalling) a memory is often understood as partial stimulation of an assembly and potentially of some other neurons [56]–[58]. By ways of its learnt connections the assembly produces a filling-in and generates a spatially quite complete excitation pattern including most of its members (so-called pattern completion). According to the literature [1], [14], [56]–[60] this represent the behaviorally relevant recall activity. Therefore, only a randomly selected subset of assembly-neurons receives a stimulation (we used here about with some outliers). The resulting network activity clearly shows a filled-in spatial assembly structure (Figure 2 C; Please note that due to the partial stimulus all units of the assembly are stronger active than the control ones. Thus we can assume that the assembly is completed.), where, however, sometimes strongly active neurons are neighbors of weakly active ones. For such constellations the different activities induce a dissimilar weight dynamic. Consider a pair of mutually connected neurons (see hatching in panel C). The weakly active neuron (but still more active than controls) induces a small synaptic plasticity term and synaptic scaling is weak, too. By contrast, the synaptic scaling term for the strongly active neuron is large and, thus, dominates the dynamics. As a consequence, the corresponding weight shrinks substantially (Figure 2 C, inset, yellow bars; see also Text S1 for equations).

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Figure 2. Spatial structure of activity and weights during learning, consolidation and recall.

(A) A local learning input (region marked by purple squares) leads to growth of all input driven weights. Mean weights are plotted, which naturally are smaller for border or corner neurons as they do not get inputs from outside. (B) Before consolidation, weights have decayed but will be recovered fully by a global and weak consolidation stimulus given to the whole network. (C) Recall stimulates only some of the input neurons. Nonetheless, activity is filled in and the memory pattern is completed. Note, an imbalanced activation induces a disparate development of weights between strongly and weakly active connected neurons, for example those marked by hatching in panel C. One weight shrinks substantially (see inset in C, yellow bars).

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We remark that such network structures with generic lateral inhibition admit separation of different assemblies from each other if learning stimuli do not overlap too much. On the other hand - as soon as overlap exists - activation imbalances, as described above, may lead to interference and competition between different memories. The consequence of this will be discussed in conjunction with the paradox of memory loss during recall [36] at the end of this study.

Analyzing STS- and LTS-domains

The difference between STS- and LTS-synapses in Figure 1 is a non-linear phenomenon, which is due to a saddle-node bifurcation and as such robust against changes in the stimulation patterns, representing different learning protocols. We tested a range of different input strengths and pulse protocols (Figure 1 D). Generally, for small external inputs the resulting synaptic weights depend roughly linear on the intensity (Figure 1 E) with a sudden jump to high values above a certain input intensity. The critical value, where this transition takes place, is insensitive to details in the pulse protocol (indicated by the strong weight differences shown in Figure 1 F1,F2).

The mechanism inducing this phenomenon is readily understood by investigating the dynamics of this system in more detail. We first analytically calculated the characteristic Weight-Input curve of this system.

In the following we will show in an abbreviated form the analytical calculations (see Text S1 for more details). We assume that the long-range inhibition separates the circuit into two (or more) subnetworks: (i) the externally stimulated local patch(es) and (ii) the unaffected control units. This enables us to average Equation 1 over all units within such a subnetwork. To calculate the fixed point of the resulting mean field differential equation we set it equal to zero and solve it. As result we receive the weight-nullcline of the system (The weight-nullcline is a set of states where weights do not change under the given dynamics.):(2)with as averaged value of variable . Equation 2 describes the resulting strength of the synaptic weights within a subnetwork given the dynamic of plasticity and scaling and a mean neuronal activation . As the maximal activation of each unit can not exceed (given by the input-output function ), the maximal possible synaptic weight is given by . The resulting weight-activity function in the phase space is shown in Figure 3 B,C (blue line) for the parameters used in Figure 1. Of course, the course of the function depends on the used synaptic plasticity rule (the numerator in Eq. 2), but it also shows that the LTP-term () dominates and that additional plasticity mechanisms (e.g., LTD [34] or short-term plasticity [61]) do not alter the basic dynamic (see Figure S1 in Text S1).

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Figure 3. Combination of plasticity with synaptic scaling generates two distinct weight domains representing short-term and long-term storage.

(A) Both domains (STS and LTS) arise from a bifurcation (see main text and Text S1) between two stable fixed point domains (continuous lines). Depending on the input, weights either continuously grow (control and STS-domain) or suddenly jump to a high value (LTS-domain). Between both domains there is a transition range (dashed line). Blue dots show results from numerical simulations (Figure 1), which match the analytical curve. (B) Fixed points are defined by the intersection between activity- (green) and weight- (blue) nullclines. As long as all three fixed points (green stable; red unstable) exists the phase space is divided into two attractor regimes which are also indicated by arrows (). (C) Higher frequencies shift the activity-nullcline (red line) upwards which results in only one attractor regime (LTS; ). (D) Passive weight decay happens for all synapses as long as there is no consolidation stimulus present. Dashed parts of the curve indicate that LTS-synapses can be consolidated.

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The average activity within a subnetwork induces certain synaptic strengths (Equation 2). In turn, the mean external input (multiplied by the input weight ) and the average recurrent synaptic weights themselves adapt the average activity. The resulting fixed point of this dynamic is calculated by the mean field differential equation of the membrane potential (Eq. 4). This yields the activity-nullcline (In analogy to the weight-nullcline, the activity-nullcline is a set of states where activities do not change.):(3)with membrane resistance and average excitatory () and inhibitory () number of connections per unit within the subnetwork. As and are the only topology-related parameters in this equation (and Eq. 2), the described dynamics are independent of the detailed topology (see Figure S2 in Text S1). The activity-nullcline follows roughly the sigmoidal shape of the activation function (Eq. 5). Furthermore, it shows that external inputs of different intensity delivered to the circuit change the neuronal activation (see green line in Figure 3 B for compared to the red line in panel C for ) and, therefore, (via Eq. 2) the synaptic weights. The direct influence of the external input on the synaptic weights within a subnetwork can be assessed by calculating the intersections between both nullclines. These intersections are the fixed points of the whole subnetwork (activity as well as weights). The resulting fixed point equation has no closed-form solution and, therefore, has to be solved numerically. Direct simulations of the whole circuit (Euler-method) match our theoretical predictions (Figure 3 A).

Specifically, we find a saddle node bifurcation where different fixed points are reached for low as compared to high input intensities. For the particular setting displayed in Figure 3, a continuous regime of fixed points for the weights exists for firing rates below approximately (Short-Term Storage, STS; green, Figure 3 A), while above this frequency, the system jumps to a fixed point regime with substantially larger weights (Long-Term Storage, LTS; red, Figure 3 A). The gray area below STS represents the range of weights found for the randomly stimulated control neurons (targets of the yellow neurons in Figure 1 A). Note, to obtain this curve we assumed that the circuit consists of several roughly independent subnetworks. This means that in one circuit different fixed points are reached at different spatial locations. For instance, in Figure 1 C after local stimulation the (local) patch is in the LTS-regime (about in Figure 3 A) while the control units are weakly stimulated and, therefore, they are in the gray control regime (about ) with small synaptic weights. The bifurcation is essential for the dynamics discussed here. Using different parameter values for the system does not change the fixed point curve significantly (see, e.g., Figure 4 B and Figure S3 in Text S1 compared to the used setting shown in Figure 4 A and Figure 1 B,C). However, if one parameter is changed dramatically an adequate adaption of the other parameters can still guarantee the desired circuit dynamics (see Figure 4 C,D). Thereby, the range of parameters remains in a physiological regime.

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Figure 4. Robustness of bifurcation structure.

(A) : The vs. function of the fixed points of the system as already shown in Figure 3 A. For simplicity we show here only the curve without indicating the different storage domains. (B) : Changing, for instance, the desired firing rate parameter of the synaptic scaling term does not induce significant changes in the vs. function. The overall circuit dynamics are the same as shown in Figure 1 (see Figure S3 in Text S1). This holds for negative values (not shown), too. (C) : Only a dramatically different value induces changes in system's dynamic. Here, a pole emerges for small input intensities. To avoid this pole and maintain the desired dynamic the background input could be increased () to keep the system on the right side of the pole. Alternatively, other parameters could be adapted. For instance, (D1) the steepness of the neuronal output function () or (D2) the inflexion point () have to be decreased.

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The emergence of the (desired) form of the vs. function can be explained by the changing locations of the nullclines in the phase space (Figure 3 B,C). For small input frequencies, the nullclines intersect at three different points and, therefore, two stable and one unstable fixed points exist (green and red markers in Figure 3 B). As weights gradually start to grow from low values, the system gets trapped in the lower stable fixed point in the STS-domain. For high input frequencies only one stable fixed point exists which is in the LTS-domain (Figure 3 C). As soon as the strong external input ends, only the lower fixed point exists and the weights start to decay and, without further inputs, reach control values after maximally ten days (Figure 3 D and Text S1). However, brief consolidation inputs prevent this as discussed next.

Consolidation of memory

Bifurcation analysis also helps to understand why synapses with values in the upper fixed point regime (LTS-synapses) respond to global and unspecific consolidation inputs while others do not. Weight changes strongly differ for differently strong initial weights when presenting a single consolidation stimulus (Figure 5 A and Figure S6 in Text S1). Weights above the bifurcation threshold (dashed line) are increased substantially, while those under the threshold are almost unaffected (close beneath threshold they rather decrease due to the lateral inhibition, see Figure 1 A, top). This phenomenon is robust against the duration of the consolidation stimulus (Figure S4 in Text S1). As a consequence, while all weights decay after learning, consolidation will recover those above bifurcation threshold. Hence, consolidation must not come too late, or also those weights might have dropped beneath threshold from which they cannot be recovered (Figure 5 B). Note, this phenomenon is not “history dependent”, which means it does not matter whether learning or consolidation had driven the weights into the LTS-regime before decay has set in (Figure S5 in Text S1).

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Figure 5. Consolidation qualitatively relies on synaptic strength and temporal protocol.

(A) The global activation signal (, ) induces the recovery of those synaptic weights which are above the bifurcation threshold (LTS-regime; dashed line). For synapses just under the threshold consolidation mildly decreases the weights. Decrease arises from the network inhibition (network as in Figure 1 A). (B) Recovery of weights occurs only if the consolidation stimulus occurs early enough (C1; upper panel). When the stimulus is too late (C2; lower panel), weights have dropped into the STS-regime and cannot be recovered. (C) Weight recovery is robust to changes of interval and duration of the consolidation signal (, and ). For intervals up to 24 h relatively short consolidation signals – either delivered as one or as many pulses – suffice for recovery.

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We remark that our model solely captures dynamic network effects and that we do not attempt to model systems consolidation, which relies on complex and little-understood physiological processes. It appears, however, important that the here observed dynamic properties of such a network allow synapses to maintain (and regain) their stability such that systems consolidation or other processes may find a stable substrate to operate on. The wide parameter range within which this happens (Figure 5 C) supports this argument, because recovery is robust and stable. Only if the consolidation input is too short or too late, forgetting sets in.

As consolidation is a sleep-induced effect [5], [6], [10], little is known about the actual activity characteristics of the consolidation process. Input intensities required for consolidation are similar to those for initiation (similar to Figure 1 E), but emphasis lies on the fact that for consolidation the whole network is stimulated in an unspecific way and that the consolidation stimuli can be shorter (in Figure 1 B,C about 15 minutes of total duration). Additionally, similar to during sleep induced activations (e.g., spindles or ripples [6]), the memory-related cell assembly is reconstructed (“replayed”; see, e.g., [62] for review) during the consolidation input (Figure 2 B).

Stimulus-dependent destabilization of memory

The recall of a previously well-learnt memory item may lead to the paradoxical phenomenon that this memory will be less well remembered than a newly learnt one. In the literature, this phenomenon is widely interpreted as memory destabilization or rather disruption [4], [11]–[13], [37], [63] and has been found in some studies [11], [64], but not in others [65], [66]. Thus, the question arises what the dynamical processes are that underlie it and especially also why memory destabilization/disruption depends on details of experimental protocols. In one specific experimental paradigm [36] destabilizing happens due to the interference of a new memory item with the previously learnt first memory, but only if the first memory was recalled before the second was learnt. In this protocol the first memory is impaired, while the new one is now susceptible to consolidation. In the following, we show that combined plasticity and scaling also naturally accounts for this paradox. We compare the experimental paradigm with the collective dynamics of our model system and highlight reasons for the ambivalence about the emergence of this phenomenon [11], [64]–[66].

In a series of elegant experiments, Walker et al. [36] have shown that destabilization of memory happens during a motor learning task. In a control experiment (Figure 6, Protocol 1) human subjects were first trained only on one motor sequence (learning, L1, blue, day one) and then tested once on day two (recall, R2) and day three (recall, R3). Significant improvement in accuracy was observed at day two, but not at day three (Figure 6 A). In the second control experiment (Figure 6, Protocol 2) subjects had been trained on the first sequence on day one (L1, blue) and on a different, second sequence on day two (L2, red), hence 24 h later. Testing was done on day three (R3, blue and red) and performance had improved for both sequences equally (Figure 6 B, blue and red bars). Both observations (panels A and B) were explained [36] by the overnight consolidation (C1, C2) of the memory. In the third experiment (Figure 6, Protocol 3) subjects learnt the first sequence on day one and were – as above – tested on day two (R2, blue) showing the same clear improvement (Figure 6 C, left blue bar). Immediately after testing they had to learn sequence two (L2, red). When re-tested on the third day (R3, blue and red) performance had significantly improved for sequence two but dramatically dropped for sequence one (Figure 6 C, right blue and red bars). This indicates that the second memory interferes with the first but only when the first is activated before the second was learnt.

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Figure 6. Learning new memory can induce disruption of previously learnt memory in experiment and model.

(A–C) Experimental results recompiled from Walker et al. [36] showing different memory characteristics in human subjects for a motor learning task over three days. Bar plots (insets) show the relative performance change compared to the previous measured data point. The main panels show the absolute performance change over time. Blue represents the first and red the second memory; ‘L’ is learning, ‘C’ consolidation, and ‘R’ recall of a memory at the respective day. (D–F) Performance indices of the model are calculated across all neurons of the memory-related sub-population. For details see in Materials and Methods. (G–I) Temporal weight development averaged over sub-population. Dashed arrows indicate consolidation at “night”. (A) Consolidation leads to significant performance improvement after one night for a single learnt memory (blue). (B) Later learning of a second memory (day 2, red) results to improvement of both (blue and red). (C) Recall of the first (R2) before learning the second memory (L2) induces strong performance decrease for the first (R3, blue) but not for the second memory (R3, red). (D–F) Performance indices (black dots) from the model at the respective points in time follow the same characteristics as human performance in (A–C). (G,H) Consolidation leads to recovery of the corresponding weights in both control protocols but (I) not for the first (blue) memory when using protocol 3, which leads to a massive reduction of LTS-synapses. (J–L) Fraction of memory-related synapses in the LTS-domain.

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In our model setting, we performed an identical set of experiments, i.e., with the same learning and testing sequences as used for the human subjects. The model was set up with two cell assemblies, partially overlapping at a corner. Assembly one (blue) was trained on one input sequence and assembly two (red) on another sequence. For recall – as explained above (Figure 2) – we stimulate only a randomly selected subset of 30% of the original neurons. Connectivity and all other parameters were the same as before (Figure 1). Training of either sequence leads to increased synaptic weights which are in the LTS-domain, hence, large enough to allow for consolidation. Consolidation stimuli, C1 and C2, were applied “at night”, where we briefly (three times 15 min) stimulated the whole network (similar to the procedures in Figure 1), as indicated by the dashed arrows in panels G–I. In these panels one can also see the development of the synaptic weights for the first (blue) and the second (red) cell assembly for all three experiments. Performance indices of the model (Figure 6 D–F) are similar to those for the human experiments and we find that data points for the two control experiments match (Figure 6 A,D and B,E). Moreover, also the non-trivial effect on memory disruption is robustly reproduced by the model (Figure 6 C,F). The weight growth normally happening at consolidation C2 is only visible in the control protocols (Figure 6 G,H). By contrast, the readout that happens for protocol 3 at R2 effectively prevents the first memory from consolidation (Figure 6 I).

This phenomenon based on the intrinsic competitive effect arising from activation imbalances already discussed for Figure 2 (see inset in panel C) above. This can be seen in panel G here (see box with magnification), as the recalls R2 and R3 yield a reduction of the average weight curve, without inducing transitions from the LTS- to STS-regime. Learning the second memory acts for the first assembly “like a recall”, due to the partial overlap between assemblies. This is visible in panel H (box). Thus, learning a second memory can reduce the average weights of the first one. In panel H all weights are far above threshold and both assemblies can be consolidated. This is different for the last experiment (panel I). Recall R2 together with learning the other sequence L2 pushes the blue curve down more strongly (see box) than in panels G and H such that it has dropped under the bifurcation threshold when consolidation C2 happens. Close beneath threshold we remember that consolidation acts disruptive (see negative parts of the curve in Figure 5 A), which leads to a further weight decrease at time point C2. Panels J–L show the time courses of the fraction of synapses of each cell assembly that are in the LTS-domain, which corresponds to the above discussed effects. We remark that we have set all parameters in this simulation purposefully so that we can in panel I exactly depict the critical bifurcation point, where at C2 the red weights are just above threshold while the blue ones are just below and the first memory is disrupted. This is meant to emphasize that the transition from the LTS- to the STS-regime, which is a qualitative change, is sensitive to the experimental parameters. This might underly the fact that destabilization, which leads to an actual memory disruption, is not always found in real experiments [65], [66]. While recall and learning of other memories can robustly destabilize a memory, it is the relation of the weight-values relative to the bifurcation threshold, which can give rise to memory disruption (or not). A detailed parameter analysis of the destabilization phenomenon, confirming its robustness, is provided in the supplemental information (Figure S7 in Text S1). This analysis shows that only, if weights are too big or stimulation for recall is too broad and not competitive enough, transitions from the LTS- to the STS-domain do not happen as the system will not travel through the bifurcation. We remark that several recalls briefly after each other affect the same subset of synapses and, therefore, a destabilized memory can not be destabilized further by applying more recalls.

More specifically, we observe that the overlap between the cell assemblies, related to the fraction of reactivated neurons during recall, is the most critical factor which determines whether one assembly can be destabilized (Figure 7). Zero overlap - trivially so - leads to no disturbance (not shown), small overlap represents the situation which is most strongly susceptible to the disruption of a long term memory (Figure 7, left rows), where more synapses move from the LTS- to the STS-domain than vice versa. By contrast, for a large overlap both assemblies drive each other up into the LTS-domain (Figure 7, right rows). Intuitively this makes sense. Large overlap means that both memories are very similar, hence they might as well begin to couple themselves in an associative (hebbian) way. For small overlap the (dis-)similarity of the memories might rather be “confusing” and an agent (animal/human) might benefit from forgetting one of them not being able to decide whether they are the same or different. It would be interesting to investigate this from a psychophysical point of view. We expect that memory similarity is the crucial factor which determines the capabilities of the system for memory maintenance versus destabilization.

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Figure 7. Transitions of synapses between LTS- and STS-regime for different degrees of cell assembly overlap related to the fraction of reactivated neurons during recall.

Columns present the fraction or overlap of activated neurons (randomly chosen) in percent of assembly size, rows show how many synapses are in the LTS- or STS-domain (STS: green; LTS: red) before (top) and after (bottom) recall. The middle row shows how many synapses have actually changed their role during the recall. Duration of recall is .

https://doi.org/10.1371/journal.pcbi.1003307.g007

Network

The network consists of a circuit (Figure 1 A) with units. Each unit receives an external input with fixed weight . Furthermore, each unit has plastic excitatory connections to its nearest-neighbors (purple area in Figure 1 A regarding blue unit) and constant inhibitory connections to its nearest and next-nearest neighbors (bluish gray and purple area in Figure 1 A). We remark that the specific layout of this topography is not relevant for the results obtained here (see Figure S2 in Text S1), as long as there is a competition between local excitation and longer-ranging inhibition.

Each neuron in the circuit is defined by its leaky membrane potential which changes according to(4)with membrane time constant , resistance , and external input given by with unchanging input weights . The input is modulated by a noise term drawn each time step from a normal distribution with mean zero and standard deviation . In all simulations the (abstract) membrane potential ranges from values about to .

The membrane potential is non-linearly transformed to a firing rate by a sigmoidal-function:(5)where is the maximum firing rate, the steepness of the sigmoidal function, and its inflexion point. All parameters combined specify the input-output behavior of the unit.

Only the excitatory synapses in the second layer are modified using the “Synaptic Plasticity and Synaptic Scaling” (SPaSS)-rule [31]:(6)where defines the plasticity rate and the ratio between plasticity- and scaling rate. The desired ‘target’ firing rate of synaptic scaling is given by . A detailed analysis of the properties of this rule is provided elsewhere [31], [32].

All equations are solved analytically in a mean field approach (see Results section and Text S1) and numerically with the Euler method (). In the following, we provide the parameters used (if not stated otherwise). For numerical simulations, we set , , , thus, the circuit is a 2-d grid. The inhibitory and projection weights are proportional to the maximal possible weight: and with . The neuronal parameters are , , , , and . The here shown results are independent of . Although, a smaller value would be biological more reasonable, we took as this avoids numeric instabilities (). The plasticity parameters are , , and . To avoid boundary effects, we used periodic boundary conditions resulting in a toroidal network topology.

Learning and recall protocol for reconsolidation experiment

In Walker et al. [36] training and recall of memory items differ in the number of blocks each consisting of 30 seconds task followed by 30 seconds rest. Here we use 36 blocks for a training session and 10 blocks for recall. Throughout the task a stimulus of intensity is given to the memory-related neurons (). Consolidation signals consist of three blocks with 15 minutes whole network stimulation () followed by 15 minutes pause. Every time step gaussian noise is added to the external stimuli as mentioned above but with a standard deviation of .

For Walker et al. [36] as well as for model results all values in the insets of Figure 6 are average values over 10 trials. Data points (black dots) in the main panels have been calculated from the bar plots by us; connecting lines are for graphical reasons only. Performance indices of the model are calculated as time- and space-averages of the synaptic weights across all neurons of the respective sub-populations. The time averages have been obtained over five blocks. These are the five last task blocks used for recalls or learning (the learning pulses define the value).