A mechanistic basis for systems memory consolidation

The suggested parallel pathway theory (PPT) relies on a parallel structure of feedforward connections onto the same output area: a direct, monosynaptic and an indirect, multisynaptic pathway. We propose that memories are initially stored in the indirect pathway and are subsequently transferred to the direct pathway via Hebbian plasticity. Because the indirect pathway is multisynaptic, it transmits signals with a longer time delay than the direct pathway (Fig 1A). A timing-dependent plasticity rule allows the indirect pathway to act as a teacher for the direct pathway.

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Fig 1. A mechanistic basis for systems memory consolidation.

(A) Circuit motif for the parallel pathway theory. Cue-response associations are initially stored in an indirect synaptic pathway (blue) and consolidated into a parallel direct pathway (red). (B) Hippocampal connectivity. The entorhinal cortex projects to CA1 through an indirect pathway via DG-CA3 and the Schaffer collaterals (SC, blue arrow), and through the direct perforant path (PP_CA1, red arrow). (C) Model of consolidation through STDP. Left: before consolidation, a strong SC input (middle, blue vertical bar) causes a large EPSP and triggers a spike in CA1 (bottom, black vertical bar). A weak PP_CA1 input (top, red) that precedes the SC input is potentiated by STDP. Right: after consolidation through STDP, the PP_CA1 input (top) can trigger a spike in CA1 by itself (bottom). (D-E) Consolidation in a single integrate-and-fire CA1 cell receiving 1000 PP_CA1 and 1000 SC excitatory inputs. (D) PP_CA1 activity consists of independent poisson spike trains; the SC activity is an exact copy of the PP_CA1 activity, delayed by 5 ms. (E) Consolidation of a synaptic weight pattern from non-plastic SC synapses to plastic PP_CA1 synapses. Left and middle: normalized synaptic weights before and after consolidation. Right: time course of correlation between SC and PP_CA1 weight vectors during consolidation (mean ± SEM for 10 trials). (F) Failure of consolidation of a synaptic weight pattern from non-plastic PP_CA1 to plastic SC synapses; panels as in E.

https://doi.org/10.1371/journal.pcbi.1009681.g001

The proposed mechanism can be exemplified in the hippocampal formation, by considering direct and indirect pathways to area CA1. CA1 receives a direct, monosynaptic pathway from the entorhinal cortex (EC), which is called perforant path (PP_CA1, Fig 1B, red; [29]). In addition, EC input is relayed to CA1 via the classical trisynaptic pathway via dentate gyrus (DG) and CA3, reaching CA1 through the Schaffer collaterals (SC; Fig 1B, blue; [29]).

As in earlier theories, we assume that the indirect pathway via CA3 is involved in the original storage of memories [30, 31], an assumption that is supported by experiments, e.g. [32–34]. We neglect, for simplicity, any encoding-related change in the direct pathway, even though in animals this pathway might also show some, putative much lower, plasticity during memory acquisition. This simplification does not affect our proposed mechanism on the consolidation-related transfer of memories.

We assume encoding in such a way that a memory can be recalled by a specific neural activity pattern in EC—a cue—that triggers spikes in a subset of CA1 cells through this indirect pathway via the SC, representing the associated response. The same cue reaches CA1 also through the direct pathway via the PP_CA1. We assume that this direct input from EC initially fails to trigger spikes because the synaptic weight pattern in the PP_CA1 does not match the cue. However, PP_CA1 inputs that are activated by the cue precede the spikes in CA1 pyramidal cells that are triggered by the indirect pathway by 5–15 ms [35] due to transmission delays. Presynaptic spikes preceding postsynaptic spikes with a short delay favor selective long-term potentiation by spike timing-dependent plasticity (STDP, Fig 1C) [36–38]. Consequently, cue-driven PP_CA1 synapses onto activated CA1 cells are strengthened until the memory that was initially stored in the indirect pathway can be recalled via the direct pathway alone. The indirect pathway thus acts as a teacher for the direct pathway.

To illustrate this mechanism, we used a simple integrate-and-fire neuron model (for details, see Methods) of a CA1 cell that receives inputs through the SC and the PP_CA1. We also considered the two pathways to contain the same number of synapses and transmit identical spike patterns apart from a 5-ms delay in the SC (Fig 1D). Consolidation then corresponds to copying the synaptic weight pattern of the SC to the PP_CA1. In line with our hypothesis, such a consolidation was indeed achieved by STDP in the PP_CA1 synapses (Fig 1E). A consolidation in the opposite direction, i.e., from the PP_CA1to the SC cannot be achieved by STDP because the temporal order of spiking activity is reversed and hence does not favour synaptic potentiation (Fig 1F). Note that in this simple example, the EC-to-DG/CA3 synapses don’t store any memory, but only introduce the transmission delay. In the following, we will show that all synapses of the indirect pathway can be involved in the original storage of memories.

To understand the conditions under which the suggested PPT can achieve a consolidation of associative memories, we performed a mathematical analysis, which shows that consolidation is robust to differences in the neural representation in the two pathways and illustrates its dependence on the temporal input statistics in the two pathways. Readers who are less interested in the mathematical details are welcome to jump to section “Consolidation of spatial representations”, where we show in simulations that the mechanism is robust to neuronal complexities; in subsequent sections, we also show that the mechanism accounts for lesion studies in rodents, and that it can be hierarchically iterated.

Effects of temporal input statistics on systems memory consolidation.

The constants A and B, which were defined in Eq (6) as A ≔ ∫dτ L(τ)f(τ) and B ≔ ∫dτ L(τ)g(τ − D), play an important role for the learning dynamics. As already elaborated, the sign of A determines stability while B should be positive to obtain consolidation. Sign and magnitude depend on the interplay between the learning window L and the temporal input statistics, characterized by the correlation functions f and g defined in Eqs (4) and (5), respectively. For the assumed separable statistics, f is fully determined by the autocorrelation of the input in the direct path, and f(τ) is therefore symmetric in time τ.

A first interesting observation is that for the special case of an antisymmetric learning window L, we obtain A = 0 for symmetry reasons. Mathematically, this implies that the first term of the learning dynamics in Eq (6)—the dependence of the change of the weights W in the direct path on their actual value—vanishes. Intuitively, the balance of potentiation and depression in an antisymmetric learning window implies that the direct path, although able to drive the postsynaptic neuron, is causing equal amounts of potentiation and depression in all of its synapses. On average, synaptic changes are caused only by the indirect pathway with weights V, which therefore acts as a supervisor for the learning dynamics of W in the direct path. A thorough analysis under which conditions STDP can be used for supervised learning has been provided elsewhere [42, 43], and the results of this analysis are applicable in the present case. Functionally, the depressing part of an STDP learning window serves to neutralize the impact of the direct pathway on its own learning dynamics, effectively creating a supervised learning scenario.

Another interesting observation relates to the magnitude of the terms A and B, which is determined by the time scale on which the inputs change (reflected, e.g., in the time constants of the decay of the correlation functions f and g). Let us assume that both correlation functions f(τ) and g(τ) are maximal for τ = 0 and that they decay to 0 for large |τ|; such conditions are reasonable for most correlation structures. We also assume that the learning window has the typical structure of potentiation for causal timing, L(τ) > 0 for τ > 0, and depression for acausal timing, L(τ) < 0 for τ < 0 [36, 37, 44]. Then the delay D > 0 in the indirect path shifts the maximum of the cross-correlation g(τ − D) into the potentiating part of the learning window (Fig 2B) while the maximum of f(τ) remains in the transition region of potentiation and depression (Fig 2A). The following three observations can be made concerning the constant B as defined by the integral in Eq (6):

1. (1). If the cross-correlation g has a narrow enough peak (i.e., narrower than the time scale of the learning window and the delay D), B is positive, suggesting that consolidation can occur (Fig 2B). The sharp localization of g corresponds to rapidly changing input signals.
2. (2). If the decay time constant of the cross-correlation g is large compared to that of the learning window, the depressing component of the learning window has more impact and reduces the constant B and thus the efficiency of consolidation (Fig 2C). In the case where the learning window is dominated by depression, B can even get negative for large time constants of g, abolishing consolidation altogether.
3. (3). If the delay D along the indirect path is much longer than the decay time constant of the learning window, we obtain B ≈ 0, meaning that consolidation is abolished (Fig 2D). In other words, the delayed correlations between the two pathways are too large to be exploited by STDP. This will limit the ability to consolidate from too long indirect paths into shortcuts.

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Fig 2. Interaction of temporal correlations and the STDP learning window.

The weight dynamics of the direct path [Eq (6)] is driven by inputs from the direct and indirect paths: weight changes are determined by the integrated products of the STDP learning window L with the autocorrelation f [Eq (4)] and the cross-correlation g [Eq (5)], respectively. (A) Examples of a learning window L(τ) and an autocorrelation f(τ), both plotted as a function of the “relative timing” τ. For separable statistics, f is symmetric. If the learning window L has a stronger negative part for τ < 0 and a weaker positive part for τ > 0, the coefficient A ≔ ∫dτ L(τ)f(τ) is typically negative. (B)–(D) Learning window L as in (A) and three example cross-correlations g. (B) The indirect path primarily induces potentiation in the direct path if B ≔ ∫dτ L(τ)g(τ − D) > 0. This is the case if (i) the delay D between the paths is positive, (ii) the learning window is positive for positive delays, and (iii) the time scale of the decay of cross-correlations g is shorter than the delay D and the width of the learning window L. These three conditions favor consolidation. (C) If the cross-correlation g decays on a time scale that is much longer than the width of the learning window and the delay D, the indirect path can drive both potentiation and depression, and consolidation is weaker (i.e., the coefficient B is smaller) than for shorter correlations. (D) If the delay D between the direct and the indirect paths is longer than the width of the learning window L, the indirect path cannot induce systematic changes in the weights of the direct path (coefficient B ≈ 0), and consolidation is ineffective.

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Consolidation of spatial representations

The mathematical analysis of the PPT makes two key predictions. First, it suggests that STDP in a parallel direct pathway achieves consolidation by performing a linear regression between inputs in the direct and the indirect pathways [Eqs (7) and (8)]. Therefore, the proposed mechanism should generalize to situations in which the cue representations in the direct and indirect pathways differ. Second, the theory suggests that consolidation is most effective when the correlation time constants of the input during consolidation is matched to the coincidence time scale of STDP (Fig 2B). In the following, we will show in simulations that those predictions hold and, moreover, that the mechanism is robust to neuronal nonlinearities.

To begin with, we show that the mechanism is robust to differing cue representations in the two pathways and to weaker correlations among them [45]. To this end, we used place cell representations [46] for the SC input from CA3 and grid cell representations [47, 48] for the PP_CA1input from EC (Fig 3A). Moreover, we show that the suggested mechanism is compatible with the biophysical properties of CA1 neurons, which receive inputs in different subcellular compartments. To this end, we simulated a multicompartmental CA1 pyramidal cell (Fig 3B) that was endowed with active ion channels supporting backpropagating action potentials and dendritic calcium spikes (Fig 3C, Methods).

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Fig 3. Consolidation of spatial representations.

(A) Replay of PP_CA1 and SC activity during sleep. 500 PP_CA1 inputs and 2500 SC inputs are spatially tuned on a linear track with periodic grid fields (top, red) and place fields (bottom, blue). Spiking activities are independent Poisson processes (10 spikes/s) inside place/grid fields, otherwise silent. SC activity is delayed by 5 ms. (B) Multi-compartmental model of a reconstructed CA1 pyramidal neuron (see Methods). PP_CA1 and SC inputs project to distal apical tuft dendrites (red dots) and proximal apical and basal dendrites (blue dots). (C) Active neuron properties. Top: somatic sodium spike (black) propagates to the distal tuft and initiates a dendritic calcium spike (red) and further sodium spikes. Bottom: dendritic calcium spike leads to bursts of somatic spikes. (D) Spatial tuning before consolidation. SC provides place field-tuned input to the CA1 cell (left, blue), which yields spatially tuned spiking activity (right, blue); PP_CA1 input is not spatially tuned (left, red), and (alone) triggers low and untuned spiking activity (right, red). (E) Somatic and dendritic activity during consolidation. During replay, SC input generates backpropagating sodium spikes (black vertical lines) that generate dendritic calcium spikes (red). (F) After consolidation. Spatial tuning is consolidated from the indirect SC pathway into the direct PP_CA1 pathway. Left: spatial tuning of total PP_CA1 input (red) approaches theoretically derived PP_CA1 input tuning (magenta; see Methods). Right: CA1 output is place field-tuned through either SC or PP_CA1 input alone. (G) Evolution of correlation between actual and optimal PP_CA1 input tuning (see F) for replay speeds corresponding to hippocampal replay events (black) and real-time physical motion (grey). Position in D, E, and F normalized to [0, 1].

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The use of spatial representations in the input pathways allows us to consider simple forms of memories in a navigational context in which a given location on a linear track is associated with the activity of a given CA1 cell. Effectively, such an association generates a CA1 place cell. In line with the PPT, we assumed that the spatial selectivity of this CA1 place cell is initially determined solely by the indirect pathway via the SC, i.e., by place cell input from CA3. The goal of systems memory consolidation is then to transfer this spatial association to the direct input, which reaches the CA1 cell via the PP_CA1 derived from grid cells in EC. In other words, place-cell input should supervise grid-cell input to develop a place-cell tuning. Note that we use the spatial setup primarily as an illustration of the theory. We do not make claims regarding the temporal development of CA1 place cells in vivo, which is not fully understood [49–51].

SC place field inputs were modelled by synapses that were active only in a small region of the track, whereas individual PP_CA1 grid cell inputs were active in multiple, evenly spaced regions along the track (Fig 3A). In terms of the theory, the cue representation in the two pathways is now different, but correlated, because the same location is encoded. The SC and PP_CA1 inputs projected to proximal and distal dendrites, respectively (Fig 3B, [52]). Synapses were initialized such that the SC input conductances were spatially tuned and resulted in place field-like activity in the CA1 cell while the PP_CA1 input had no spatial tuning (Fig 3D).

During consolidation, SC and PP_CA1 input to the CA1 cell consisted of replays of previously encountered sequences of locations [13, 14], with a replay speed 20 times faster than physical motion [13]. During replay, the SC input led to somatic spikes, which in turn triggered backpropagating action potentials that caused calcium spikes in the distal dendrites where the PP_CA1 synapses arrive (Fig 3C and 3E, [53]). Through synaptic plasticity, PP_CA1 synapses active in the place field of the neuron were potentiated. Over time, the PP_CA1 input adopted the spatial tuning of the SC input (Fig 3F, left) and reproduced the original SC-induced place field output (Fig 3F, right) with high correlation (Fig 3G). The fact that the spatial tuning of two inputs is not perfectly matched does not contradict with theoretical results, which merely state that the direct input should attain the best possible linear approximation of the indirect pathway. In the present setting, this approximation is bounded by the finite range of frequencies of the entorhinal inputs (in analogy to reconstructing a high-frequency signal, e.g. a narrow peak, with a finite set of Fourier components), which causes the ringing next to the target peak in Fig 3F (left). In summary, the PPT mechanism therefore consolidated associations even though the spatial representations in the two pathways differed and although the two pathways targeted different neuronal compartments with different numbers of synapses in the CA1 neuron with complex morphology.

The theory also predicts that consolidation is most effective when the correlation time in the input is matched to the time scale of STDP (Fig 2B). In line with this prediction, consolidation failed when replay speed was reduced to that of physical motion (Fig 3G) because the time scale of rate changes in place and grid cell activity is then much longer than the delay between the two pathways and the time scale of STDP (Fig 2C). Accelerated replay during sleep [13] hence supports systems memory consolidation within the PPT by aligning the time scales of neural activity and synaptic plasticity [54], and this alignment is similar to the effect of phase precession during memory acquisition [55].

Finally, we note that the theoretical analysis relied on a separability assumption for the statistics in the two pathways; cf. Eqs (4) and (5). This condition is not fulfilled for sequence replay during consolidation because the time-delayed covariance of different place cells depends on the relative spatial location of their place fields; such correlations are non-separable even for slower replay or during memory acquisition with real-time physical motion. The observation that consolidation was successful nevertheless illustrates that the separability assumption does not need to be fulfilled for the PPT to achieve a successful consolidation.

Consolidation of place-object associations in multiple hippocampal stages

Ultimately, to consolidate memories into neocortex, they have to move beyond the PP_CA1. Notably, the PP_CA1 is itself part of an indirect pathway from EC to the subiculum (SUB) that is shortcut by a direct connection from EC to SUB (referenced as PP_SUB; Fig 4A, left; [29]). This suggests that the PPT can be reiterated to further consolidate memories from the PP_CA1 to the PP_SUB and beyond.

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Fig 4. Consolidation of place-object associations in multiple hippocampal stages.

(A) Structure of the extended model. PP_SUB: perforant path to the subiculum. Each area (EC, DG-CA3, CA1, SUB) contains object-coding and place-coding populations. Open arrows: all-to-all connections between these areas. (B) Decoding of consolidated associations. Top: The location of a platform in a circular environment is stored as an object-place association in the SC (thick diagonal arrows in A, right). Middle: Platform position probability maps given the platform object cue, inferred from the CA1 output resulting from SC or PP_CA1 alone, at different times during consolidation (see section “Consolidation of place-object associations in multiple hippocampal stages” in Methods). Bottom: Platform-in-quadrant probabilities (±SEM) given PP_CA1 input alone during consolidation. Quadrant with correct platform position (target quadrant) in orange. (C) Consolidation from SC to PP_CA1 and to PP_SUB over four weeks. Each day, a new association is first stored in SC and then partially consolidated. An association on day 0 is monitored in SC, PP_CA1, and PP_SUB. Panels as in B. (D) Effects of PP_CA1 lesions on memory consolidation, model and experiment (data with permission from [27]). Histograms of time (±SEM) spent in quadrants at different delays after memory acquisition (“probe”). Dashed lines at 25% are chance levels. T: target quadrant; Left, Right: adjacent quadrants; O: opposite quadrant. Top: Control without lesion. Middle: Lesion before memory acquisition. Bottom: Lesion 21 days after memory acquisition.

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To illustrate this idea, we considered a standard paradigm for memory research in rodents: the Morris water maze [56]. In the water maze, the rodent needs to find a submerged platform (object), i.e., it must store an object-place association. Thus this paradigm requires neural representations of objects (such as the submerged platform) and places. We hence constructed a model in which subregions of the hippocampal formation included neurons that encode places and neurons that encoded the identity of objects (Fig 4A, right).

For simplicity and computational efficiency we switched to a rate-based neuron model (Methods). An object was chosen from a set of 128 different objects and placed in a circular open field environment (Fig 4B, top). As motivated by experiments [32–34], we implemented object-to-place associations in our model by enhancing, as before, synaptic connections in the SC, but now between object-encoding neurons in CA3 and place-specific neurons in CA1 (Fig 4A, right). Here, we did not consider place-to-object associations. These are less relevant for the water maze task, where the task is to recall the location of a given object—the platform—rather than to recall which object was encountered at a given location. We tested object-to-place associations stored in the SC by activating the object representation in EC—as a memory cue—and determining the activities in CA1, triggered by the SC alone. From these activities we inferred a spatial probability map of the recalled object location (Fig 4B; Methods).

We first stored a single object-place association in the SC. During a subsequent consolidation cycle—representing one night—place and object representations in EC were then randomly and independently activated. Consistent with our previous results, the object-place association was gradually consolidated from the SC to the PP_CA1: after one night of consolidation, the correct spatial probability map of an object location was inferrable from CA1 activity triggered by the PP_CA1 alone (Fig 4B).

To track the consolidation process over longer times, we assumed that a new random object-place association is stored in the SC every day. This caused a decay of previous SC memory traces due to interference with newly stored associations (Fig 4C, [57, 58]). During the night following each day, associations in the SC were partially consolidated into the PP_CA1, such that the consolidated association could be decoded from the PP_CA1 after a single night, but previously consolidated associations were not entirely overwritten. As a result, object-place associations were maintained in the PP_CA1 for longer periods than in the SC, thus extending their memory lifetime (Fig 4C). Eventually, a given PP_CA1 memory trace would also degrade as new interfering memories from the SC are consolidated. However, as noted above, the PP_CA1 itself is part of an indirect pathway from EC to the SUB, for which there is in turn a parallel, direct perforant pathway PP_SUB. The association in the PP_CA1 (and SC) could therefore, in turn, be partially consolidated into the PP_SUB, further extending memory lifetime (Fig 4C). Note that the extension of memory lifetime is supported in the model by a reduced plasticity (i.e. halved learning rate in Eq (33)) in PP_SUB compared to PP_CA1.

The model suggests that the PP_CA1 serves as a transient memory buffer that mediates a further consolidation into additional shortcut pathways downstream. This hypothesis is supported by navigation studies in rats. Using PP_CA1 lesions, Remondes and Schuman [27] have shown that the PP_CA1 is not required for the original acquisition of spatial memories, but that it is critically involved in their long-term maintenance. However, lesioning the PP_CA1 21 days after acquiring a memory did not disrupt spatial memories, suggesting that the PP_CA1 is not the final storage site (Fig 4D) and further supporting the idea that the PP_CA1 is important to enable a transition from short-term to long-term memories.

To test whether our model could reproduce these experimental results, we simulated PP_CA1 lesions either before the acquisition of an object-place association or 21 days later. Assuming that the rat’s spatial exploration is determined by the probability map of the object location [59], the model provided predictions for the time spent in different quadrants of the environment, which were in quantitative agreement with the data for all experimental conditions (Fig 4D). Our model thus suggests that a hierarchical reiteration of parallel shortcuts—the central circuit motif of the PPT—could explain these experiments.

Similar to lesioning the PP_CA1, we predict that lesioning PP_SUB also has an impact on memory consolidation: PP_SUB should act as a transient memory buffer but on a longer timescale than PP_CA1. In general, lesioning a pathway with a set of synapses that cover a specific range of time scales such that there is a “gap” should result in an impairment of consolidation if the lesion is done before the memory has “moved on”. To illustrate this idea in more detail, we study in the next section a model with many stages in a hierarchy.

Consolidation from hippocampus into neocortex by a hierarchical nesting of consolidation circuits

Given that shortcut connections are widespread throughout the brain [25, 60, 61], we next hypothesized that a reiteration of the PPT can also achieve systems consolidation from hippocampus into neocortex. To test this hypothesis, we studied a network model (Fig 5A), in which the hippocampus (now simplified to a single area) receives input from a hierarchy of cortical areas, representing, e.g., a sensory system. It provides output to a different hierarchy of areas, representing, e.g., the motor system or another sensory system.

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Fig 5. Consolidation from hippocampus into neocortex by hierarchical nesting of consolidation circuits.

(A) Schematic of the hierarchical model. The hippocampal formation (HPC) is connected to cortical input circuit 1 and output circuit 1. Increasing numbers indicate circuits further from the HPC and closer to the sensory/motor periphery. Each direct connection at one level (e.g., dark blue arrow between input 1 and output 1) is part of the indirect pathway of the next level (e.g., for pathways from input 2 to output 2). Learning rates of the direct connections decrease exponentially with increasing level (i.e., from blue to red). (B) Memories gradually propagate to circuits more distant from the HPC. The correlation of the initial HPC weights with the direct pathways is shown as a function of time and reveals a memory wave from HPC into neocortex. The maximum of the output circuits follows approximately a power-law (black curve). Noise level indicates chance-level correlations between pathways. (C) Consolidated memories yield faster responses (from sensory periphery, e.g., Input 8, to system output) because these memories are stored in increasingly shorter synaptic pathways.

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The network also contained shortcut connections that bypassed the hippocampus. As in the previous section, new memories were stored in the hippocampus but not in any other indirect connection in the hierarchy. The repeated storage of new memories every day leads to a decay of previously stored hippocampal memories. But memories are also consolidated by Hebbian plasticity in parallel pathways; for details, see Methods.

Tracing a specific memory over time revealed a gradual consolidation into the cortical shortcut connections, forming a “memory wave” [10] that travels from hippocampus into neocortex (Fig 5B). By exponentially decreasing the shortcut learning rate with distance from the hippocampus, a power-law decay of memories can be observed in the union of all shortcuts, e.g., by reading out the shortcut with the strongest memory trace at any moment in time (Fig 5B). This observation is in line with a rich history of psychological studies on the mathematical shape of forgetting curves [28]. Note that for the readout we tried to make as few assumptions as possible by letting all pathways contribute on an equal footing. Taking the maximum over the pathways (as well as the mean) generates a power law. Notably, we achieved memory retention times of years through only a small number (∼5) of iterations of the PPT. Finally, we found that memory retrieval accelerates during consolidation (Fig 5C), in line with consolidation studies for motor skills [62]. In our consolidation model, the time to recall decreases because the path from peripheral input to output becomes shorter through the use of more direct (peripheral) shortcut connections (Fig 5A and 5B).

The predicted consolidation-mediated decrease of the time to recall critically depends on the utilized plasticity rule (STDP), which uses timing of input and output of neurons, and on our assumption that memories are initially acquired in an indirect pathway with a longer delay than direct pathways. While this assumption is reasonable for declarative memories that are initially stored in the hippocampus and then consolidated in sensory or motor areas towards neocortex, the underlying computational reasons for such a strategy are unknown. The strategy of the initial storage of memories in a pathway with a longer transmission delay could be related to the Complementary Learning Systems Theory (CLST) [11, 63] if the initial storage needs some preprocessing, e.g., to achieve representations that are suited for one-trial learning, e.g. population-sparse representations [9]. In general, our results do not imply that the reduction of delay is a central goal of systems memory consolidation or that it is even necessary. Reduction of delay may, however, be a nice side effect of systems memory consolidation with timing-based plasticity rules [64]. And such a reduction of delay does not need to be restricted to declarative memories but also could apply to, e.g., motor skill learning or habit formation.

Theory requirements and predictions

In addition to the anatomical motif of shortcut connections and Hebbian synaptic plasticity, the parallel pathway theory relies on four further requirements during the consolidation phase, which can also be considered as model predictions.

1. (1). Temporal correlations between the inputs from the two input pathways are necessary during consolidation, and these correlations should be similar to the ones during storage and recall. For example, a consolidation from hippocampus into neocortex would require correlations between cortical and hippocampal activity, as reported in [65]. Similarly, a consolidation of spatial memories within the hippocampal formation (including the medial entorhinal cortex, MEC) during replay would require correlations between activity in MEC and hippocampus; in particular, the same locations should be replayed, but represented by grid cells in MEC and by place cells in CA3 and CA1, as in Fig 3. A significant but weak correlation between the superficial layers of MEC (which provides input to the hippocampus) and CA1 was indeed observed [45]. Furthermore, pyramidal cells in the superficial layer III (projecting to CA1, “direct path”) and stellate cells in the superficial layer II (projecting to DG, which projects to CA3, “indirect path”) are expected to be correlated due to a strong excitatory feedforward projection from pyramids to stellates [66]; reviewed in [51]. Coordinated grid and place cell replay was also observed in [67] but there CA1 and deep layers of MEC (which receives the hippocampal output) were studied.
2. (2). The direct pathway should be plastic during consolidation, while the stored associations in the indirect path remain sufficiently stable (in contrast to the model in [24]). In practice, this requires the degree of plasticity to differ between periods of storage and consolidation (e.g., due to neuromodulation [68, 69]), in a potentially pathway-dependent manner. In other words, the requirement is that the content of a memory should not be altered much while creating a backup.
3. (3). Plasticity in the shortcut pathway should be driven by a teaching signal from the indirect pathway. This can be achieved by STDP in combination with longer transmission delays in the indirect pathway, as suggested here, but other neural implementations of supervised learning may be equally suitable [42, 43, 70].
4. (4). Within the present framework, a systematic decrease in learning rates within the consolidation hierarchy (Fig 5) is needed to achieve memory lifetimes on the order of years. That is, synapses involved in later stages of consolidation should be less plastic during consolidation periods such as sleep, as also suggested by [10] and [24]. Furthermore, Roxin and Fusi elegantly showed in [10] that a multistage memory system confers an advantage (in terms of memory lifetime, memory capacity, and initial signal-to-noise ratio) compared to a homogeneous memory system with the same number of synapses, which provides a fundamental computational reason for the existence of a memory consolidation processes at the systems level. However, to be able to exploit this advantage, an efficient mechanism to transfer memories across stages is necessary. The proposed PPT explains how memories can be transferred in a biologically plausible way in a multistage memory system.
   Conceptually related to models of systems-memory consolidation with a systematic decrease in learning rates across a hierarchy of networks are models of synaptic memory consolidation with complex synapses that can assume many different states and a decrease of plasticity across a hierarchy of states. In such models of synaptic memory consolidation, also a power-law forgetting has been achieved [8, 71]. Synaptic and systems memory consolidation models are different but not mutually exclusive.

What limits systems memory consolidation?

Our account of systems memory consolidation explains how memories are re-organized and transferred across brain regions. However, certain forms of episodic memory remain hippocampus-dependent throughout life [21].

In the context of the present model, this restriction could result from different factors. The PPT simplifies memory engrams by replacing multisynaptic by monosynaptic connections whenever possible. However, a shortcut pathway may not be present anatomically, or it may not host an appropriate representation for a given cue-response association in question. For example, it may be difficult to consolidate a complex visual object detection task into a shortcut from primary visual cortex (V1) to a decision area because the low-level representation of the visual cue in V1 may not allow it [72, 73]. The same applies to tasks that require a mixed selectivity of neural responses [74]. Such tasks cannot be fully consolidated into shortcuts with simpler representations of cues and/or responses that do not allow a linear separation of the associations. On the basis of similar arguments, early work suggested that the hippocampus could be critical for learning tasks that are not linearly separable [75].

Within the present framework, the consolidated memory is in essence a linear approximation of the original cue-response association, as indicated in the theoretical analysis around Eqs (7) and (8). The resulting simplification of the memory content could underlie the commonly observed semantization of memories and the loss of episodic detail [20, 21]. Such a semantization could already occur in the earliest shortcut connections [76], but could also gradually progress in a multi-stage consolidation process.

Relation to phenomenological models of systems consolidation

The basic mechanism of our framework explains memory transfer between brain regions, which is in line with the Standard Consolidation Theory (SCT) [11, 19]. Our theoretical framework is closely related to the Complementary Learning Systems Theory (CLST) [11, 63], which posits that slow and interleaved cortical learning is necessary to avoid catastrophic interference of new memory items with older memories [77]. In our model, later—presumably neocortical—shortcut connections have lower learning rates to achieve longer memory retention times. Interleaved learning could be achieved by interleaved replay [78–80] during consolidation. Thereby, the results of CLST can be directly applied to learning in shortcuts in our model, such as the rapid neocortical consolidation of new memories that are in line with a previously learned schema [17, 63, 81].

Limitations of memory transfer between brain regions—as discussed above—can impair the consolidation process, resulting in memories that remain hippocampus-dependent throughout life. Hence, our theoretical framework is also in agreement with the Multiple Trace Theory (MTT) [16] and the Trace Transformation Theory (TTT) [20, 21]. The MTT postulates that memories are re-encoded in the hippocampus during retrieval, generating multiple traces for the same memory. Our model maintains multiple memory traces in different shortcut pathways, even without a retrieval-based re-encoding. The consolidation mechanism of the PPT, however, could also transfer a specific memory multiple times if it is re-encoded during retrieval. If neocortex extracts statistical regularities from a collection of memories [11], the consolidation of such a repeatedly re-encoded memory could then lead to a gist-like, more semantic version of that memory in neocortex [16, 21, 82], as emphasized by the TTT.

The premise of our model is that memories are actively transferred between brain regions. This premise has recently been subject to debate [83–85], following the suggestion of the Contextual Binding (CB) theory. The CB theory argues that amnesia in lesion studies and replay-like activity can be explained by simultaneous learning in hippocampus and neocortex, together with interference of contextually similar episodic memories [83]. Note, however, that our framework does not exclude a simultaneous encoding in neocortex and hippocampus, which can be combined with active consolidation [1, 86].

Hence, our mechanistic approach is in agreement with and may allow for a unification of several phenomenological theories of systems consolidation.

Consolidation of non-declarative memories

Given that shortcut connections are widespread throughout the central nervous system [25, 60], the suggested mechanism may also be applicable to the consolidation of non-declarative memories, e.g., of perceptual [4] and motor skills [5], fear memory [87] or to the transition of goal-directed to habitual behaviour [88].

Several studies have suggested two-pathway models in the context of motor learning [89–92]. In particular, Murray and Escola [92] recently used a two-pathway model to investigate how repeated practice affects future performance and leads to habitual behaviour. While their model does not incorporate an active consolidation mechanism or multiple learning stages, the basic mechanism is the same: A fast learning pathway from cortex to sensorimotor striatum first learns a motor skill and then teaches a slowly learning pathway from thalamus to striatum during subsequent repetition.

Limitations of the model and future directions

The present work focuses on feedforward networks and local learning rules. Hence, the model cannot address how systems memory consolidation affects the representation of sensory stimuli and forms schemata that facilitate future learning [17, 81] because representation learning typically requires a means of backpropagating information through the system, e.g., by feedback connections [93]. The interaction of synaptic plasticity with recurrent feedback connections generates a high level of dynamical complexity, which is beyond the scope of the present study. Our framework also does not explain reconsolidation, that is, how previously consolidated memories become labile and hippocampus-dependent again through their reactivation [94, 95].

On the mechanistic level, the PPT predicts temporally specific deficits in memory consolidation when relevant shortcut connections are lesioned, that is, a tight link between the anatomical organisation of synaptic pathways and their function for memory. These predictions may be most easily tested in non-mammalian systems, where connectomic data are available [96].

The PPT could provide an inroad to a mechanistic understanding of the transformation of episodic memories into more semantic representations. This could be modelled, e.g, by encoding a collection of episodic memories that share statistical regularities and studying the dynamics of statistical learning and semantisation in the shortcut connections during consolidation. Such future work may allow us to ultimately bridge the gap between memory consolidation on the mechanistic level of synaptic computations and the behavioural level of cognitive function.

Consolidation in a single integrate-and-fire neuron

For the results shown in Fig 1E and 1F we used a single integrate-and-fire model neuron that received excitatory synaptic input. The membrane potential V(t) evolved according to (9) with membrane time constant τ_m = 20 ms, resting potential V_rest = −70 mV, and synaptic reversal potential E_syn = 0 mV. When the membrane potential reached the threshold V_thresh = −54 mV, the cell produced a spike and the voltage was reset to −60 mV during an absolute refractory period of 1.75 ms.

The total synaptic conductance g_syn(t) in Eq (9) is denoted in units of the leak conductance and thus dimensionless (parameters are taken from [97]). The total synaptic conductance was determined by the sum of 1000 Schaffer collateral (SC) inputs and 1000 perforant path (PP_CA1) inputs. Activation of input i (where i denotes synapse number) leads to a jump g_i > 0 in the synaptic conductance: (10) All synaptic conductances decay exponentially, (11) with synaptic time constant τ_syn = 5 ms. The PP_CA1 inputs were activated by mutually independent Poisson processes with a mean rate of 10 spikes/s. The activity patterns of the SC fibers were identical to those of the PP_CA1 fibers but were delayed by 5 ms.

The synaptic peak conductances or weights, g_i, were either set to a fixed value or were determined by additive STDP [98]. A single pair of a presynaptic spike (at time t_pre) and a postsynaptic spike (at time t_post) with time difference Δt ≡ t_pre − t_post induced a modification of the synaptic weight Δg_i according to (12) with τ_STDP = 20 ms. L(Δt) is the learning window of STDP [98]. Hard upper and lower bounds were imposed on the synaptic weights, such that for all i, where the dimensionless maximum synaptic weight was . Parameters and A⁻ = 1.05 ⋅ A⁺ with η = 0.005 determine the maximum amounts of LTP and LTD, respectively.

Synaptic weights were initialized to form a bimodal distribution, such that it agrees with the steady state weight distribution resulting from additive STDP, when presynaptic input consists of uncorrelated Poisson spike trains [98]. Specifically, half the weights were sampled from an exponential distribution with mean , the other half as minus that same exponential distribution.

The dynamics were integrated numerically using the forward Euler method, with an integration time step of 0.1 ms.

Consolidation of spatial representations in a multi-compartment neuron model

The results presented in Fig 3C–3G relied on numerical simulations of a conductance-based compartmental model of a reconstructed CA1 pyramidal cell (cell n128 from [99]). Passive cell properties were defined by the membrane resistance R_m = 30 kΩ cm² with reversal potential E_L = −70 mV, intracellular resistivity R_i = 150 Ωcm, and membrane capacitance C_m = 0.75μF/cm². Dendrites were discretized into compartments with length smaller than 0.1 times the frequency-dependent passive space constant at 100 Hz. Three types of voltage-dependent currents and one calcium-dependent current, all from [100], were distributed over the soma and dendrites. Gating dynamics of the currents evolved according to standard first-order ordinary differential equations. The steady state (in)activation functions x_∞ and voltage-dependent time constants τ_∞ for each gating variable (i.e., x = m, h, n; see below) were calculated from a first-order reaction scheme with forward rate α_x and backward rate β_x according to x_∞(V) = α_x(V)/(α_x(V) + β_x(V)) and τ_x(V) = 1/(α_x(V) + β_x(V)) where V was the membrane potential. All used current densities and time constants were selected for a temperature of 37°C (see [100]).

A fast sodium current, I_Na, was distributed throughout the soma ( pS/μm²) and dendrites ( pS/μm²), except from the distal apical dendritic tuft, (13) with reversal potential E_Na = 60 mV. The dynamics of activation gating variable m and inactivation gating variable h were characterized by (14) Here and in the following, we dropped units for simplicity, assuming that the membrane potential V is given in units of mV.

The steady-state inactivation function was defined directly as (15)

A fast potassium current, I_Kv, was present in the soma ( pS/μm²) and throughout the dendrites ( pS/μm²), (16) with reversal potential E_K = −90 mV and with activation gating variable n characterized by (17)

A high-voltage activated calcium current, I_Ca, was distributed throughout the apical dendrites ( pS/μm²) with an increased density ( pS/μm²) for dendrites distal from the main apical dendrite’s bifurcation, (18) with reversal potential E_Ca = 140 mV and with activation gating variable m and inactivation gating variable h characterized by (19)

A calcium-dependent potassium current, , was similarly distributed throughout the apical dendrites ( pS/μm²) with an increased density ( pS/μm²) beyond the main bifurcation of the apical dendrite, (20) with activation gating variable n characterized by (21) with [Ca²⁺] in μM.

Internal calcium concentration in a shell below the membrane surface was computed using entry via I_Ca and removal by a first-order pump, (22) with Faraday constant F, depth of shell d = 0.1 μm and with [Ca²⁺]_∞ = 0.1μM, and τ_R = 80 ms. To account for dendritic spines, the membrane capacitance and current densities were doubled throughout the dendrites. An axon was lacking in the cell reconstruction and was added as in [100].

Excitatory synaptic inputs were distributed over the membrane surface. Upon activation of a synapse, the conductance with a reversal potential of 0 mV increased instantaneously and subsequently decayed exponentially with a time constant of 3 ms. The PP_CA1 provided 500 inputs that were distributed with uniform surface density throughout the distal apical tuft dendrites; the SC provided 2500 inputs, distributed uniformly over basal dendrites and proximal apical dendrites [52].

All inputs were spatially tuned on a 2.5 m long linear track over which the simulated rat walked. The PP_CA1 inputs showed periodic, grid field-like spatial tuning with periodicity ranging from 2 to 6 grid fields along the entire track with random phase: , where is the Heaviside step function, r is the mean firing rate within the grid field, k is the spatial frequency, and ξ_i is the random spatial phase offset for neuron i (for i = 1, …, 500). The 2500 SC inputs showed place field-like tuning, having single, 25 cm long place fields distributed uniformly random along the track. When the virtual rat was within the place or grid field of an SC or PP_CA1 fiber, respectively, the input was activated as an independent Poisson process with a mean rate of r = 10 spikes/s. Outside of the place/grid fields the fibers were quiescent. Simulations of the consolidation phase considered replay of the rat walking back and forth along the linear track, with running speeds increased, compared to realistic speeds, by a factor 20 (5 m/s; [13]). SC input activity to the CA1 cell was delayed by 5 ms with respect to the PP_CA1 input [101], accounting for the extra processing stages involved for information reaching CA1 from the entorhinal cortex through DG and CA3, compared to the direct entorhinal PP_CA1 input.

The PP_CA1 and/or SC inputs showed additive STDP, operating in the same manner as defined around Eq (12). Post-synaptic spikes were defined as local voltage crossings of a threshold at −30 mV. The maximum synaptic weight for the SC inputs was 400 pS and 140 pS for the PP_CA1 inputs.

The reference tuning curve shown in Fig 3F (PP_CA1 inputs theory) was computed by adding up all grid field tuning functions that had an active field in the SC-encoded spatial position (i.e., halfway along the linear track).

Simulations were carried out with a fixed time step of 25 μs using the NEURON simulation software [102].

Consolidation of place-object associations in multiple hippocampal stages

The results related to Fig 4 show the acquisition and consolidation of place-object associations in a hippocampal network model. Every day a virtual animal learns the position of one of many possible objects in a circular open field environment. The simulations show that during a subsequent sleep phase, replay of the hippocampal activity that is associated with runs through this environment allows for the consolidation of the place-object association. We call the imprinting of a new memory and the subsequent memory consolidation phase a consolidation cycle. In the simulations, a place-object association learned at time t = 0 is tracked for N_cycle consolidation cycles, i.e., nights after memory acquisition. Between consolidation cycles, the memory in the system is assessed as described below.

Model architecture.

The model consists of four neuronal layers: entorhinal cortex (EC), dentate gyrus/CA3 (DG-CA3; note that the dentate gyrus is not explicitly included as a separate area), CA1, and the subiculum (SUB). Each layer consists of a population of place-coding cells and a population of object-coding cells. The connectivity is depicted in Fig 4A: EC projects to DG-CA3, which connects to CA1 (through the SC pathway), which in turn connects to the SUB. EC provides also shortcut connections to CA1 (PP_CA1 pathway) and the SUB (PP_SUB pathway).

The SC, PP_CA1, and PP_SUB pathways consist of four different connection types among populations of neurons that represent either place or object: (i) from object (populations) to object (populations), (ii) from place to place, (iii) from object to place, and (iv) from place to object. For simplicity, the pathway from CA1 to the SUB consists only of place-to-place and object-to-object connections, because we never store object-place or place-object associations in this pathway. The pathway from EC to DG-CA3 was not explicitly modelled. Instead, we assumed that the same location (of the virtual animal) is represented in both areas, but with a grid cell code and a place cell code, respectively. We assumed that all connections have the same transmission delay, which is equal to one time step D = ΔT = 5 ms in the simulation (see Table 1 for parameter values). In practice, this meant that the activities in the SC pathway and the connection from CA1 to the SUB each had a transmission delay D relative to the activities in the connections from EC to DG/CA1 and from EC to SUB.

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Table 1. Parameters for simulations shown in Fig 4.

https://doi.org/10.1371/journal.pcbi.1009681.t001

Activities of neurons in each layer were described as firing rates and were determined by a linear model, (23) (24) where x_EC(t) and x_CA3(t) are the activities in the input layers EC and DG-CA3, respectively, and y_CA1(t) and y_SUB(t) represent the activities in the output layers CA1 and SUB, respectively. Time is denoted by t. The symbols W_PP-CA1 and W_PP-SUB denote the weight matrices of the pathways from EC to CA1 and from EC to SUB, respectively. The matrices V_SC and V_CA1-SUB summarise the weights from DG-CA3 to CA1 and from CA1 to SUB, respectively, which mediate the transmission delay D. Eqs (23) and (24) are identical in structure to Eq (1) except that now the output is a vector (and not a scalar) and the synaptic weights are a matrix (and not a vector).

As already mentioned above, each neuron in a layer is assumed to primarily encode either place or object information (see Fig 4A). To simplify the mathematical analysis, we turn to a notation where we write a layer’s activity vector z (where z = x_EC, x_CA3, y_CA1, or y_SUB) as a concatenation of place and object vectors: (25) where the number of place- and object-coding cells is identical, dim(z^place) = dim(z^object) = N, hence dim(z) = 2N. Correspondingly, the weight matrices M (where M = W_PP-CA1, W_PP-SUB, V_SC, or V_CA1-SUB) are composed of four submatrices, connecting the corresponding feature encoding sub-vectors (place-place, place-object, object-place, and object-object): (26) Associations between objects and places were initially stored in V_SC as described below. To achieve a consistency in the code for places and objects, the weights in V_SC and V_CA1-SUB that connect neurons coding for the same feature (i.e., place-place or object-object) were set proportional to identity matrices I, (27) (28) The scaling factors and ensure that these pathways had similar impact as the other pathways projecting to CA1 cells and SUB cells, respectively, and is twice as large as to account for the fact that only in the CA1-SUB pathway the object-place and place-object connections were set to zero. The matrices W_PP-CA1 and W_PP-SUB, which represent shortcuts, were plastic during a consolidation cycle and evolved according to the learning rule described below. Their initial values were chosen as a random permutation of an equilibrium state, taken from a long running previous simulation.

Consolidation in a hierarchical rate-based network

Fig 5 demonstrates the consolidation of memories in a hierarchy of connected neural populations. In the model, signals flow along distinct neocortical neural populations to the hippocampal formation (HPC) and back into neocortex (black arrows in Fig 5A). Shortcut connections exist between the neocortical populations (colored arrows in Fig 5A). All connections carry the same transmission delay D.

Every day new memories are imprinted into the weight matrix representing the HPC. The model describes the transfer of the memories into neocortex during N_cycle consolidation phases, of which there is one per night (for all model parameters and values, see Table 2). In contrast to the model for Fig 4, we do not consider object-place associations, but directly analyse correlations between a stored memory weight matrix and the weight matrices that describe the neocortical shortcut connections.

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Table 2. Parameters for simulations in Fig 5.

https://doi.org/10.1371/journal.pcbi.1009681.t002

Theoretical analysis of hierarchical consolidation

As outlined in the Results and illustrated in Fig 5, the suggested consolidation mechanism can be hierarchically iterated and leads to power law forgetting when the learning rates in the various pathways are suitably chosen. To get a theoretical understanding of this behaviour, let us consider the architecture shown in the Fig 6A, which is a generalized version of Fig 5A. The network consists of a hierarchy of N + 1 input layers and N + 1 output layers. For mathematical simplicity, the network is assumed to be linear (in contrast to the model described in Fig 5A, which was nonlinear due to biologically motivated weight constraints), and the representation in the input layers is assumed to be the same, i.e., the weight matrices between the input layers (indicated in black in Fig 6A) are all simply the identity matrix (in contrast to the model described in Fig 5A where the identity matrices were also scaled). Similarly, we also assume that all weight matrices between the output layers are also the identity matrix. The mathematical derivations presented in the following can be generalized to arbitrary weight matrices both in the input and the output pathways, but we prefer to treat the simple case to avoid cluttered equations and to make the theoretical approach more accessible.

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Fig 6. Mathematical analysis of the hierarchical consolidation network.

(A) The mathematical analysis is performed for a network consisting of N + 1 input and N + 1 output layers. All output layers (except output layer 0) weight the input from the previous layer with a factor α and the input via the shortcut pathway with a factor 1 − α, to ensure that activity does not rise as increasingly many pathways converge onto the output layers. Input layer i is hence connected to output layer i through a shortcut connection with weight matrix (1 − α)W_i (except for the bottom-most layers i = 0, for which no factor 1 − α is required). All connections between input layers are set to the identity matrix I, and all connections between output layers are set to αI, for notational simplicity in the derivations. The math can be generalized to arbitrary connection matrices, as long as the network is linear. Each connection introduces a synaptic delay of D. The multi-synaptic pathway from input layer i to output layer i via shortcut connection j ≠ i has a total delay of (2(i − j) + 1) ⋅ D, so the difference in delays between the pathway through shortcut i and shortcut j is D_ij = 2(i − j) ⋅ D. (B) The similarity O_i of the weight matrix W₀ (in which memory traces are initially stored) and the shortcut connection W_i as a function of the time elapsed after storage (colored lines), and their maximum (black line). Simulations shown for D = 2 ms, α = 0.8, η_i = 2⁻ⁱ and STDP time constant τ_STDP = 40 ms.

https://doi.org/10.1371/journal.pcbi.1009681.g006

We assume that due to newly acquired memories during the day, the weight matrix W₀(t) (earlier called V_HPC) that represents the memory trace in the hippocampus is varying in time, with an exponentially decaying autocorrelation function with time constant , where tr denotes the trace of a square matrix.

All other pathways that project from an input layer to an output layer are plastic according to STDP. To derive the learning dynamics for these pathways, we first have to calculate the activity y_i in the i-th output layer, (50) where x_j denotes the activity in input layer j and c_ij denote weighting factors that determine the impact of the jth pathway, i.e. the indirect pathway via W_j, on output layer i. These weighting factors are needed, because we would like to keep the weight matrices on a similar scale, but avoid that the activity increases from one output region to the next, because more synaptic pathways converge onto “later” output layers. The symbol D_ij = 2D(i − j) (defined only for i ≥ j) denotes the total additional delay that is accumulated on the connection from the i-th input layer to the i-th output layer that traverses the j-th direct “shortcut” pathway, relative to the direct shortcut from input layer i to output layer i. For simplicity, we assumed that all connections have the same delay D. In a very similar way as in Eq (3), the learning dynamics of the weight matrix W_i in the direct path can be written as (51) where η_i denotes the learning rate for the i-th pathway. For simplicity, we will assume that the different components of the input signal vector x_i(t) are uncorrelated amongst each other, and have identical temporal autocorrelations that are also independent of the layer index: , where I is the identity matrix. The learning dynamics then simplify to (52) with A(D) ≔ ∫L(τ)f(τ − D) dτ.

To measure the degree to which a memory trace that is stored in the weight matrix W₀ at time t = 0 is still present in the j-th shortcut pathway at a later time t, we compare the weight matrix W_j(t) at time t to the weight matrix W₀(0) at time t = 0. We quantify the correlation of these two matrices by calculating the summed overlap of the column vectors: (53)

Note that the overlaps O_i(t) are real numbers, and that their temporal dynamics for the shortcut connections (i.e., for all i > 0) are dictated by the dynamics of the weight matrices in the network: (54) (55) (56) To capture the exponential decay of the initially stored memories in the “hippocampal” weight matrix W₀ due to the storage of new memories, the set of dynamical equations is completed by (57) Note that the dynamics of the overlaps O_i form a linear dynamical system.

To show that this mathematical description shows a power-law behavior akin to the simulated system in Fig 5, we simulated the equations with the following parameter choices. Consistent with the exponential decay of the learning rates in the simulations, we chose the learning rates as η_i = 2⁻ⁱ. The weighting factors c_ij were chosen based on the assumption that output layer i (for i > 0) receives a fraction α of its input from the output layer i − 1 below, and a fraction 1 − α via its direct shortcut connection (associated with the weight matrix W_i). Taking into account that the signal reaching layer i through shortcut connection j traverses several of these weighting stages (Fig 6A), this choice yields c_ij = α^i−j for j = 0 and c_ij = α^i−j(1 − α) for j > 0. Note that , so the activity level in different output layers should be similar. Finally, we assume that each synaptic transmission generates a fixed delay D and that the autocorrelation function f(τ) decays much more quickly than the STDP learning window. In this case, we can approximate .

For the simulations illustrated in Fig 6, we chose τ_STDP = 40 ms as the time constant of an exponentially decaying STDP learning window for positive delays τ > 0, and we set A⁺ = 1 in Eq (12). Furthermore, we used D = 2 ms. As shown in the Fig 6B, the maximum of the overlaps O_j indeed approximates a power law decay.