^{*}

^{†}

^{‡}

^{‡}

Edited by: Don Kulasiri, Lincoln University, New Zealand

Reviewed by: Céline Kuttler, Université de Lille, France; Yoshiyuki Asai, Okinawa Institute of Science and Technology, Japan

*Correspondence: David B. Kastner

This article was submitted to Systems Biology, a section of the journal Frontiers in Neuroscience

†Present Address: David B. Kastner, Department of Psychiatry, University of California, San Francisco, CA, USA

‡These authors have contributed equally to this work.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Reconsolidation of memories has mostly been studied at the behavioral and molecular level. Here, we put forward a simple extension of existing computational models of synaptic consolidation to capture hippocampal slice experiments that have been interpreted as reconsolidation at the synaptic level. The model implements reconsolidation through stabilization of consolidated synapses by stabilizing entities combined with an activity-dependent reservoir of stabilizing entities that are immune to protein synthesis inhibition (PSI). We derive a reduced version of our model to explore the conditions under which synaptic reconsolidation does or does not occur, often referred to as the boundary conditions of reconsolidation. We find that our computational model of synaptic reconsolidation displays complex boundary conditions. Our results suggest that a limited resource of hypothetical stabilizing molecules or complexes, which may be implemented by protein phosphorylation or different receptor subtypes, can underlie the phenomenon of synaptic reconsolidation.

Reconsolidation describes a process for the alteration of memories, and highlights the dynamic nature of information processing and storage in the brain. Reconsolidation represents the phenomenon that recently triggered memories are susceptible to degradation whereas memories that have not been retrieved are spared from degradation (Nader et al.,

Recent reviews (Nader and Hardt,

Although cognitive and conceptual models have elucidated functional roles for reconsolidation (Blumenfeld et al.,

Computational models of consolidation involve a cascade of different processes acting on different time scales (Fusi et al.,

We consider a postsynaptic neuron that receives input from

In the following, we put forward a generic extension of such models that endows synapses with synaptic reconsolidation-like dynamics and yields a possible explanation for the slice experiments of Fonseca et al. (_{j} that represents the state of consolidation at synapse _{j} > 0, signifies that the synapse is in a consolidated, “strong” state, whereas a negative value indicates that the synapse is not consolidated, or in a “weak” state. In the following, a consolidated synapse will be also referred to as a “big” synapse.

A consolidated synapse interacts with hypothetical stabilizing entities, _{j} (_{j} > 0), and thereby stabilize its “big” state. The rate at which this synapse gets bound is _{1}_{A}_{j}), where _{1} is the constant binding rate per stabilizing entity; _{A} is the number of available entities _{A} of available unbound entities to zero during application of PSI.

^{*} (see Equation 2), and only when it is bound can it remain in the consolidated state. ^{*} (see Equation 2). An unstable big synapse (left) can bind a stabilizer coming from two different pools, a PSI susceptible pool (green rectangles, left, ^{*} from Equation 2). A stabilized synapse (right) is more likely to release the stabilizer into the PSI immune pool (solid arrow) than into the PSI susceptible pool. All synapses on the postsynaptic neuron, which is modeled as a leaky integrate-and-fire neuron, use the same pool of stabilizing entities. For

To capture the combined effect of PSI duration and low-frequency stimulation (LFS) reported in Fonseca et al. (^{*}, which does not degrade on the time scale during which PSI is applied. In this way, the pool of immune entities ^{*} is resistant against PSI, and forms an effective reservoir of stabilizing entities for consolidated synapses. Like the original form, the immune form can bind to an unbound big synapse, and thereby stabilize the synapse even in the presence of PSI. The binding rate of a big synapse with ^{*} is _{3} is a constant rate and ^{*} entities.

To capture the effect of LFS on the stability of consolidation, as reported in Fonseca et al. (^{*}, respectively. Specifically, LFS yields an increased synapse specific input,

α = 2, 4. In Equation (1), _{0} is the minimal unbinding rate, _{0} is the maximal unbinding rate,

A big synapses (i.e., _{j} > 0) will eventually decay and transition to a value _{j} < 0 if it is not bound to a stabilizing entity, see “extended write-protected model” and “extended state-based model” below. In this way, the model can be seen as an activity-dependent creation of a PSI immune reservoir of entities ^{*} that are useful for the stabilization of consolidated synapses.

The total number of stabilizing entities available in the postsynaptic cells is _{A, tot} stabilizers we do not need to keep track of each stabilizing entity but only the total numbers _{A} and _{A} is decremented by 1 if a second random number _{A} is incremented by 1 with probability

See Table

^{exc} |
0 mV | τ_{thr} |
2 ms | τ_{ampa} |
5 ms | τ_{adapt} |
250 ms | ||

^{rest} |
−70 mV | ϑ^{rest} |
−50 mV | τ_{nmda} |
100 ms | ^{spike} |
10 | ||

^{inh} |
−80 mV | ϑ^{spike} |
100 mV | β | 0.5 | _{−} |
0.05 | ||

τ_{m} |
20 ms | _{w} |
3 | ||||||

τ_{w} |
200 s | _{wT∕Tz} |
3.5 | _{−} |
2 × 10^{−4} |
_{up} |
1 s^{−1} |
||

τ_{T} |
200 s | _{Tw} |
1.3 | _{+} |
5 × 10^{−4} |
_{down} |
1/2000 s^{−1} |
||

τ_{z} |
200 s | _{zT} |
0.95 | τ_{x} |
16.8 ms | θ | 0.01 | ||

_{w} |
3 | τ_{γ} |
600 s | τ_{y} |
33.7 ms | ||||

σ | 10^{−2} s^{−1∕2} |
ϑ_{γ} |
0.37 | τ_{triplet} |
114 ms | ||||

α | 0.017 min^{−1} |
β | 0.067 min^{−1} |
τ_{e} |
0.017 min^{−1} |
τ_{l} |
0.01 min^{−1} |
τ_{r} |
1 × 10^{−4} min^{−1} |

_{b} |
1 | _{0} |
5 × 10^{−4} s^{−1} (wp) |
_{20} |
5.7 × 10^{−2} (wp) |
φ | 72 s | ||

_{1} |
1/10,000 s^{−1} (wp) |
0.02 s^{−1} (sb) |
5.8 × 10^{−2} (sb) |
ϑ_{big} |
−80 s^{−1} |
||||

1/3,000 s^{−1} (sb) |
2 × 10^{−3} s^{−1} (wp) |
_{40} |
4.5 × 10^{−2} |
η | 10 s | ||||

_{3} |
1/1,750 s^{−1} (wp) |
0.25 s^{−1} (sb) |
_{A, tot} |
2 × 10^{4} |
ω | 5.4 s | |||

1/30 s^{−1} (sb) |
3 × 10^{−3} |
τ_{A} |
150 s |

To understand the dynamics of synaptic reconsolidation, we compare simulations of the full stochastic model with an effective model of the stabilizing entities (^{*}) inside the postsynaptic cell. Besides providing analytical insight into the reservoir dynamics, the reduced model also allows us to rapidly explore the parameter space for cellular reconsolidation so as to trace the “boundary conditions” observed in experimental data. The reduced model is derived as follows: first, we note that it is sufficient to study the mean of the number of stabilizers, _{A} = 〈_{A}〉 and

Here,

of _{big} synapses with _{A, tot} stabilizers. The unbinding rates ^{A}(

Here, ω is a fixed increase that occurs with each pulse of stimulation during LFS captured by the sum of δ functions.

The number

Let us first consider the case with intact protein synthesis. After LTP induction, synapses in the “big” state rapidly bind to a stabilizing entity, since _{ASyn}(_{big}. More formally, Equation 3 predicts under the stated assumptions _{ASyn} = _{big}. Equation 4 then implies that the reservoir obeys the simple dynamics _{0} so that the increase is slower. Note that the initial increase in the slope is due to the fast rise of the number of consolidated synapses _{big} from near zero to its constant maximal value (Figure

In contrast, in the case of PSI, the pool of non-immune stabilizers degrades immediately. As a consequence, we cannot exactly replace _{ASyn}(_{A} = 0, Equations 3 and 4 can be combined to _{ASyn} is roughly constant and we find that the reservoir decays like _{ASyn} is still roughly given by _{big}). The activity dependence of

We stated above that with intact protein synthesis Equation 3 lets us deduce that _{ASyn} = _{big}, i.e., each big synapse is in contact with a stabilizing entity. For rapid changes, however, this is not correct. Moreover, if not enough stabilizers are available not all of the big synapses can be stabilized, and _{big} will itself not be constant. Empirically, the decay of _{big} follows that of _{ASyn} with a small delay:

where φ is the temporal delay. The negative change in _{ASyn} has to be steeper than a (negative) threshold ϑ_{big} for it to manifest itself in a decay in _{big}, and

where η is a time window over which the change in _{ASyn} is computed.

The numerical values of φ and η were determined empirically. See Table

A postsynaptic neuron is connected to a random number of presynaptic Poisson neurons drawn from a binomial distribution with a mean of 200. The variability in the number of presynaptic neurons is included to ensure that our results were not dependent upon an exact number of presynaptic neurons, and can occur over a range of connectivity.

The postsynaptic neuron is modeled as a leaky integrate-and-fire (LIF) neuron with conductance based synapses and adaptation (Gerstner et al.,

where ^{exc} and ^{exc} (^{adapt} and ^{adapt}) are the excitatory (adapting) conductance and reversal potential respectively. A spike is emitted when the potential reaches the threshold ϑ. After a spike, ^{rest} and ϑ is set to ϑ^{spike} to implement refractoriness. The threshold then relaxes back to its rest value according to

The excitatory conductance, ^{exc}, has a fast and a slow component, representing α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) conductance, and N-methyl-D-aspartate (NMDA) conductance, respectively. The two components are combined as ^{exc} = β^{ampa}+(1 − β)^{nmda}, where β is the relative contribution of AMPA. The time course of the conductance of the AMPA receptor channel is given a by a first order low-pass filter

where Δ_{j} is the plastic synaptic weight of the connection from presynaptic neuron _{j}(^{th} spike from the presynaptic neuron

The voltage dependence of NMDA receptors is neglected for the sake of computational efficiency.

For the simulations presented here, there are no inhibitory presynaptic neurons, and therefore no γ-amionbutyric acid (GABA) receptor simulation. However, the postsynaptic neuron has an adaptive component with spike-triggered self inhibition, where the adaptation conductance increases by an amount, ^{spike}, with each spike of the postsynaptic neuron, otherwise relaxing exponentially to zero as described by

where

The initial conditions for the neuron model were: ^{rest}, ϑ(0) = ϑ^{rest}, and all conductances (^{ampa}(0), ^{nmda}(0), and ^{adapt}(0)) were initialized to 0. See Table

The low frequency stimulation (LFS) is modeled by prescribed spiking activity of the presynaptic neurons. During LFS the presynaptic spike trains are jittered versions of a periodic spike train with rate 0.1 Hz, meaning that a spike shifts around the periodic time _{0} by a delay drawn from a Gaussian distribution with mean of 0 and a standard deviation of 3 ms. HFS occurs in the same manner as LFS, but the frequency of stimulation is 100 Hz for 60 s.

We extend the model of Ziegler et al. (

The external input _{j} = _{0}[_{−} + (_{j} + 1)(_{+}−_{−})∕2], where _{+} = _{w}_{−}. The second equation determines the state of the tag at the synapse,

And the third equation determines the state of the scaffold of the synapse,

For all of these equations _{j}(

The coupling parameter terms _{Tw}, _{wT}, _{zT}, and _{Tz} determine the strength of the interactions between the variables. The gating variable _{j} couples the weight and the tag, and depends upon a low-pass filtered version of the plasticity-induction stimulus,

That is, _{j} switches if γ_{j} surpasses the threshold ϑ_{γ}. The factor κ = 1

The second gating variable

The constants _{up} and _{down} determine the sensitivity of

The input dependent terms

is driven at the moment of postsynaptic spiking,

is independent from LTP inductions and occurs in the model at the moment of presynaptic spikes _{j}(

where ξ^{α} ∈ {^{+}, ^{triplet}, ^{−}} and τ_{α} is the respective decay constant. Using the above formulation of the triplet rule as the foundation (Ziegler et al.,

where [_{+} =

In contrast to the original model by Ziegler et al., the equation for the consolidation variable _{j}, Equation 16, includes an additional coupling term with strength _{b} ≥ 0. This term destabilizes the consolidated state when the synapse is not bound to a stabilizing entity. To that end, we introduced a binary variable _{j} that is unity if the synapse is bound and zero if it is unbound. If _{j} = 1 the novel term vanishes and Equation 16 becomes identical to that in Ziegler et al. (_{j} = 0 the novel term tilts the potential to the left such that the right potential well (corresponding to the consolidated state) becomes shallower or even vanishes, thereby destabilizing the consolidation. The Heaviside function in Equation 16 ensures that the destabilization is only effective when the synapse is in the consolidated state.

Transitions between _{j} = 0 and _{j} = 1 (binding and unbinding) occur according to the kinetics given by Equation 2. The transition rates for unbinding

where τ_{A} is the decay constant.

The delayed decay of consolidation discussed in the last paragraphs of the “Results” section, is achieved by an activity-dependent intensity of the noise associated with the weight (Equations 14 and 15) as follows:

This ensures that the noise is bigger in the presence of activity (as long as the activity is greater than θ).

One third of synapses were initialized to their big, consolidated, state, meaning _{j}(0) = 1, _{j}(0) = 1, and _{j}(0) = 1, the remaining synapses were initialized as unconsolidated, meaning _{j}(0) = −1, _{j}(0) = −1, and _{j}(0) = −1. All of the synapses that were initially consolidated were initialized as bound by stabilizing entities (i.e., initialized in the _{j}(0) = 0, and all of the STDP parameters (^{−}(0)) were initialized to 0. See Table

We extend the model of Barrett et al. (

where _{i}(

The state-based model works through the changes in the transition rates between the seven states in response to different stimulus paradigms, such as HFS to cause LTP, or LFS to cause long-term depression (LTD). In the absence of any LTP or LTD inducing stimulus, the weak and strong basal states (states 3 and 4) transition back and forth by a rate of α (weak to strong) and β (strong to weak). The intermediately stable weak and strong states (states 2 and 5) both transition back to the basal states with a fixed rate of τ_{e}, and the stable weak and strong states (states 1 and 6) both transition to the basal states with a fixed rate of τ_{l}.

The original paper (Barrett et al.,

Following HFS, α becomes instantaneously very large (_{0}), moving all synapses that are in the weak basal state to the strong basal state. The rate from the strong basal state (state 4) to the intermediately stable strong state (state 5) becomes non-zero, following the time course:

And the transition rate from the intermediately stable strong state (state 5) to the stable strong state (state 6) becomes non-zero, following the time course:

The induction of LTD follows a similar pattern, just in response to LFS; however, that part of the model was not relevant to this study.

The transition from state 6 to 7 occurs through the binding of a stabilizer, and the transition back from state 7 to 6 occurs through the unbinding of a stabilizer. The dynamics of the stabilizer and its binding and unbinding was identical to that described for the extended write protected model, just with different values for some of the parameters. We selected a faster decay rate for the transition from state 6 to state 4, τ_{l}, making state 6 far less stable than in the original model. State 7 now has the longest decay rate, τ_{r}, making it the most stable state.

Since the state based model does not have any activity dependence, we used the averaged input values ^{A} from the reduced model, see Equation 25, at each point in time to determine the unbinding rates.

See Table

All simulations were run in Igor Pro (WaveMetrics). The leaky integrate-and-fire neuron for the write protected model was simulated with a time-step of 0.1 ms, the synapse model with the reconsolidation extension was simulated with a time step of 100 ms, and the state based model was simulated with a time step of 1 s. We used the Euler method for the integration of the deterministic neuron dynamics (Sections Neuron Model and Stimulation Protocols), and we used the Euler-Maruyama method for the integration of the stochastic differential equations in Section Extended Write-Protected Model.

All of the code for the simulations has been placed in a github repository, which can be found at

To model synaptic reconsolidation we took advantage of the finding that the molecular machinery for reconsolidation is distinct from that of consolidation (Taubenfeld et al.,

Experimentally, reconsolidation paradigms involve PSI combined with activation of synapses. The synaptic activation during reconsolidation paradigms is significantly weaker than during an LTD protocol. However, once synapses in the write-protected model reside in the consolidated state, the only way to transition back to the unconsolidated state is in response to a stimulation protocol for LTD. Therefore, this model has no capacity to undergo reconsolidation since no amount of PSI combined with activity that is insufficient to cause LTD would cause its synapses to decay from a consolidated state. The same observation holds for other models of consolidation (Fusi et al.,

We analyzed the results of Fonseca et al. (

This simple model, combined with the write-protected consolidation model reproduces the results of Fonseca et al. (

In response to the first paradigm, where PSI occurred alone without any stimulation, consolidation persisted in the model throughout the entire duration of the experiment, and was indistinguishable from the control condition without PSI (Figure

The stimulation for the third paradigm was almost identical to that of the second paradigm except that the initial low frequency stimulation continued for more than 1 h after LTP induction. Consolidation persisted in response to this paradigm with and without PSI (Figure

Finally, the fourth paradigm was the simplest, and most comparable to standard experiments of slice-based consolidation. This paradigm had continuous low frequency stimulation with an extended period of PSI. Consolidation persisted in response to this final paradigm, indistinguishable from the control condition without PSI (Figure

The extended write-protected model works as follows: the state of each synapse results from the interaction of three bistable systems, which are related to the weight, tag, and scaffold. Once the scaffold transitions from its “small” to its “big” state, it supports the tagging-related and the weight-related variables to also remain in their high state. This high state with “big” scaffold corresponds to the consolidated state of the synapse. In our extension of the model, the scaffold can remain “big” only if it is bound to a stabilizing entity. If the scaffold remains unbound, the consolidated “big” state is unstable and it will jump back to its small state after some time (Figure

To understand the different behaviors of the model in response to the different stimulus paradigms we need to track the development of the reservoir of PSI immune stabilizing entities (Figure

The difference between the responses to the first stimulation paradigm (Figure

The responses to the second stimulation paradigm (Figure

The “boundary conditions” for reconsolidation refer to the conditions under which reconsolidation does or does not occur (Eisenberg et al.,

We distinguish two phases of the experiment. First, the rising phase, corresponding to the first 60 min, was simulated with the complete stochastic model. Because during this phase the stimulation was identical for all simulations (20 min @ 0.1 Hz

The reduced model describes a system of two coupled differential equations for the number of bound synapses and the size of the reservoir of PSI immune entities. Two inputs drive it. First, an activity-related input captures the average joint spiking activity of the pre- and postsynaptic neuron pair (Figure

Importantly, the reduced model predicted the presence or absence of synaptic reconsolidation in the full model when tested on 5 arbitrary combinations of the durations of PSI and stimulation (data not shown). This allowed us to explore a broad range of combinations of PSI and stimulation to map out the boundary conditions for reconsolidation. This broad exploration would have taken a prohibitive amount of time if run on the full model, since the full model requires the stochastic simulation of the full neural dynamics including spike generation, and the dynamics of the variables that relate to the weight, tag, and scaffold for every synapse. All simulations began with a period of HFS and then a brief period of background low frequency stimulation. The PSI and stimulation were always centered on the same point in time such that an increase in duration would cause the beginning and ending to occur earlier or later (Figure

The write-protected model is not the only model of slice-based consolidation. Another useful model is the state-based model (Barrett et al.,

The state-based model describes each synapse as residing in one of six states (Figure

We extended the state-based model to capture the four stimulation paradigms studied in Fonseca et al. (

Both extended models deviated from the results described by Fonseca et al. (

The absence of the delayed decay in our models occurs because once the reservoir is empty (Figure

We were able to capture the delayed decay found by Fonseca et al. (

We have put forward a model that extends two different existing models of consolidation to enable them to capture reconsolidation at the synaptic level. Our model implements, using simple kinetics, two key features of slice-based reconsolidation: stabilization of consolidated synapses through the binding of stabilizing entities (Figure

Given the critical nature of the evolution of the reservoir for reconsolidation we developed a reduced two-dimensional model of the reservoir dynamics (Figure

In designing the model, we sought simplicity while still maintaining a connection to the underlying biology. Such biophysically inspired models have proven useful in connecting the molecular and cellular level to the computation performed by neurons and circuits (Ozuysal and Baccus,

Protein degradation plays a critical role in consolidation (Fonseca et al.,

In deciding upon the dynamics of the stabilizing entity, we used the finding that there seems to be limited, shared resources for consolidation (Fonseca et al.,

The existence of a reservoir of immune stabilizing entities seemed a plausible and parsimonious way to capture the experimental results; however, we are not aware of an exact biologic corollary to such a reservoir in this system. Posttranslational modifications, such as phosphorylation, provides enhanced stability for proteins (Cohen,

In keeping with our goal of simplicity, we chose to have relatively simple dynamics for the binding and unbinding. This limits the model in its robustness to perturbation. For instance, if the duration of PSI were a little longer in Figures

DK, LZ, and WG designed the research; LZ contributed unpublished reagents; DK and TS performed the research; DK, TS, and WG wrote the paper.

This work was supported by the European Research Council (Grant Agreement no. 268689, MultiRules), by a Swiss Government Excellence Scholarship as part of a Fulbright Award (DK) and by NIH R25MH060482 (DK).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank F. Zenke, S. Mensi, C. Pozzorini, A. Seeholzer, M. Deger, and J. Fitzgerald for helpful discussions, and we thank F. Dunn and O. Gozel for helpful comments on the manuscript.