Synaptic Consolidation Research Articles

Forget me not: The effect of doxycycline on human declarative memory

Investigations into neuroprotective drugs are in high demand for the treatment of neurodegenerative diseases, such as multiple sclerosis or Alzheimer's disease, but also psychiatric disorders, such as depression, trauma, and substance use. One potential drug class being investigated are tetracyclines impacting on a variety of neuroprotective mechanisms. At the same time, tetracyclines like doxycycline have been suggested to affect human fear and spatial memory as well as reducing declarative memory retention. Based on the assumed necessity for synaptic consolidation in hippocampus-dependent learning, we hypothesised declarative memory may be similarly impaired by doxycycline as fear and spatial memory. Therefore, in this study we investigate the potential diminishing effects of doxycycline on consolidation of declarative memory in healthy humans. Additionally, to test for effect specificity we assessed motor memory, sustained attention, and processing speed. We administered a neuropsychological test battery in three independent randomized placebo-controlled double-blind trials (RCTs), in which healthy young volunteers (total N = 252) either received a single oral dose doxycycline (200 mg, n = 126) or placebo (n = 126) in a between-subject design. We found no evidence for a detrimental effect of doxycycline on declarative memory; instead, doxycycline improved declarative learning (p-value=0.022, Cohen's d=0.15) and memory consolidation (p=0.040, d=0.26). Contrarily, doxycycline slightly reduced motor learning (p=0.001, d=0.10) but subtly strengthened long-term motor memory (p=0.001, d=0.10). These results suggest that doxycycline can improve declarative learning and memory without having long term negative effects on other cognitive domains in healthy humans. Our results give hope to further investigate doxycycline in neuroprotective treatment applications.

Recalled through this day but forgotten next week?—retrieval activity predicts durability of partly consolidated memories

Abstract Even partly consolidated memories can be forgotten given sufficient time, but the brain activity associated with durability of episodic memory at different time scales remains unclear. Here, we aimed to identify brain activity associated with retrieval of partly consolidated episodic memories that continued to be remembered in the future. Forty-nine younger (20 to 38 years; 25 females) and 43 older adults (60 to 80 years, 25 females) were scanned with functional magnetic resonance imaging during associative memory retrieval 12 h post-encoding. Twelve hours is sufficient to allow short-term synaptic consolidation as well as early post-encoding replay to initiate memory consolidation. Successful memory trials were classified into durable and transient source memories based on responses from a memory test ~6 d post-encoding. Results demonstrated that successful retrieval of future durable vs. transient memories was supported by increased activity in a medial prefrontal and ventral parietal area. Individual differences in activation as well as the subjective vividness of memories during encoding were positively related to individual differences in memory performance after 6 d. The results point to a unique and novel aspect of brain activity supporting long-term memory, in that activity during retrieval of memories even after 12 h of consolidation contains information about potential for long-term durability.

Advancing Neuro-Inspired Lifelong Learning for Edge with Co-Design

Lifelong learning, which refers to an agent's ability to continuously learn and enhance its performance over its lifespan, is a significant challenge in artificial intelligence (AI), that biological systems tackle efficiently. This challenge is further exacerbated when AI is deployed in untethered environments with strict energy and latency constraints. We take inspiration from neural plasticity and investigate how to leverage and build energy-efficient lifelong learning machines. Specifically, we study how a combination of neural plasticity mechanisms, namely neuromodulation, synaptic consolidation, and metaplasticity, enhance the continual learning capabilities of AI models. We further co-design architectures that leverage compute-in-memory topologies and sparse spike-based communication with quantization for the edge. Aspects of this co-design can be transferred to federated lifelong learning scenarios.

Learning with filopodia and spines: Complementary strong and weak competition lead to specialized, graded, and protected receptive fields.

Filopodia are thin synaptic protrusions that have been long known to play an important role in early development. Recently, they have been found to be more abundant in the adult cortex than previously thought, and more plastic than spines (button-shaped mature synapses). Inspired by these findings, we introduce a new model of synaptic plasticity that jointly describes learning of filopodia and spines. The model assumes that filopodia exhibit strongly competitive learning dynamics -similarly to additive spike-timing-dependent plasticity (STDP). At the same time it proposes that, if filopodia undergo sufficient potentiation, they consolidate into spines. Spines follow weakly competitive learning, classically associated with multiplicative, soft-bounded models of STDP. This makes spines more stable and sensitive to the fine structure of input correlations. We show that our learning rule has a selectivity comparable to additive STDP and captures input correlations as well as multiplicative models of STDP. We also show how it can protect previously formed memories and perform synaptic consolidation. Overall, our results can be seen as a phenomenological description of how filopodia and spines could cooperate to overcome the individual difficulties faced by strong and weak competition mechanisms.

3 Towards Detecting a Pre-clinical Signature of Dementia: Accelerated Forgetting in Healthy Older Samples -Implications for Methodology, Future Ageing Studies and Early Identification of Dementia

Objective:Accelerated long-term forgetting (ALF) has been reported in healthy older individuals, and is a possible early marker for risk of developing Alzheimer’s disease (AD). The Verbal Associative Learning & Memory Test (VALMT; McGibbon & Jansari, 2013) addresses methodological weaknesses in existing clinical tests and has detected ALF in epilepsy within an hour. We used VALMT to investigate learning and forgetting in healthy older participants.Participants and Methods:Older (60-69yrs) and Younger (19-31yrs) participants were compared. Using VALMT, unrelated word-pairs were learned to criterion, then cued-recall was tested at delays of 5, 30 and 55 minutes. Unique pairs were tested at each delay. Subjective memory complaints data were gathered, and the Wechsler Memory Scale Logical Memory test (WMS-LM; a standard clinical measure) was administered.Results:VALMT identified a significant difference in delayed recall between Younger and Older groups by 55 minutes (d = 1.32). While 'fast-learning’ Older participants scored similarly to Younger participants, 'slow-learning’ Older participants were impaired at all delays. Forgetting rates suggested degradation of memory starts during early synaptic consolidation rather than later system-level consolidation. Increased subjective memory complaints were associated with reduced VALMT scores. By contrast, WMS-LM failed to identify significant differences between any groups, and did not correlate with memory complaints.Conclusions:We conclude VALMT may be better able than WMS-LM to identify subtle impairments in healthy older adults within a single clinical visit, and VALMT results better reflect subjective experience. Older slow-learners forget faster and report more subjective memory complaints, which may indicate a group at risk of developing AD.

Improvements in operant memory of Aplysia are correlated with age and specific gene expression.

The transcription factor Aplysia CCAAT/enhancer binding protein (ApC/EBP) is expressed as an immediate early gene in the cAMP responsive element binding protein (CREB) mediated gene cascade, and it has essential functions in the synaptic consolidation of memory following a learning event. Synaptic consolidation primarily involves morphological changes at neuronal synapses, which are facilitated through the reorganization of the actin and microtubular cytoarchitecture of the cell. During early nervous system development, the transmembrane synaptic protein teneurin acts directly upon neuronal presynaptic microtubules and postsynaptic spectrin-based cytoskeletons to facilitate the creation of new synapses. It is reasonable to hypothesize that teneurin may also be linked to learning-induced synaptic changes and is a potential candidate to be a later gene expressed in the CREB-mediated gene cascade downstream of ApC/EBP. To assess the role of ApC/EBP and teneurin in learning and memory in the marine snail Aplysia californica, young (age 7-8 months) and aged (age 13-15 months; aging stage AII) siblings of Aplysia were trained in an operant conditioning paradigm-learning food is inedible (LFI)-over 2 days, during which they learned to modify the feeding reflex. Aged Aplysia had enhanced performance of the LFI task on the second day than younger siblings although far more aged animals were excluded from the analysis because of the initial failure in learning to recognize the inedible probe. After 2 days of training, ApC/EBP isoform X1 mRNA and teneurin mRNA were quantified in selected neurons of the buccal ganglia, the locus of neural circuits in LFI. Teneurin expression was elevated in aged Aplysia compared to young siblings regardless of training. ApC/EBP isoform X1 expression was significantly higher in untrained aged animals than in untrained young siblings but decreased in trained aged animals compared to untrained aged animals. Elevated levels of ApC/EBP isoform X1 and teneurin mRNA before training may have contributed to the enhancement of LFI performance in the aged animals that successfully learned.

An analytical theory of curriculum learning in teacher–student networks* *This article was also accepted in NeurIPS2022 Proceedings.

In animals and humans, curriculum learning—presenting data in a curated order—is critical to rapid learning and effective pedagogy. A long history of experiments has demonstrated the impact of curricula in a variety of animals but, despite its ubiquitous presence, a theoretical understanding of the phenomenon is still lacking. Surprisingly, in contrast to animal learning, curricula strategies are not widely used in machine learning and recent simulation studies reach the conclusion that curricula are moderately effective or even ineffective in most cases. This stark difference in the importance of curriculum raises a fundamental theoretical question: when and why does curriculum learning help? In this work, we analyse a prototypical neural network model of curriculum learning in the high-dimensional limit, employing statistical physics methods. We study a task in which a sparse set of informative features are embedded amidst a large set of noisy features. We analytically derive average learning trajectories for simple neural networks on this task, which establish a clear speed benefit for curriculum learning in the online setting. However, when training experiences can be stored and replayed (for instance, during sleep), the advantage of curriculum in standard neural networks disappears, in line with observations from the deep learning literature. Inspired by synaptic consolidation techniques developed to combat catastrophic forgetting, we propose curriculum-aware algorithms that consolidate synapses at curriculum change points and investigate whether this can boost the benefits of curricula. We derive generalisation performance as a function of consolidation strength (implemented as an L 2 regularisation/elastic coupling connecting learning phases), and show that curriculum-aware algorithms can yield a large improvement in test performance. Our reduced analytical descriptions help reconcile apparently conflicting empirical results, trace regimes where curriculum learning yields the largest gains, and provide experimentally-accessible predictions for the impact of task parameters on curriculum benefits. More broadly, our results suggest that fully exploiting a curriculum may require explicit adjustments in the loss.

Neuromodulator-dependent synaptic tagging and capture retroactively controls neural coding in spiking neural networks

Events that are important to an individual’s life trigger neuromodulator release in brain areas responsible for cognitive and behavioral function. While it is well known that the presence of neuromodulators such as dopamine and norepinephrine is required for memory consolidation, the impact of neuromodulator concentration is, however, less understood. In a recurrent spiking neural network model featuring neuromodulator-dependent synaptic tagging and capture, we study how synaptic memory consolidation depends on the amount of neuromodulator present in the minutes to hours after learning. We find that the storage of rate-based and spike timing-based information is controlled by the level of neuromodulation. Specifically, we find better recall of temporal information for high levels of neuromodulation, while we find better recall of rate-coded spatial patterns for lower neuromodulation, mediated by the selection of different groups of synapses for consolidation. Hence, our results indicate that in minutes to hours after learning, the level of neuromodulation may alter the process of synaptic consolidation to ultimately control which type of information becomes consolidated in the recurrent neural network.

A multifarious exploration of synaptic tagging and capture hypothesis in synaptic plasticity: Development of an integrated mathematical model and computational experiments

The synaptic tagging and capture (STC) hypothesis not only explain the integration and association of synaptic activities, but also the formation of learning and memory. The synaptic pathways involved in the synaptic tagging and capture phenomenon are called STC pathways. The STC hypothesis provides a potential explanation of the neuronal and synaptic processes underlying the synaptic consolidation of memories. Several mechanisms and molecules have been proposed to explain the process of memory allocation and synaptic tags, respectively. However, a clear link between the STC hypothesis and memory allocation is still missing because the encoding of memories in neural circuits is mainly associated with strongly recurrently connected groups of neurons. To explore the mechanisms of potential synaptic tagging candidates and their involvement in the process of memory allocation, we develop a mathematical model for a single dendritic spine based on five essential criteria of a synaptic tag. By developing a mathematical model, we attempt to understand the roles of the potentially critical molecular networks underlying the STC and the essential attributes of a synaptic tag. We include essential memory molecules in the STC model that have been identified in earlier studies as crucial for STC pathways. CaMKII activation is critical for the setting of the initial tag; however, coordinated activities with other kinases and the biochemical pathways are necessary for the tag to be stable. PKA modulates NMDAR-mediated Ca2+ signalling. Similarly, PKA and ERK crosstalk is essential for Ca2+ - mediated protein synthesis during l-LTP. Our theoretical model explains the quantitative contribution of Tags and protein synthesis during l-LTP in synaptic strength.

Prior fear learning enables the rapid assimilation of new fear memories directly into cortical networks

Long-term memory formation involves the reorganization of brain circuits, termed system consolidation. Whether and how a prior fear experience influences system consolidation of new memories is poorly understood. In rats, we found that prior auditory fear learning allows the secondary auditory cortex to immediately encode new auditory memories, with these new memories purely requiring the activation of cellular mechanisms of synaptic consolidation within secondary auditory cortex. Similar results were obtained in the anterior cingulate cortex for contextual fear memories. Moreover, prior learning enabled connections from these cortices to the basolateral amygdala (BLA) to support recent memory retention. We propose that the reorganization of circuits that characterizes system consolidation occurs only in the first instance that an event is learned, subsequently allowing the immediate assimilation of new analogous events in final storage sites.

MFS: A Brain-Inspired Memory Formation System for GAN

Generative adversarial networks (GANs) are subject to catastrophic forgetting when learning stream of data. Inspired by the knowledge of neuroscience, this article develops a memory formation system (MFS) to establish memory for GANs. MFS is composed of three modules, including the identifier, weights upgrade (WU), and weights reactivate (WR). These modules simulate the function of encoding, consolidating, and retrieving in memory formation of human. Identifier creates indexes to label continuous tasks and these indexes are used as a cue when the corresponding tasks are recalled. WU and WR work as synaptic consolidation and system consolidation, respectively. In WU, a novel method, weight saliency measure (WSM) is proposed to measure the saliency of weight. Valuable weights are protected and invaluable weights are released for update when GAN is trained for the new task. In WR, traditional data-replay methods are improved by selecting <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${k}$ </tex-math></inline-formula> representatives in each class and their feature vectors are restored. When pseudo data of this class are regenerated, only the samples with distance less than a threshold can be used in future training. Experimental results based on the testing of continual image generation and continual 3-D reconstruction show that MFS can establish a memory system to handle the catastrophic forgetting problem effectively.

Accelerated forgetting in healthy older samples: Implications for methodology, future ageing studies, and early identification of risk of dementia.

Accelerated long-term forgetting (ALF) has been reported in healthy older individuals, and is a possible early marker for risk of developing Alzheimer's disease (AD). The Verbal Associative Learning and Memory Test (VALMT) addresses methodological weaknesses in existing clinical tests and has detected ALF in epilepsy within an hour. We used VALMT to investigate learning and forgetting in healthy Older participants. Older (60-69 years) and Younger (19-31 years) participants were compared. Using VALMT, unrelated word pairs were learnt to criterion, then cued-recall tested at delays of 5, 30, and 55 min. Unique pairs were tested at each delay. Subjective memory complaints data was gathered, and the Wechsler Memory Scale Logical Memory test (WMS-LM; a standard clinical measure) was administered. VALMT identified a significant difference in delayed recall between Younger and Older groups by 55 min (d = 1.32). While "fast-learning" Older participants scored similarly to Younger participants, "slow-learning" Older participants were impaired at all delays. Forgetting rates suggested degradation of memory starts during early synaptic consolidation rather than later system-level consolidation. Increased subjective memory complaints were associated with reduced VALMT scores. By contrast, WMS-LM failed to identify significant differences between any groups, and did not correlate with memory complaints. We conclude that VALMT may be better able than WMS-LM to identify subtle impairments in healthy older adults within a single clinical visit, and VALMT results better reflect subjective experience. Older slow-learners forget faster and report more subjective memory complaints, which may indicate a group at risk of developing AD.

Memory Recall: A Simple Neural Network Training Framework Against Catastrophic Forgetting.

It is widely acknowledged that biological intelligence is capable of learning continually without forgetting previously learned skills. Unfortunately, it has been widely observed that many artificial intelligence techniques, especially (deep) neural network (NN)-based ones, suffer from catastrophic forgetting problem, which severely forgets previous tasks when learning a new one. How to train NNs without catastrophic forgetting, which is termed continual learning, is emerging as a frontier topic and attracting considerable research interest. Inspired by memory replay and synaptic consolidation mechanism in brain, in this article, we propose a novel and simple framework termed memory recall (MeRec) for continual learning with deep NNs. In particular, we first analyze the feature stability across tasks in NN and show that NN can yield task stable features in certain layers. Then, based on this observation, we use a memory module to keep the feature statistics (mean and std) for each learned task. Based on the memory and statistics, we show that a simple replay strategy with Gaussian distribution-based feature regeneration can recall and recover the knowledge from previous tasks. Together with the weight regularization, MeRec preserves weights learned from previous tasks. Based on this simple framework, MeRec achieved leading performance with extremely small memory budget (only two feature vectors for each class) for continual learning on CIFAR-10 and CIFAR-100 datasets, with at least 50% accuracy drop reduction after several tasks compared to previous state-of-the-art approaches.

Primary cilia are required for the persistence of memory and stabilization of perineuronal nets

SummaryIt is well established that the formation of episodic memories requires multiple hippocampal mechanisms operating on different time scales. Early mechanisms of memory formation (synaptic consolidation) have been extensively characterized. However, delayed mechanisms, which maintain hippocampal activity as memories stabilize in cortical circuits, are not well understood. Here we demonstrate that contrary to the transient expression of early- and delayed-response genes, the expression of cytoskeleton- and extracellular matrix-associated genes remains dynamic even at remote time points. The most profound expression changes clustered around primary cilium-associated and collagen genes. These genes most likely contribute to memory by stabilizing perineuronal nets in the dorsohippocampal CA1 subfield, as revealed by targeted disruptions of the primary cilium or perineuronal nets. The findings show that nonsynaptic, primary cilium-mediated mechanisms are required for the persistence of context memory.

Role of adult hippocampal neurogenesis in the antidepressant actions of lactate

In addition to its role as a neuronal energy substrate and signaling molecule involved in synaptic plasticity and memory consolidation, recent evidence shows that lactate produces antidepressant effects in animal models. However, the mechanisms underpinning lactate’s antidepressant actions remain largely unknown. In this study, we report that lactate reverses the effects of corticosterone on depressive-like behavior, as well as on the inhibition of both the survival and proliferation of new neurons in the adult hippocampus. Furthermore, the inhibition of adult hippocampal neurogenesis prevents the antidepressant-like effects of lactate. Pyruvate, the oxidized form of lactate, did not mimic the effects of lactate on adult hippocampal neurogenesis and depression-like behavior. Finally, our data suggest that conversion of lactate to pyruvate with the concomitant production of NADH is necessary for the neurogenic and antidepressant effects of lactate.

Memory consolidation and improvement by synaptic tagging and capture in recurrent neural networks

The synaptic-tagging-and-capture (STC) hypothesis formulates that at each synapse the concurrence of a tag with protein synthesis yields the maintenance of changes induced by synaptic plasticity. This hypothesis provides a biological principle underlying the synaptic consolidation of memories that is not verified for recurrent neural circuits. We developed a theoretical model integrating the mechanisms underlying the STC hypothesis with calcium-based synaptic plasticity in a recurrent spiking neural network. In the model, calcium-based synaptic plasticity yields the formation of strongly interconnected cell assemblies encoding memories, followed by consolidation through the STC mechanisms. Furthermore, we show for the first time that STC mechanisms modify the storage of memories such that after several hours memory recall is significantly improved. We identify two contributing processes: a merely time-dependent passive improvement, and an active improvement during recall. The described characteristics can provide a new principle for storing information in biological and artificial neural circuits.

Herpes Simplex Virus Type 1 Neuronal Infection Triggers the Disassembly of Key Structural Components of Dendritic Spines.

Herpes simplex virus type 1 (HSV-1) is a widespread neurotropic virus. Primary infection of HSV-1 in facial epithelium leads to retrograde axonal transport to the central nervous system (CNS) where it establishes latency. Under stressful conditions, the virus reactivates, and new progeny are transported anterogradely to the primary site of infection. During the late stages of neuronal infection, axonal damage can occur, however, the impact of HSV-1 infection on the morphology and functional integrity of neuronal dendrites during the early stages of infection is unknown. We previously demonstrated that acute HSV-1 infection in neuronal cell lines selectively enhances Arc protein expression - a major regulator of long-term synaptic plasticity and memory consolidation, known for being a protein-interaction hub in the postsynaptic dendritic compartment. Thus, HSV-1 induced Arc expression may alter the functionality of infected neurons and negatively impact dendritic spine dynamics. In this study we demonstrated that HSV-1 infection induces structural disassembly and functional deregulation in cultured cortical neurons, an altered glutamate response, Arc accumulation within the somata, and decreased expression of spine scaffolding-like proteins such as PSD-95, Drebrin and CaMKIIβ. However, whether these alterations are specific to the HSV-1 infection mechanism or reflect a secondary neurodegenerative process remains to be determined.

Overcoming Long-Term Catastrophic Forgetting Through Adversarial Neural Pruning and Synaptic Consolidation.

Enabling a neural network to sequentially learn multiple tasks is of great significance for expanding the applicability of neural networks in real-world applications. However, artificial neural networks face the well-known problem of catastrophic forgetting. What is worse, the degradation of previously learned skills becomes more severe as the task sequence increases, known as the long-term catastrophic forgetting. It is due to two facts: first, as the model learns more tasks, the intersection of the low-error parameter subspace satisfying for these tasks becomes smaller or even does not exist; second, when the model learns a new task, the cumulative error keeps increasing as the model tries to protect the parameter configuration of previous tasks from interference. Inspired by the memory consolidation mechanism in mammalian brains with synaptic plasticity, we propose a confrontation mechanism in which Adversarial Neural Pruning and synaptic Consolidation (ANPyC) is used to overcome the long-term catastrophic forgetting issue. The neural pruning acts as long-term depression to prune task-irrelevant parameters, while the novel synaptic consolidation acts as long-term potentiation to strengthen task-relevant parameters. During the training, this confrontation achieves a balance in that only crucial parameters remain, and non-significant parameters are freed to learn subsequent tasks. ANPyC avoids forgetting important information and makes the model efficient to learn a large number of tasks. Specifically, the neural pruning iteratively relaxes the current task's parameter conditions to expand the common parameter subspace of the task; the synaptic consolidation strategy, which consists of a structure-aware parameter-importance measurement and an element-wise parameter updating strategy, decreases the cumulative error when learning new tasks. Our approach encourages the synapse to be sparse and polarized, which enables long-term learning and memory. ANPyC exhibits effectiveness and generalization on both image classification and generation tasks with multiple layer perceptron, convolutional neural networks, and generative adversarial networks, and variational autoencoder. The full source code is available at https://github.com/GeoX-Lab/ANPyC.

A Psychological Foundation for Team-Based Learning: Knowledge Reconsolidation.

Although team-based learning is a popular instructional approach, little is known about its psychological foundation. In this Perspective, the authors propose a theoretical account of the psychological mechanisms through which team-based learning works. They suggest a knowledge reconsolidation hypothesis to explain how the distinct phases of team-based learning enable students to learn. Knowledge reconsolidation is the process whereby previously consolidated knowledge is retrieved from memory with the purpose of actively consolidating it again. Reconsolidation aims to preserve, strengthen, and adjust knowledge that is already stored in long-term memory. This process is generally considered an important reason why people who reactivate what they have previously learned many times develop knowledge structures that are extremely stable and easily retrieved.The authors propose that 4 psychological mechanisms enable knowledge reconsolidation, each of which is tied to a distinct phase of team-based learning: retrieval practice, peer elaboration, feedback, and transfer of learning. Before a team-based learning session, students engage in independent, self-directed learning that is often followed by at least one night of sleep. The latter is known to facilitate synaptic consolidation in the brain. During the actual team-based learning session, students are first tested individually on what they learned, then they discuss the answers to the test with a small group of peers, ask remaining "burning questions" to the teacher, and finally engage in a number of application exercises.This knowledge reconsolidation hypothesis may be considered a framework to guide future research into how team-based learning works and its outcomes.

Optimal Stimulation Protocol in a Bistable Synaptic Consolidation Model.

Synaptic changes induced by neural activity need to be consolidated to maintain memory over a timescale of hours. In experiments, synaptic consolidation can be induced by repeating a stimulation protocol several times and the effectiveness of consolidation depends crucially on the repetition frequency of the stimulations. We address the question: is there an understandable reason why induction protocols with repetitions at some frequency work better than sustained protocols—even though the accumulated stimulation strength might be exactly the same in both cases? In real synapses, plasticity occurs on multiple time scales from seconds (induction), to several minutes (early phase of long-term potentiation) to hours and days (late phase of synaptic consolidation). We use a simplified mathematical model of just two times scales to elucidate the above question in a purified setting. Our mathematical results show that, even in such a simple model, the repetition frequency of stimulation plays an important role for the successful induction, and stabilization, of potentiation.