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Edited by: Hava T. Siegelmann, Rutgers University, USA

Reviewed by: Paolo Del Giudice, Italian National Institute of Health, Italy; Gianluigi Mongillo, Paris Descartes University, France; Alberto Bernacchia, Jacobs University Bremen, Germany

*Correspondence: Misha Tsodyks

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.

Human memory can store large amount of information. Nevertheless, recalling is often a challenging task. In a classical free recall paradigm, where participants are asked to repeat a briefly presented list of words, people make mistakes for lists as short as 5 words. We present a model for memory retrieval based on a Hopfield neural network where transition between items are determined by similarities in their long-term memory representations. Meanfield analysis of the model reveals stable states of the network corresponding (1) to single memory representations and (2) intersection between memory representations. We show that oscillating feedback inhibition in the presence of noise induces transitions between these states triggering the retrieval of different memories. The network dynamics qualitatively predicts the distribution of time intervals required to recall new memory items observed in experiments. It shows that items having larger number of neurons in their representation are statistically easier to recall and reveals possible bottlenecks in our ability of retrieving memories. Overall, we propose a neural network model of information retrieval broadly compatible with experimental observations and is consistent with our recent graphical model (Romani et al.,

Human long-term memory capacity for names, facts, episodes and other aspects of our lives is practically unlimited. Yet recalling this information is often challenging, especially when no precise cues are available. A striking example of this deficiency is provided by classical studies of free recall, where participants are asked to recall lists of unrelated words after a quick exposure (Murdock,

Several influential models of recall were developed. Some of them were driven by the description of behavioral aspects (Glenberg and Swanson,

According to the influential “search of associative memory” (SAM) model, items presented for recall acquire a set of mutual associations when stored temporarily in working memory buffers (Raaijmakers and Shiffrin,

We consider a Hopfield neural network of _{i} is an independent random variable having a gaussian distribution with mean zero and variance ξ_{0} and τ is a constant^{1}

The gain function is:

Each of the

Memory μ is recalled if the average firing rate of neurons corresponding to memory μ (_{thresh}. This threshold is chosen so that two memories are never recalled simultaneously. If in a given time interval, e.g., from time 0 to _{1}, μ_{2}, μ_{3}.. at different times, we say that the network has “retrieved” these memories in a time

A slight modification of the model allows to account for short-term associations as in the SAM model. For example, temporal contiguity is the tendency to recall neighboring presented items in temporal proximity. To account for this effect we add a new term to the connectivity matrix _{ij}:

The new part δ_{ij} consists of two terms which respectively connect a given memory μ with the memories presented immediately before and after it (μ − 1 and μ + 1) (Sompolinsky and Kanter, _{+} and _{−}.

We analyze the network in the absence of noise (ξ_{0} = 0) and temporal contiguity (_{+} = _{−} = 0). To quantify the degree of memory activations we introduce the “overlaps” defined as in Amit and Tsodyks (

While ^{0}(^{μ}(

At a fix point of the network dynamics (Equation 2) the synaptic currents can be expressed via the values of the overlaps:
^{P} possible realizations of vector _{i} that are denoted by a random vector ^{P} where each component is indipendent from any other being 1 with probability _{i} = ^{μ} = 1 (Curti et al.,

The cardinality of a vector is defined as

The probability for each vector

The fixed point solutions can then be characterized in the limit _{v} as:

This system determines the fixed points of the network in the meanfield limit. It cannot be solved in general but for a given

the currents to each population _{v} + θ > 0 if ^{μ} = 1;

the currents to each population that doesn't belong to the active memory μ are below threshold _{v} + θ < 0 if ^{μ} = 0;

This two conditions define our ansatz for a single memory state. From this definition it follows that in the state of single memory the the only overlap ^{μ}. Similarly we define the ansatz for the intersection between two or more memories. In this state only two overlaps

To study the influence of finite size effects and noise on the dynamics of the network we simulate the dynamic of a network of ^{5} neurons. To achieve this goal we simplify the system in Equation (2). This is a dimensionality reduction of the network that reduces the number of simulated units. All the neurons that have the same vector _{i} (i.e., are in the same population _{i} = _{v}(_{v}(

The vectors ^{P} equations instead of the ^{μ∈{1..P}}. Although in principle the system has 2^{P} equations, in practice, due to the finite size of the network and its sparse connectivity, there are much less populations since _{v} = 0 for most ^{5} neurons. Indeed taking ^{5} of the original system in Equation (2) to the ≈1000 of the reduced one of Equation (17).

Simulations are run according to Equation (17) employing the parameters in Table _{trials}. For each simulation the network is initialized in the state of a single, randomly chosen memory μ. In this state all the populations _{ini} while the others are initialized to a zero rate. In the model the transitions between memories are triggered by oscillations of the variable φ. This oscillates sinusoidally between the values φ^{max} and φ^{min}. The oscillations have a period τ_{o} which is much larger than τ so that the network is undergoing an adiabatic process. Integrations of Equation (17) are performed with the Euler method with a time step of _{o}.

N | Number of neurons | 100,000 |

P | Number of memories | 16 |

f | Sparsity | 0.1 |

τ | Decay time | 0.01 |

κ | Excitation parameter | 13, 000 |

φ^{max} |
Max inhibition parameter | 1.06 |

φ^{min} |
Min inhibition parameter | 0.7 |

γ | Gain function exponent | 2/5 |

θ | Gain function threshold | 0 |

τ_{o} |
Oscillation time | 1 |

_{tot} |
Total time | 450 |

Integration time step | 0.001 | |

_{+} |
Forward contiguity | 1500 |

_{−} |
Backward contiguity | 400 |

ξ_{0} |
Noise variance | 65 |

_{thresh} |
Recall threshold | 15 |

_{trials} |
Number of trials | 10, 000 |

_{ini} |
Initial rate | 1 |

The data analyzed in this manuscript were collected in the lab of M. Kahana as part the Penn Electrophysiology of Encoding and Retrieval Study. Here we analyzed the results from the 141 participants (age 17–30) who completed the first phase of the experiment, consisting of 7 experimental sessions. Participants were consented according the University of Pennsylvanias IRB protocol and were compensated for their participation. Each session consisted of 16 lists of 16 words presented one at a time on a computer screen and lasted approximately 1.5 h. Each study list was followed by an immediate free recall test. Words were drawn from a pool of 1638 words. For each list, there was a 1500 ms delay before the first word appeared on the screen. Each item was on the screen for 3000 ms, followed by jittered 800–1200 ms inter-stimulus interval (uniform distribution). After the last item in the list, there was a 1200–1400 ms jittered delay, after which the participant was given 75 s to attempt to recall any of the just-presented items. Only trials without errors (no intrusions and no repeated recalls of the same words) were used in the analysis.

We analyze this dataset to validate our model. We investigated several aspects of the dataset as described in Katkov et al. (

For each pair of consecutive reported items we compute the IRT by the difference of their times of retrieval. This is the quantity shown on the y-axis of

The main principle of recall that was suggested in Romani et al. (^{1} = … = ^{Q} = ^{active}. The overlaps have all the same values as all the active neurons in the intersection of ^{active} and ^{inactive} are respectively the value of the overlap for an active and inactive memory and ^{0} denotes the average activity of the network.

Based on this analysis, we simulate the network while modulating the inhibition to cause the transitions between these two states (see Section 2 for details of simulations). We also add noise in order to trigger the transitions to the intersections between two attractors when inhibition rises. To mimic the experimental protocol (see Section 2), we simulate multiple recall trials where random samples of 16 items are selected for each trial. One sample epoch of simulations is shown in Figures

Each of the colored line in Figure _{thresh} we regard the corresponding memory as retrieved. We note that the precise sequence of retrieved items is not predictable for a given list of presented words, as it strongly depends on the first item being recalled (here assumed to be chosen randomly) and is sensitive to noise.

The effect of the oscillations is to modulate the overall activity in such a way that at each cycle the state of the network can potentially move from one attractor to another. The details of the underlying dynamics are shown in the plot of Figure

Although a switch between different states of the network is induced at every oscillation cycle, not always the state of the network shifts toward a new memory (Figures

Since the recall of subsequent memories is a stochastic process triggered by noise in the input, we perform multiple simulations to characterize the average accumulation of recalled memories with time (Figure

Here we study the dependence of the recall process on the statistics of memory representations as defined by the memory patterns introduced in Section 2 (see Equation 4). In particular we consider the effects of representation size (number of neurons encoding a given item) and the size of intersections between the representations of two memories (number of neurons encoding both of the items). The representation size higly influences the probability of recall for a given memory. Our simulations show that simulating the network many times with items having a randomly drawn size, the probability to recall an item is monotonically increasing with the size of the corresponding representation (Figure

Intersections between memory representations play a crucial role in our model of recall. In Romani et al. (

We now focus on factors which influence the recall performance, namely the number of items that can be retrieved in a given time window, between time 0 and time

The performance of the network is limited as item representations that control the retrieval dynamics are random and hence same items are recalled numerous times before the network can retrieve a new memory. It is known however that the order of recall is not completely random, e.g., words that have neighboring positions in the list have a tendency to be recalled in close proximity (Sederberg et al.,

_{+}. _{+} ranges between the fixed value of _{−} = 400 and 2500. The number of memories is

Another crucial element of the model is the noise that causes the recall dynamics to escape the short loops and retrieve new items. We thus computed the network performance for increasing noise levels (Figure

We presented a neural network model of information retrieval from long-term memory that is based on stochastic attractor dynamics controlled by periodically modulated strength of feedback inhibition. The model provides a more realistic implementation of the mechanisms behind associative recall based on neuronal representations of memory items, as proposed in Romani et al. (

In classical models of recall, such as SAM (Raaijmakers and Shiffrin,

Our network model is based on the basic assumption that when a word is recalled, a corresponding neuronal ensemble that represents this word in long-term memory is temporarily activated. The issue that we dont explicitly address is how the words that are presented for recall are selected, or primed and why other word representations are not reactivated (excluding rare instances of erroneous recall of words from previous lists). In the spirit of Kahanas TCM model (Howard and Kahana,

Another simplifying unrealistic assumption of the model concerns the statistics of long-term representations that are taken as random uncorrelated binary vectors of fixed average sparsity. Real statistics of word representations is not clear but can be safely assumed to be much more complicated, possibly reflecting the rich semantic associations between words and the frequency of their usage. With our assumptions, overlaps between different representations exhibit Gaussian distribution with variance to mean ratio decaying in the limit of infinitely large networks. Considering the effects of overlap distribution in this limit requires an extended mean-field analysis that will be presented elsewhere.

Very often the same attractor is repeatedly activated before noise causes the transition to a new one, and it can still be activated again at a later time. Since participants are instructed to only recall each word once, we assume that they suppress the report of a word after it is already recalled. In some experiments, subjects are explicitly instructed to report a word as many times as it comes to mind during a recall. Comparing the model to the results of such experiments could be of interest for a future work.

We considered modulated inhibition as a driving force for transitions between network attractors. Other mechanisms could potentially play this role, e.g., neuronal adaptation or synaptic depression. We believe that oscillatory mechanism is more plausible as it allows the system to regulate the transitions by controlling the amplitude and frequency of oscillations. The oscillations of network activity could correspond to increased amplitude of theta rhythm observed in human subjects during recall (Kahana,

At the current level of realism, we propose to view our model as a platform for further development of realistic neural network models of information retrieval and other related types of cognitive tasks. Future modifications should include effects of positional order on recall, or positional chunking, i.e., the tendency to divide the presented lists on groups of contiguous words (Miller,

MT and SR designed the study; SR developed and simulated the model; MT, MK, and SR performed a mathematical analysis, SR and MK performed data analysis; all the authors wrote the paper.

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 are grateful to M. Kahana for generously sharing the data obtained in his laboratory with us. The lab of Kahana is supported by NIH grant MH55687. MT is supported by EU FP7 (Grant agreement 604102) and the Foundation Adelis. A part of this work related to network recall performance (Section 3.4) was supported by The Russian Science Foundation No.14-11-00693.

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