A remarkable development (and implementation, too) of Ebbinghaus' theory is the study by Georgiou et al. (2021). Basically, Georgiou, Katkov, and Tsodyks proposed a model which is strength-dependent retroactive interference between the memories. Hence, only if a stronger memory is acquired after the weaker one, then the weaker one is erased. The model results in powerlaw retention curves with exponents that very slowly decline toward -1. The asymptotic value for all realistic time lags that can be measured experimentally.

In our opinion, Murre et al. (2013) showed a stunning example of an application of Ribot's law to modeling amnesias. Their model assumes that memory processes can be decomposed into a number of processes that contain memory representations. Memory processes has a wide range of variability, from milliseconds (extremely short-term processes) to decades (very long-term processes). A memory representation could be thought of as consisting of one or more traces, such a representation can be viewed as neural pathways, any of which suffices to retrieve the memory. This trace generation is governed in a random way. Each trace in a process generates traces of its representation in the next higher process, for example, through long-term potentiation (LTP) in the hippocampus (Abraham, 2003) or neocortex (Racine et al., 1995). LTP is a stable facilitation of synaptic potentials after high-frequency synaptic activity, is very prominent in the hippocampus and is a leading candidate memory storage mechanism. We remark that a trace can be overwritten by different traces or by neural noise; in these cases, the trace is lost. As a consequence, it can no longer generate new traces in higher processes. The authors hypothesize that first, all traces in a process share the same loss probability; second, higher processes in the chain have lower decline rates. If the hippocampus undergoes a lesion at time t_l, then no more memories will be formed after that. In addition, no more consolidation from hippocampus-to-cortex happens.

If r(t_l) denotes the intensity of a particular memory at the time of the lesion, after t_l, a decline of the memory intensity, with neocortical decline rate a₂, will be observed, the equation representing this case is given by

where τ is the time elapsed since the lesion. Interestingly, the authors introduce the case partial lesion of the hippocampus, this means that they leave the size of the lesion as a free parameter. The lesion parameter is denoted as λ, λ ranges from 0 to 1, extremes included. If the lesion parameter is 0, no lesion is present; on the opposite, if λ = 1, there is a complete lesion. In case of a partial lesion, the Ribot gradient is equal to

This is the most general form of the model, based on the Ribot gradient, proposed by the authors. Generally, the tests of retrograde amnesia provide recall probabilities as a function of time elapsed, such a probability is denoted as p(t). Mathematically speaking, an observed recall probability p(t) can be transformed into an intensity r(t) by taking − ln(1 − p(t)), where ln is the natural-based logarithm.

The sensory memory store has a large capacity, but for a very brief duration, it encodes information from any of the senses (principally from the visual and auditory systems in Humans), and most of the information is lost through decay. The threshold above mentioned is strictly linked to the attention. Indeed, attention is the first step in remembering something; if a person's attention is focused on one of the sensory stores, then the data are likely to be transferred to STM (for more details, see for example Goldstein, 2019).

If the information passes the selection in the first stage (sensory memory) of selection, then it is transferred to the short-term store (also short-term memory). As with sensory memory, the information enters short-term memory decays and is lost, but the information in the short-term store has a longer duration, approximately up to 30 s when the information is not being actively rehearsed (Posner, 1966). A key concept in this model is the memory rehearsal, and it is a term for the role of repetition in the retention of memories. It involves repeating information over and over in order to get the information processed and stored as a memory.

It should be noted that a (continuous) rehearsal acts as a sort of regeneration of the information in the memory trace, thus making it a stronger memory when transferred to the long-term store (see Section 2.3.3). Differently, if maintenance rehearsal (i.e., the repetition of the information) does not occur, then information is forgotten and lost from short term memory through the processes of displacement or decay. Once again, a thresholding procedure occurs.

In terms of capacity, the short-term store has a limit to the amount of information that it can held in, quantitatively from 5 to 9 chunks (7 ± 2).¹

The long-term memory is in theory a sort of unlimited store, where the information could have a permanent duration. In the authors' model, the information that is stored can be transferred to the short-term store where it can be manipulated.

Information is postulated to enter the long-term store from the short-term store after the thresholding process. As Atkinson and Shiffrin modeled it, transfer from the short-term store to the long-term store is occurring for as long as the information is being attended to in the short-term store. The longer an item is held in short-term memory, the stronger its memory trace will be in long-term memory. Atkinson and Shiffrin based their observations on the studies by Hebb (1961) and Melton (1963), which show that repeated rote repetition enhances long-term memory. There is also a connection with the Ebbinghaus' studies on memory that shows how forgetting increases for items which are studied/repeated fewer times (Ebbinghaus, 1913).

Remarkably, simple rote rehearsal is not the stronger encoding processes; indeed, in author's opinion, the new information to information which has already made its way into the long-term store is a more efficient process.

The authors used a mathematical description of their proposal. Such a mathematical formalization is well detailed in Atkinson et al. (1967). In short, the memory buffer may be viewed as a state containing those items which have been selected from the sensory buffer for repeated rehearsal. Once the memory buffer is filled, each new item which enters causes one of the items currently in the buffer to be lost. It is assumed that the series of study items at the start of each experimental session fills the buffer and that the buffer stays filled thereafter. The size of the memory buffer is denoted by r, which is defined as the number of items which can be held simultaneously, depends upon the nature of the items and thus must be estimated for each experiment. It is also assumed that a correct response is given with probability one if an item is in the buffer at the time it is tested. Every item is selected by the sensory buffer (namely, it undergoes a thresholding process) to be entered into the memory buffer. The authors assume that the items are examined at the time they enter the sensory buffer. The items can be already in the buffer, i.e., their stimulus member can already be in the buffer or their stimulus member can not currently be in the buffer. The former case is denoted by the Authors as O-item (or “old” item), while the latter as N-item (“new” item). When an O-item is presented for study, it enters the memory buffer with probability one; the corresponding item, which was previously in the buffer, is discarded. When an N-item is presented for study, it enters the buffer with probability α, such a probability is function of the particular scheme that a subject is using to rehearse the items currently in the buffer. It is an N-item enter, the probability that such an event occur is α, then some item currently in the buffer is lost. Of course, the probability that an N-item fails to enter the buffer is 1 − α; in this case, the buffer does not undergo any change and the item in object decays and is permanently lost from memory.

The memory buffer is arranged as a push-down list. The newest item that enters the buffer is placed in slot r, and the item that has remained in the buffer the longest is in slot 1, i.e., the slot where the oldest item is. If an O-item enters slot r, the corresponding item is lost. Then, the other items move down one slot if necessary, retaining their former order. When an N-item is presented for study and enters the buffer (with probability α), it is placed in the r^th slot. The item currently in slot j has a probability κ_j to be discarded (or knocked out, the term used by the authors), the following condition must hold: κ₁ + κ₂ + κ₃ + ... + κ_j + ...κ_r = 1, with r ≥ j. When the j^th item is discarded, each item above the j^th moves down one and the new item enters the r^th slot. The simplest form of κ_j is κj=1r; in this case, the item to be knocked out is chosen independently of the buffer position.

At this point, let us focus on long-terms storage (LTS).

LTS can be viewed as a memory state where the information accumulates for each item. The authors made a few assumptions:

In case of a subject undergoes a test on an item, the subject gives the correct response if the item is in the sensory or memory buffer, but if the item is not in either of these buffers, the subject searches LTS. This LTS search is called the retrieval process. In this regard, two important observations should be made: First, it is assumed that the likelihood of retrieving the correct response for a given item improves as the amount of information stored concerning that item increases. Second, the retrieval of an item gets worse; the longer the item has been stored in LTS. In other words, there is some sort of decay in information as a function of the length of time information has been stored in LTS.

After these assumptions and observations, it is then possible to specify the probability of a correct retrieval of an item from LTS. If the amount of information stored at the moment of test for an item is zero, then the probability of a correct retrieval should be at the guessing level. As the amount of information increases, the probability of a correct retrieval should increase toward unity. The authors define p_ij as the probability of a correct response from LTS of an item that had a lag of i trials between its study and test and that resided in the buffer for exactly j trials. Hence, such a probability can be mathematically written as

where g is the guessing probability; for example, if an experiment is made up of 26 response alternatives, then the guess probability is 126.

As already mentioned previously, the Atkinson-Shiffrin memory model is an influential model. It is no surprise to note that several models have been developed on its basis. In the following, we provide a chronological history of such developments.

The Search of Associative Memory (SAM) model by Raaijmakers and Shiffrin, was proposed in 1981 and described in Raaijmakers and Shiffrin (1981); the likelihood of remembering one of the remaining words is lower than if no cues are given at all when free recall of a list of words is prompted by a random subset of those words. SAM utilizes interword connections extensively in retrieval, a mechanism that has been overlooked by prior thinking, to predict this effect in all of its forms.

The SAM model for recall (Raaijmakers and Shiffrin, 1981) is extended by assuming that a familiarity process is used for recognition. The recall model, proposed in 1984 by Gillund and Shiffrin (1984), proposes probabilistic sampling and recovery from an associative network that is dependent on cues. The recall model postulates cue-dependent probabilistic sampling and recovery from an associative network. The recognition model, proposed by Gillund and Shiffrin, is strictly linked to the recall model because the total episodic activation due to the context and item cues is used in recall as a basis for sampling and in recognition to make a decision. The model predicts the results from a new experiment on the word-frequency effect.

In 1997, Shiffrin and Steyvers (1997), proposed the REM model (standing for retrieving effectively from memory) developed to predict places explicit and implicit memory, as well as episodic and general memory, into the framework of a more complex theory that is being created to explain these phenomena. The model assumes storage of separate episodic images for different words, each image consisting of a vector of feature values.

Mueller and Shiffrin (2006) presented the REM-II model, and this model is based on Bayesian statistics. REM-II models the development of episodic and semantic memory. Semantic information is represented by the model as a collection of these features' co-occurrences, while episodic traces are represented as sets of features with varying values. Feature co-occurrence approaches the complexity of human knowledge by enabling polysemy and meaning connotation to be recorded inside a single structure. The authors present how knowledge is formed in REM-II, how experience gives rise to semantic spaces, and how REM-II leads to polysemy and encoding bias.

The SARKAE (Storing and Retrieving Knowledge and Events) model proposed by Nelson and Shiffrin (2013), which represents a further development of the SAM model, describes the development of knowledge and event memories as an interactive process: Knowledge is formed through the accrual of individual events, and the storage of an individual episode is dependent on prior knowledge. To support their theory, the authors refer to two experiments that provide data to support the theory: These experiments involve the acquisition of new knowledge and then testing in transfer tasks related to episodic memory, knowledge retrieval, and perception

Lastly, we would like to point out that there are also models that are in contrast with the Atkinson and Shiffrin's original model; among these, there is a dynamic model by Cox and Shiffrin (2017), that consider that memory is cue-dependent, such a model is in line with MINERVA (see Section 2.5).

Anderson assumed that the holding capacity of short-term memory is limited. Therefore, the stability of information in STM partially depends on the amount of information stored in STM and in general will decline as the information load increases. Some characteristics of the information could influence its efficient storage in STM and the capacity of the learner to effectively organize and transmit the information to long-term memory (LTM). Two properties of stimulus information considered by Anderson in this first approximation are (1) the information quality (β) and (2) the information quantity (ρ). Information quality is defined as the abstractness of the information. Then, Anderson introduces S, which represent the activity of the central nervous system associated with storage of information and its stability in short-term memory. The magnitude of this activity will decline as the load of information increases; in other words, the stability of information in STM decreases as STM holding capacity begins to reach saturating levels. The rate of decrease in stability with time will be proportional to the amount of activity accumulated. As a consequence, this mathematically is equivalent to write

Equation (7) is the more general form for describing the rate of decrease in stability. Indeed, it should be considered that the rate of decrease in stability should be less for learners with higher intellectual ability than those of lower ability. Moreover, the rate of decrease in stability should be increased; the more abstract the information and the greater the rate of presentation (the larger the progression density). Both of these factors contribute to the cognitive demand placed on the learner. Hence, Anderson proposed the following refined statement that represents the instantaneous rate of change in stability:

where α is a constant of proportionality, β is the content quality (i.e., the abstractness), δ is the content quantity (progression density), and κ is the learner's intelligence quotient properly scaled, are constant too. By integrating Equation (8) the analytical form of S is obtained:

where S₀ is the initial value of S at t_o and t is time since the start of the learning experience. This is a decreasing exponential function representing the rate of decay in stability of information in STM as information load increases with time. Equation (8) is, therefore, a time-dependent function representing CNS stability. In psychological terms, it is a prediction of the amount of residual short-term memory holding capacity at a point in time after onset of the learning experience. The amount of STM information storage capacity depends on the amount of information already stored in STM and the complexity of the incoming information as represented in part by the variables β and ρ in the rate coefficients of the equation. In addition, to the stability of information in STM, the amount of instability in CNS associated with uncertainty in encoding novel stimulus material must be considered.

As learning progresses and behavior becomes more differentiated, initial instability associated with the new learning task will decrease. Let I represent activity in the CNS associated with instability of the system and λ the coefficient of decay of I with time. Hence, the instantaneous rate of decay in instability of CNS for information encoding, which is related to the amount of activity I through the instability coefficient λ can be mathematically written as

The integration of Equation (10), with the initial condition I(t = 0) = I₀, provides

At any point in time, the capacity of the CNS to encode information will be equivalent to the difference between the stability function and the instability function or

Equation (12) represents the net encoding capacity of STM at an arbitrary point in time t.

Then, Anderson introduce CNS activity correlated with information gain, called N. Then, he wrote

In Equation (13), it is clear how the instantaneous rate of increase in information is directly proportional to N and κ, the intelligence of the subject, inversely related to β, and the abstraction of stimulus information, and δ, the progression density. ᾱ is a constant of proportionality. Theoretically speaking, the gain function represents the amplification of CNS activity associated with the elaboration of information in memory through active memory processes of reorganization of information in LTM.

By solving the following Cauchy problem,

The obtained solution is

The product of the gain function, Equation (15), and the modulation factor, Equation (12), yields the composite equation:

where N_t is the net information's gain at time t. With appropriate choice of constants (α and ᾱ) and properly scaled variables (δ, β, κ, and λ), the equation yields learning curves that can be empirically tested in relation to data obtained in human learning experiments. The composite equation, therefore, represents the total information gain (N_t) at a point in time (t) and is the product of the subjects' capacity to generate interrelationships among units of information in LTM (G factor), and the amount of immediate net STM encoding capacity (M factor).

Anderson proposed some implemented versions of the original model. For example, in Anderson (1986), he included coefficients which represent the motivational state of the learner. In particular, two coefficients were included: the first is an exponential coefficient in the gain function representing largely a change rate of learning associated with varying motivation, while the second is an initial factor in the gain equation change in motivation at the outset of a learning task. This permits modeling of the effects of variations in motivation on the rate and amount of information in a learning task.

We remark that some criticisms to Anderson's model were moved by Preece and Anderson (1984). Preece suggests that Anderson's data could better, or “more parsimoniously”, represented by a learning model proposed by Hicklin (1976). In response to this critique, Anderson stated that several mathematical models have been created to forecast human learning curves, with a significant portion of these models being dependent on learner-specific characteristics. These models, however, do not take into account variations in the information input or the complexity of the information, such as the interaction between short- and long-term memory. Therefore, more complex models are required to explore more natural learning scenarios where information receipt occurs, and the Anderson model is designed to do just that.

MINERVA 2 bears some similarity to MINERVA 1 (see Hintzman and Ludlam, 1980) but is applicable to a much wider variety of tasks. An experience (or event) is represented as a vector, whose entries (which represent the features, i.e., a configuration of primitive properties that activate so that an experience occurs) belongs to the set {+1, 0, −1}. The values +1 and −1 occur about equally often, so that over a large number of traces, the expected value of a feature is 0. In a stimulus or event description, a feature value of 0 indicates that the particular feature is irrelevant. In an SM trace description, a value of 0 may mean either that the feature is irrelevant or that it was forgotten or never stored. In learning, active features representing the present event are copied into an SM trace. Each such feature has probability L of being encoded properly, and with probability 1 − L the tract feature value is set at O. If an item is repeated, a new trace is entered into SM each time it occurs. The authors define P(j), it represents the feature j of a probe or retrieval cue, and T(i, j), a mathematical object (see Hintzman and Ludlam, 1980), which is the corresponding feature of memory trace i. T(i, j) must be statistically compared to P(j), that is why T(i, j) is a function both of the trace i and the probe j. The similarity of trace i to the probe is computed as

where N is the total number of features that are nonzero in either the probe or the trace.

S(i) can be viewed a sort of correlation index: If S(i) = 0, then the probe and trace are orthogonal, if S(i) = 1, they perfectly match, taking on both positive and negative values. The activation level of a trace, A(i), is a positively accelerated function of its similarity to the probe. In the study's simulations,

Raising the similarity measure to the third power increases the signal-to-noise ratio, in that it increases the number of poorly matching traces required to overshadow a trace that closely matches the probe. It should be noted that if trace i was generated randomly (by a process orthogonal to that generating the probe), then the expected value of A(i) is 0 and the variance of A(i) is quite small. Thus, A(i) should be very near to 0 unless trace i fairly closely matches the probe.

There is a rich literature regarding the developments as well as implementations of the Hinztman model. For example, the ATHENA model (see Briglia et al., 2018) as an enactivist² mathematical formalization of Act-In model by Versace et al. (2014), within MINERVA2 non-specific traces: ATHENA is a fractal model which keeps track of former processes that led to the emergence of knowledge; in this way, it can process contextual processes (abstraction manipulation). An interesting characteristic of ATHENA is that it is a memory model based on an inference process that is able to extrapolate a memory from very little information (Tenenbaum et al., 2011). As a consequence, ATHENA accounts for the subjective feeling of recognition, unlike MINERVA2 (for details see Benjamin and Hirshman, 1998). As a final remark, it should be noted that Nelson and Shiffrin (2013) considered that this process should be implemented in SARKAE, as suggested and described by Cox and Shiffrin (2017).

The act of remembering involves accessing stored information from experiences that are no longer in the conscious present. In order to model remembering, it is necessary therefore define the representation that is being remembered. Mathematically, a static image can be represented as a two-dimensional matrix, which can be stacked to form a vector. Memories can also unfold over time, as in remembering speech, music, or actions. Although one can model such memories as a vector function of time, theorists usually eschew this added complexity, adopting a unitization assumption that underlies nearly all modern memory models. The unitization assumption states that the continuous stream of sensory input is interpreted and analyzed in terms of meaningful units of information. These units, represented as vectors, form the building blocks (units) of memory and both the inputs and outputs of memory models. Scientists interested in memory study the encoding, storage, and retrieval of these units of memory.

Let fi⃗∈ℝN represent the memorial representation (vector) of item i in the scalar space ℝ^N. The N elements of the vector fi⃗ are denoted by fi(1),fi⃗(2),...,fi⃗(N), that represent information in either a localist or a distributed manner. According to localist models, each item vector has a single, unique, non-zero element, with each element thus corresponding to a unique item in memory. Hence, the localist representation of item i can be viewed as a vector fi⃗(j), whose elements f_i(j) are defined such that

The last case represents the unit vectors.

Differently, according to distributed models, the features representing an item distribute across many or all of the elements. In this case, a probability p of assuming scalar 1 must be introduced. In detail, consider the case where f_i(j) = 1 with probability p and f_i(j) = 0 with probability 1 − p. The expected correlation between any two such random vectors will be zero, but the actual correlation will vary around zero. The same is true for the case of random vectors composed of Gaussian features as is commonly assumed in distributed memory models (see for example Kahana et al., 2005).

Encoding is the set of processes where a subject (the learner) records information into memory. The subject does not simply record sensory images but, rather, creates the multidimensional (i.e., vectorial) representation of items as well as produce a lasting record of the vector representation of experience. To this aim, it needs to introduce another mathematical tool able to describe how the brain record a lasting impression of an encoded item or experience since the only vector is not enough to do that. Such a mathematical tool is the matrices. Mathematically, the set of items in memory form a matrix, that is basically an array, where each row represents a feature or dimension, and each column represents a distinct item occurrence. The matrix encoding item vectors f1⃗,f2⃗,f3⃗,...,ft⃗ can be represented as follows:

where the first column of the matrix represents the entries (i.e., the elements) of vector f1⃗, the second column the entries of vector f2⃗ and so on. The multitrace hypothesis implies that the number of traces can increase without bound. In summary, the multitrace theory positing that new experiences, also including repeated ones, add more columns to the growing memory matrix M¯¯ described in Equation (22). Nevertheless, without positing some form of data compression, the multitrace hypothesis creates a formidable problem for theories of memory search.

This theory, in contrast with the view that each memory occupies its own separate storage location, states that memories blend together in the same manner that pictures may be combined (as happens in morphing). From a mathematical point of view, this translates in simply summing the vectors representing each image in memory. Then, there are at least two techniques to be used to deal with such a sum: first, averaging the sum of features, but in this way, information about the individual exemplars are discarded; second, defining a composite storage model to account for data on recognition memory, as proposed by Murdock (1982). This model specifies the storage equation in the following way:

where mt⃗ is the memory vector and ft⃗ represents the item studied at time t. The variable 0 < α < 1 is a forgetting parameter, and B¯¯t is a diagonal matrix whose entries B¯¯t(i,i) are independent Bernoulli random variables (i.e., a variables that take the value of 1 with probability p and 0 with probability 1 − p). The model parameter, p, determines the average proportion of features stored in memory when an item is studied.

If the same item is repeated, then it is encoded again. Indeed, some of the features sampled on the repetition could not be previously sampled; hence, repeated presentations will fill in the missing features, thereby differentiating memories and facilitating learning. It is possible to consider the feature of the studied items as independent and identically distributed normal random variables as done by Murdock (1982).

Rather than summing item vectors directly, it is better first expanding an item's representation into a matrix form and then sum the resultant matrices since if not there would be a substantial loss of information. Although this is beyond the scope of this study, we note that this operation forms the basis of many neural network models of human memory (Hertz et al., 1991). In this case, the entries of vector f⃗ represent the firing rates of neurons, then the vector outer product f⃗·fT⃗ forms a matrix M¯¯ whose entries are M¯¯i,j=f(i)f(j). Incidentally, this matrix exemplifies the Hebbian learning. However, this treatment could be interpreted as oversimplified since Hopfield network is not considered. The matrix M¯¯ should represent connections between neurons in the network, which itself defines transitions of the network state, and the fixed point of the dynamic is desired memory. We refer interested readers to Hopfield (2007) and related references.

If an item has already encoded and it is encountered again, we often quickly recognize it as being familiar. To create this sense of familiarity, the brain must somehow compare the representation of the new experience with the contents of memory. Such a research could be lead in series or in parallel. In the former case, the target item is compared to each stored item memory until a match is found. This process is generally slow. In the latter case, the research is in parallel, meaning by this that a simultaneous comparison of the target item with each of the items in memory. This second process is faster. Nevertheless, there is a point of attention to be considered: when an item is encoded in different situations, the representations will be very similar but not identical. Summed similarity models present a potential solution to this problem. Rather than requiring a perfect match, we compute the similarity for each comparison and sum these similarity values to determine the global match between the test probe and the contents of memory. There are a few similarity models, one of the simplest summed-similarity model is the recognition theory first proposed by Anderson (1970) and finally elaborated by Murdock (1989). The model elaborated by Murdock is called TODAM (Theory of Distributed Associative Memory). In this model, subjects store a weighted sum of item vectors in memory as detailed in Equation (23). In order to establish if a (test) item was already encoded, it is necessary that the dot product between the vector characterizing the item and the memory vector exceeds a threshold. Specifically, the model states that the probability of finding a perfect match (we denote this case with “OK”) between the test item (called g⃗) and one of the stored memory vectors is

The TODAM embodies the direct summation model of memory storage. Such a summation model of memory storage implies that memories form a prototype representation. Hence, each individual memory contributes to a weighted average vector whose similarity to a test item determines the recognition decision. However, some criticisms are moved to this approach. Indeed, studies of category learning indicate that models based on the summed similarity between the test cue and each individual stored memory provide a much better fit to the empirical data than do prototype models (Kahana and Bennett, 1994). Some alternative approaches (see for example Nosofsky, 1992) represent psychological similarity as an exponentially decaying function of a generalized distance measure. That is, they define the similarity between a test item, g⃗, and a (fixed) studied item vector, fi*⃗, where i^* is any fixed value between 1 and L, as

where N is the number of features, γ indicates the distance metric (γ = 2 corresponds to the Euclidean norm), and τ determines how quickly similarity decays with distance. Equation (25) can be generalized to L items, by considering the encoding item vectors fi⃗, i = 1, ..., L vectors and the corresponding memory matrix M¯¯=(f1⃗,f2⃗,f3⃗,...,fL⃗). Then, the generalized equation is obtained by summing the similarities between g⃗ and each of the stored vectors in memory,

The summed-similarity model generates an “OK” match if S exceeds a threshold.

We remark that g⃗ can play the role either of target (i.e., g⃗=fi⃗ for some value of i) or probe, in this last case g⃗∉M¯¯.

Another relevant point in the study of memory encoding is temporal coding, associations are learned not only among items but also between items and their situational, temporal, and/or spatial context (see for example, some fundamental studies such as Carr, 1931). The idea of temporal coding was developed more recently in 1970 by Tulving and Madigan (1970). Specifically, these authors distinguished temporal coding from contemporary interpretations of context. Differently from this, subsequent research brought these two views of context together: this is the case shown in Bower's temporal context model (Bower, 1972). According to Bower's model, contextual representations constitute a multitude of fluctuating features, defining a vector that slowly drifts through a multidimensional context space. These contextual features form part of each memory, combining with other aspects of externally and internally generated experience. Because a unique context vector marks each remembered experience, and because context gradually drifts, the context vector conveys information about the time in which an event was experienced. By allowing for a dynamic representation of temporal context, items within a given list will have more overlap in their contextual attributes than items studied on different lists or, indeed, items that were not part of an experiment (see Bower, 1972). It is possible to implement a simple model of contextual drift by defining a multidimensional context vector, c⃗=[c(1),c(2),...,c(N)], and specifying a process for its temporal evolution. To this aim, it needs specify a unique random set of context features for each list in a memory experiment or for each experience encountered in a particular situational context. However, contextual attributes fluctuate as a result of many internal and external variables that vary at many different timescales. An alternative approach proposed by Murdock (1997), is to write down an autoregressive model for contextual drift, such as

where ϵ⃗ is a random vector whose elements are each drawn from a Gaussian distribution, while each item presentation is represented by i indexes. The variance of the Gaussian is defined such that the inner product ϵi⃗·ϵi⃗ equals one for i = j and zero for i ≠ j. Accordingly, the similarity between the context vector at time steps i and j falls off exponentially with the separation: ci⃗·ci⃗=ρ|i-j|. This means that the change in context between the study of an item and its later test will increase with the number of items intervening between the study and the test, producing the classic forgetting curve. In terms of the study of memory and in continuity with the above sections, it is possible to concatenate each item vector with the vector representation of context at the time of encoding (or retrieval) and store the associative matrices used to simulate recognition and recall in our earlier examples. An alternative way is directly associate context and item vectors in the same way that we would associate item vectors with one another.

These models are quite recent, therefore, as far as we know, there are no developments published in the literature yet.

Maybe Alzheimer's disease (AD) is the most popular neurological disease affecting memory (Eustache et al., 1990), and the most common form of dementia (Jack, 2012). It is a progressive, degenerative, and fatal brain disease, in which synapses connections in the brain are lost. The evidence suggests that women with AD display more severe cognitive impairment relative to age-matched males with AD as well as a more rapid rate of cognitive decline (Dunkin, 2009).

Semantic dementia (SD) designates a progressive cognitive and language deficit, primarily involving comprehension of words and related semantic processing, as described in a very pioneering work by Pick (1904). These patients lose the meaning of words, usually nouns, but retain fluency, phonology, and syntax. Semantic dementia is distinguishable from other presentations of frontotemporal dementia (see Section 2.7.3) and Alzheimer's disease (see Section 2.7.1) not only by fluent speech and impaired comprehension without the loss of episodic memory, syntax, and phonology but also by empty, garrulous speech with thematic perseverations, semantic paraphasias, and poor category fluency.

Frontotemporal dementia is an uncommon type of dementia that causes problems with behavior and language. It is result of damage to neurons in the frontal and temporal lobes of the brain. Many possible symptoms can result, including unusual behaviors, emotional problems, trouble communicating, difficulty with work, or difficulty with walking.

Talking about neurodegenerative diseases one relevant case of interest is autobiographical amnesia (Piolino et al., 2003). There are different theories regarding long-term memory consolidation that can be applied to investigate pathologies involving memory. For example, according to the standard model of systems consolidation (SMSC) (Squire and Alvarez, 1995), the medial temporal lobe (MTL) is involved in the storage and retrieval of episodic and semantic memories during a limited period of years. An alternative model of memory consolidation, called the multiple trace theory (MTT), posits that each time some information is presented to a person, it is neurally encoded in a unique memory trace composed of a combination of its attributes (Semon, 1923). In other words, it suggests that the capacity of the MTL to recollect episodic memories is of a more permanent nature. Piolino et al. (2003), to test these models, studied three groups of patients with a neurodegenerative disease predominantly affecting different cerebral structures, namely, the MTL (patients in the early stages of Alzheimer's disease) and the neocortex involving either the anterior temporal lobe (patients with semantic dementia) or the frontal lobe (patients with the frontal variant of frontotemporal dementia, fv-FTD). Then, they compared these groups of patients (the cardinality of the three set of patients was nearly the same) with control subjects using a specific autobiographical memory task designed specially to assess strictly episodic memory over the entire lifespan.

This task considers the ability to mentally travel back in time and re-experience the source of acquisition by means of the remember/know paradigm. The outcome was interesting since all three groups of patients produced strongly contrasting profiles of autobiographical amnesia regardless of the nature of the memories in comparison with that of the control group. In details, temporally graded memory loss in Alzheimer's disease, showing that remote memories are better preserved than recent ones; in semantic dementia, memory loss is characterized by a reversed gradient,³ while memory loss without any clear gradient was found in fv-FTD. By focusing on episodic memories (see Section 1), the authors found that they were impaired, whatever the time interval considered in the three groups, though the memory loss was ungraded (i.e., no temporal gradient was detected) in Alzheimer's disease and fv-FTD and temporally graded in semantic dementia, sparing the most recent period.⁴ A deficit of autonoetic consciousness⁵ emerged in Alzheimer's disease and fv-FTD but not in semantic dementia though beyond the most recent 12-month period. The authors remarked that the sematic dementia group could not justify their subjective sense of remembering to the same extent as the controls since they failed in providing contextual information, spatial or temporal details, etc. The results demonstrated that autobiographical amnesia varies according to the nature of the memories under consideration and the locus of cerebral dysfunction. The analysis was carried on by considering both the two competing models for long-term memory consolidation above described (i.e., SMSC and MTT), the authors observed that new insights based on concepts of episodic memories in the early of 2000s challenge the standard model and tend to support the MTT instead.