A model of working memory for encoding multiple items and ordered sequences exploiting the theta-gamma code

A column consists of the feedback arrangement of four neural populations, according the schema depicted in Fig. 1. These are pyramidal neurons, excitatory interneurons, GABA-ergic inhibitory interneurons with slow synaptic dynamics, and GABA-ergic inhibitory interneurons with fast synaptic dynamics. A more detailed description of the column is provided in previous papers of the authors (Ursino et al. 2010; Cona et al. 2011, 2012). It is worth noting that we adopted the same parameters for each cortical column in each layer, i.e., differences in rhythms between one layer and another originate from the presence of feedback synapses among the columns, produced by the training procedure. Indeed, with the present parameter values a single column, if isolated from the others, and stimulated by a constant external input, produces an intrinsic oscillation in the alpha band (see Fig. 3).

Just in the case of layer WM, which implements working memory (possibly located in a structure of the frontal lobe) we added a further auto-excitation loop for pyramidal neurons (i.e., a group of pyramidal neurons can auto-excite themselves via parameter C_pp in Fig. 1). This loop is necessary to keep information in memory during the maintenance period of a working memory task, and is reset to zero whenever a new input updates the memory content. From a neurophysiological perspective, the excitatory closed loop among pyramidal neurons included in WM (and not used in the other layers) can be ascribed either to reverberations within local cortical circuits, or to long-range circuits (Guo et al. 2017; Zylberberg and Strowbridge 2017).

Finally, it is worth noting that each cortical column can receive two different inputs, the first directed toward pyramidal neurons, with an excitatory function, and the second toward fast inhibitory interneurons, with an inhibitory role. Both inputs can be affected by noise. The output of each column is the spike density of pyramidal neurons (ranging between 0 and 5, as in previous models, see Jansen and Rit (1995)).

We used a four-layer structure in our model, arranged as in Fig. 2, but with two possible alternative uses: the first, named “sequence ordering working memory”, aims to recover a previously learned sequence of items in an assigned temporal order, by exploiting a theta-gamma coupling mechanism. The second, named “semantic working memory”, aims to maintain several objects simultaneously in memory, by desynchronizing them in the gamma range. Since some works in the literature (see Discussion) suggest that the theta rhythm is important especially in sequence ordering, in the second model use we do not generate the theta rhythm. In other words, we implement two different networks in this work, to solve two distinct problems. However, the two networks exhibit the same topology and share most internal parameters, and differ only for what concerns a few synapses. It is possible that these networks represent processing in different parts of the hippocampus. Indeed, some experimental evidences suggest that the hippocampus is functionally differentiated between the dorsal (posterior) and ventral (anterior) areas (Moser and Moser 1998) and that distinct hippocampal-cortical connections are one mechanism by which the hippocampus can represent different kinds of knowledge (Frank et al. 2019).

The two kinds of networks differ only for what concerns two aspects: first, the synapses between WM and L1 are three-times stronger in the “semantic” modality than in the “sequence ordering” modality: in this way, the pyramidal neurons in L1 are excited to saturation and a theta rhythm is not produced. This difference agrees with the presence of two parallel streams in the hippocampus (one named “what” stream and other “where” stream) as described in Preston and Eichenbaum (2013). In particular, the latter authors assert that the medial prefrontal cortex, in the “what” streams, is positioned to influence the retrieval of specific object representations via its particularly strong connections to perirhinal and lateral entorhinal cortex, whereas connections are smaller in the “where” stream (see Fig. 1 in their work). Second, feedback synapses from L3 to L2 are trained only in the “sequence ordering” modality, to produce a sequence of items from an initial cue, but are set at zero in the “semantic” modality. All other aspects of the model are equal in the two modes.

Each layer consists of an array of LxM identical cortical columns (hence, we can have LxM different features) arranged in a regular lattice. As said before, the only difference among the columns is the presence of an auto-excitatory loop in pyramidal neurons of layer WM. In the present simulations we used L = M = 20 (hence, we have a total of 400 distinct features).

In the following, three different kind of synapses will be used, to connect columns within a layer or among different layers:

Values of the time constants for these three synaptic mechanisms are about 7.7 ms, 6.8 ms and less than 1 ms, respectively.

The different speeds of the synaptic mechanisms in points i and ii, compared with point iii, can be ascribed to the presence of differences in AMPA receptors in the postsynaptic membrane. Another stronger possibility is that faster dynamics are mediated by gap junctions.

As specified above: (i) lateral excitatory synapses of type W are created among cortical columns within the same objects in layer L1, via Hebbian potentiation, to allow reconstruction of lacking information; (ii) lateral inhibitory synapses of type K are created among units within the same object in layers L2 and L3, via Hebbian potentiation, to allow synchronization in the gamma range; (iii) faster lateral inhibitory synapses of type A are created among columns in different objects in layers L2 and L3, via anti-Hebbian potentiation, to allow desynchronization; (iv) in case of a sequence of objects, excitatory synapses of type W are created from units in one object in layer L3 to units of the subsequent object in layer L2, via Hebbian potentiation, to allow the reconstruction of a sequence starting from the first item.

Training consists in the presentation of the objects to the layers, one after the other, as specified below, with excitatory inputs so strong that all cortical units reach a saturation level (i.e., the columns do not oscillate during training). The Hebb and anti-Hebb rules are then applied after the units have reached a steady state level.

To realize the connections described above, we implemented two different training steps, the first valid for both networks, to memorize individual objects, the second for the sequence ordering network only, to memorize those objects in an assigned sequence.

(i) During the first step we provided each object separately to layers L1, L2 and L3; this is obtained by stimulating the corresponding pyramidal neurons in layer L1, and both the pyramidal neurons and the fast inhibitory interneurons in L2 and L3. Synapses of type W are then created in L1 using the Hebb rule (see Eqs. (11) and (12) in the Appendix), synapses of type K are created in L2 and L3 using the Hebb rule (see Eqs. (13) and (14) in the Appendix), whereas synapses of type A were formed with an anti-Hebbian rule (see Eqs. (15) and (16) in the Appendix). This procedure required 2000 epochs.

(ii) Only if a sequence of N objects must be memorized in an assigned temporal order (i.e. one needs to store and recover an ordered list of items) a further training step was performed, involving layers L2 and L3 together. This is coherent, thinking to a task in which first we memorize some items, and then memorize a particular sequence for the same items. In particular, naming the objects in the list as: Obj_1, Obj_2, …, Obj_N during each training step the following inputs are provided:

layer2: Obj_k; layer3: Obj_(k-1) (with k = 2, 3, …N).

To this end, an input was provided to the corresponding pyramidal neurons in the two layers. By applying the Hebb rule (see Eqs. (17) in the Appendix) hetero-associative synapses are created from Obj_(k-1) in layer L3 to Obj_k in layer L2. This procedure required 1000 epochs.

After the step ii, the network can be used in a modality named “sequence-ordering working memory”. The objective here is to recover an entire sequence, starting from a first incomplete item, and to nest this sequence within the on phase of each theta cycle.

To recapitulate all previous considerations, a list of the main functions introduced in the model, the necessary model components for these functions, and possible neurobiological evidences in the literature, with reference to the PFC-hippocampus interactions, is presented in Table 1.

In order to illustrate how an object can be maintained in working memory, even after removal of the corresponding input, Fig. 5 shows the input to the network, and the activity of pyramidal neurons in layers WM and L1, after the separate presentation of two objects (in this particular example object 1 is presented per 50 ms between the instants 0.005 and 0.055 s, and the object 2 is presented between 0.405 and 0.455 s, upper panel). Both objects are incomplete, i.e., they lack 30% of their pixels. The second and third panels of the figure represent the average activity of all pyramidal neurons coding for the object 1 (blue line) and for the object 2 (red line). Since the maximal activity in the model saturates at a value 2e₀ = 5 (see the Appendix), an average activity as great as 5 means that all neurons are simultaneously active in a given object, while an average activity as large as 4 means that 80% of neurons in that object are excited, and a value close to zero means that all neurons are silent. As it is clear form Fig. 5, 70% of neurons in the object 1 are initially excited in response to the input and this information is transmitted to layer L1 through the feedforward synapses. Thanks to the presence of auto-associative synapses in L1, trained with the Hebb rule, layer L1 is able to recover all lacking information. The activity in layer L1 oscillates with a theta rhythm (approximately 5 cycles/s) and during each cycle the overall object is reconstructed in this layer. It is worth noting that information is permanently maintained in memory in layer WM, thanks to the presence of auto-excitation between pyramidal neurons. At the presentation of the second object (instant 0.405 s) this auto excitation is reset to zero, thus allowing a refresh of the memory, so that a second object is reconstructed in L1 and maintained in memory in WM.

Figure 6 shows how the model can work in the “sequence ordering” modality, i.e. after a sequence was learned during the step iii of the training procedure. After a brief 50 ms presentation of object1 (between the instants 0.005 and 0.055 s), the network is able to reconstruct the initial portion of the sequence, from object 1 to object 6: each object is coded by an individual period in the gamma cycle (note the different colors, representing activity within different objects) nested within the lower-frequency theta cycle. The sequence is then maintained in memory, and repeated periodically, until a new input is given to the network to refresh the WM content. In particular, the brief presentation of the second object (between the instants 0.605 and 0.655 s) allows the reconstruction of the sequence from object 2 to object 8, and the quick presentation of the object 3 (between the instants 1.205 and 1.255 s) produces the reconstruction of a sequence from object 3 to object 9. The well-known phenomenon of phase precession (similar to that observed in the rat’s spatial cells during movement) is evident looking at the phase of the different objects within the theta cycle.

From the figure the role of the different layers is evident: just a single object is reconstructed in layer L1 (by recovering all lacking information), where the theta rhythm is formed. A noisy gamma sequence is first produced in layer L2, and significantly polished in L3, where the reconstruction of all subsequent objects in the sequence turns out almost perfect.

The previous simulation illustrated the functioning of the model in the “sequence ordering” modality, when a sequence of events can be reconstructed in the correct order, starting from the first element of the list (and, moreover, the first element can shift in time). Figure 7 shows the model working in the “sematic memory” modality (i.e., step ii was not performed in the training procedure, and synapses between L1 and WM are stronger). In this particular example, four objects are simultaneously given as an input, and the model is able to desynchronize their activity in the gamma range, so that objects appear separately (that is, in time division) in layer L3.

An aspect deserves attention. In this modality no sequence order was previously memorized (i.e., we did not train the feedback synapses from L3 to L2), and so we were not requiring any particular order to be reproduced. Nevertheless, the model “devises” a particular order (in the specific example, the order is “object 1—object 2—object 4—object 3”) and repeats this order constantly, after a brief transient. This order can change as a consequence of the particular noise realization.

This kind of behavior (i.e., a correct desynchronization with a fixed although arbitrary order) can be observed, with the assigned synapse values, when either two, three or four objects are simultaneously given as input. We also tried to desynchronize a greater number of simultaneous objects with the same values of synapses, but, if the number of objects was greater than four, not all objects could be desynchronized correctly.

However, we observed that the capacity to desynchronize objects improves significantly if the strength of the synapses A in layers L2 and L3 (i.e., synapses \({A}^{{L}_{2},{L}_{2}}\) and \({A}^{{L}_{3},{L}_{3}}\) see the Appendix) is increased. In particular, up to nine objects could be desynchronized rather often by multiplying the previous synapse values by a factor in the range 1.25–1.7 (A_max = 0.15–0.20). Some examples, concerning the desynchronization of 5, 6, 7, 8 and 9 objects, given simultaneously as inputs, is illustrated in Fig. 8, where we used A_max = 0.20 for all simulations. In this figure, for brevity, only the average activity in layer L3 is presented. As it can be seen, all objects appear at least once during the simulation period. However, there is no fixed sequence; the objects emerge in an unpredictable order and with a different occurrence. Furthermore, the greater the number of objects given as input, the lower the frequency of the emerging rhythm, which decreases to approximately 20 Hz (i.e., in the beta range) when nine objects must be desynchronized together. We ascribe this reduction in frequency to an increased competition among the objects, which requires more time to be resolved. However, objects were not always recognized perfectly even when using the increased values of A synapses: in some occasions, especially concerning the desynchronization of eight or nine objects, one object failed to appear during the overall simulation period.

Some recent experimental studies report that, in subjects affected by Alzheimer disease, an anomalous behavior of the neural networks related to memory (hippocampal neural networks) is linked to dysfunctions of the interneurons—in particular, the inhibitory interneurons involved in the generation of gamma oscillations (GABAfast) (Verret et al. 2012; Palop and Mucke 2016). Furthermore, some results prove that, starting from the pathological condition, correcting the activity of these neurons impacts on the state of the hippocampal network, making its activity more similar to that found in healthy subjects (Park et al. 2020). In order to analyze these conditions, we simulated the recovering of a sequence (i.e., the same situation as in Fig. 6) after a reduction of parameter C_ff, which represents the auto-inhibition of the fast GABAergic interneurons. This parameter, as demonstrated in our previous works (Ursino et al. 2010) is closely involved in the generation of the gamma rhythm. Results, shown in the upper panel of Fig. 9 (concerning the layer L3 only) show that, after a reduction of this parameter to 1/4 of its basal value, the network becomes unable to recover a correct sequence. Just the initial two or three patterns in the list are recovered, and they occur repeatedly inside the same theta cycle.

The second test concerns schizophrenia. It has been reported in the literature that subjects suffering from this pathology exhibit alterations in the AMPA receptors (Zeppillo et al. 2020) (in the present model these may be associated with the synapses named A), as well as an anomalous coupling between theta and gamma rhythms in the prefrontal areas during working memory (Barr et al. 2017). To simulate a similar pathological condition, we used the network in both its modalities (“sequence ordering” and “semantic”), after a reduction in the strength of synapses A to 1/4 of their original value. Results shown in the mid panel of Fig. 9 demonstrate that, in this condition, the network is unable to recover a correct sequence of objects: unlike the previous case, however, now we can observe too many objects superimposed together; in particular, the first items of the sequence restarts again within the same theta cycle, superimposing with the subsequent items of the sequence in a confusing way. A similar superimposition of items is evident also in the modality “semantics” (bottom panel of Fig. 9), where we gave five objects as input to the network.

As a last simulation, we analyzed the model behavior when the network does not receive any external input, but just a uniform noisy excitation is given to layer L1. This condition wishes to simulate a circumstance in which the subject is isolated from the environment, but can “imagine” or “dream” autonomously, by spontaneously recovering and recombining the information stored in layers L1, L2 and L3. In particular, we analyzed the network in the modality “sequence recovering”, i.e., when the synapses from L3 to L2 were previously trained to memorize a list of items.

The simulation in Fig. 10 refers to a condition in which all pyramidal populations in layer L1 receive an input noise with uniform distribution ranging between 80 and 160. This signifies that layer L1 is disconnected from the working memory, and is “over excited” uniformly. As it can be seen from the figure, which represents a typical simulation, thanks to the presence of much noise the network can spontaneously recover several patterns in L1 (it is worth noting, however, that now the dynamics in L1 is slower, down to approximately 2 Hz). Just one of these patterns, however (generally, the first active in L1), wins the competition in L2, causing a sequence to start. Some aspects deserve attention. First, the sequence “imagined” can change from one moment to another. Second, in these simulations we used the objects in the second configurations of Fig. 4 (bottom panel), and the network was previously trained to independently learn two list of objects: “1–2–3–4-5” and “6–7–8–9–10”. Moreover, the object 6 in the second list has a portion in common with object 4 of the first list, and the object 10 in the second list has some elements in common with object 2 of the first list. We can see that sometimes the network links the two lists in L3 to form a longer hybrid list. In particular, we can observe that sometimes object 7 in the second list is evoked with a small delay after object 5 in the first list, and object 3 in the first list is evoked with a small delay after object 10 in the second list, i.e., a different list is linked at the end of the previous one. For instance, the first gamma sequence in the simulation is composed of the objects “3–4–5” + “7–8–9” (the first three items belonging to the first list, the other three to the second list) and the third gamma sequence is “4–5” + “7–8–9–10”. We can explain this behavior if we assume that the appearance of object 4 in the first list evokes a small activity for the object 6 in the second list too (since these two objects have some common features). This is not sufficient to evoke the entire object 6 in layer L3, but its partial activity favors the excitation of object7 in L2 (which is the subsequent object in the second list). As a consequence, when the first list terminates, this object is facilitated to appear. Similarly, the last sequence in the simulation consists of the objects “7–8–9–10” + “3–4–5” (the first four items belonging to the second list, the other three to the first list). In this case, the appearance of object 10 in the second list evokes a small activity for the object 2 in the first list, and the residual activity of object 2 in L3 favors the emergence of object 3 in L2.

The previous considerations can explain most of the combinations observed during our simulation. However, it is also possible that a list emerges after another in a more independent and less predictable way, as in the case of the second gamma sequence of Fig. 10 (“4–5” + “8–9–10” which is a delayed version of the third sequence lacking the object 7), or the case of the fourth gamma sequence (“6–7–8–9–10” + “1–2” which can only be explained by a random appearance of object1 at the end of the other list). These patterns are representative of what is occurring in most simulations.

Briefly, the following main conclusions can be drawn: (i) the network, deprived from any external input and disconnected from the WM layer, in the presence of a small uniform excitation and much noise, can autonomously evoke some of the learned lists; (ii) even more important, the network can link some lists together, frequently exploiting the similarity among items; (iii) in any case, it is necessary that one list terminates to allow the concatenation with another one. These aspects will be further commented in the Discussion and lines for future improvements will be given.

Working memory denotes the ability to maintain information in the brain during short time periods for immediate use: it involves the initial encoding of information, the retrieval of WM items from external cues or stimuli, and their maintenance during a delayed time until a goal is achieved. WM is crucial for many goal-directed cognitive functions and exhibits different characteristics, depending on the specific problem that has to be dealt with. Traditional models assume that WM is based either on persistent firing of some neurons in the prefrontal cortex (PFC), whose activity is maintained via recurrent positive connections until a new stimulus reset them (Funahashi et al. 1989; Compte et al. 2000) or via short-term potentiation of synaptic connections (Sandberg et al. 2003; Mongillo et al. 2008). However, data collected in the last decade evoke a much more complex scenario. First, the interaction between working memory and long term memory has become a topic of much current interest (Burgess and Hitch 2005). As argued by Burgess and Hitch (2005), models of short term memory and long term memory exhibit a crucial link, so that a complete separation among them is often not possible. Second, there is a wide consensus that WM does not only implicates the PFC, but involves several different areas, in particular the hippocampus and the parieto-occipital regions (Johnson et al. 2017). Finally, and more important, many experiments, both in rodents and humans, suggest that WM activity crucially depends on brain rhythms, in the theta, alpha and gamma bands. The role of neural oscillations in memory has become a crucial topic in recent neuroscience research.

Aim of the present work was to develop an original mathematical model to investigate the role of theta and gamma rhythms in memory. Our attention was especially focused on WM, although it is not always possible to distinguish between long term and short term memory within the present theoretical framework. Indeed, as pointed out by Manohar et al. (2019) whether a model describes short term or long term memory (hence WM, or semantic or episodic memory) depends on the duration of the synaptic changes incorporated on the Hebbian rules, i.e. on short term vs. long term potentiation, a problem that we do not explicitly address in the present model. Actually, a preponderant role of gamma-theta coupling has been demonstrated not only in studies concerning WM tasks (Lundqvist et al. 2011; Chaieb et al. 2015; Rajji et al. 2017; Tamura et al. 2017; Bahramisharif et al. 2018; Reinhart and Nguyen 2019), but also associative tasks (Köster et al. 2018), long term spatial memory (Vivekananda et al. 2021), and episodic memory (Hsieh and Ranganath 2014), thus stressing the probable presence of similar underlying mechanisms for different memory types.

Our model is based on a few basic assumptions which, although realistic, still require a future validation. What is important in the present study is the proposed architecture and the basic mechanisms incorporated, while individual details can be the subject of future improvements. Fundamentally, we demonstrated that a single multilayered structure, which makes use of neural mass models and Hebbian mechanisms, can solve different kinds of working memory problems, either involving an ordered sequence of items or segmentation of several independent items, exploiting theta and gamma oscillatory patterns. Moreover, we demonstrated that some alterations in synaptic mechanisms can potentially induce anomalies in the way memory is restored and managed, simulating pathological conditions. Finally, the free network, isolated form the environment in the presence of much superimposed noise can exhibit a dreaming or imagination behavior. All these aspects indicate that the model can represent a promising tool to investigate several fundamental problems in cognitive neuroscience, within a single theoretical framework.

At present we have not a unique indication on where the individual layers of the model can be located, but some hypotheses can be formulated on the basis of the present knowledge.

The Layer WM implements the core of working memory, where the different items are initially evoked and auto-sustained and is likely located in the PFC. This region has been traditionally considered a fundamental location for the short-term goal oriented memory (Fuster and Alexander 1971). A basic assumption in our model is that the memory content is maintained thanks to a positive reverberating loop, which is reset as soon as the memory content must be updated. This idea finds some support in the literature, and can be explained via a BG-thalamus gating mechanism (Bolkan et al. 2017; Nir-Cohen et al. 2020).

Layer L1, where the theta rhythm originates and the items are reconstructed, could be ascribed to the hippocampus, for instance involving the lateral entorhinal cortex and CA3. There are various aspects in the literature that support this model subdivision. The hippocampus is an evident candidate for the generation of theta rhythm (Bastiaansen and Hagoort 2003; Mitchell et al. 2008), as also shown by a large literature on hippocampal theta oscillations in rats (Vinogradova 1995; Buzsáki 2002, 2005). The medial PFC is bilaterally connected to the hippocampus through the perirhinal and lateral entorhinal cortex. Experiments in which neural activity is simultaneously recorded in the rodent mPFC and in the hippocampus have demonstrated that mPFC neuronal spiking occurs at a specific phase of hippocampal theta oscillations (Hyman et al. 2003; Siapas et al. 2005; Jones and Wilson 2005; Gordon 2011; Kim et al. 2011) that this phase relationship is especially prominent after learning (Kim et al. 2011), and that PFC spiking activity is best phase-locked to hippocampal theta oscillations occurring in approximately the past 50 ms (Siapas et al. 2005). The latter results agree with the patterns shown in Fig. 5, where, in the “sequence ordering” modality, the network exhibits a theta phase-lock between the WM and L1 after learning: the theta rhythm originates in L1 and is transmitted back to WM with a 40–50 ms time delay. A bidirectional flow of information between the PFC and the hippocampus has been recently observed in context-driven memory trials (Place et al. 2016). Moreover, the hippocampus CA3 region has been traditionally considered an ideal place for auto associative completion (Marr 1971; Treves and Rolls 1994; Hasselmo et al. 1995) due to the presence of abundant recurrent collaterals (Li et al. 1994) and Hebbian synaptic modification (Bliss and Collingridge 1993). Of course, alternative hypotheses can also be viable, and can be tested in different future models. For instance, theta rhythms may be generated in multiple sources of the brain, not just in the hippocampus, and are known to propagate through the Default Mode Network (see Hsieh and Ranganath 2014). The latter observation however does not contradict our simulations: in our model theta rhythm actually propagates from L1 to downstream layers and, likely, spreads toward other brain areas too (not included in the model), where the evoked memory patterns are utilized and manipulated (but see also Zhang et al. 2020; Yuan et al. 2021 for recent studies on the subject).

A particular aspect of our model is that the theta rhythm is generated directly within the L1 auto associative layer as a consequence of strong Hebbian learning. A more traditional hypothesis is that the theta rhythm is generated in the septum, which possibly acts as a pacemaker for theta activity (Pignatelli et al. 2012) and from there is transmitted to hippocampal neurons (Roux and Uhlhaas 2014). However, more recent data have shown that oscillations in the theta range can be recorded in hippocampal preparations in vitro, thus confirming that the hippocampus itself can act as a theta oscillator (Cataldi and Vigliotti 2018).

Basically, in our model two conditions are necessary to generate a clear theta rhythm: a moderate external input (in our case coming from the WM layer) and excitatory auto-associative synapses induced by Hebbian learning. As consequence, a group of pyramidal neurons in the same item is maximally excited, and strongly excites GABAergic inhibitory interneurons with slow synapse dynamics. The latter, in sequence, inhibits pyramidal neurons causing a decrease in the overall activity to zero. This model mechanism agrees with the ideas (Roux and Uhlhaas 2014; Colgin 2016) that “In the hippocampus, theta oscillations are generated by an interplay of glutamatergic and GABAergic neurons” and represents an important testable hypothesis of our model, requiring further verification.

However, an important characteristic of our theoretical framework is that theta rhythms are generated only when the network works in the modality “sequence ordering”. In the “semantic memory” modality the theta rhythm is practically absent. Of course, this implies the existence of two alternative circuits linking WM and L1: the first, with weaker connections, devoted to temporal sequencing, based on a robust theta, and the other, with stronger connections, devoted to the segmentation of items and their semantics recognition without any assigned order. This is a further model prediction requiring future analysis. Cohen (2011) suggests that the relationship between theta activity and memory encoding might depend on connectivity between the hippocampus and PFC. The possibility of a different connectivity weight between the PFC and two alternative pre-processing hippocampal pathways (named “what” and “where” by the authors) is discussed in Preston and Eichenbaum (2013).

Several works support the existence of alternative circuits in the PFC-hippocampus pathways functionally differentiated along its dorsoventral axis and the ventral (anterior) hippocampal formation (Moser and Moser 1998; Hoge and Kesner 2007; Farovik et al. 2010). Furthermore, recent data suggest that the theta rhythm is prominent only during WM problems which involve temporal order, but plays a less important role in tasks which do not require any temporal sequencing (Roux and Uhlhaas 2014; Hsieh et al. 2011; Heusser et al. 2016).

The layer L2 in our model, probably located in the hippocampus CA3 area, is essential for the production of the gamma rhythm, involving GABAergic interneurons with faster kinetics. This modeling requisite is confirmed by several studies which underline the critical role of fast-spiking parvalbumin (FS) interneurons in the emergence of cortical gamma activity (Bartos et al. 2007; Cardin et al. 2009; Whittington et al. 2011). In the present model, the gamma code is generated downstream of the theta rhythm (specifically in layers L2 and L3), and involves both Hebbian and anti-Hebbian mechanisms. The Hebbian mechanism produces strong glutamatergic synapses from pyramidal neurons in an object to fast GABAergic interneurons in the same object, generating a gamma rhythm and solving the binding problem. Anti Hebbian mechanisms are used to create much more rapid connections from pyramidal neurons in an object to fast inhibitory interneurons in other objects, to solve the segmentation problem.

An important question concerns the nature of the rapid synapses (named A in the model) working with a very fast time scale: they might involve very fast AMPA receptors or more probably gap junctions. Schmitz et al. (2001) provided evidence that axons of hippocampal principal cells are electrically coupled, and can represent a mechanism for very fast electrical communication. The idea that principal cells in the hippocampus establish electrical synapses with each other, implicated in network oscillations and synchronization, is further supported by additional recent studies (Molchanova et al. 2016; Ixmatlahua et al. 2020). Furthermore, recent experiments suggest that the strength of gap junctions can be modified in an activity-dependent manner, similar to that of chemical synapses (Cachope et al. 2007; Turecek et al. 2014; Wang et al. 2015). Pernelle et al. (2018), using a computational model, demonstrated that gap-junction plasticity can play a role to regulate oscillations and transmit information in a network of inhibitory and excitatory neurons. These results confirm several assumptions of the present model but, of course, require further analysis.

However, we must mention a further alternative hypothesis for fast desynchronization in L2 and L3. In fact, Kwon et al. (2018) showed that Schaffer collateral from CA3 to CA1 innervate both excitatory pyramidal cells and inhibitory interneurons, but with different connectivity rules. Hence, it is also possible that a very fast inhibition, to desynchronize objects, is provided by a feedforward link from a previous layer to inhibitory interneurons in a downstream layer. This may represent an alternative desynchronization mechanism which requires attention in future work.

Finally, a feedback external loop has been introduced in L2, to synchronize the gamma rhythm with the on phase of the theta rhythm originating in L1, as observed in hippocampus place cells. Evidence for a similar loop can be found in the literature, in particular involving the so-called Papez’s circuit which involves the hippocampal formation, the mammillary bodies and the anterior thalamus. There is large evidence that the theta rhythm resonates throughout this circuit and that this circuit can have a role for coordinating hippocampo-cortical activity and to control mnemonic functions (Vertes et al. 2001; Dillingham et al. 2021). A further similar mechanism may involve the medial septum, which acts through a dishinibition of the hippocampus, i.e., a mechanisms very similar to the one proposed in the present work (Kocsis and Kaminski 2006; Salib et al. 2019).

Layer L3 in our model has two functions: to further improve segmentation and, in the sequence ordering mode, to produce a sequence of items via Hebbian feedback connections. It is not easy and probably not unique to determine the relative location of layers L2 and L3. In the classic model by Lisman et al. (2005), where however the position of the hetero- and auto-association networks are inverted compared with ours, a similar mechanism was ascribed to the dentate gyrus and CA3. According to an hypothesis developed in Yamaguchi et al. (2007) (see Fig. 1 in that paper), a similar mechanism may involve CA3, CA1 and the superficial layers of the entorhinal cortex. In our opinion CA1 is a good candidate for the sequential order modality. In fact, the existence of feedback from CA1 to CA3 neurons is well documented, via the participation of the entorhinal cortex (Eichenbaum 2017). Sandler et al. (2015) demonstrated the presence of a Granger causality not only from CA3 to CA1 (feedforward) but also from CA1 to CA3 (feedback). Hoge and Kesner (2007) observed that rats with CA1 lesions displayed a profound deficit in remembering the order of the visual object presentations, suggesting that the CA1 is critical for processing temporal information. Other studies suggesting that CA1 area is selectively required for temporal coding are summarized in Mankin et al. (2012).

An important aspect of our model is that it can be used to investigate, although at a preliminary stage, additional important neurocognitive problems directly connected with memory. In particular, we focused attention on two main questions: the role of gamma rhythms during imagination and dreaming, and the alterations in the memory encoding and retrieval occurring as a consequence of synaptic alterations (mimicking pathological conditions such as Alzheimer or schizophrenia).

The present network, isolated from the external world and stimulated with noise in the L1 layer, can randomly replay some of the sequences previously memorized. Moreover, if some of these sequences exhibit a few superimposed features (i.e., items are not completely orthogonal) they can be recombined in a new and creative manner, so that the terminal portion of a sequence is linked to portions of another one. This result opens interesting perspectives for comprehension of some fascinating neurocognitive problems, such as those involved in dreaming or imagination. For instance, place cells in the rat hippocampus not only fire during an experience, but also later ‘replay’ sequences in a similar order or in reverse order during sleep (Skaggs and McNaughton 1996; Dragoi and Buzsáki 2006; Foster and Wilson 2007) or can even construct never-experienced new path-sequences (Gupta et al. 2010).

Finally, with the present model we simulated two different alterations in model synapses, which can have implications in neurological disorders. Indeed, results in the literature suggest that theta-gamma coupling is impaired in schizophrenic subjects, and this modification can determine WM dysfunctions (Cho et al. 2006; Basar-Eroglu et al. 2007; Barr et al. 2010, 2017; Berger et al. 2016). A common hypothesis is that these alterations depend on a deficiency in GABAergic parvalbumin interneurons (Lewis et al. 2012): these findings suggest a new model of cortical dysfunction in schizophrenia in which inhibition is decreased. Accordingly, we tested two different dysfunctions concerning fast inhibitory interneurons in our model. First, we decreased the strength of the synapses A, that target into fast interneurons and are essential for correct desynchronization of items, thus producing a smaller inhibition in the network compared with the normal case. The consequence is that the model, in the “sequence ordering” modality, becomes unable to recover a sequence correctly, producing a complex and confused mixing of various items (Fig. 9 middle panel) with evident alterations in sequential WM; in the “semantic” modality, the network cannot segment items correctly (Fig. 9 bottom panel), thus producing a confused superimposition of features in different objects. Although still at an oversimplified stage, these preliminary simulations can provide some cues on the origin of some positive symptoms in schizophrenia, such as hallucinations, incoherent thinking, distortions. Another possible alteration simulated with the model concerns a reduction of parameter C_ff, which represents the auto-inhibition of the fast GABAergic interneurons. After this change, the network loses the capacity to evoke a sufficient number of items (Fig. 9 upper panel when only three items are recalled), i.e., it works in a condition of memory deficiency. This behavior can simulate some aspects of Alzheimer disease. Various recent studies underline the existence of deficits in the GABA–ergic interneurons in the Alzheimer too (Xu et al. 2020).

Of course more sophisticate combinations of parameter changes can be performed in future works, with a more detailed comparison with the neurobiology, to emphasize the possible use of the model in the comprehension of neurological deficits.

Our model makes use of various hypotheses which, although justified by the present knowledge on the prefrontal cortex-hippocampus interactions, still need a validation. Based on these hypotheses, the results lead to some testable predictions, which can be verified in future studies.

(i) the model assumes the presence of very fast inhibitory mechanisms, working on a time scale smaller than gamma, essential to desynchronize objects. In the “semantic modality” the model can segment three or four objects simultaneously quite easily, using a phase desynchronization in the gamma band. This agrees with the classic number of items assumed in WM. A weakening of this mechanism is associated with poor segmentation and confusion among objects; an increase in this mechanisms can allow the solution of more complex segmentation problems (involving up to nine objects). It is possible that a control in the strength of synapses incorporates a sort of attentional mechanism: the higher the number of items, the stronger the attention devoted to that particular task. The synapse arrangement of these mechanisms is anti-Hebbian. A test should study the presence of ultra-fast inhibition, the effect of its impairment and possible the arrangement of these synapses (anti-Hebbian).

(ii) The theta rhythm plays a different role in sequence ordering problems and in scene segmentation problems. Different WM tasks would exhibit a different dependence on this rhythm. A test can analyze the dependence of the theta power on the role of temporal ordering in the assigned task.

(iii) The model assumes that brain rhythms in the hippocampus are affected by Hebbian (and anti-Hebbian) mechanisms, hence by past experience and by the process of object storage. Hence, the storage of objects can modify these rhythms, by strengthening or weakening them. An alternative hypothesis is that these rhythms are intrinsic (i.e., independent of previous memorization). These alternative hypotheses can be checked by studying the dependence of rhythm power and frequency on the task and the storage requirements.

(iv) Model results show that the frequency of the gamma rhythm decreases when a greater number of objects must be simultaneously segmented in a given semantic problem. The dependence of frequency on the task complexity can be tested.

(v) The same structure (in our case the pre-frontal cortex – hippocampus complex) can solve different memory problems in different segregated circuits, as a consequence of a different learning of feedback synapses and different potentiation of feedforward synapses.

In the present work we made use of a neural mass model which, although largely used in practice, is of course a simplification of the real dynamical behavior of spiking neurons. A fundamental problem, which requires detailed analysis in future work, is the relationship between the sigmoidal formulation in the NMM and a biophysical description of spiking neurons. Briefly, the capacity to produce oscillations, their frequency content and amplitude, and transmissibility in a NMM crucially depend on the position of the various populations on the sigmoidal relationship. It will be worthwhile in future work to simulate the same connections and possibly similar waves with populations of spiking neurons to test similarities and differences. Spiking neurons offer the advantage of a greater physiological reliability and likely the possibility of more complex pieces of behavior, at the cost of increased computational time and, above all, of a greater difficulty to summarize the results into a simple synthetic frame. The possibility to work out with multiscale models, which move from the neuronal level to the population levels and back to neurons will be an essential subject of research in future studies.

In this work we demonstrated that the model can solve different problems in WM using brain rhythms and a four layer structure. However, this general framework is quite flexible and can be modified in future work to meet additional data and accommodate new ideas. For instance, a frequent idea is that a gamma rhythm is generated in the brain upstream of the prefrontal region (and not in downstream layers as in our model) and transmitted to the PFC from the input connections (for instance from occipital visual regions to frontal regions, which in turn exhibit a top-down influence, (see Fries 2015). However, this choice would require a modification of the present auto-associative net, since in this case auto-association would necessitate that the gamma cycles are synchronized before the involvement of a theta rhythm.

A problem in our model is that, if a given object or item appears entirely in two different sequences, the model has no mechanism to choose whether to continue on one path or on another. For this reason, we never simulated such a condition: all the objects memorized in this paper, although not orthogonal, exhibit only a limited percentage of common features (less than 20%). As suggested by Alexander et al. (2020), the possibility to disambiguate multiple overlapping spatiotemporal trajectories requires the presence of more sophisticate cells, which differentially code the context of prior or future behavior. This important question may be the subject of future model improvements. In particular, the possibility to manage overlapping trajectories can strongly improve the model’s ability to simulate dreaming, imagination and pathological behavior.