The communication-through-coherence (CTC) hypothesis states that a sending group of neurons will have a particularly strong effect on a receiving group if both groups oscillate in a phase-locked (“coherent”) manner (Fries, 2005, 2015). Here, we consider a situation with two visual stimuli, one in the focus of attention and the other distracting, resulting in two sites of excitation at an early cortical area that project to a common site in a next area. Taking a modeler’s perspective, we confirm the workings of a mechanism that was proposed by Bosman et al. (2012) in the context of providing experimental evidence for the CTC hypothesis: a slightly higher gamma frequency of the attended sending site compared to the distracting site may cause selective interareal synchronization with the receiving site if combined with a slow-rhythm gamma phase reset. We also demonstrate the relevance of a slightly lower intrinsic frequency of the receiving site for this scenario. Moreover, we discuss conditions for a transition from bottom-up to top-down driven phase locking.
The communication-through-coherence (CTC) hypothesis (Fries, 2005, 2015) states that a sending group of neurons may communicate a message effectively to a receiving group if both groups oscillate and the oscillations in the two groups are phase-locked (“coherent”) to each other. Emphasizing its possible role with respect to the attentional gating of information flow (Moran & Desimone, 1985), this mechanism has also been referred to as selective attention through selective synchronization (see the review in Womelsdorf & Fries, 2011). Assuming that the oscillations are in the gamma range (30–90 Hz), the rationale behind this hypothesis is that the gamma oscillations are given by recurring windows of maximal excitability resulting from rhythmically decaying synchronous inhibitory effects (see, e.g., the discussion of the gamma-generating PING mechanism in Börgers and Kopell, 2005). In case of coherence, the inputs from the sending group may then arrive at times of maximal excitability of the receiving group, implying communication as an effect of the incoming signals that is not blocked by inhibition.
Experimental evidence for this way of communication between different cortical areas has been described in Womelsdorf et al. (2007) and Bosman et al. (2012); (see also Ray & Maunsell, 2015), for a critical evaluation of these and other experimental findings). The former reference focuses on the phase relations between the sending and receiving groups; the latter illuminates the presence of frequency differences. Correspondingly, theoretical modeling with a view on the experimental results described in Womelsdorf et al. (2007) focused on these phase relations as indicators of the effective strength of the couplings (Tiesinga & Sejnowski, 2010; Buehlmann & Deco, 2010; Akam & Kullmann, 2010; Gielen, Krupa, & Zeitler, 2010; Eriksson, Vicente, & Schmidt, 2011; Wildie & Shanahan, 2012). Here, in contrast, we concentrate on the possible role of the frequency differences that were observed in Bosman et al. (2012). (Actually, apart from phase and frequency differences, the coherence of the incoming pulse train is a third candidate for the crucial property that triggers the selective synchronization; see the discussion in Börgers & Kopell, 2008, and our related remarks in section 2.) Going beyond a consideration of gamma rhythms only, our discussion includes considering a possible role of cross-frequency couplings of the fast (gamma) rhythm with a slow (theta) rhythm. In the following, we present a model that confirms a working of such cross-frequency couplings in accordance with the mechanistic proposal put forward in Bosman et al. (2012; see also the related discussion in the recent overview article given by Fries, 2015).
In section 2, we briefly describe several approaches to modeling the CTC mechanism. Note that the models listed in this section appeared earlier than the experimental results described by Bosman et al. (2012). Correspondingly, these works do not discuss a proposal put forward in Bosman et al. (2012) that relates fast rhythm frequency differences and (slow rhythm) resets. In contrast, this letter focuses on a modeling approach to confirm this proposal, as stated in section 3. The dynamical character of interareal coupling hierarchies (Bastos et al., 2015) is emphasized in section 4. This is done to prepare the discussion of a corresponding phenomenon observed in the context of our simulations. This phenomenon may be interpreted as switching from bottom-up to top-down information flow as a consequence of increasing the frequency of the receiving site above the frequencies of the sending sites. In section 5, we present the model that we use for our simulations. The model is based on theta models for the units of the network. The theta model was shown to be equivalent to a quadratic integrate-and-fire model (as explained in Börgers & Kopell, 2005). Section 6 describes our simulation results. These are then illuminated from a theoretical perspective in section 7. For the theoretical discussion, we use a linear integrate-and-fire model. Finally, the summary and outlook are given in section 8.
2 Earlier Models
Our modeling and analysis in sections 5 and 7 is inspired by Wang (2002) and the work of Börgers and Kopell (2005, 2008). The latter work discusses the CTC hypothesis by studying a receiving model in terms of a theta model. Such a model is related to a quadratic integrate-and-fire model (see section 7.1 in this letter). While Börgers and Kopell (2005) dealt with the generating mechanism and stability of the gamma oscillation with a constant input, the study in Börgers and Kopell (2008) uses a pulse train for the external input. Interpreting such a pulse train as input from an oscillating sending site, the latter study is then related to the CTC hypothesis by defining the pulse train as a superposition of two different pulse trains that correspond to two sending sites. Simulations as well as theoretical arguments have then confirmed that entrainment, corresponding to selective synchronization, occurs between the receiving site and the one of the two sending sites that is more coherent. Crucially, it was demonstrated that this entrainment is enforced through the coupling to inhibitory units at the receiving site.
Using Hodgkin-Huxley-like models for the neurons, Paul Tiesinga and coworkers studied various aspects of neurophysiologically observed phase relations between periodic excitatory and inhibitory neurons and their consequences for synchronization at sending and receiving sites. In particular, it was demonstrated in José, Tiesinga, Fellous, Salinas, and Sejnowski (2001, 2002), Tiesinga, Fellous, Salinas, José, and Sejnowski (2004), Buia and Tiesinga (2006), and Mishra, Fellous, and Sejnowski (2006) that gain modulation and gating of signals may result from such phase differences. A review of the resulting studies, together with a discussion of their relation to possible mechanisms for CTC, has been given in Tiesinga and Sejnowski (2010).
Networks of excitatory and inhibitory integrate-and-fire (IaF) neuron models were used by Buehlmann and Deco (2008), following an approach given in Brunel and Wang (2001), to study the relation between rate modulation and synchronization, leading to a reproduction of the neurophysiological experimental results given in (Fries, Reynolds, Rorie, & Desimone, 2001; Womelsdorf, Fries, Mitra, & Desimone, 2006). This approach bears analogy to our approach in two respects. First, with respect to the local interaction of excitatory and inhibitory units, selective and different excitatory pools are considered that interact with a common inhibitory pool, as proposed in Brunel and Wang (2001). Second, inputs to the excitatory pools are modeled with Poisson spike trains, and the attentional bias is identified with an additional Poisson spike train to the attended site. Effectively, this corresponds to a higher spike rate entering the attended site—a property that has some relation to our encoding of the attentional bias through a slightly higher gamma frequency (as motivated by the results presented in Bosman et al., 2012).
Masuda (2009) chose leaky-integrate-and-fire neuron models to model two sending and the receiving populations, where all three populations were driven with the same gamma frequency but independent phases. In contrast to the studies mentioned above, inhibitory interneurons were neglected. Stimuli entered the sending population and competed for representation in the receiving population. In accordance with the neurophysiological results reported in Womelsdorf et al. (2006), this competition was decided by appropriately adjusting the gamma phases of one of the sending and the receiving population. The input from the winning sending population neither had to be stronger (Börgers, Epstein, & Kopell, 2005) or more coherent (an essential ingredient in Börgers & Kopell, 2008).
Another approach that uses a filtering dynamics of the receiving neurons was studied by Akam and Kullmann (2010). The units of the sending networks were modeled with an exponential IaF neuron dynamics, and several of these were connected to a receiving network. Each sending network was given a circular topology corresponding to an encoding of different features (e.g., orientation). Considering the case of one synchronously oscillating sending network with the other senders being asynchronous, the conditions were studied that allowed the receiving network to reproduce the input from the oscillating network while neglecting the other inputs. As an essential ingredient, the dynamics of couplings to the receiving network were based on connections that correspond to the balanced inhibitory couplings discussed by Vogels and Abbott (2005). This also allowed reading out signals at specific frequency bands in case of multiple inputs oscillating at different frequencies, realizing a kind of frequency-division multiplexing of the neural inputs.
Gielen et al. (2010) studied various models of a single neuron (leaky and quadratic IaF, Hodgkin-Huxley) with two sinusoidally oscillating inputs with different frequencies. (In contrast, the inputs in Börgers & Kopell, 2008, were modeled as pulse trains where width of the pulses reflected coherence of the input.) Gielen et al. (2010) found phase locking of the receiving neuron to the incoming oscillation with larger amplitude. Corresponding results were also confirmed for the case where the receiving site is a PING system, that is, a system where the excitatory neuron is coupled to an inhibitory partner. For the latter case, each of the neurons was modeled with a quadratic IaF model, relating this case to the study in Börgers and Kopell (2008).
The coupling of two oscillating sites was modeled as a linear superposition of intrinsic and coupling harmonics by Eriksson et al. (2011). Assuming the same frequency for the intrinsic and coupling oscillations, the dependency of the power correlation on the uncoupled phase difference was studied as the phase difference before the interaction is the crucial parameter for the CTC hypothesis. The power correlation refers to the local field potential/or multi-unit activity at the two sites, motivated by the corresponding measurements by Womelsdorf et al. (2007). Confirming the experimental results (Womelsdorf et al., 2007, figure 3), the theoretical and simulation work found that a particular (“good”) phase relation precedes strong power correlation.
Also Wildie and Shanahan (2012) use a model of two sending sites and a receiving excitatory site. They distinguish two cases they refer to as “bottom up” and “top down.” For the latter case there is an external input to the target group that generates the gamma rhythms, while the former case assumes an internal generation of gamma rhythm in the sending groups. In accordance with the foregoing work, simulations of both cases confirm that entrainment of stimulus and target is correlated with signal transmission. Here, the competition between the different sending sites that tend to synchronize with the target is determined by different coherences.
3 Hypothesis Regarding the Role of Frequency Differences and Resets (Bosman et al., 2012)
Studying again the case of two sending sites (in macaque monkey area V1) and one receiving site (in area V4), Bosman et al. (2012) found another confirmation for selective synchronization in accordance with the CTC hypothesis. Going beyond the earlier studies, however, this work took a closer look at the involved frequencies. In particular, it was found that “attention to their respective driving stimulus led to a slight but highly consistent increase in the frequency of the gamma-band activity” (compare the related Figures 1I and 1J in Bosman et al., 2012). This observation and the following related proposal formulated in Bosman et al. (2012) is the motivation for the work we present in this letter.
Bosman et al. (2012) proposed that the observed gamma-frequency differences play a mechanistic role in establishing the selective synchronization. Related to this, they emphasized that an additional ingredient would be needed for establishing this role of the frequency difference. This additional ingredient is a theta-rhythmic reset of the gamma phase across the involved areas. Such a reset was described before and related to a microsaccadic rhythm in Bosman, Womelsdorf, Desimone, and Fries (2009). This hypothesis was further described by Fries (2015).
The rationale for this need of reset is the following (see the discussion of the “third scenario” in Bosman et al., 2012). After each reset, due to the frequency difference at the attended site, the input from this site will enter the receiving site earlier than the input from the unattended site. In consequence, inhibition at the receiving site will be excited, and when the slower input from the unattended site arrives, it will be attenuated because of the already excited inhibition. Clearly such a scenario would not work without resets, as this reset ensures the temporal ordering, that is, the earlier arrival of the input from the sending site that is driven by slightly higher frequency.
Beginning with section 5, we study this heuristically described scenario in the framework of a model that captures essential ingredients of this proposal. Note also a related recent modeling discussion of the relevance of the frequency differences given by Lowet et al. (2015). These authors use network models of circular topology to study the relevance of frequency differences as they arise, for example, due to contrast differences in an input image. In contrast, the following discussion is closer to the experimental setup of two sending and one receiving sites as it was studied by Bosman et al. (2012) and in earlier experiments (Moran & Desimone, 1985; Fries et al., 2001; Womelsdorf et al., 2007).
4 Frequency Differences between Sending and Receiving Sites May Determine the Direction of Information Flow
In section 3, we emphasized the relevance of frequency differences between sending sites. In this section, we comment on frequency differences between sending and receiving sites. In particular, this is done as a preparation for our comments on related simulation results in section 6.5.
Bosman et al. (2012) observed that the receiving site has a slightly lower frequency than the sending sites (see their Figure 5). Interestingly, we find a corresponding property with our simulation in section 6: selective synchronization requires a slightly lower intrinsic frequency of the receiving site (in the context of the model in section 5, this corresponds to the parameter choice in equation A.1, where determines the difference between the intrinsic frequency at site and sites and ). With no selective synchronization is observed (see also the remarks in section 6.1 and the simulation result, case V.1, in section 6.5). This may be taken as a hint on the functional relevance of the observed frequency difference.
The model given in section 5 allows studying bidirectional flows of information, that is, we also allow for couplings from the receiving sites back to the sending sites. The question then arises as to what actually constitutes the forward direction. Given that the receiving site requires the slightly lower intrinsic frequency in comparison with the sending site, one may speculate that this property defines the direction of information flow (in a sense that is defined in section 6.5). Correspondingly, one may expect that lifting this frequency above the frequency of the sending sites as a result of a stronger stronger excitation of the corresponding neurons may imply a kind of reversal of the flow of excitation. In fact, this is what we observe with the simulations in section 6.5 (cases V2 to V4; see the more detailed discussion there). Here, we add that hierarchies between cortical areas are in fact not invariant but may depend on actual tasks or different stages of a task (see Figure 3 in Bastos et al., 2015). Thus, it may be of interest to find the conditions that modify the direction of information flow (or, at least, modify the strengths of causality between the different levels of the hierarchy). In that respect, the results of section 6.5 indicate that frequency differences may also play a role in determining the direction of information flow.
5 The Model
5.1 Coupling Architecture
In accordance with the experimental paradigm that Bosman et al. (2012) used, we assume there are two sending sites in the same area, here referred to as sites and in area 1, and one receiving site in another area, here referred to as site in area 2. At the receiving and each sending site, we model the competition between excitatory units through interactions with inhibitory units in a manner that is reminiscent of, for example, the architecture chosen in Wang (2002, Figure 1A). For completeness, we include excitatory and inhibitory units also at the sending sites, thereby allowing for a dynamic generation of gamma oscillations also at the sending sites and a study of recurrent couplings that project from area 2 back to area 1. (See Figure 1 for an illustration of the different sets of units and couplings.)
5.2 Dynamics of the Sending Sites
The dynamics of the other sending site, , and , is obtained through replacing the indices in equations 5.1 and 5.2 with and letting the indices run over the corresponding ranges defined through , , and .
5.3 Dynamics of the Receiving Site
For the subnetwork that describes the receiving site , we assume that there are excitatory and inhibitory units. Moreover, the excitatory units are split into and units () that describe the channels that receive input from sites and , respectively. The corresponding indices are chosen to be and , where . This corresponds to an architecture of the receiving site that has some similarity with an architecture Wang (2002, Figure 1A) used to model decision processes.
5.4 Comments on the Chosen Architecture
With respect to the explicit form of the model just given, we add three comments before going on to include phase resets.
First, note that the different sites , , and in the described model have different inhibitory pools. Thus, locality is defined through the range of each inhibitory pool. This is in accordance with the spatial structure of the gamma oscillation as reviewed, for example, in Maris, Fries, and van Ede (2016).
Second, we let sites and project to two nonoverlapping sets of receiving units at site . This is the architecture that may also be found, for example, in the discussion given in Wang (2002, Figure 1A). Future considerations may also consider overlapping sets of receiving excitatory units. It may well be that this would require including a discussion of additional properties. For example, one may think of the discussion by Börgers and Kopell (2008), where two sending units project to the same receiving unit and the degree of coherence of each sending site is decisive for selective synchronization.
Third, we included only intra-areal couplings, not inter-areal ones. Excitatory input to inhibitory units plays a particularly important role in the context of so-called balanced inhibition (see, e.g., the review in Vogels, Rajan, & Abbott, 2005). With respect to (inter-areal) couplings , an accompanying chain of couplings (let the first arrow now correspond to inter-areal and the second to intra-areal couplings) may have a stabilizing role: higher or lower excitations in the path are compensated through, correspondingly, higher or lower inhibition due to . For this discussion, we do not need such compensating effect. Moreover, as discussed, for example, by Börgers and Kopell (2005), a constituting mechanism of the gamma (PING) rhythm is that the inhibitory units are driven by their excitatory local partner. A strong inter-areal drive to the inhibitory units may therefore imply a breakdown of the gamma rhythms that has similarity with the phase walk-through discussed in Börgers and Kopell (2005). Here, we concentrate on the conditions for selective synchronization, and for simplicity, we restrict the discussion to the inter-areal drive of the excitatory units. Nevertheless, a more complete discussion may also include the inter-areal drive of the inhibitory units. This study is, however, beyond the scope of the discussion in this letter.
5.5 Slow-Rhythm Gamma Phase Reset
As we stated in section 3, an essential ingredient of the mechanism described by Bosman et al. (2012) is the slow rhythm (theta rhythm) phase reset. Therefore, we also want to study the effect of gamma phase resetting on the selection properties of the network we have described. While we used the intricate dynamics for modeling the gamma rhythm, the phase resetting will be included with a simple initialization procedure. The condition without resetting corresponds to random initial phases, while the condition with resetting corresponds to aligned initialization of the phases to a common value (see the details in the context of our examples in section 6).
The slowness of this rhythm implies that the resetting does not interfere with gamma dynamics for several gamma cycles. Thus, to study a slow-rhythm gamma phase reset, it is sufficient to consider only one moment of resetting (here, the initial starting of the simulation) and study the effect of this on the gamma dynamics for the next few gamma cycles (until a new resetting would be applied).
We demonstrate the relevance of this kind of resetting in the context of simulating the model in the next section.
In this section, we illustrate the workings of the model presented in section 5 through examples that we refer to as cases I to V.
With case I, we demonstrate how a slightly higher frequency of one of the sending sites implies a selective synchronization of this site with the receiving site if we assume an initial gamma phase resetting. The effect of the other sending site is then suppressed.
Cases II and III serve to demonstrate what causes the selective synchronization and suppression in case I. Therefore, cases II and III use the same set of parameters and initial values as in case I except for using the same frequencies for the sending sites in case II and using again a slightly higher frequency for one of the sending sites but no phase resetting in case III. We also demonstrate the robustness to noise in the initial phase resetting with three examples, referred to as cases IV.1 to IV.3, which describe the transition between cases I and III.
Finally, we use four examples, cases V.1 to V.4, to demonstrate the relevance of a slightly lower intrinsic frequency of the receiving site in comparison to the sending sites. This includes a demonstration of a reversal of the information flow (with a meaning described in the context of the examples) in case that this frequency is lifted above the frequencies of the (originally) sending sites.
6.1 Case I: Selective Synchronization and Suppression
For case I, we assume a slightly higher frequency at site compared to site and an initial phase reset (implemented as described in section 5.5). The resulting dynamics for an example with the parameters given in the appendix is shown in Figure 2.
As a result of the resetting, the different frequencies of and are not relevant for the first volley of spiking; they do, however, affect the spiking for the next volley. As explained in section 5.5, the resetting stands for an initial common zero phase of the model units. Due to the external drive given by the currents (see equations A.1 in the appendix) and the couplings from and to , the excitatory units at sites and fire and imply also a firing of the excitatory units at site . Moreover, the couplings to the corresponding pools of inhibitory units imply the spiking of the inhibitory units at sites , , and .
It is the firing of the inhibitory pool at the receiving site (IC) that is crucial for the observed selective synchronization and the resulting gating effect. Due to the higher frequency of the sending site , it is the input from at that initiates the inhibitory effect in , while the input from arrives at only when the inhibitory effect is already initiated (see the plots of the relevant in Figure 3). In consequence, the excitatory input from is not able to initiate spiking at ; its effect is suppressed at the receiving site. This is what constitutes the gating based on selective synchronization. It implies the phase locking of the sending site and the receiving site as demonstrated with the example displayed in Figure 2.
Note also that we assumed a slightly lower intrinsic frequency for the receiving site, expressed through in equations A.1. With vanishing , no entrainment as observed with Figure 2 occurs. This is demonstrated with case V.1 below. The necessity of having a lower intrinsic frequency may well correspond to the lower frequency at the receiving site in area V4 as observed in Bosman et al. (2012, Figure 5; see the discussion in section 6.5).
6.2 Case II: Relevance of the Frequency Differences
With case II, we demonstrate that the the gamma-frequency difference between and used in case I is crucial for the observed gating effect. This is done by using for case II the identical parameters and initializations that were used in case I except for making the frequencies at sites and identical (see equations A.1 in the appendix). As a result, we find that the suppression of the excitatory effect of input from that was observed in Figure 2 is no longer present (see Figure 4). Now the input from may arrive at times when the inhibitory effect is not yet excited. There is no longer a temporal advantage of site compared to site , and both excitatory inputs may have their effect, none of the two being suppressed.
6.3 Case III: Relevance of the Resetting Process
With case III, we reestablish the frequency differences between and that have been used in case I. Now, however, we no longer apply the phase resetting. All other parameters and initializations are the same as with case I. Due to replacing the initial resetting with some random initialization, the times of arrival of the excitatory inputs from and are random processes, and we no longer observe the reliable suppression of input from and the selective synchronization of activity in and . Instead, some random initialization may even lead to an early selection of the input from and not A (see Figure 5). This may be taken as an illustration of the relevance of the resetting process that links the arrival times of inputs from and at to the common phase resetting at and .
6.4 Cases IV: On the Robustness to Noise
These cases IV of gradually noisier resetting may be taken as an illustration of what we observed with several simulations: in the context of the model of section 5, the selective synchronization process is not easily destroyed with noise in the resetting process. Even if a noisy resetting causes an early firing of the receiving site that is connected to the lower-frequency sending site, the higher frequency of the other site may compensate for this “early firing” (see case IV.2) and establish the same selective synchronization as for the case without noise (i.e., case I). Nevertheless, the initial resetting supports the selective synchronization, its absence may abolish it as demonstrated with cases III and IV.3.
6.5 Cases V: Switching the Synchronization from a Bottom-Up to a Top-Down Driven Selection
The model given in section 5 includes recurrent couplings from the receiving to the sending sites (see Figure 1). This allows us to study an interesting question: Is it possible to reverse the role of sending and receiving sites, that is, in some sense, reverse the flow of information from bottom-up to top-down? Indeed, with the cases V of this section, we demonstrate that if enough excitation drives site , then the units at the (originally) receiving site will dominate the choice for selective synchronization, realizing a top-down control of the selective synchronization process.
7 Understanding the Role of Frequency Differences and Phase Resets
We now want to gain some analytical insight into the mechanism observed with the simulations in section 6. This can only be done with drastic simplifications, leaving some issues as open questions. Nevertheless, the following discussion will picture the processes in a manner that should help to understand what was observed in the simulations. Inspired again by the work of Börgers and Kopell (2008), we relate the theta model to a quadratic integrate-and-fire (IaF) model and thereby justify a simplification to a linear IaF model in section 7.1. This is then used in section 7.2 for the analytical discussion.
7.1 Relation to Integrate-and-Fire Models
7.2 An Analytical Approach
For simplicity, we restrict the discussion to three linear IaF units that are assumed to represent the receiving site , that is, two excitatory units—one receiving input from site , the other from site —and an inhibitory unit. In this section, the denominations , , and refer to these units, respectively. Each of the excitatory units and drives the inhibitory unit, and the inhibitory unit acts back on both excitatory units.
8 Summary and Outlook
Using a network architecture similar to the one described by Wang (2002; see Figure 1A), we modeled aspects of the situation that was studied in a physiological experiment to investigate the neuronal machinery underlying covert visual attention with respect to two stimuli in the visual field (Bosman et al., 2012). Following Börgers and Kopell (2005, 2008), we used theta models for the dynamics of the coupled units of the network. Going beyond the latter work, we also modeled the inhibitory pool as a set of dynamical units.
Using this neural architecture and dynamics, we confirmed a proposal that was formulated by Bosman et al. (2012). These authors proposed that a resetting resulting from a slower rythm, for example, the theta rhythm discussed in Bosman et al. (2009), may be crucial for the selective synchronization when combined with gamma frequency differences: “After the reset, the attended V1 gamma and the unattended V1 gamma partly precess relative to the slightly slower V4 gamma. The attended V1 gamma is of higher frequency than the unattended V1 gamma and therefore precesses faster. Correspondingly, in each gamma cycle, the attended V1 input enters V4 before the unattended V1 input. The earlier entry together with feedforward inhibition makes the attended V1 input entrain V4 at the expense of the unattended V1 input.” Moreover, as Bosman et al. (2012) hypothesized (see the third scenario in their discussion), “There is one crucial additional ingredient to this scenario, namely a theta-rhythmic gamma phase reset across V1 and V4.”
There are three components of this scenario and in the context of our model, we demonstrated that all of these are relevant. First, there is a need to have a “slightly slower V4 gamma” frequency. Second, the attended V1 gamma frequency should be “of higher frequency than the unattended V1 gamma.” Third, the reset at a substantially lower frequency than gamma (so that it does not interfere with the gamma dynamics) is crucial as a starting point for the above scenario.
With our case I simulation, we demonstrated that a dynamics with these three ingredients indeed leads to a selective synchronization analogous to the one observed in the neurophysiological experiment (see Figures 2 and 3). In the context of case I, we mentioned that choosing a slower frequency for the receiving (V4, here area 2) site is necessary. (Setting in equations A.1 leads to a breakdown of the selective synchronization demonstrated with case I.) With case II, we demonstrated the relevance of the slightly higher gamma frequency of the attended (V1, here area 1) site in comparison to the unattended site. With cases III and IV, we demonstrated the relevance of the initial reset. The cases V served to demonstrate the relevance of frequency differences between sending and receiving sites. These may determine whether the dominating influence on the selective synchronization decision is bottom-up or top-down driven. Using an abstraction based on a linear integrate-and-fire model, we also discussed parameter dependencies for the selective synchronization process. The analytical discussion confirmed (see equation 7.27) that the discussed selection process that is based on frequency differences and phase resetting requires an inhibitory activity which is sufficiently synchronized.
Recent neurobiological experimental work found that the gamma rhythm is modulated as a function of the phase with respect to the slower cycle (Lowet et al., 2015). In this study, we focused on the resetting aspect of the slower rhythm, corresponding to considering only a momentary effect of the slower rhythm resulting in a phase resetting that was found to be essential for the selective synchronization. Going to a more realistic modeling of the slower rhythm will need to incorporate additional features of the more complex cross-frequency interplay of the slow and fast rhythms, eventually even incorporating additional rhythms like beta/alpha, where gamma may be dominating in the superficial cortical layers, while beta/alpha rhythms may contribute through the deeper layers (Buffalo, Fries, Landman, Buschman, & Desimone, 2011). Correspondingly, an interplay that takes the laminar profiles into account, as observed by van Kerkoerle et al. (2014), should be subject to future studies. Moreover, another recent study by Cannon and Kopell (2015) illuminates the mechanisms of inhibition-based rhythms under periodic forcing from a theoretical point of view, and it would be of interest to relate these insights to the results presented here. The work by Cannon and Kopell (2015) also addresses the possibility of a reversal of directed entrainment by changing the relative drive; see their discussion in section 9.3 and (Battaglia, Witt, Wolf, & Geisel, 2012). This reversal is related to the change from bottom-up to top-down of information flow as we studied it in this letter.
Finally, as another outlook on possible future work, we point to a potential relevance of the discussed mechanisms in a wider context of information processing. There has been recent progress toward studying the experimental paradigm of “free viewing of natural images.” Clearly, this poses new challenges also for the theoretical modeling that may also have to incorporate neural correlates of object recognition. Related to this, it is of interest to see that, based on oscillatory network properties (Burwick, 2007, 2008b; see also (Burwick, 2008a)), recent modeling approaches have combined a gamma-like mechanism with recognition processes, resulting in an attention-based selective attention of attentional targets while suppressing distracting influences (Burwick, 2011; Blaes & Burwick, 2015). The corresponding oscillatory network has been modeled with phase and amplitude dynamics, and it would be of interest to see whether it may be extended to spiking dynamics like the one discussed here. The commonality between the discussion in Blaes and Burwick (2015) and our discussion in this letter is the crucial effect of gamma-frequency differences and initializing resets. This raises the expectation that such frequency differences in the gamma range and resets related to slower rhythms as hypothesized by Bosman et al. (2012) and studied here from a theoretical perspective may also be relevant for the more realistic “free viewing of natural images” scenario.
Appendix A: Simulation Parameters and Initializations
In section 6, we consider a network with . Assuming the usual ratio of excitatory and inhibitory neurons in the cortex, we set , and . Moreover, we split the set of receiving excitatory units into and units so that .
The intra-areal couplings are chosen to be , , and , while (see Börgers et al., 2005, for a motivation of the latter choice). The inter-areal couplings are given by and . Thus, we allow for substantial recurrency. No inter-areal couplings between excitatory and inhibitory units are included.
The parameters of the synaptic dynamics in equation 5.3 are given by , , and , .
As we argued in section 5.5, the initial phases at time in cases I and II are set to equal values in order to mimic the effect of the slow-rhythm phase reset. Only in case III are the phases drawn randomly from 0 to with equal probability.
The result of any random process is determined only once and then applied to all cases I to III (except for the initial random phases that are used only in case III). This is done to ensure that the observed differences between the cases can be traced back to the described differences in frequencies (case I versus case II) and initial values (case I versus case III) and not to different results of random processes.
Appendix B: Parameter Choices for the Figure 10
Figure 10 serves to illustrate the arguments of section 7.2. Therefore, the following parameters have been chosen to imply an illustration that is helpful to make these arguments clear: , , , , and . The two cases (left and right column of panels) have different driving frequencies related to site . With , we chose for the left and for the right column (frequencies in Hz, times in ms).
It is a pleasure to thank Pascal Fries for valuable discussions of the neurobiological background. We thank Jonathan Cannon for proposing to use our model to study the reversal of directed entrainment from bottom-up to top-down.