Abstract

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.

1  Introduction

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.

Figure 1:

The coupling architecture used for our model in section 5. Two sending sites ( and ) in area 1 (e.g., V1) project to a receiving site in area 2 (e.g., V4). Arrows indicate excitation; reversed arrow heads indicate inhibition. The role of the various parameters is described in section 5.2 for the sending sites ( and ) and in section 5.3 for the receiving site . Comments on the phase-resetting part are given in section 5.5.

Figure 1:

The coupling architecture used for our model in section 5. Two sending sites ( and ) in area 1 (e.g., V1) project to a receiving site in area 2 (e.g., V4). Arrows indicate excitation; reversed arrow heads indicate inhibition. The role of the various parameters is described in section 5.2 for the sending sites ( and ) and in section 5.3 for the receiving site . Comments on the phase-resetting part are given in section 5.5.

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.)

The dynamics of the units is described by the theta model dynamics that is equivalent to a quadratic integrate-and-fire model (Börgers & Kopell, 2005; we explicitly describe this equivalence in section 7.1).

5.2  Dynamics of the Sending Sites

In this section, we first describe the dynamics of site and then explain how to obtain the analog description for site . We assume that site consists of excitatory units and inhibitory units. The states of the excitatory and inhibitory units are described with phases and where the range of indices is and . Following the discussion of the theta model in Börgers and Kopell (2005, 2008), the dynamics of the excitatory units at the sending site is taken to be
formula
5.1
for , while the dynamics of the inhibitory units is
formula
5.2
for . The and are timescales, and the and are external inputs. The parameter parameterizes the coupling strength among the excitatory units at site , while describes the effect of the inhibitory units on these excitatory units and describes the effect of the excitatory units on the inhibitory units. The parameter gives the strength of projection toward site (more on this in section 5.3). Here, we also allow for recurrent projections from area 2 to area 1, parameterized with a constant that multiplies and enters equation 5.1 correspondingly. The corresponding range specified through is defined in section 5.3. (See Figure 1 for an illustration of the role of these parameters.) All of the above and the following parameters are assumed to be positive.

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 .

The variables are dynamical and describe the gating (the synaptic opening and closing of the neurons). Their dynamics is given by
formula
5.3
where is a placeholder for the different sites and is the corresponding index for the units at this site. There are two timescales in equation 5.3 that determine the rise and decay times of the gating: and , respectively.

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.

Using again a theta model description, the states of the excitatory and inhibitory units are described with phases and , respectively, where for the excitatory units and for the inhibitory units. The dynamics of the excitatory units that receive input from site is then given by
formula
5.4
for , while the ones that receive input from site are, correspondingly, described with
formula
5.5
for , where . The is a timescale, and the are external inputs (i.e., not inputs from sites and ). The is the coupling strength for the input from the sending area. The parameter describes the coupling strength among the excitatory units of each of the two groups, while describes the effect of the inhibitory units on the excitatory units (see Figure 1).
What makes the excitatory units part of the same location is that they share a common pool of interneurons (as, for example, with the model described in Wang, 2002, Figure 1A). With respect to the inhibitory units, we concentrate on the local, internal couplings and neglect any feedforward input from the sending sites. The dynamics of the inhibitory units at site is then given by
formula
5.6
for . The is a timescale, and the are external inputs. The parameter describes the coupling strength among the inhibitory units, while describes the effect of the excitatory units on the inhibitory units (see again Figure 1 for an overview on the different couplings). The dynamics of the gating variables was described above with equation 5.3.

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.

6  Simulations

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.

Figure 2:

Case I, where phase resetting is applied at time and the sending site has a higher frequency than site . Displayed are the spikings of the excitatory (top panel) and inhibitory (bottom panel) units at sites , , and . The spiking of each unit is shown through bars that indicate . Due to the lower frequency at site , the excitations from arrive only when already initiated the inhibition at site (through the couplings with ). As a result, the effects of the sending site on the receiving site are suppressed and the inhibitory pool phase-locks to the rhythm of the sending excitatory units .

Figure 2:

Case I, where phase resetting is applied at time and the sending site has a higher frequency than site . Displayed are the spikings of the excitatory (top panel) and inhibitory (bottom panel) units at sites , , and . The spiking of each unit is shown through bars that indicate . Due to the lower frequency at site , the excitations from arrive only when already initiated the inhibition at site (through the couplings with ). As a result, the effects of the sending site on the receiving site are suppressed and the inhibitory pool phase-locks to the rhythm of the sending excitatory units .

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 .

The crucial effect of the higher frequency is then obvious with the next volleys of spiking. To gain some insight into the dynamics, we define the following kinds of total currents,
formula
6.1
formula
6.2
for and . The currents for , where , are obtained by replacing on the right-hand side of equation 6.1. Without couplings, the excitatory (inhibitory) theta model units () spike if (). Therefore, with couplings present, the currents we have defined provide some insight into the readiness of the units to spike.

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.

Figure 3:

The total inputs at site as defined in equation 6.1 (with the analog definition for the units receiving input from site B) and equation 6.2. The broken line displays the input to the inhibitory units, , being identical in both panels. The inputs from sites and are shown as solid lines in the bottom and top panel, respectively. Note that these plots show the averaged over the corresponding units. It may be seen from the top panel that following the resets at the initial time and due to the lower frequency at site , the inputs from arrive at times when the inhibitory pool is already excited, resulting in the corresponding suppression of spikes as observed with the upper panel in Figure 2.

Figure 3:

The total inputs at site as defined in equation 6.1 (with the analog definition for the units receiving input from site B) and equation 6.2. The broken line displays the input to the inhibitory units, , being identical in both panels. The inputs from sites and are shown as solid lines in the bottom and top panel, respectively. Note that these plots show the averaged over the corresponding units. It may be seen from the top panel that following the resets at the initial time and due to the lower frequency at site , the inputs from arrive at times when the inhibitory pool is already excited, resulting in the corresponding suppression of spikes as observed with the upper panel 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.

Figure 4:

Case II, where the model parameters and initilizations are the same as in case I except for using the same frequencies at sending sites and . In consequence, the excitatory input from at is no longer suppressed. The figure shows the spiking of the units with the same conventions as in Figure 2.

Figure 4:

Case II, where the model parameters and initilizations are the same as in case I except for using the same frequencies at sending sites and . In consequence, the excitatory input from at is no longer suppressed. The figure shows the spiking of the units with the same conventions as in Figure 2.

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 .

Figure 5:

Case III, where the model parameters and initializations are the same as in case I except for not using the initial phase resetting. Instead, the initilization of the phase is randomly chosen. In consequence, the selection process at the receiving site is randomized, and the figure shows an example where the input from causes excitation of spikes at the receiving site while the excitatory input from is suppressed at early cycles in spite of the higher frequency at sending site . The figure shows the spiking of the units with the same conventions as in Figure 2.

Figure 5:

Case III, where the model parameters and initializations are the same as in case I except for not using the initial phase resetting. Instead, the initilization of the phase is randomly chosen. In consequence, the selection process at the receiving site is randomized, and the figure shows an example where the input from causes excitation of spikes at the receiving site while the excitatory input from is suppressed at early cycles in spite of the higher frequency at sending site . The figure shows the spiking of the units with the same conventions as in Figure 2.

6.4  Cases IV: On the Robustness to Noise

The resetting process in case I was implemented through setting the initial phases to zero, , where is again the placeholder for the different sites and is the corresponding index for the units at each site. In contrast, some random intialization was used for case III, , where was a random number that was uniformly drawn from . Thus,
formula
6.3
where for case I and for case III. In order to demonstrate the robustness of the selective synchronization process of case I to a gradual increase of noise in the resetting process, we now also give results for intermediate values. In fact, we observe that the selective synchronization of case I is still intact for values as large as (see the case IV.1 as displayed in Figure 6, top panel). With , the random initialization is large enough to allow for an earlier spiking at site . However, given the higher frequency of the sending site , the units at site take over again and lead to a suppression of further spiking at site for several cycles (see case IV.2 as displayed in Figure 6, middle). With the randomly induced earlier spiking at the “wrong” site is then large enough to destroy the selective synchronization as observed with the case III (see case IV.3 in Figure 6, bottom).
Figure 6:

Cases IV, distinguished by the parameter , which describes the transition between case I () and case III (). The three panels correspond to case IV.1 (top) with , the case IV.2 (middle) with , and the case IV.3 (bottom) with (see the discussion in section 6.4). The spiking of the inhibitory units is not shown.

Figure 6:

Cases IV, distinguished by the parameter , which describes the transition between case I () and case III (). The three panels correspond to case IV.1 (top) with , the case IV.2 (middle) with , and the case IV.3 (bottom) with (see the discussion in section 6.4). The spiking of the inhibitory units is not shown.

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.

As we already remarked in section 4, the slightly lower frequency at site supports the selective synchronization observed with case I. This may be interpreted as a bottom-up process in the sense that the decision for selective synchronization is triggered by the frequency difference at the early stage (area 1 in Figure 1). In the following, we denote for the indices that describe excitatory units at site and for the indices that describe the excitatory units at site . A corresponding notation may be introduced for the currents . Case I (and also cases II to IV) used for each , thereby establishing a drive to the excitatory units at site that is slightly smaller than the drive to the sending units at and . Its relevance is easily demonstrated by comparison with case V.1 that is given by the modification
formula
6.4
For case V.1 all other parameters and initializations are unchanged. The resulting dynamics of the excitatory units is displayed in Figure 7 (top panel). We find that the selective synchronization is lost. With case I, the lower drive at site (area 2 in Figure 1) implied that the rhythm at site is driven by the incoming activation from area 1. Due to the slightly higher frequency of the bottom-up influence, it is the bottom-up influence and the frequency difference between input from and that generates the selective synchronization. With case V.1, however, the oscillation at site seems to be driven by its internal dynamics, and the difference between the external drives at sites and does not generate the selective synchronization that is observed with case I.
Figure 7:

Cases V.1 (top) and V.2 (bottom) demonstrate the role of the slightly lower intrinsic frequency used with case I. Case V.1 shows that the selective synchronization is lost if the intrinsic frequency of the receiving site is no longer lower in comparison to the sending sites, that is, case V.1 differs from case I only by using in equations A.1. Keeping the value but returning to a lower intrinsic frequency at site through increasing the local inhibition with , case V.2 shows that this reestablishes the selective synchonization. The spiking of the inhibitory units is not shown. See also the discussion in section 6.5.

Figure 7:

Cases V.1 (top) and V.2 (bottom) demonstrate the role of the slightly lower intrinsic frequency used with case I. Case V.1 shows that the selective synchronization is lost if the intrinsic frequency of the receiving site is no longer lower in comparison to the sending sites, that is, case V.1 differs from case I only by using in equations A.1. Keeping the value but returning to a lower intrinsic frequency at site through increasing the local inhibition with , case V.2 shows that this reestablishes the selective synchonization. The spiking of the inhibitory units is not shown. See also the discussion in section 6.5.

This interpretation is in accordance with the observation made with case V.2. For this case, we go back to a slightly lower frequency through increasing the inhibitory effect on the excitatory units at site (this implies that the excitatory units need longer to recover from inhibition and therefore the rhythm is slowed):
formula
6.5
The corresponding dynamics is displayed in Figure 7 (bottom panel). We find that this modification reestablishes the selective synchronization. In fact, for the remaining cases, V.3  and V.4 , we keep the latter modification. With these cases, we study the effect of selectively increased drive at site C, and the higher value of serves to stabilize the resulting oscillation.
Let us now study the effect of increasing the drive at site . For example, going to will generate a dynamics that is qualitatively similar to case V.1, that is, no selective synchronization is reached. More interesting is an increase of the drives that distinguishes between sites and . We first look at
formula
6.6
The resulting excitatory dynamics is displayed in Figure 8 (top panel). We again find selective synchronization between sites and where the doubled spikes at site are due to the increased drive at these units. For this case, the higher frequencies at in comparison to are compatible with the higher frequency at the lowerlevel . Therefore, in order to see the top-down influence, we need to consider a case where the frequency difference at area 2 is incompatible with the difference at area 1. Such a case may be studied by choosing
formula
6.7
The resulting dynamics is displayed in Figure 8 (bottom panel). In contrast to case V.3 we now find selective synchronization between sites and . As it is now the higher drive to the units that dominates the selection for synchrony, and not the higher drive to the units at , one may interpret this selective synchronization to be driven by top-down influences. In this sense, the frequency difference at area 2 overrules the frequency difference at area 1.
Figure 8:

Cases V.3 (top) and V.4 (bottom) demonstrate that a higher drive at target site , corresponding to higher intrinsic frequencies ( in equations A.1), makes the frequency differences at site dominating for the selective synchronization decision. The double spike volleys at site indicate the higher drive to the excitatory units and in cases V.3 (top) and V.4 (bottom), respectively. One finds that the decision between selective synchronizations (case V.3) and (case V.4) is now determined by the frequency differences at site . (The drawn ellipsoids emphasize examples of the selective synchronization.) This may be seen as a top-down-driven situation in contrast to the bottom-up driven case I. The spiking of the inhibitory units is not shown. See also the discussion in section 6.5.

Figure 8:

Cases V.3 (top) and V.4 (bottom) demonstrate that a higher drive at target site , corresponding to higher intrinsic frequencies ( in equations A.1), makes the frequency differences at site dominating for the selective synchronization decision. The double spike volleys at site indicate the higher drive to the excitatory units and in cases V.3 (top) and V.4 (bottom), respectively. One finds that the decision between selective synchronizations (case V.3) and (case V.4) is now determined by the frequency differences at site . (The drawn ellipsoids emphasize examples of the selective synchronization.) This may be seen as a top-down-driven situation in contrast to the bottom-up driven case I. The spiking of the inhibitory units is not shown. See also the discussion in section 6.5.

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

Consider the theta model (),
formula
7.1
where is the reversal potential of the synapse. This is related to the quadratic integrate-and-fire (IaF) model,
formula
7.2
for the membrane potential if the two variables and the parameters are related through the variable transformations (Börgers & Kopell, 2005)
formula
7.3
The standard reset that defines the moment of spiking ( is the initial time) for the IaF model is
formula
7.4
Equations 7.1 to 7.3 show that the theta model and the quadratic model are equivalent except for the resetting in equation 7.4. The letter amounts to “cutting out” the range that defines the spike for the theta model (see Figure 9).
Figure 9:

The relation between theta and the membrane potential as described with equation 7.3. For the theta model, the moment of spiking is identified with and the width of the spiking with the range . For the integrate-and-fire model the resetting condition in equation 7.4 eliminates just the range of theta that corresponds to this pulse of spiking, effectively shrinking the time width of this pulse to zero. (The indicated position at rest is exact for and is shifted to neighboring positions on the circle in case of while spiking occurs for .)

Figure 9:

The relation between theta and the membrane potential as described with equation 7.3. For the theta model, the moment of spiking is identified with and the width of the spiking with the range . For the integrate-and-fire model the resetting condition in equation 7.4 eliminates just the range of theta that corresponds to this pulse of spiking, effectively shrinking the time width of this pulse to zero. (The indicated position at rest is exact for and is shifted to neighboring positions on the circle in case of while spiking occurs for .)

Instead of analyzing the quadratic IaF model, we now consider as a simplified model the linear version of the IaF model, following Börgers and Kopell (2008). This is done to obtain some qualitative insight into an analog of the processes observed in section 6. The linear IaF model has the following form:
formula
7.5
where is the reciprocal of the membrane time constant and is again defined such that is the time of spiking. The value of can be obtained as follows. Note first that equation 7.5 is solved by
formula
7.6
where we reformulated the current as , where is the charge that is applied during the pulse duration . Given the reset condition of equation 7.4, the requirement for spiking at time () is therefore
formula
7.7
implying
formula
7.8

In section 7.2, we use this identity to discuss a situation that is akin to the dynamics that we observed in section 6.

7.2  An Analytical Approach

As we stated in the discussion of section 6, the difference between frequencies and at sites and , respectively, plays a crucial role, and so we quantify this by considering the dependencies on
formula
7.9
In the following, we sketch a situation that has some analogies with the dynamics studied in sections 5 and 6 in order to illuminate the role played by when the frequency difference is combined with initial resetting.

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.

In section 6 and with respect to case I, we illustrated how opening and closing of synaptic channels due to excitatory inputs from sites and relates to the spiking of the units at site (compare Figures 2 and 3). Analyzing excitatory and inhibitory inputs as complicated as the ones in Figure 3 is beyond the current discussion, and so we apply a simplification by modeling the driving effects for the three units through assuming an IaF dynamics that is adapted to the present architecture:
formula
7.10
formula
7.11
formula
7.12
with a boundary that is given in equation 7.16 below. The boundary conditions are
formula
7.13
The describe the inputs from and while the describe internal interactions at site . The positive and negative signs in front of the indicate excitation and inhibition, respectively. The rationale behind equations 7.10 to 7.12 is explained in the following (see Figure 10 for an illustration). The parameter choices used for Figure 10 may be found in the appendix B.
Figure 10:

Illustration of the role that frequency differences of sending sites and together with initial resettings have at site for suppressing (left column) or not suppressing (right column) the spiking of an excitatory unit that receives input from , corresponding to cases I and II (Figures 2 and 4). The top, middle, and bottom rows illustrate the dynamics of , , and (understood as integrate-and-fire units in the context of section 7), respectively. Note that the left and right column differ only in the top row, where the left panel is based on a larger frequency difference, resulting in a larger delay (case I), while the frequencies are assumed to be closer for the right panel (case II). See the discussion in section 7.2 for further explanations.

Figure 10:

Illustration of the role that frequency differences of sending sites and together with initial resettings have at site for suppressing (left column) or not suppressing (right column) the spiking of an excitatory unit that receives input from , corresponding to cases I and II (Figures 2 and 4). The top, middle, and bottom rows illustrate the dynamics of , , and (understood as integrate-and-fire units in the context of section 7), respectively. Note that the left and right column differ only in the top row, where the left panel is based on a larger frequency difference, resulting in a larger delay (case I), while the frequencies are assumed to be closer for the right panel (case II). See the discussion in section 7.2 for further explanations.

With this discussion, we are interested only in the dynamics up to the decision of whether the spiking of is suppressed due to internal inhibition (left panels of Figure 10) or not suppressed (right panels). In particular, we assume an initial resetting at sites and and so that (the case of absent common reset is discussed in the context of equation 7.28). Based on these assumptions, the input from to is set to arrive at , while the input from to arrives at . Modeling each input as a pulse of charge and duration , we therefore have
formula
7.14
with vanishing values for other times . See Figure 10 where these currents have been displayed in the middle and top rows, respectively, as rectangular boxes for two choices of , corresponding to a larger (left) and smaller (right) difference in the driving frequencies of sites and .
It is a defining property of gamma oscillations (more precisely, of its PING rhythm understanding; Börgers & Kopell, 2005) that the spiking of the excitatory cell elicits a spiking of the inhibitory cell and a subsequent recurrent inhibition of the excitatory cell. Modeling this sequence of events is implemented with equations 7.10 to 7.12 through the following choices of the parameters. These have to be defined together with the timing of the spiking. For unit , the moment of spiking is given by solving equation 7.10 (with as inhibition is not elicited before the spiking of ). The analog of equation 7.6 gives
formula
7.15
Then, defining the spike time and solving for it in analogy to equations 7.7 and 7.8 leads to
formula
7.16
This spike time is then taken as the moment of starting the activation of the inhibitory unit . Again we simplify this activation through modeling it as an idealized “rectangular” input, now in equation 7.12, with a form that mimics the currents in equations 7.14:
formula
7.17
and for other times. The time course of the inhibitory membrane potential up to the spiking of unit is then given by
formula
7.18
and the corresponding inhibitory spike time is
formula
7.19
The membrane potential time courses and the resulting moments of spikings of and are illustrated with the middle and bottom row panels of Figure 10.
What effect does the frequency difference have? We first assume that is large enough so that the recurrent effect of the inhibitory spiking suppresses a possible spiking of the unit , that is, we use the condition:
formula
7.20
To make this inequality more explicit, we still have to define the quantity . In case that the above inequality is violated, the is the spike time of unit . Therefore, in order to derive an explicit expression for it, we may set in equation 7.11 and, in analogy to the foregoing discussions, we obtain
formula
7.21
for
formula
7.22
and the spike time is then obtained to be
formula
7.23
Whenever is large enough so that inequality 7.20 holds, inhibition arrives at unit before it may spike. This results in the suppression of its spiking as observed in case I of the simulations in section 6. For simplicity, let us now assume
formula
7.24
Then and, using equations 7.9 and 7.19, inequality 7.20 reads as
formula
7.25
where is the geometric mean of the frequencies and and is the width of the excitatory pulse acting on the inhibitor unit measured in units of the membrane time constant . In order to have a selective suppression analogous to the ones observed with case I in section 6, we need a frequency difference large enough to obey inequality 7.25 (see the upper left panel in Figure 10). With a violation of this condition, there may be a spike elicited at sites and and no selective suppression occurs in analogy to case II (see the upper right panel in Figure 10).
To understand the meaning of inequality 7.25, it is helpful to recall the dynamics of the inhibitory unit during the application of the internal excitation expressed with equation 7.12, now using the conventions specified above:
formula
7.26
Note that an inhibitory spike () is elicited only if . This also ensures that the denominator of the above inequality is positive, . Inequality 7.25 places a condition on the frequency difference (left-hand side) related to the internal dynamics of site (right-hand side). When increasing the membrane time constant (while keeping constant) the intrinsic dynamics of the inhibitory unit gets slower, and as a result the inhibitory effect sets in at a later time. Thus, a larger frequency difference is needed to allow for selective synchronization. This is reflected through a corresponding increase of the right-hand side of inequality 7.25. Moreover, there are two dependencies on the excitatory pulse that excites the inhibition. First, given a particular width of this pulse, increasing the transmitted charge will lead to an earlier onset of the inhibitory effect. Thus, less frequency difference is needed, and this is reflected by the fact that the logarithm is monotonically decreasing for an increasing with as , making the right-hand side smaller. Second, given a particular value of the transmitted charge , an increase (decrease) of the pulse width will lead to a later (earlier) onset of the inhibitory spike as is obvious already from equation 7.26. A larger (smaller) frequency difference is then needed for the selective synchronization and, correspondingly, we find that the logarithm on the right-hand side of inequality 7.25 gets larger (smaller) with an increase (decrease) of the pulse width (in the range ).
Notice also another lesson that may be learned from inequality 7.25. For a given frequency difference and charge , there is an upper limit for the decoherence of the inhibitory activation, expressed through the temporal width of the current:
formula
7.27
For large enough , this condition would be violated and the selective synchronization would be abolished. This is more evidence for the need to deal with synchronized pulses in order to realize selective suppression.
With case III in section 6, we observed that a lack of initial phase resetting may also destroy the selective suppression at the site driven by the lower frequency. In the context of this analytic treatment, the lack of initial phase reset may be easily traced by replacing
formula
7.28
in equation 7.9, where  is supposed to describe some deviation from the resetting of and to a common phase. Following the analog of the above derivation, the would then show up in inequalities 7.20 and 7.25 and may violate this condition. In consequence, the selective synchronization of the receiving site with the target and the suppression of the distracting influence would become a matter of chance if is chosen randomly.

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 .

For defining the external inputs we introduce time periods , and, following Börgers and Kopell (2005, equation 4), define the currents through
formula
A.1
where in cases I and III to V, while in case II. Thus, in cases I and III to V, the frequency (driven by the currents) at site is slightly higher than the frequency at site . The timescales are chosen . The dependency of the currents on the time periods in equations A.1 is motivated by the properties of the theta model as described in Börgers et al. (2005, section 2.1). For we allow for a suppressing contribution that implies a kind of threshold for activation and corresponds to a slightly slower gamma frequency at the receiving site. Special values of in cases V.1 to V.4 are specified in section 6.5. We also include random contributions , where are drawn from a standard normal distribution.

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 , .

Apart from the heterogeneity introduced with the randomness in equations 4.1, we introduce a second source of randomness by using random spiking. Following Börgers et al. (2005, ch. 8), we define random spiking times,
formula
A.2
and . These are exponentially distributed with an expected spike time of 50. At each spiking, the phase of the spiking unit is reset to and the associated gating variable is set to . Slightly modifying the procedure described in Börgers et al. (2005), we apply this reset (that is, the random spiking) only if the total input to the units is not negative (and thereby not inhibitory) at the randomly generated moments. Here, the notion of total input refers to the terms multiplying in equations 5.1, 5.2, and 5.4 to 5.6. This restriction avoids a complete decoupling of the random spiking from the underlying dynamics.

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).

Acknowledgments

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.

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