Phase difference and attention

It is well known that voluntary orienting of attention to a specific location, or to a specific feature or object, enhances processing of that location or stimulus [21]. One way this might work is for there to be enhanced information transmission about the attended stimulus between the cortical areas that are processing the sensory information. In the light of the CTC hypothesis, it is then reasonable to suggest that such enhanced information transmission could result from synchronization of neural oscillations between the relevant cortical areas. A study by Saalman and colleagues [22] reported that oscillations in the alpha band (around 10 Hz), originating in the pulvinar nucleus of the thalamus, were closely tied to top-down attentional processing in the cortex. Moreover, a more recent study by Eredath and colleagues [23] demonstrated a causal role of pulvinar input in synchronizing neural activity between V4 and Lateral Intra-Parietal (LIP) cortex by reversibly inactivating the dorsal pulvinar.

There is, in fact, a large literature on the role of alpha frequency oscillations in cognitive processing, including in memory, attention, and other cognitive functions (e.g., [24–26]). Alpha oscillations have been argued to modulate large-scale communication in the brain [27, 28]. Motivated by the general importance of alpha oscillations in neural processing, and more specifically by the results of [22], Quax and colleagues [29] created a spiking neuron network model of two communicating cortical areas that received noisy sine-wave alpha input as if from the pulvinar nucleus. They showed that the alpha input organized the gamma oscillations in the cortical areas, increasing phase coherence, information transmission, and Granger causality between the areas, consistent with the CTC hypothesis. Surprisingly, the alpha inputs to the two cortical areas in their study had maximum effects for the specific phase difference of Δϕ = −π/2 between the 10 Hz oscillations sent to the two cortical areas.

Intrigued by this finding, Greenwood and Ward [30] wondered what was special about −π/2. They proved a lemma that showed that, in fact, phase coherence, defined as the phase locking index of [31], between summed sine waves, say α₁ + γ₁ and α₂ + γ₂, similar to those in the study of [29], was maximal when the phase difference in radians, Δϕ, between α₁ and α₂ equalled that between γ₁ and γ₂, i.e., in the case of the results of [29], −π/2. The finding of [29], that maximal effects of alpha inputs to gamma-oscillating cortical areas were found when Δϕ = −π/2, pointed to the possibility that there was a similar phase difference, Δψ, between the gamma oscillations in the two modelled cortical areas. Greenwood and Ward then found that the results of [29] could be replicated using a firing-rate-based macro-model of cortical neural activity to generate the gamma oscillations while retaining the noisy sine wave input for the alpha oscillations. Their cortical rate model comprised a coupled Excitatory-Inhibitory, or EI, pair of linear stochastic differential Eqs (1), (2) and (3) (see Methods for the equations). This model is similar to a linearized version of the firing rate model originated by Wilson and Cowan [32]. The linear model of [30] produces quasi-cycle oscillations rather than the limit cycle oscillations produced by the original, nonlinear, Wilson-Cowan model. Quasi-cycles are noisy periodicities resulting from damped oscillations sustained by noise, e.g., [33]. In a subsequent part of their study, Greenwood and Ward implemented a similar rate model with EI pairs also generating the 10 Hz oscillations supposedly emanating from the pulvinar, replacing the noisy sine waves used by [29] with quasi-cycles. Their replication of the results of [29] using this method to produce oscillations completed the expression of a cortex-pulvinar model entirely in terms of quasi-cycles.

Fig 1 displays illustrative (both for [30] and the current work) single realizations of quasi-cycle oscillations for the excitatory processes in the two feed-forward-connected gamma-oscillating ‘cortical’ areas. These are the E processes of the EI pairs, where each E process is reciprocally connected to its respective I process (see Methods for equations). In the corresponding models, noisy alpha-frequency sine waves have been added to the increments of the inhibitory processes in each area, and are thus transmitted to the respective E processes through their E-to-I connection. The figure also illustrates the two main measures of signal transmission employed by [29, 30]. Maximum signal onset effect, ‘max sigoe’ in the figure, is defined as the difference between the amplitude of the positive part of the excitatory oscillation in the period from signal onset until 275 ms after signal onset in cortical area 2 minus that in cortical area 1. Extended signal onset effect, ‘extended sigoe,’ is defined as the mean positive excursion during the 50 ms on each side of the maximum in cortical area 2 minus that in cortical area 1. Apparent in the figure is, first, the transient burst of larger oscillations at signal onset, larger in V2E than in V1E, and, second, that the difference is larger, and the transient burst lasts longer, when the phase offset, Δϕ, of the alpha sine waves matches that of V1E and V2E, Δψ. It is important to note that the burst of larger oscillations does not last for the entire duration of the signal; this feature will be discussed in the Results section. Also, one can see in the figure that the oscillations in V1E and V2E are more coherent with each other when the phase offsets match (ρ = 0.64) than when they do not (ρ = 0.51).

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Fig 1. Representative results of [30] (and the current study) demonstrating single realizations of gamma frequency ‘cortical’ quasi-cycle excitatory processes (V1E and V2E), with noisy alpha sine waves added to their respective, connected, inhibitory processes (V1I and V2I).

The V2E process is comprised of its own oscillations plus 0.5 times that of V1E with a lag of Δψ = 0 radians. Signal = 20 onset occurs at 5 x 10⁴ timepoints and continues until end of the run. The location and extent of maximum signal onset effect, ‘max sigoe,’ and mean signal onset effect, ‘extended sigoe,’ are indicated in the figure. See text for how the onset effects are computed. (A) Phase offset of the alpha waves is Δϕ = 0; max sigoe = 24.5, extended sigoe = 8.3, phase coherence between V1E and V2E ρ = 0.64. (B) Phase offset of the alpha waves is Δϕ = π; max sigoe = 15.35, extended sigoe = 3.76, ρ = 0.51.

https://doi.org/10.1371/journal.pcbi.1011431.g001

Fig 2 demonstrates the effects of alpha-gamma phase offset matching on phase coherence of the combined alpha-gamma oscillations, as well as on maximum and extended signal onset effects, over a range of phase offsets of the added alpha-frequency inputs to V1I and V2I, as in [29, 30]. Clearly the maximum of each of these measures occurs when the phase offset of the alpha waves approximately matches the phase offset of the gamma waves, near 0 in Fig 2. As mentioned, these results closely mirror those of [29], but with quasi-cycle oscillations rather than populations of spiking neuron models and sine waves. These results of [30] suggest that it is the oscillations that are essential in this situation, rather than the details of the neuron spiking that underlies the oscillations. The rate model in [30] has no spiking, only oscillations, but retains the results of [29].

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Fig 2. Results of [30] demonstrating the effects of phase offsets of alpha (Δϕ) and gamma (Δψ) oscillations on (C) phase coherence of the combined oscillations, (D) maximum signal onset effect, and (E) mean signal onset effect.

Fig 1 displays representations of these effects. The gamma phase offset was Δψ = 0. Thus, the maximum phase coherence and signal onset effects occur near Δϕ = 0. Reproduced from Fig 4 of [30] with permission.

https://doi.org/10.1371/journal.pcbi.1011431.g002

Another recent study has also implicated phase differences between oscillations as important in information transfer between cortical areas [34]. In a context similar to that of [29], communicating populations of model neurons were simulated for two cortical areas. Here the frequencies of the oscillations in the sending and receiving areas, as well as the time delay (translated to a phase delay) between them, were varied. Again, in this study, an optimal phase difference was discovered; in this case information transmission and coherence were maximum at a phase delay of π between the oscillations in the two areas. In the light of these additional findings, we undertook to elaborate the model of CTC studied in [29, 30], aiming to ascertain both its generality and usefulness as well as to reveal its limitations.

Selective attention via phase offset

Here we describe an elaborated model based on that studied in [29, 30], where attentional selection is controlled by pulvinar input to communicating cortical areas (here assumed to be V1 and V2 in the visual system) as in [22, 35]. The model operates to focus attention on one out of several stimuli that are present at once, whether in a ‘top-down,’ voluntary, fashion or in a ‘bottom-up,’ involuntary, fashion, as attention does in the brain [21]. Fig 3 displays a conceptual picture of this model, which is an elaboration of Fig 2C in [30]. Within Fig 3 are two copies of Fig 2C in [30], with parts labelled X and Y. The two parts appear as separate if the inputs, signals X and Y, and the TRN are removed, although in actual cortex there might be connections between them. The two copies are representatives of the many ‘channels’, or subsystems, in each sensory cortical region, that are tuned to different features of stimulus inputs, including, in vision, e.g., position in space, colour, shape, and so forth. So, in Fig 3, channels X and Y represent groups of neurons in V1 that respond to the features of signals X and Y, respectively. The elaboration consists of the addition of circuitry involving the Thalamic Reticular Nucleus (TRN) and its interaction with the pulvinar generators of 10 Hz oscillations.

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Fig 3. Structural model of top-down and bottom-up attentional selection.

EI pairs are red (excitatory, E) and blue (inhibitory, I) disks; red arrows are excitatory, blue arrows are inhibitory, black arrows represent input signals, or stimuli,—they add increments in neural activity to the processes to which they connect. Top-down signals, green arrows, excite thalamic reticular nucleus (TRN) inhibitory neurons in the same way as do inputs from V1. The (visual) signals excite the V1 neurons that are tuned to them, e.g., vertical (say, signal X) or horizontal (say signal Y) lines on the retina excite V1 neurons tuned to vertical or horizontal lines respectively via tuned neurons in the LGN. See text for more detailed explanation.

https://doi.org/10.1371/journal.pcbi.1011431.g003

The elaborated system displayed in Fig 3 operates in top-down mode as described in [29, 30]. First, as in [29, 30], it is assumed, and implemented by parameter choices, that cortical areas oscillate at approximately 40 Hz, and pulvinar inputs to cortex oscillate at approximately 10 Hz. In the model of [30] both oscillations are produced internally to cortex and pulvinar as noisy quasi-cycles as defined in (1), although other generators of neural oscillations, such as PING models, are not being ruled out. Next, it is assumed that the phase offset, Δψ between V1 and V2 gamma oscillations, arising from the synaptic delay between V1 and V2 neurons, is fixed at the same value for all signals to which V1 neurons are tuned. The phase offset, Δϕ, of the 10 Hz pulvinar inputs to V1 and V2 is fixed at approximately the same value as the phase offset between V1 and V2. Thus, the phase offsets between alpha and gamma oscillations approximately match, meaning, as shown in [29, 30], that there is relatively high coherence and information transmission between V1 and V2 when corresponding areas of V1 and V2 receive 10 Hz oscillations from the pulvinar. The TRN contains only inhibitory neurons and is connected to all of the 10 Hz oscillators in the pulvinar. Thus, the TRN can modulate alpha input to V1 and V2 by inhibiting the output of particular 10 Hz oscillators to V1 and V2. For example, in Fig 3 each of the two signals, X and Y, will excite their respective E processes of C_1X, C_1Y in V1, which in turn excite both their corresponding E processes C_2X, C_2Y in V2, and their opposite inhibitory processes in TRN, TY and TX. The inhibitory processes in TRN inhibit each other as well [36], so there is a competition between incoming stimuli implemented in the TRN. Exciting the inhibitory TRN processes will inhibit activity in the relevant pulvinar neurons, P_1Y and P_2Y for signal X and P_1X and P_2X for signal Y, to the extent permitted by the internal TRN competition and any top-down control signals. Note that when there are many objects in the visual field, as is usual, all of the areas of the TRN corresponding to those objects or locations will be inhibited except for the attended object or location.

In a typical top-down visual attention scenario, there would be a variety of stimuli available in the visual scene, say A,B,C,…X and Y. One of these, at some location and extent in the visual field, would be selected based on salience given by current goals, previous interactions, memory, etc. Top-down control signals implementing ‘voluntary’ attentional selection (re these goals, search targets, etc.) would affect the TRN neurons by strongly exciting those TRN neurons connected to cortical neurons tuned to stimuli other than the desired stimulus. In Fig 3 the to-be-attended stimulus initially is signal Y (via solid green arrow). Thus, the TRN would inhibit the 10 Hz pulvinar oscillations from going to those (unselected) tuned neurons in V1, C_1X, that are responding to signal X (and also any other stimuli in the visual field). This would allow the 10 Hz and 40 Hz phase-offset match to induce optimal coherence and signal transmission between C_1Y and C_2Y in V1 and V2 for the selected signal Y. If signal Y is selected in this way, then phase coherence and signal transmission would remain suboptimal for signal X and any other signals because of lack of 10 Hz input, and signal Y would be ‘attended.’

There is a nuance that should be mentioned here: when visual objects possessing a particular feature, e.g., colour red, are to be attended but are intermixed spatially with other objects with a different feature, say green, attention appears to operate by inhibiting the distractors rather than by exciting the targets [37]. The simple model of Fig 3 does not address this situation, although the basic elements of the inhibitory TRN, spatial maps of simple feature locations in cortex (e.g., a retinotopic map of colours—[38]), and connections from cortex to TRN are present. This would allow multiple locations to be excited in TRN, thus inhibiting multiple locations in pulvinar, and reducing input levels for the inhibited locations. If the features are not simple and represented in multiple locations in a retinotopic map, however, but rather are complex parts of objects distributed over spatial locations, a more sophisticated version of our system must be called upon. Presumably the relevant representations of these features are located in higher areas of the visual system, and thus attending to them could be a somewhat different process, although still possibly mediated by alpha-frequency oscillations [28].

Competition between TRN neurons would mediate between signals of different strengths and top-down selected signals for precedence, as well as between signals in different sensory modalities (and primary sensory cortical areas). Top-down attention can overcome some differences in signal strength as depicted in our example (signal X arrow is thicker, e.g., [39]). Some signals, however, are simply too strong, e.g., a very bright flash of light. Let us assume that signal X suddenly flashes brightly, indicated by the X line being dashed. In this case the the bottom-up, ‘involuntary,’ or ‘automatic,’ mechanism might pull attention away from whatever top-down signal was being selected at the time [21]. We assume that, as in behavioural studies [21], this pulling of attention to a new stimulus typically is followed by reorienting top-down attention to the new stimulus until its relevance is ascertained. This is not a necessary action, however, and bottom-up attentional enhancement effects are typically observed substantially earlier than in the case of top-down orienting, around 100-200 ms after such a flash, indicating that the orientation of top-down attention is not required to produce the attention effect. Even though attention currently is being paid to signal Y through the top-down mechanism just described, the brightening of signal X will have several effects. First, it will produce a very strong signal onset effect, in spite of the fact that no pulvinar 10 Hz oscillation is being sent there because of the allocation of top-down attention to signal Y (see Results for a supporting simulation result). This will be transmitted to executive areas of the brain responsible for sending signals to implement top-down attention. Also, because the very strong signal X wins the competition between the TRN neurons, and this will be reinforced after some time by a top-down signal exciting TY (indicated by the dashed green arrow), 10 Hz input from P_1Y and P_2Y to C₁Y and C₂Y will be terminated. Thus, 10 Hz oscillations from P_1X and P_2X now will be injected into the 40 Hz oscillating C₁X and C₂X in V1 and V2, respectively. The roughly equal phase offsets between 10 Hz and 40 Hz oscillations for signal X now will optimize the phase coherence and signal transmission between V1 and V2 for signal X, as described in [30]. Because the 10 Hz input from P_1Y and P_2Y to C₁Y and C₂Y has been blocked by TY, however, the coherence and signal transmission between relevant cortical areas become lower (but not 0) for signal Y than for signal X. The large signal onset effect of signal X at V2 (and likely further up the visual system), is presumed to be what it means for signal X, the very strong signal, to ‘pull,’ or ‘draw,’ attention to itself in a bottom-up manner. There is a wealth of evidence that this effect is ubiquitous, both in the brain and in behaviour [21].

Our contribution here is the above description of a possible mechanism for the interaction between the pulvinar, the TRN, and the cortex, to select an object of attention. The question arises: to what extent is this mechanism dependant upon oscillations? Our conception of an implementation is in terms of quasi-cycles, which we now introduce.

Computational implementation of the model

As in [30], here we use an EI rate model that produces quasi-cycle oscillators that are described in detail in [40] based on a model of [41]: (1) where (2) (3)

This model is a description of the time evolution of the local field potentials (LFPs), closely related to firing rate, of two coupled populations of neurons, excitatory, V_E(t), and inhibitory, V_I(t). The V_E and V_I processes defined by (1), (2) and (3) will be called the E and I processes. We refer to a stochastic process defined by a model of the form (1) as an EI pair. The parameters in represent the synaptic coupling strengths, S_ij, and time constants, τ_i, between the excitatory and inhibitory neuron populations, σ_i are noise amplitudes, and W_i are standard Wiener noise processes. In this model, when the eigenvalues of are complex, −λ ± iω, with 0 < λ ≪ ω, the system (1) has sustained (noisy) oscillations, called quasi-cycles, distributed in a narrow band around ω. The parameters in can be adjusted to give a wide range of quasi-oscillation frequencies within the range relevant to brain activity related to mental processes [40]. The parameter values used in to produce approximately 40 Hz oscillations in the EI pairs are listed in Table 1. The left hand side of the diagram in Fig 3, without the TRN or signals (or, similarly, the right hand side by itself), was the subject of [30]. Simulations in [30] used (1) to produce independent quasicycles of central frequency gamma in the EI pairs denoted C_1X, C_2X, and of central frequency alpha in the EI pairs denoted P_1X, P_2X in Fig 3.

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Table 1. Parameters of used in 40 Hz EI pair simulations.

https://doi.org/10.1371/journal.pcbi.1011431.t001

In this paper we have opted to implement simulations of the components of the elaborated model in the simplest possible way, given that their main effects have already been demonstrated in both populations of individual spiking neuron models [29] and of generic rate models like (1) [30]. Both types of models emphasize periodic alpha and gamma oscillations, although the population model of [29]exhibits the oscillations in bursts of spiking organized by a sinusoidal input, whereas the rate model studied in [30] and here exhibits interacting quasi-cycles. The process defined by (1) can be factored into a rotation (sinusoid) multiplied by a 2D Ornstein-Uhlenbeck process [33], and so is essentially a noisy sinusoid [33].

Each of the two signal channels in the model of Fig 3 is computed as follows. The channels of the cortical processes (V1 and V2) are modelled as (4) (5) where is the matrix of coefficients that generates 40 Hz oscillations, damped but sustained by noise, are 2D vector processes, (C_1E, C_1I)′, (C_2E, C_2I)′, for EI pairs, as in (1), (2), (3), oscillating at natural frequency 40 Hz but with , , added incrementally, and where b₁, b₂ are the amplitudes of the alpha sine waves injected into C_1I, C_2I. Further, represents the input to C_2E from C_1E delayed by Δψ, where Δψ is the phase difference between C_1E, C_2E caused by adding a fraction, c₂₁, of C_1E to C_2E at a time delay that implements the phase offset between V1 and V2. The signal in (4) is modelled as a constant input, i.e., a step function, as in [29, 30], except where indicated. Although common to both in vivo and in vitro studies, this way of introducing a signal to the model is problematic because LGN input to V1 actually consists of structured bursts of spikes [42]. This point will be discussed in Sensitivities and Limitations.

As we demonstrated in [30], modelling the pulvinar 10 Hz inputs as noisy sine waves or as EI pair quasi-cycles gives rise to similar results. Here we use noisy sine waves for simplicity and to enable the most precise phase offset of the alpha inputs. The inputs from the noisy sine waves oscillating at approximately 10 Hz are: (6) (7) where M_α is the amplitude of the alpha sine waves and Δϕ is the α₁ to α₂ phase difference. The white noise, S_α(t), added to the alpha sine waves has standard deviation s_α.

Sample paths of processes , defined by Eqs (4) and (5) were simulated using the Euler-Maruyama algorithm, with the current values of the noisy 10 Hz sine waves added to the rhs of the SDE at each time point.

Table 2, which is the main focus of this paper, displays the results of a large number of simulations with strategically chosen representative values for the parameters of Eqs (4)–(7). The simulations were run for 100,000 time points at 0.00005 sec per time point, so for 5 sec. A signal was added at time point 50,000 and remained on until time point 100,000. As in [30], phase coherence, ρ, was calculated as , [31], where θ is the instantaneous phase of the V1 or V2 oscillation computed from the analytic signal via the Hilbert transform, and N = number of time points over which ρ is computed. Here N = 100,000 time points. Signal onset effects, as in [29, 30], were measured as (1) the maximum (max) of the positive part of the oscillation in V2 minus that in V1 between time points 50,000 and 55,000 (250 ms after signal onset), and (2) the mean positive part of the oscillation over the interval max-1000 and max+1000 timepoints (50 ms).

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Table 2. Results of simulations of model in Fig 1 using selected values of the parameters appearing in Eqs (4)–(7).

Parameter values (Δϕ, Δψ in radians) are in the leftmost seven columns, output values in the rightmost four columns. “sig noise” is signal noise; “max sigoe,” the maximum signal onset effect, is the maximum of the positive part of the oscillation at V2 minus that at V1, i.e., max(V₂₊ − V₁₊), between signal onset and signal onset + 250 ms; ‘extended sigoe,’ the extended signal onset effect, is the mean of the positive part of the oscillation between time at maximum −50 ms and time at maximum + 50 ms at V2 minus that at V1; MI is mutual information between the phases of V1 and V2 after signal onset until the end of the run. All output values in the table, ρ, max sigoe, extended sigoe, and MI, are the means of 10 realizations, with standard errors approximately 0.01, 0.95, 0.32, and 0.02 resp.

https://doi.org/10.1371/journal.pcbi.1011431.t002

Mutual information, MI, was computed between V1 and V2 using a Matlab function [43] as (8) where H(V2) is the entropy of V2, and H(V2|V1) is the entropy of V2 conditional on V1. MI was computed separately for the interval before signal onset and for the interval after signal onset, for the raw time series of V1 and V2, and also for their phases and amplitudes. Phases and amplitudes were computed from the analytic signal and the Hilbert transformed time series. As the results for the various MI computations were similar, only those for phase-based MI for the interval after the signal onset are reported here.

Attentional selection mechanism

The first seven rows of Table 2 further develop the results of [30], and illustrate, for selected parameter values, how the model of Fig 3 works for the bottom-up and top-down attentional selection scenarios. Row 1 is an example where a strong signal, X = 20, wins the competition in the TRN and thus prevents α oscillations from the pulvinar from reaching V1 and V2 neurons tuned to the weaker signal, Y = 10. For the stronger signal, α oscillations at the matching phase offset, 0 in this case, would reach the areas of V1 and V2 that respond to that signal, causing a fairly large phase coherence value mean ρ = 0.69 (mean over 10 realizations), a strong signal onset enhancement of ‘max sigoe’ = 17.6, and also a strong extended signal enhancement effect of ‘extended sigoe’ = 6.5. These values are consistent with those for the matching phase offset case in [30] (see Fig 1 for an illustrative single realization and Fig 2 for average results over several different phase offsets).

For the weaker signal, Y, however, no α oscillations from the pulvinar reach V1 or V2 because of inhibition from the TRN. This situation is portrayed in Row 2 of Table 2. There, all but the signal strength being the same, V1-V2 phase coherence, ρ is substantially lower, as are the maximum and extended signal enhancement effects, consistent with both [29] and with [30]. Mutual information, ‘MI,’ between V1 and V2 during the signal period is also lower when there are no α oscillations sent to V1 and V2 for signal Y. Making Δϕ = π ≠ Δψ = 0, on the other hand, even with the fairly strong signal X = 20, reduces both ρ and MI and the extended signal enhancement effect (Row 5), similar to results of [29, 30] (see Figs 1 and 2). Note that reducing the magnitude, M_α, of the α oscillations to 2 from 6, as in Row 7, has only a small effect on ρ and mutual information, and no effect on signal enhancement relative to Row 1 values. So at least some input from the pulvinar would seem to be sufficient for the attentional effects we see.

Row 3, compared with Row 1 of Table 2, shows the effects of top-down attention selecting one of two signals. If the two signals have equal magnitude, then the selected one (Row 1) still shows greater phase coherence, signal onset effects, and mutual information than does the unselected one (Row 3), again because the TRN has been signalled to inhibit α coming from the area of the pulvinar connecting to the unselected neurons in V1 and V2.

On the other hand, if a very strong signal, say X = 30, suddenly appears, attention can be pulled away from its current focus by very large signal onset effects, as in Row 4, despite the fact that, at the moment the strong signal appears, no pulvinar 10 Hz oscillations are going to the neurons tuned to the strong signal. Subsequently, however, 10 Hz oscillations would be sent to those neurons because of the mutual effects of the strong signal and top-down signal exciting TY, thus inhibiting P_1Y, P_2Y, and releasing P_1X, P_2X from inhibition by TX.

Thus, our elaborated model uses the same basic mechanism, TRN inhibition of pulvinar α oscillations, to accomplish both top-down and bottom up attentional selection. In the next section we will see that the results displayed in Table 2, Rows 6-18, and those in Figs 4–6, point to a number of important sensitivities and limitations of this model.

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Fig 4. Results of simulations of model in Fig 3 using selected values of the parameters appearing in Eqs (4)–(7).

The graphs depict the effects of cortical connectivity and cortical noise on the four outcome variables. Each point represents the average of 10 realizations. The standard errors are approximately the size of the data points or smaller and thus are not displayed. Parameter values other than the one changed are those in Row 1 of Table 2. The arrow points to an outcome listed in Table 2 for the indicated value of the changing parameter. MI = mutual information.

https://doi.org/10.1371/journal.pcbi.1011431.g004

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Fig 5. Results of simulations of model in Fig 3 using selected values of the parameters appearing in Eqs (4)—(7).

The graphs depict the effects of signal amplitude and signal noise on the four outcome variables. Each point represents the average of 10 realizations. The standard errors are approximately the size of the data points or smaller and thus are not displayed. Parameter values other than the one changed are those in Row 1 of Table 2. The arrow points to an outcome listed in Table 2 for the indicated value of the changing parameter. MI = mutual information.

https://doi.org/10.1371/journal.pcbi.1011431.g005

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Fig 6. Results of single realizations of simulations with gamma oscillations only in the V1 process.

The V2E process is comprised of random noise plus 0.5 times that of V1E with a lag of Δψ = 0 radians (9), whereas the V2I process is comprised of random noise only (10). Note that V2E and V2I are not connected in these simulations. Signal = 20 onset occurs at 5 x 10⁴ timepoints and continues until end of the run. (A) No alpha input to V1 or to V2 processes: max sigoe = -36.8, extended sigoe = -5.7, phase coherence between V1E and V2E ρ = 0.8, MI = 2.8. (B) Alpha input only to V1I process: max sigoe = -35.4, extended sigoe = -7.3, ρ = 0.76, MI = 1.9.

https://doi.org/10.1371/journal.pcbi.1011431.g006

Sensitivities and limitations

In evaluating a model such as the one studied here, one is interested in its behaviour not only at the optimal parameter values, but also elsewhere in the parameter space. A useful model will be sensitive to parameter changes in a way that mirrors the behaviour of the phenomenon itself, while not oversensitive to small changes to which the phenomenon would be robust. In our present case these questions present difficult challenges. To begin to meet these challenges and further explore the parameter space of the model studied here, we extended the simulations illustrated in rows 6-18 of Table 2 and in Figs 4–6. Not surprisingly, the “success” of any model depends on the balance of several effects. When balance fails, pathologies are known to arise. Some of the sensitivities we are about to discuss could point to a source of pathologies in attention, or in neural transmission more generally, or they could simply arise from limitations of the model.

As we have shown, the elaborated model of selective attention can account for some basic attentional phenomena, in particular the association of phase coherence with information transmission between neural areas, as in the CTC hypothesis. There are, however, important limitations to the model, which we will illustrate using Table 2 and Figs 4–6. Some are obvious: we are modelling at a macro scale, our EI equations are rate approximations, parameter values are not derived from empirical data, the details of both cortical and thalamic (TRN and pulvinar) connectivity are missing, and so forth. Importantly, it seems we and others (e.g., [29]) have found a sweet spot in parameter space within which the indicated mechanism operates. But parameter values outside that sweet spot yield a somewhat different picture. For example, the connectivity between V1 and V2 is critical. Reducing the connectivity from 0.5 to 0.1, as in Row 6 of Table 2 and Fig 4, yields moderate phase coherence and unchanged mutual information, but negative signal onset enhancement. Increasing the V1-V2 connectivity while leaving everything else unchanged, as in Row 8 and Fig 4, leads to values of ρ and MI comparable to those for Row 1, but accompanied by much larger values for signal enhancement. Finally, increasing the amplitude of the γ quasi-cycles by a factor of 10, to 0.3, by increasing the noise in the SDEs for V1E and V2E, as in Row 12 and Fig 4, leads to a relatively small value for ρ and MI, but a very large value for signal enhancement. Abolishing the α input under those conditions, Row 13, gives rise to a large value of signal enhancement but still accompanied by relatively small ρ and MI. In both of these latter cases, the centroid of the distribution of phase differences between V1 and V2 was shifted by the noise to -2 radians, not matched to the 0 phase offset of the α frequency. These values are clearly inconsistent with CTC. They are also inconsistent with the idea that phase offset-matching of 10 Hz input to cortex is the most important factor in affecting signal transmission between cortical areas.

Another aspect needing attention in both our and other models is the question of how input signals (stimuli) should be modelled. Some authors, e.g., [29, 30], have used a constant, i.e., step function, signal. Others have used a burst of Poisson or Gaussian pulses (basically, noise), and still others have used a constant plus some noise. Some, e.g., [44], have used a signal modelled on actual output from the LGN as input to V1. It does appear, however, that using as input the output from a noisy integrate and fire neuron (with a constant input to the model neuron) provides a good approximation of LGN output statistics, and has been shown to give better orientation selectivity than does LGN output modelled as an inhomogeneous Poisson point process [42]. Use of a constant as our input signal represents another limitation of our and others’ model results. In most of our simulations we have used a step function as a signal, as in [29]. When we increase the amplitude of this step function, as in Fig 5, we find an increase in the signal onset effects as well as in phase coherence, ρ, and in mutual information. When we input a noisy signal, defined as a constant plus Gaussian noise (called ‘signal noise’), to our EI modelled V1, as in Row 14 of Table 2 and Fig 5, we see little difference in signal onset effects, ρ, or MI compared to those in Row 1, but a decrease in signal onset effects and mutual information when the signal noise reaches higher levels. Without α input however, Row 15, these effects are only slightly reduced. So noise, both cortical and in the signal, if present, is apparently playing a role in the signal enhancement effect. Note that both our quasi-cycle model, here and in [30], and the neuron population model of [29], involved noisy (stochastic) processes.

Conclusions

We have elaborated a model of attentional selection in which alpha oscillations generated in the pulvinar nucleus of the thalamus organize gamma oscillations in cortical areas so that they are more coherent than without the alpha input. Thus, they transfer information about signals more efficiently than without that input. This model describes how an attention system can select a particular stimulus to emphasize, either through a top-down signal from other cortical areas, or by preferring more salient stimuli in a bottom-up fashion. There are significant limitations to the model as it exists, however. In particular, the model works well only in a limited parameter range, so it would be very sensitive to physiological conditions. Moreover, there is no consensus on how to characterize stimuli (or signals) in such models. Some researchers simply add a constant to the input, as is often done in physiological studies in vitro. Others, including us, have argued that signals should be inputs that increase the amplitude of the LFP, for example outputs of leaky integrate and fire neurons [42].

The relation between coherence and communication is not as simple as it has seemed from the CTC hypothesis. First, through input from a third area (e.g., the pulvinar), coherence can be increased even though there is no direct communication at all between the monitored areas. This can fool data analysis that only focuses on, e.g., phase coherence between pairs of cortical areas. Second, coherence between cortical areas can be increased simply by one area sending spikes to another: coherence through communication (CTC′). Third, it is not clear how to measure communication in these models or in experimental data. Some studies measure information transmission by focusing on the onset of the stimuli, whereas others focus on the entire time the stimulus is present, with somewhat different results.

Finally, it is becoming clear that alpha oscillations could perform many roles in the brain, from attention selection to timing to long-range coordination. It is important that these roles and their individual circumstances be clarified. Future modelling and empirical work might be able to determine when and how the different suggested mechanisms perform. But it will be important to remember the scope and limitations of these mechanisms. It is also important to determine whether the ubiquitous neural oscillations play a dynamic and/or causal role in any of these mechanisms, or simply arise epiphenomonally from the spiking of neurons as they transmit information throughout the brain.