To model the period between cue and target presentation in the Egly-Driver task (Fiebelkorn et al., 2013), we introduced simulated mdPul input—consisting of bursts of α frequency (12 Hz) excitation arriving during half of each period of a (4 Hz) θ rhythm—targeting LIP granular layer RS and FS cells, and FEF visuomotor RS cells. Under these conditions, the model reproduced the nested rhythmic dynamics observed experimentally (Figure 2B): during mdPul excitation (the putative good θ phase), FEF visuomotor cells produced a β 2 rhythm, and LIP produced a γ rhythm; during mdPul quiescence (the putative poor θ phase), LIP produced a β 1 rhythm. Note that in the absence of additional visual input to FEF visual cells (i.e. when no target was present on screen), these cells and FEF decision cells (not shown in Figure 2) were relatively quiet. Below, we explain the circuit mechanisms of these rhythms and their interactions in detail for each module.

The LIP module (Figure 3A) combined two previous laminar models of working memory and attention (Kramer et al., 2008; Lee and Whittington, 2013). The first of these was a model of a β 1 rhythm observed in rodent parietal slices (Kramer et al., 2008), shown to be useful for retention and manipulation of cell assemblies after the fading of a sensory input (Gelastopoulos et al., 2019). The model mimics experimental data (Roopun et al., 2008) in which adequate stimulation (sensory in vivo, excitatory agonist kainate in vitro) produced a γ rhythm in the superficial layers and an independent β 2 (∼ 25 Hz) rhythm in the deep layers. Following a sufficient period of kainate application in vitro, the removal of kainate (modeling the disappearance of the sensory signal) caused both deep and superficial layers to switch to a β 1 (∼ 15 Hz) rhythm (Roopun et al., 2008). A single period of this β 1 rhythm concatenated one γ period and one β 2 period (Roopun et al., 2008; Kramer et al., 2008), so that the duration of the β 1 period was the sum of the γ and β 2 periods.

Both the γ period and the β 2 period comprising the β 1 are initiated by rebound from inhibition; this allows the β 1 to remain active even at low levels of excitatory input (Kramer et al., 2008). As a consequence of the unusual physiology producing it, this rhythm exhibits unique functional properties: a superficial γ-rhythmic cell assembly excited by an initial stimulus maintains persistent activity as part of a β 1 rhythm after the cessation of the initial stimulus; and multiple cell assemblies can be added and subtracted from this ‘working memory buffer’ without interference (Gelastopoulos et al., 2019). Thus, this model is especially relevant to our task, which involves maintaining the cue location in working memory for up to 1.6 ⁢ s , until target appearance.

Of note, the prior excitation required to produce the β 1 rhythm experimentally did not change the β 2 or γ rhythms during the period of excitation, but was hypothesized to produce plasticity altering interlaminar interactions. Such plasticity would occur in vivo either more rapidly during stimulation or in response to repeated stimulation. In computational models (Kramer et al., 2008; Gelastopoulos et al., 2019), the parameters were set as if this plasticity had already occurred; thus the shift from independent γ and β 2 rhythms to a β 1 rhythm required only a change model inputs.

The second inspiration for the LIP module was a model of the effects of (attentional) cholinergic modulation on the potentiation of γ power in the superficial layers of sensory cortex (Lee and Whittington, 2013). In this model, the excitation-inhibition balance in the granular layers is such that FS cells dominate RS cells, largely damping RS cell responses to excitatory input. In the presence of cholinergic modulation, β rhythms delivered to deep layers activate SOM interneurons, which in turn produce slowly decaying inhibition that suppresses granular layer FS cells. This results in disinhibition of granular layer RS cells, changing the excitation-inhibition balance sufficiently that RS cells drive FS cells to produce a γ rhythm. This input layer γ rhythm forms from the interaction of RS and FS cells, dependent on the FS cells receiving slow inhibition from deep layer SOM interneurons. In turn, granular γ rhythms drive γ rhythmicity in the superficial layers.

As in the working memory model, the circuitry of our LIP module produced a β 1 ‘concatenation’ rhythm in the absence of input from mdPul or FEF (i.e. in the poor θ phase, Figure 3B–D). We interpreted this β 1 as a ‘post-excitation’ rhythm, since in our model it always occurred following excitation—both that of cue presentation and that of the (prior) good θ phase. As in previous models, the excitation and the resulting plasticity were absorbed into the parameters of the model, which are set to the ‘post-excitation’ state throughout the simulation.

As in the previous model of attention (Lee and Whittington, 2013), β 2 input to deep SOM interneurons in our LIP module partially suppressed granular FS cells, leading to disinhibition of granular RS cells. (Granular FS cells remained active, but came under the control of granular RS cells.) When this disinhibition was combined with increased excitatory input to the granular layer, a γ rhythm resulted, which then propagated to the superficial layer. During the good θ phase, the LIP module received exactly these two inputs—a β 2 input from FEF exciting deep SOM interneurons, and an α input from mdPul exciting the granular layer (Figure 3A). Thus, as in the previous model (Lee and Whittington, 2013), FEF beta input to the deep layers of LIP created the potentiated γ in the superficial layers of LIP during the good θ phase (Figure 3B and C, E; also see Robustness and Flexibility of the LIP Module).

The α-rhythmic input that excites the RS and FS cells of the LIP module’s granular layer differed from the tonic input used previously (Lee and Whittington, 2013), but the result was the same. We modeled mdPul input excitation as a series of AMPA-like excitatory pulses arriving at α frequency. Since mdPul neurons are unlikely to fire in complete synchrony, the rise and decay time of the input synapses to LIP was increased (from t r ⁢ i ⁢ s ⁢ e = 0.1 ⁢ m ⁢ s and t d ⁢ e ⁢ c ⁢ a ⁢ y = 0.5 ⁢ m ⁢ s in AMPA synapses within the network, to t r ⁢ i ⁢ s ⁢ e = 2 ⁢ m ⁢ s and t d ⁢ e ⁢ c ⁢ a ⁢ y = 10 ⁢ m ⁢ s in input synapses, see Appendix 2—figure 2A ), although we found the superficial γ (but not its frequency) to be robust to changes in input synapse rise and decay times (Appendix 2—figure 2; see Robustness and Flexibility of the LIP Module for more details on the parameter-dependence of the superficial γ rhythm).

The FEF visuomotor module consisted of RS cells reciprocally inhibited by SOM cells (Figure 4A). (FEF visuomotor module decision cells were active only when a target was present (see Mechanisms of Target Detection), and thus did not play a role in model behavior during the cue-target delay.) RS and SOM cells received excitatory inputs from LIP, and RS cells received excitatory inputs from mdPul. In line with experimental data (Fiebelkorn et al., 2019c) (and the behavior of our LIP module), synthetic LIP inputs to the FEF visuomotor module were γ frequency during the good θ phase and β 1 frequency during the poor θ phase. As in the LIP module, mdPul inputs at α frequency arrived only during the good θ phase. Excitatory noise, representing asynchronous inputs, also arrived from mdPul during the poor θ phase.

During the poor θ phase, the excitation-inhibition (E-I) balance between SOM and RS cells was such that the SOM cells (activated by tonic excitation) silenced the RS cells. During the good θ phase, RS cells, stimulated by mdPul, fired and took control of SOM cell activity. The timescale of SOM inhibition determined the frequency of the resulting RS-SOM rhythm, a β 2 . This module exhibited a motif similar to that which created a γ in LIP: when SOM interneurons were too active, rhythms did not appear; when RS cells were stimulated, an altered E-I balance allowed rhythms (driven by RS cell firing) to emerge. Slow-decaying pulvinar stimulation (representing an input relying on a combination of NMDA and AMPA receptors) enabled up to two β 2 cycles to take place after each mdPul excitatory impulse. In the poor θ phase, the asynchronous input from pulvinar elicited a few spikes from RS cells, but this RS cell activity was not strong enough to initiate β 2 oscillations. Figure 4B shows a raster plot from a two second-long simulation of the isolated FEF visuomotor cell module, and a plot of its spectral power as a function of θ phase. Section Robustness and Flexibility of the FEF Visuomotor Module explores the dependence of FEF visuomotor module dynamics on model inputs and parameters.

We varied the functional connectivity from LIP to the FEF visual module by altering the strength of the chemical synapses from superficial LIP RS cells to FEF visual cells. Increasing LIP→FEF connectivity increased both hit rates (Figure 7A (i)) and false alarm rates (Figure 7A (ii)). (We considered a trial to contain a ‘false alarm’ whenever more than half of the decision cells fired in any 25 ⁢ m ⁢ s -long window outside target presentation.) Computing the D’ sensitivity index, which combines both hit and false alarm rates, revealed that intermediate values of LIP→FEF connectivity resulted in the highest levels of task performance (i.e. both high hit rates and low false alarm rates, Figure 7A (iii)).

The LIP→FEF connectivity was also related to the 8 Hz rhythmicity observed in the dependence of the hit rate on the cue-target interval. For sufficiently high values of the conductance of LIP→FEF chemical synapses, the temporal profile of hit rates exhibited a dominant 8 Hz component (Figure 7B), while for small values of the LIP→FEF conductance, the hit rates exhibited robust 4, 8, and 12 Hz periodicity. Varying LIP→FEF conductance revealed that, while a hit rate peak at the beginning of the good θ phase (at 0 degrees) was common across 4 and 8 Hz hit rate rhythms, 8 Hz rhythms resulted from the addition of another, smaller peak in hit rate at the beginning of the poor θ phase (Figure 7B (i) & (ii)). This was a result of LIP superficial RS spikes participating in β 1 oscillations, which excited FEF visual neurons, and rescued target detection in the poor θ phase. Plotting the number of decision cell spikes during target presentation as a function of the phase of the LIP β 1 oscillation in the poor θ phase revealed a stronger influence of LIP β 1 phase on decision cell spiking for a medium than for a low value of the LIP→FEF synaptic conductance (Appendix 1—figure 2A). The LIP→FEF synaptic conductance also influenced the dependence of the hit rate on the LIP β 1 phase (Appendix 1—figure 2B).

The dominance of the 8 Hz rhythm in hit rate for high LIP→FEF connectivity was also dependent on the shape of mdPul input to LIP and FEF. When mdPul input had the same maximal conductance across the entire good θ phase, the power of the 8 Hz component to the hit rate rhythm was much lower, and it was comparable in size to the 4 Hz component in the hit rate rhythm for all values of LIP→FEF connectivity (Appendix 1—figure 3A). An increase in 8 Hz power with the strength of LIP→FEF connectivity was still observed, however (Appendix 1—figure 3A). Notwithstanding this difference in the dominance of 8 Hz rhythmicity, the dependence of network sensitivity on LIP→FEF connectivity (and in particular the value of g L ⁢ I ⁢ P → F ⁢ E ⁢ F resulting in peak sensitivity) remained unchanged for constant mdPul input (Appendix 1—figure 3B).

In behavioral data (Fiebelkorn et al., 2013), an 8 Hz rhythm in hit rate was observed for trials in which the target and cue appeared within the same object, while a 4 Hz rhythm in hit rate was observed for trials in which the target and cue appeared within different objects. Our model is consistent with the hypothesis that the deployment of attention towards a cued object moderately increases the strength of LIP→FEF connectivity in neural populations representing that object, resulting in both an increase in performance (as reflected by an increase in D’) and a change in the rhythmicity of behavior.

One question that naturally arose from the previous results was whether the frequency of LIP and FEF rhythms is significant for task performance. To answer it, we modified the synaptic time constants of inhibition in FS cells and SOM cells ( τ d,FS and τ d,SOM , respectively), varying the frequencies of inhibition-paced rhythms across the β and γ ranges. We then analyzed the temporal profile of hit rates as a function of the cue-target interval, as in the previous section.

We found the sensitivity of target detection (as measured by D’) was highest for τ d,SOM ≃ 20 ms, corresponding to observed β 2 frequencies of ∼30 Hz, and for τ d,FS ≃ 5 ms, corresponding to observed γ frequencies of 40 Hz and above (Figure 8A). The peak in sensitivity at these time constants resulted from shifts in both hit rates and false alarm rates (Figure 8B & C). We describe the mechanisms of these changes in detail below. A key point is that the activation of FEF visual module SOM cells at a γ timescale, that is, faster than their β time constant of inhibition, provides a blanket of inhibition that prevents VIP cells from spiking outside of target presentation and reduces false alarms in the network during the good θ phase.

As shown in the previous sections, our model reproduced both the neural dynamics and the behavioral results observed during rhythmic sampling in spatial attention (Fiebelkorn et al., 2019c). In order to be relevant to the brain, our model must be somewhat robust to parameter changes. However, since LIP and FEF are involved in cognitive tasks other than rhythmic attention, it should also be flexible enough to allow different rhythms to appear as conditions are varied. This section will address both these issues by looking at the conditions necessary for oscillations and target detection in the different submodules of the model.

Similarly to LIP, the activity of the FEF visuomotor module depended on rhythmic input received from LIP and mdPul, as well as on tonic excitation to its RS and SOM cells. Pulvinar input was required for the network to achieve a β 2 rhythm during the good θ phase—without mdPul input, RS cells spiked only once during each good θ phase (Appendix 3—figure 1A). The frequency of mdPul input also played a role in generating sustained β 2 oscillations: input at θ or slower frequencies did not provide enough excitation to RS cells for them to take control of SOM cells during the entire good θ phase (Appendix 3—figure 1B). Model behavior was preserved when mdPul input arrived at frequencies faster than α, but not when the width of mdPul excitatory pulses was varied significantly (Appendix 3—figure 1B, C). The frequency of the alternation of the good and poor θ phases was required to be less than half of mdPul α frequency for the network to achieve a β 2 rhythm in the good θ phase (Appendix 3—figure 1D).

LIP input to RS and SOM cells also influenced the activity of the FEF visuomotor module. Simply removing this input decreased SOM inhibition of RS cells, causing them to fire at a rhythm faster than the β 2 experimentally observed during the good θ phase, and also during the poor θ phase (Appendix 3—figure 2A). For the same reason, LIP input at a frequency in the α range or lower led to FEF visuomotor cells exhibiting a faster rhythm in the good θ phase (Appendix 3—figure 2B). LIP input at a β 2 or γ frequency was required to achieve the desired β 2 frequency in FEF visuomotor RS cells. Increasing the frequency of LIP input above the γ range caused FEF visuomotor RS cells to fire at a frequency faster than β 2 , provided that the input was strong enough. Because FEF visuomotor RS cell spiking could occur even if LIP was still producing the β 1 rhythm characteristic of its poor θ phase dynamics, our model suggests that the transition from the poor to the good θ phase is likely initiated by FEF, which then recruits LIP in a top-down fashion.

Even at experimentally observed frequencies, the strength of LIP input influenced FEF visuomotor rhythms (Appendix 3—figure 2C). A too-weak input did not provide enough excitation to FEF SOM cells to silence FEF during the poor θ phase, and resulted in FEF to producing a rhythm faster than β 2 during the good θ phase, while with a too-strong input FEF RS cell spiking followed the LIP input rhythm (at a β 1 frequency during the poor θ phase, and a γ frequency in the good θ phase). Importantly, although this behavior is undesirable for the rhythmic attention task we are modeling here, the activation of the FEF visuomotor module by LIP input in the absence of mdPul stimulation may represent a bottom-up attentional state, in which the network responds to the appearance of a highly salient target in a bottom-up fashion.

Finally, just as LIP activity depended on the tonic excitation given to its input layer RS and FS cells, FEF visuomotor activity depended on the tonic excitation given to its RS and SOM cells (Appendix 3—figure 3).

While simpler versions of the FEF visual module were able to detect changes in contrast, they exhibited some disadvantages that motivated our modeling choices for this module (Figure 12). A model including only RS and FS cells (Figure 12A) exhibited a γ burst during target presentation in response to strong-enough excitation of RS cells, even without the disinhibition provided by VIP cells. However, since RS cells have a larger capacitance than VIP cells, the strength of the target input had to be at least 1.5 times larger than the background stimulation from LIP to result in γ rhythmicity (Figure 12A (ii) & (iii)), which we interpreted as a high contrast target. Thus, the VIP and SOM cell circuit in the visual module played an important role in the detection of near-threshold contrast changes in our simulations.

In contrast, simulations of a model including only one interconnected network of RS, FS, VIP, and SOM cells (as opposed to two clusters of cells with different inter- and intra-cluster connectivity) were unable in our hands to satisfactorily exhibit robust target detection (Figure 12B). Robust target detection depended on VIP cells being silenced by SOM cells when no target was present, and SOM cells being silenced by VIP cells during target presentation, during both the good and the poor θ phase (that is, with background inputs from LIP in the γ or β 1 range). We were unable to achieve this behavior for any of the conductances of VIP→SOM inhibitory synapses or levels of tonic excitation to VIP and SOM cells tested. An example of an unwanted behavior in a model without clusters is shown in Figure 12B (ii) & (iii), where the inhibition from SOM to VIP was strong enough to ensure VIP cells didn’t spike outside target presentation in the good θ phase (that is, when LIP input was in the γ range and SOM cells are also active at a γ rhythm), but not in the poor θ phase (when LIP input and SOM cell spiking were at β 1 frequency). Increasing the inhibition from VIP to SOM led to an inability of VIP cells to spike at target presentation. Separating SOM and VIP cells into two clusters with differential intra- and inter-cluster connectivity enabled a finer tuning of the inhibition between them (in particular, VIP cells received inhibition from only half of the SOM population, rather than all-or-nothing SOM inhibition), which helped achieve the dynamics necessary for task performance under a variety of different constraints.

The rhythms in our network did not operate independently. The FEF visuomotor β 2 oscillation—activated by pulvinar excitation of FEF visuomotor cells in the good θ phase—elicited γ oscillations from LIP by effectively disinhibiting LIP granular layer pyramidal cells. LIP γ oscillations in turn acted to enhance sensory processing in the network by stimulating FEF visual cells, making them more likely to detect a target. This increased hit rates during the good θ phase, as did the direct excitation of decision cells by FEF visuomotor β 2 rhythms. In the context of our task (in which the LIP granular layers did not receive very strong inputs), the FEF β 2 rhythm was necessary for the production of LIP γ. In contrast, the FEF RS cell spiking characteristic of the good θ phase could occur even if LIP was still producing the β 1 characteristic of its poor θ phase dynamics. Thus, our results suggest that in this task the FEF visuomotor module leads the rhythmic attention process in a top-down manner, and our model makes the prediction that the good θ phase is associated with the appearance of β 2 oscillations in FEF before γ oscillations in LIP. (Note that, while Figure 3 appears to support this hypothesis, rigorous testing, even in silico, would require addressing analytical challenges to measuring the precise onset times of rhythmic dynamics.)

Under different stimulation parameters, our model is also able to generate LIP γ oscillations without FEF β 2 input, provided that the tonic excitation to the LIP granular layers is strong enough. In particular, this means that a very salient stimulus, by providing strong excitation to the LIP input layers, could lead to enhanced sensory processing in a bottom-up fashion. FEF visuomotor cells are able to produce a β 2 or a γ depending on the strength of inputs from LIP; thus, FEF could participate in different (frequency-specific) functional networks depending on whether visuomotor circuits are activated by highly salient as opposed to near-threshold stimuli.

An important recent rate-based computational model (Jaramillo et al., 2019) suggests that pulvinar functions by modulating the gain of feedforward (FF) cortico-cortical connectivity and by altering the balance between feedforward and feedback (FB) cortico-thalamo-cortical (CTC) pathways. This work draws on the fact that the pulvinar interacts with the cortex via two pathways: in addition to feedback pathways in which projections return to the deep layers of the cortical area from which input arrives, pulvinar participates in feedforward pathways in which cortical input originating in one area (A) activates projections to the granular layers of a different cortical area (B), ‘boosting’ the direct connections from area A to area B. In this previous model, it is the latter cortico-thalamo-cortical (CTC) feedforward pathway through which pulvinar excitability (fixed for the duration of a trial) changes the strength of cortico-cortical connectivity (Jaramillo et al., 2019).

In our model, the gain of feedforward connectivity (as indexed by the conductance of synapses from LIP→FEF) plays a similar role in tuning the network’s sensitivity for target detection, representing the difference between cued and uncued locations. However, the mechanisms by which pulvinar affects the dynamics of LIP and FEF are quite different. In the current work, pulvinar input has its own temporal dynamics, and induces disparate dynamical responses in FEF and LIP, helping to create a β 2 rhythm in FEF and a γ rhythm in LIP during the good θ phase. We suggest that these dynamic changes, rather than biasing a competition between FF and FB information transmission, allow the pulvinar to increase the precision of both FF and FB information transmission between FEF and LIP, effectively optimizing the sensitivity of the network to target appearance (at the cued location) during the good θ phase. This increase in the precision of information transmission is due to the deployment of different rhythmic dynamics in the good and poor θ phases: in the good θ phase, an FEF β 2 mediates top-down information transmission as well as lowering the threshold for and changing the temporal dynamics of target detection, while an LIP γ mediates bottom-up information transmission; in the poor θ phase, information transmission between FEF and LIP (i.e. spiking in FEF deep layers and LIP superficial layers) occurs at α frequency (which also changes the threshold and temporal dynamics of target detection in FEF). Altering the timescales of rhythmic dynamics of either the FEF β 2 or the LIP γ decreases the sensitivity of the network. Thus, the dynamics revealed by our model are more complex and more finely tuned to the task at hand than those discussed in the previous model.

This prior work also addressed how pulvinar activity could alter rhythmicity in cortex. In that work, V4 rhythms at lamina-specific frequencies were modeled via E-I networks with lamina-dependent timescales of inhibition, giving rise to a superficial γ and a deep α. Feedback connections arising from the thalamus target interneurons in deep layers (Cruikshank et al., 2010; Zhou et al., 2018; Audette et al., 2018). Thus, in this previous work, pulvinar feedback pathways inhibit deep layer excitatory cells, resulting in less α; and the disinhibition of deep layer E-I networks explains experimental observations that pulvinar lesion results in an increase in α rhythmicity in V4 (Zhou et al., 2016). In contrast, in our model, it is top-down β from FEF that disrupts LIP α/ β 1 , even though pulvinar input to LIP granular layers also contributes to the disruption of LIP’s concatenation β 1 rhythm. Our model suggests that pulvinar lesion might indirectly increase low-frequency ( β 1 ) power in LIP, just as it increases α power in V4, although through a different mechanism.

Finally, our results are in agreement with work suggesting that β and γ rhythms transmit top-down and bottom-up information, respectively (Buschman and Miller, 2007; Buffalo et al., 2011; van Kerkoerle et al., 2014; Bastos et al., 2015; van Pelt et al., 2016). However, they seem to contradict predictive processing accounts in which top-down β rhythms transmit predictions which serve to diminish the bottom-up sensory activity they predict (Bastos et al., 2012; van Pelt et al., 2016), since, in our model, top-down FEF β 2 rhythmicity enhances bottom-up LIP γ rhythmicity. Recent work suggests that such an account of the functions of β and γ rhythms is too simple to account for the full complexity of cortical rhythmic dynamics (Bastos et al., 2020). This apparent contradiction may also be resolved by noting that high-level representations (hypotheses) can produce top-down predictions of two types: first-order predictions of the values of given sensory variables (i.e. the content of given representations); and second-order predictions about the relative reliability or informativeness of those variables (regardless of their value Koelsch et al., 2019). These second-order predictions about reliability can be used to flexibly weight and combine the multiple sensory inputs that provide evidence relevant to a given hypothesis. For example, while visual and auditory information are both relevant to speech content, visual information may be judged more reliable in a noisy, crowded bar. While first-order predictions may be confirmed (via the canceling out of bottom-up signaling) or may result in prediction errors, second-order predictions may affect the gain of certain bottom-up channels, independent of whether those channels produce prediction errors, so that bottom-up information that is (predicted to be) more reliable has more influence on the updating of high-level hypotheses (Feldman and Friston, 2010; Kanai et al., 2015; Koelsch et al., 2019; Millidge et al., 2021).

In our model of the cued location, top-down signals from FEF function to increase the gain of LIP superficial layer spiking by disinhibiting the LIP granular layers. We suggest this could be interpreted as a second-order prediction about the relative reliability of bottom-up information from the cued location. Indeed, some accounts of attention identify it more or less explicitly with this gain modulation of bottom-up signals according to their (expected) reliability or precision (Feldman and Friston, 2010; Moran et al., 2013; Kanai et al., 2015). While our conception of attention is broader in general, we note that, in this case, attentional modulation can be interpreted as the gain modulation of bottom-up input based on its expected reliability. Thalamic as well as cortical mechanisms have been implicated in such precision-weighting, with hypotheses singling out pulvinar in particular (Kanai et al., 2015); it is tempting to speculate that driving FEF β 2 rhythms and priming LIP to respond to them may be one concrete way mdPul carries out this function.

The frequency of FEF and LIP oscillations in our model has consequences for task performance, and the values typically reported in experiments in fact produce peak levels of sensitivity in our model attention network. More precisely, increasing the oscillatory frequency of FEF visuomotor cells from β 2 to γ or reducing LIP oscillatory frequency during the good θ phase from γ to β 2 , by changing the time constants of local inhibition, led to higher false alarm rates, with little to no benefit on hit rates.

The optimization of network sensitivity by FEF β 2 and LIP γ results from the way that these rhythms and their interactions subtly and differentially alter the excitation/inhibition balance in different layers and populations of LIP and FEF during different phases of the θ rhythm (see Task Performance). These observations point to one important role of rhythms in mediating the effects of attention. However, an account of the effects of rhythms in terms of the E/I balance of FEF and LIP neuronal populations is incomplete, as it omits the temporal dynamics of both neural activity and its behavioral consequences. In both experimental data (Fiebelkorn et al., 2013; Fiebelkorn et al., 2019c) and our model, hit rate (and false alarm rate) varied not only between ‘conditions’ (i.e. cued vs. uncued object/LIP→FEF connectivity, time scale of inhibition/rhythmic frequency), but also in time. The phase of θ, FEF β 2 , LIP γ, and pulvinar α all have an effect on target detection (and its sensitivity).

We suggest that these temporal dynamics of network activity are not coincidental or epiphenomenological, but play important functional roles. The first of these is to synchronize and sequence precise monitoring and information transmission with (multiple) other computational processes occurring in other brain regions, each of which may have a different characteristic frequency (for example, β 2 for motor processes, and γ for visual processes). The second is to provide periodic windows of enhanced sensitivity, that enable precise monitoring of a given region of the visual field with acceptable temporal ‘coverage’, while also minimizing energetic cost and allowing monitoring to be interleaved with other computational imperatives, including monitoring of other regions and motor planning and execution. We further elaborate on this sequencing and segregation of cognitive functions below.

Our model replicated experimental results at the behavioral level, with hit rates that varied with the θ phase of pulvinar input to the cortex and which were increased during the good θ phase (i.e. during pulvinar stimulation). In the FEF visual module, VIP cells enabled the detection of low-contrast targets. In our simulations, increasing the LIP→FEF functional connectivity (by increasing the conductance of LIP chemical synapses onto FEF visual cells) increased first hit rates and then false alarm rates, so that intermediate values of connection strength led to peak sensitivity of target detection. These intermediate values of LIP→FEF visual module connectivity were also associated with a strong 8 Hz θ component in the hit rates, which was observed experimentally for targets appearing within the cued (but not the uncued) object (Fiebelkorn et al., 2013).

Our model thus suggests that the appearance of a cue on the screen might enhance LIP→FEF visual cell functional connectivity, leading to both increased perceptual sensitivity especially during the poor θ phase and an 8 Hz dominant frequency in the hit rate spectrum. In other words, the 8 Hz hit rates in our model did not come from the division of a 4 Hz rhythm into multiple possible target locations, as previously suggested for similar data (Landau and Fries, 2012), but were a consequence of the increased LIP→FEF connection strength for the cued object. In our results, very strong LIP→FEF connectivity led to high false alarm rates, and such excessive connectivity may be linked to hallucinations (as in the detection of a target in the absence of a visual stimulus) or other pathologies in visual perception.

In this work, the increase in LIP→FEF functional connectivity is implemented via an increase in the maximal conductance of the chemical synapses from LIP to FEF. However, there are multiple ways such an increase in drive could arise. For example, either FEF or LIP may be disinhibited by the activation of VIP interneurons in these regions, or thalamus may modulate the gain of LIP→FEF connectivity via a cortico-thalamo-cortical feedforward pathway, as described in the next section. Multiple mechanisms may be implemented in the brain, to allow multiple information streams—e.g., reward statistics from superior colliculus or learned associations from hippocampus or amygdala—to influence the attentional processing of a given visual location.

As the good θ phase is associated with improved sensory perception and behavioral performance in attention tasks (Fiebelkorn and Kastner, 2019b), one could wonder what need is served by the poor θ phase—and, indeed, the θ rhythm itself—in the first place.

One possibility is that the θ rhythm is a solution to the necessity of attending to multiple objects ‘at once’, allowing the focus of attention to switch between objects θ-rhythmically. In this account, the poor θ phase for the representation of one object or location would occur simultaneously with the good θ phase for the representation of another object or location (Fiebelkorn et al., 2013; Landau and Fries, 2012). In our model, this could be achieved with multiple copies of LIP and FEF representing one cortical columnar network for each object to be monitored; pulvinar inputs would select the attended object at any point in time, and manage switching the attentional focus.

Alternatively, the good and poor θ phases may result from an alternation between more and less specific hypotheses about the visual location of target appearance. In this case, the good θ phase would represent preferential processing of visual information from the cued object and location, but the poor θ phase would simply represent a lapse of this preferential processing, and not the preferential processing of the uncued locations (relative to the cued location). This is suggested by the fact that the hit rates at the cued and uncued locations are similar during the (cued location’s) poor θ phase (Fiebelkorn et al., 2013).

In either account, even when attention is deployed at a single location, the break in attentional focus allowed by the poor θ phase would allow attention and behavior to adapt quickly to changes in visual stimuli and their perceived salience. In this framework, the good θ phase could be described as a sampling phase for one location, while the poor θ phase could be described as an exploration phase (Fiebelkorn and Kastner, 2019b). Accounts of binocular rivalry suggest that this alternation between two visual percepts may result from periodic shifts in the levels of confidence ascribed to alternative hypotheses (Hohwy et al., 2008), and it is interesting to consider whether similar dynamics apply in the current case. The failure of a target to appear during the good θ phase may be seen as reducing confidence in the hypothesis that the target will appear at the cued location. As a result, (relative) confidence may increase in other hypotheses about the location of target appearance (either specific ones related to the uncued locations or general ones that include cued and uncued locations). When the target does not appear during the poor θ phase, confidence in these hypotheses may in turn wane, and the original hypothesis may be ‘revived’.

Finally, it has been hypothesized that the poor θ phase could enable the planning and execution of motor actions (Fiebelkorn and Kastner, 2019b). Interestingly, in the presence of adequate excitatory drive from mdPul, the FEF visuomotor β 2 that elicits LIP γ in our model can also be transformed into γ rhythmicity by α/ β 1 -frequency LIP input. If the FEF β 2 rhythm is interpreted as a ‘hold signal’ (Engel and Fries, 2010), pausing FEF γ rhythmicity and consequent attentional shifts, our model suggests that the interactions between FEF and LIP may mediate an oppositional relationship between motor engagement (operationalized by γ in FEF) and sensory processing (operationalized by γ in LIP).

These accounts of the role of the θ rhythm are far from incompatible. The θ rhythm may be a mechanism that interleaves periodic assessment of multiple hypotheses about target appearance with motor execution, with movements occurring at the transitions between good and poor θ phases, each of which represents the sampling of evidence for one particular hypothesis. Essentially, movement would simply be the gross motor component of shifts in ‘attentional focus’, sequenced in turn with preferential processing of multiple salient regions (potentially having different spatial scales) of the visual field.

A deeper understanding of the role of the θ rhythm awaits a detailed investigation of pulvinar activity and the mechanisms of θ rhythmicity and spatial selection in this task. In ongoing work, we are extending the current model to explore how circuits in the pulvinar and the superior colliculus may coordinate ongoing periodic temporal dynamics (as modeled here) between multiple identical networks representing different spatial locations and/or objects, in line with prior learning.

As described in the Introduction, our model represents a number of simplifications relative to the full dynamics of the visual attention network in the context of this task. The model presented here is intended to allow realistic insights into the mechanisms underlying observed brain dynamics and their functional consequences while avoiding excessive complexity. Thus, we have chosen to highlight the cell types and connections necessary to produce the dynamics of interest, while leaving out other cell types and connections. In particular, the spatial structure of the full Egly-Driver task is beyond the scope of the current model. The full task involves spatially informative cues activating subpopulations that respond selectively to spatial locations and/or objects. This greater mechanistic complexity, in addition to involving a detailed pulvinar model and inputs from other regions, may also involve cell types and connections omitted in the current model, for example VIP cells and excitatory connections between RS cells in LIP superficial layers. VIP cells may participate, for example, by mediating the disinhibition of spatially selective populations. Excitatory synapses among cortical pyramidal cells, which are extremely important for plasticity and formation of cell assemblies, may also be important in the selection of spatially specific subpopulations. However, in our hands, neither are necessary for the current model (a stepping stone to a more complex model of the full Egly-Driver task), in which all the cells in LIP can be thought of as part of a single, cue-relevant cell assembly (although dividing FEF visual neurons into two populations was necessary for the detection of low-contrast targets). The qualitative dynamics of the LIP module remained the same when recurrent connections between RS cells were added ( Appendix 4—figure 1) and when VIP cells were added to the superficial layer (as part of a target detection module like the one instantiated in the FEF visual module; Appendix 4—figure 2). Similarly, the addition of FS cells in the FEF visuomotor module also did not significantly alter this module’s β 2 dynamics (Appendix 4—figure 3).

We have also chosen to model relatively small networks of biophysically detailed neurons. In order for the behavior of small networks to qualitatively match the dynamics of much larger networks, their model parameters often must be different from those of the larger networks. Their dynamics also may be different, in ways that we believe are irrelevant to understanding the qualitative dynamics of the network and the functional implications of interest—for example, rhythms that are more regular than those seen in vivo. Testing the conclusions of models such as ours in larger, more detailed models, as well as in experiments, is essential. However, initial modeling studies in smaller, simplified networks are indispensable in generating initial hypotheses about the mechanisms and functions of brain dynamics.

The parameters of our model are determined in part by the incorporation of model components shown to qualitatively reproduce experimentally observed brain dynamics relevant to other physiological and cognitive contexts: a model of cholinergic attentional modulation in laminar cortex (Lee and Whittington, 2013), and models of interlaminar interactions in parietal cortex that instantiate a working memory mechanism (Kramer et al., 2008; Gelastopoulos et al., 2019). The use of these components imposes additional functional constraints on the parametrization of the model.

Aside from these functional constraints, model parameters are for the most part derived from experimental data in rodents. The differences between rodent and primate neocortex are numerous. Relative to rodents, primates have a 50% greater proportion of interneurons, due in large part to an increase in calretinin, VIP, and neurogliaform interneurons in the superficial layers of association cortex, which distinguishes association and sensory cortices in primates but not rodents (Džaja et al., 2014; Krienen et al., 2020; Hodge et al., 2019). Gene expression patterns of interneuron subtypes also differ more between cortical regions in primates than rodents, suggesting that interneurons may be more specifically tailored to different cortical regions in primates (Krienen et al., 2020).

Among these many differences between rodents and primates, there are a few that particularly influence the dynamics of our model. The increase in superficial VIP cells in primates suggests these interneurons are likely part of the circuitry of LIP superficial layers, despite their absence in our model. As discussed above, VIP cells may play a role in determining the allocation of attention between objects in the full Egly-Driver task. However, our results suggest they do not qualitatively alter the dynamics following cue presentation; in our hands, placing a target-detection circuit like the one modeled in the FEF visual module in LIP superficial layers did not alter LIP rhythms in response to low-contrast targets (Appendix 4—figure 2). In monkeys relative to rats, PV-positive basket cells have been shown to have a higher input resistance and a lower firing threshold and thus to be more excitable (Povysheva et al., 2008). In rats, PV-positive basket cells fired in response to stimulation with a substantial delay, and generated spike trains interrupted by quiescent periods (Povysheva et al., 2008), hallmarks of stuttering interneurons exhibiting a D-current (Golomb et al., 2007). Although originally parameterized using data from rat hippocampus, our model FS cells lack a D-current, and thus exhibit behavior more similar to primate than to rodent FS cells. Another between-species difference with potential consequences for dynamics in the present model is that supragranular pyramidal neurons are known to exhibit a higher h-current conductance in humans as compared to mice (Kalmbach et al., 2018). This higher h-current conductance was shown to promote transfer of θ frequencies from the dendrites to the soma of human pyramidal neurons (Kalmbach et al., 2018). In our model, a higher h-current conductance in superficial pyramidal cells might thus lead to a stronger θ resonance in these cells.

We have chosen to model FEF and LIP as driven θ-rhythmically by simulated mdPul input. As we mention in the Introduction, this is consistent with experimental observations that mdPul exerts a Granger causal influence over FEF and LIP at α frequency only during the good θ phase (Fiebelkorn et al., 2013; Fiebelkorn et al., 2019c). This α-frequency drive during the good θ phase also amounts to a θ-rhythmic drive from mdPul to FEF and LIP, which is sufficient—in our model—to induce the experimentally observed θ rhythmicity in the dynamics and function of LIP and FEF. However, we cannot rule out the possibility that FEF and LIP also exhibit θ rhythmicity or resonance, or that they (and/or the pulvinar) are paced by θ-rhythmic drive from one or more other brain regions. Indeed, neocortical SOM interneurons have been shown to form electrically coupled networks that generate oscillations at θ frequencies (Beierlein et al., 2000; Fanselow et al., 2008; Hu and Agmon, 2015; Huang et al., 2020). These oscillations occur in response to persistent activation by synaptic inputs, metabotropic glutamate agonism, and muscarinic acetylcholine agonism (Beierlein et al., 2000; Fanselow et al., 2008) as well as in response to salient visual stimuli (Huang et al., 2020), and exert a coherent inhibitory influence on cortical pyramidal cells and FS interneurons (Beierlein et al., 2000; Karnani et al., 2016; Safari et al., 2017). Coupled with a potential θ resonance in superficial RS cells induced by increased levels of h-current in humans (see above), such a θ-rhythmic SOM network could result in endogenous θ rhythmicity in LIP superficial layers. Whether and how this θ rhythmicity would change our results is an important question for future investigations.

Finally, in our FEF visual module, the interactions of inhibitory PV, SOM, and VIP interneurons within and across cortical columns produce a circuit that is particularly sensitive to low-contrast stimuli modeled as a weak input to one column and not another. The functional role of interactions between these three interneuron types has received considerable attention (Karnani et al., 2016; Yang et al., 2016; Lee et al., 2017; Dipoppa et al., 2018; Krabbe et al., 2019; Lee et al., 2019; Garrett et al., 2020; Hertäg and Sprekeler, 2020; Keller et al., 2020; Millman et al., 2020; Veit et al., 2021). This literature highlights the role of VIP interneurons in disinhibiting pyramidal cells (via inhibition of SOM cells) to produce context-dependent stimulus sensitivity. VIP-mediated disinhibition has been implicated in (i) detection of stimulus and surround differences (Keller et al., 2020), (ii) the creation of a high gain regime that increases sensitivity to weak inputs (Millman et al., 2020), (iii) the desynchronization of gamma oscillations representing spatially discontiguous visual features (Veit et al., 2021), (iv) the facilitation of auditory cortical stimulus responses by nicotinic acetylcholine receptor activation (Askew et al., 2019), (v) response reversal (Garcia Del Molino et al., 2017), (vi) the coding of prediction errors (Hertäg and Sprekeler, 2020), and (vii) the gating of information flow between cortical regions (Yang et al., 2016; Lee et al., 2017; Krabbe et al., 2019; Lee et al., 2019). These results motivate in part the circuitry of the FEF visual module in the present model. A number of other models have been constructed to explore these effects, including rate models (Millman et al., 2020; Dipoppa et al., 2018; Hertäg and Sprekeler, 2020), mean-field models (Veit et al., 2021), and spiking models (Yang et al., 2016; Lee et al., 2017; Garcia Del Molino et al., 2017; Veit et al., 2021). While a full understanding of the relationships between these models is beyond the scope of the current work, in our model, both pyramidal disinhibition via VIP-mediated SOM suppression and intercolumnar inhibition via SOM cells were required for the amplification of small inputs. The latter feature is not common in existing models of VIP-SOM interactions (but see Veit et al., 2021). Another requirement is that our VIP interneurons exhibit a steeper f-I curve than our SOM interneurons (Appendix 1—figure 1). VIP interneurons are among the interneurons with the highest input resistance and are therefore among the most excitable (Rudy et al., 2011), and cortical and thalamic inputs are greatest onto VIP interneurons as opposed to PV or SOM cells (Wall et al., 2016). Our model is thus in line with recent investigations suggesting that a stabilized supralinear network with three interneuron types having a distribution of f-I curves can mediate an input-strength dependent switch between a (VIP-mediated) high gain regime that increases sensitivity to weak inputs and a (SOM-mediated) low gain regime that stabilizes the network response to strong inputs (Millman et al., 2020). Such disinhibitory mechanisms are clearly widespread in the cortex and an important subject for future work.

The model presented in this article was adapted from Kramer et al., 2008 and Gelastopoulos et al., 2019. These laminar models reproduce somatosensory and parietal rhythmic dynamics, and the cells are reductions of detailed multi-compartment representations of rat cortical neurons (Traub et al., 2005). The current model consisted of Hodgkin-Huxley neurons, modeled with a single compartment except for LIP intrinsically bursting (IB) cells which were modeled with four compartments. The membrane potential V of each single-compartment neuron and each IB cell compartment followed an equation of the form: C ⁢ d ⁢ V d ⁢ t = - ∑ I m - ∑ I syn - I ran

where C is the membrane capacitance, I m are the membrane currents, I syn are the synaptic currents, and I ran is a random input current. Some cells are also electrically coupled with gap junctions. The membrane capacitance was equal to 0.9 ⁢ μ ⁢ F / c ⁢ m 2 for all neurons except VIP interneurons, for which it was 2 ⁢ μ ⁢ F / c ⁢ m 2 .

Our regular spiking (RS) pyramidal cell model included a leak current I L , a sodium current I Na , a potassium current I K , and a hyperpolarization activated current (h-current) I AR . The conductances of these currents (Table 1) were adapted in Traub et al., 2005 from experiments on rat hippocampal neurons (Martina and Jonas, 1997). There were 80 RS cells in our LIP superficial layer model, 20 in LIP granular layer, 20 in visuomotor FEF, and 20 in visual FEF. The 20 decision cells in visuomotor FEF were also RS cells.

Our fast spiking (FS) interneuron model in LIP and FEF included a leak current I L , a sodium current I Na , and a potassium current I K . The conductances of these currents (Table 2) were adapted in Traub et al., 2005 from experimental studies of rat hippocampal neurons ( I L , I Na , & I K ; Martina and Jonas, 1997; Martina et al., 1998). There were 20 FS cells each in the LIP superficial layer, LIP granular layer and FEF visual module.

Our SOM cell model included a leak current I L , a sodium current I Na , a potassium current I K , and an h-current I AR . The conductance for I AR was modified from experiments on guinea pig thalamic interneurons Huguenard and McCormick, 1992; all other conductances were the same as for FS cells (Table 3). There were 20 SOM cells each in the LIP superficial layer, LIP deep layer, and FEF visuomotor and visual modules.

Our VIP cell model included a leak current I L , a sodium current I Na , a potassium current I K and a potassium D-current I D with conductances taken from studies of guinea pig cortical neurons (Golomb et al., 2007, Table 4). There were 20 VIP cells in the FEF visual module.

Our IB cell model consisted of a somatic compartment, an axonal compartment, and apical and basal dendritic compartments, with electrical coupling between them (Table 5). All compartments included a leak current I L , a sodium current I Na , and a potassium current I K , whose kinetics were taken from experiments on rat hippocampal neurons (Martina and Jonas, 1997). The axonal compartment and both dendritic compartments also included a potassium M current ( I KM ). Additionally, the dendritic compartments included an h-current I AR (based originally on experiments on guinea pig thalamic interneurons; Huguenard and McCormick, 1992) and a high-threshold calcium current I CaH (parameters based on studies of rat thalamocortical neurons; Destexhe et al., 1998). There were 20 IB cells in the LIP module.

Each membrane channel’s current I was expressed in the form. I = g . m p . h q . ( V − E )

where g is the channel conductance, E its reversal potential, and m and h are the gating variables, which followed the differential equation. d x d t = 1 τ x ⋅ ( x ∞ − x ) , x ∈ { m , h }

or, in an equivalent manner d ⁢ x d ⁢ t = α x ⁢ ( V ) ⋅ ( 1 - x ) - β x ⁢ ( V ) ⋅ x , x ∈ { m , h }

with the exception of the sodium currents for which m = m ∞ ⁢ ( V ) . The expressions and values for the variables and parameters g , E , p , q , τ m and m ∞ (or α m and β m ), τ h and h ∞ (or α h and β h ) for each channel and each cell type are detailed in Tables 1–5.

The random current I ran received by each cell followed the equation: I ran = σ ran 2 ⋅ d ⁢ W

where d ⁢ W is Gaussian white noise of unit variance per m ⁢ s , and σ ran 2 is the intensity of this additive white noise. The value of the parameter σ ran 2 used for each cell type is given in Table 6. An additional random noise of amplitude 30 ⁢ μ ⁢ A / c ⁢ m 2 was provided to the FEF visuomotor module in the poor theta phase, representing arrythmic inputs from the pulvinar.

The neurons in our model interacted with each other through AMPA and GABA_A synapses as well as gap junctions. The synaptic current I syn received by each cell (or compartment) was the sum of the synaptic currents I syn , i , one for each incoming synapse. The contribution of each synapse was given by the equation. I syn , i = s i ⋅ g i ⋅ ( V - E i )

where g _i is the maximum conductance of the synapse and E i is its reversal potential. The synaptic gating variable s _i depended on the rise and decay time constants of the synapse τ r , i and τ d , i as well as the membrane potential of the presynaptic neuron V pre , i , according to the equation. s i = − s i τ d , i + 1 − s i τ r , i ⋅ 0.5 ⋅ ( 1 + t a n h V p r e ,   i 10 )

The parameters chosen for each synapse type in the network are summed up in Table 7 (for the reversal potentials, rise and decay time constants) and Table 8 (for the maximum conductances).

Similarly, the total gap junction current received by a cell I gap was the sum of all contributions I gap , i received from each individual cell or compartment contacting it, described by the equation. I gap , i = g i ⋅ ( V - V pre , i )

where g _i is the maximum conductance of the gap junction and V pre , i is the membrane potential of the other neuron or compartment involved in the gap junction. Gap junctions were present between LIP superficial RS cells with a maximum conductance g i = 0.04 , between LIP superficial SOM cells with a maximum conductance g i = 0.2 , between LIP IB cell axons with a maximum conductance g i = 0.0025 , and between the different compartments of each LIP IB cell, with conductances summed up in Table 9. Of note, the maximal conductances of these gap junctions were likely larger than what would be measured for single pairs of cells in vivo, as is appropriate for currents in a smaller network.

Some cell populations in the network received an external input current I inp , following the set of equations. I inp = s inp ⋅ g inp ⋅ ( V − E inp ) d s inp d t = − s inp τ d , i n p + ( 1 − s inp ) τ r , i n p ⋅ 0.5 ⋅ ( 1 + tanh ⁡ V inp 10 )

where g inp is the maximum conductance of the input synapse and E inp = 0 ⁢ m ⁢ V is its reversal potential. The synaptic gating variable s inp depended on the rise and decay time constants of the synapse τ r , i ⁢ n ⁢ p = 0.1 ⁢ m ⁢ s and τ d , i ⁢ n ⁢ p = 0.5 ⁢ m ⁢ s as well as the membrane potential of the presynaptic neuron V inp .

Some cell populations also received input from neuronal populations that were not modeled explicitly (i.e. with full Hodgkin-Huxley dynamics). Each neuron in these input populations was modeled as a periodic spike train with frequency f , having inter-spike intervals uniformly randomly distributed in the interval ( 0.9 ⁢ f , 1.1 ⁢ f ) , and independently chosen for each input neuron. The efferent synapses of these input populations followed the equations d ⁢ V inp d ⁢ t = 1 τ inp ⋅ ( V low - V inp ) on spike : V inp ← V high

that is, whenever a spike occurred, the membrane potentials of their synaptic targets were reset to V inp = V high = 0 ⁢ m ⁢ V before exponentially decaying to their baseline value V inp = V low = - 80 ⁢ m ⁢ V with a decay time constant of τ inp = 0.5 ⁢ m ⁢ s . To implement the sinusoidal or sawtooth shape of the θ rhythm, we associated the two volleys of mdPul input occurring during each good θ phase with different maximal conductances g inp . The frequency f and conductance g inp for each input synapse are given in Table 10.

All the simulations were performed using the Brian2 libraries for Python (Stimberg et al., 2014), using a 4th order Runge-Kutta method to solve the differential equations defining the model, and a 0.01ms timestep. All simulations were 2 s long, and the first 200 ms of each simulation was discarded to remove transients. Targets were presented at cue-target delays spaced every 50ms from 200ms to 1800ms. An ensemble of 50 simulations was computed for each delay value, with randomized noise and randomized temporal jitter in the inputs. All code is available at https://github.com/benpolletta/egly-driver-network (copy archived at Pittman-Polletta, 2023).

All analyses were performed in Python, with the exception of spectrograms in Figure 3C, Figure 4C which were computed in Matlab.

The statistics of target detection were computed for 50 simulations at each of 32 cue-target intervals (see above). A ‘hit’ was recorded if more than half of FEF decision cells spiked at least once during the 100 ⁢ m ⁢ s -long window in which the target was presented. A ‘false alarm’ was recorded if more than half of FEF decision cells spiked within a 25 ⁢ m ⁢ s -long window in which no target was presented. Given a hit rate H and a false alarm rate F , the sensitivity was calculated as D ′ = z − 1 ( 1 − F ) − z − 1 ( 1 − H ) ,

where z - 1 ⁢ ( r ) is defined as the inverse cumulative distribution function for the standard normal distribution on the interval ( 0 , 1 ) (i.e., z - 1 ⁢ ( r ) is the real number whose standard normal p -value is r ), z - 1 ⁢ ( 0 ) = z - 1 ⁢ ( ϵ ) , and z - 1 ⁢ ( 1 ) = z - 1 ⁢ ( 1 - ϵ ) for ϵ = 0.1 .

The θ phase was determined from synthetic mdPul α inputs, which had a period of 250ms and a duty cycle of 125ms. Thus, 0° occurred at the onset of the mdPul α, and 180° occurred at the offset of mdPul α. The LIP β 1 phase during the poor θ phase was determined from the timing of superficial RS cells spiking (RS cells spiking was considered to be at 0°), assuming a β 1 frequency of 10 Hz.

LIP LFPs were computed by averaging the membrane potentials of superficial RS cells; FEF visuomotor LFPs were computed by averaging the membrane potentials of visuomotor RS cells.

Power spectra were calculated using SciPy’s signal.periodogram with a flat top window. The spectrograms in Figure 3C and Figure 4C were calculated from LFPs taken from a single 2 s simulation, via convolution with complex Morlet wavelets (obtained with Matlab’s dftfilt3) with frequencies between 9 and 60 Hz; the width of these wavelets was linearly interpolated from 4 cycles at 9 Hz to 12 cycles at 60 Hz.