Above we showed that the spiking activity of neurons is coupled to multiple aspects of the motor beta rhythm during two different tasks (MC and BC), and that the form of this beta-to-rate mapping changes in a reversible, task-dependent way. For example, as beta power increases, a given neuron may increase spiking during MC but decrease spiking during BC, exhibit a reversible shift in the preferred phase of firing, or remap its sensitivity to relative phase differences between areas. This dependence on beta amplitude was well-fit by a sigmodial function (), while the dependence of spiking on beta phase followed a cosine function (), weighted by beta amplitude (). These results expand on prior findings showing cross-level coupling (CLC) between spiking and LFP phase in multivariate signals 
, here showing an additional, independent coupling to beta amplitude. Critically, this work shows that cells can exhibit task-dependent changes in this coupling. Importantly, the parameters describing this beta-to-rate mapping are stable across multiple datasets of the same task (within-task stability) but exhibit reliable changes when moving from one task to another (cross-task diversity). Furthermore, we showed that the ensemble diversity of amplitude-to-rate and phase-to-rate mappings describes a set of discrete ensemble states, where each state is defined by the rank order of instantaneous spike rates. What are the implications of these empirical findings for different hypotheses about the oscillatory control of distributed networks, especially regarding local computation in a given area and long-range communication between areas?
First, there is the question of how the observed beta-to-rate mappings arise – presumably the spike activity of a subset of presynaptic cells is the origin of the amplitude- and phase-to-rate mappings for a given neuron. Rather than speculate on these origins, here we take it as given that the beta-to-rate mapping exists and instead ask what computations are now possible that are not possible or difficult if CLC is absent. We focus on two potential mechanisms that operate over different timescales: first, we consider the impact of CLC on rate-based winner-take-all (WTA) competition mediated by recurrent synaptic inhibition. Operating over a timescale of hundreds of milliseconds, modulation of WTA dynamics via the amplitude-to-rate mapping provides one link from cross-level coupling to functional neural computation. Second, operating over a timescale of tens of milliseconds, the phase-to-rate mapping biases ensemble spike timing such that some spike timing patterns are more likely than others. Through this route, cross level coupling may modulate robust temporal coding mechanisms such as synfire chain propagation.
When evaluating different neurocomputational mechanisms, it is important to keep the anatomical facts clearly in mind in order to rule out mathematically elegant but biophysically implausible options. In this regard, the recurrent excitatory/inhibitory loops of local cortical circuits appear to provide an ideal platform for winner-take-all (WTA) dynamics 
. presents a simplified schematic of a WTA module, where multiple input paths are converted into the activation of one output path via competitive di-synaptic inhibition. In this module, two excitatory cells, E1
(red triangles), are reciprocally connected to an inhibitory cell (blue circle) that receives input from both E-cells. Both E-cells also receive independent excitatory input from outside the module. None of the cells inside the WTA module need have amplitude-to-rate mappings or any beta sensitivity whatsoever. Next, assume two cells outside the WTA module provide the external excitatory input, and that both of these cells have amplitude-to-rate mappings that intersect. For example, consider the purple and gold cells in , which have amplitude-to-rate mappings as shown in Figure S10
. For simplicity, assume these external cells providing WTA input are driven solely by their amplitude-to-rate mappings. Then for low beta amplitudes, the WTA cell E1
becomes active (), whereas high beta amplitudes cause E2
to become active (). In fact, the switch between E1 and E2 occurs at the beta amplitude value corresponding to the intersection of the amplitude-to-rate mappings for the purple and gold input cells. That is, the relative spike rate rank order of the cells providing input to the WTA module is transformed into tonic spiking along one of two possible output paths. Since the evidence presented here shows that within-task amplitude-to-rate mappings are stable, this binary output switch is tuned to a particular value of beta amplitude that is fixed for the duration of the task. Whenever beta amplitude sweeps through this value, this WTA switch changes state. By adding additional cells with amplitude-to-rate mappings that cross at other amplitude values, we can establish a linear, task-dependent sequence of binary WTA switches, each of which is tuned to or indexed by a different value of beta amplitude. Thus, each value of beta amplitude is associated with a binary vector that encodes the ensemble state.
Neuro-computational consequences of amplitude- and phase-to-rate mappings.
Why would this be useful? First, recall the 12 cells shown in . On the one hand, there are 12!
479001600 possible rank-ordered states for this set of neurons, corresponding to the number of permutations. The ability to generate sequences from such a large set of states would clearly prove computationally useful. However, it is unclear what biological mechanisms are available to quickly identify and activate an arbitrary state selected from the set of all possible ensemble states. On the other hand, if the 12 neurons have fixed baseline rates and flat amplitude-to-rate mappings, then state activation is not a problem since only one state is active at all times. Again, this case is not very computationally useful. In contrast to these extreme cases, an ensemble of neurons with a diversity of amplitude-to-rate mappings (as shown in ) has both a variety of possible states (defined by the amplitude-to-rate crossings), as well as a method for indexing each state (every beta amplitude value corresponds to one particular ensemble state). More importantly, task-dependent remapping of the amplitude-to-rate functions provide the means to select a different set of ensemble states – where again each state is indexed by beta amplitude. That is, during one task such as BC, the continuous variation in beta amplitude maps to a discrete sequence of ensemble states (16 states for the 12 neurons shown in and S2A
), while switching to another task such as MC maps the amplitude to a different sequence of ensemble states (24 states for the same set of neurons). In this view, across all tasks the continuous amplitude signal serves as an index function that establishes activation and transition probabilities for ensemble states. However, one task may require a different set of ensemble states than another – thus explaining the task-dependent remapping, as cross-level coupling parameters are tuned to evoke a desired set of ensemble states. In the example above, none of the 16 BC states or 24 MC states are shared across tasks (Figure S2I
). Task-dependent remapping thus balances the need for a diversity of ensemble states with the requirement of a simple mechanism for sequential state activation. Therefore, combining WTA dynamics with beta-to-rate mapping and remapping seems to provide a physiologically plausible mechanism for the dynamic linking of distinct sequences of ensemble states to a common, readily-accessible signal representing the overall level of population activity – namely, the beta rhythm.
These ideas are consistent with the hypothesis that the functional role of the beta rhythm is to maintain the current computational state in a local network, protecting the local population against irrelevant or contradictory input 
. That is, beta power remains high if no change in the local network state is needed, or if unwanted changes to local network state must be actively extinguished. Similarly, beta power drops when the local network state must change. The arrival of important but unexpected input may increase or decrease beta power, depending on context and task demands. In this view, beta is an active coordinating rhythm that helps to maintain or release selected patterns of ensemble activity. It is intriguing to speculate that task switching requires remapping coupling parameters in order to evoke a pre-learned sequence of WTA states, while learning involves optimization over the space of WTA sequences in a search for those sequences that prove most task effective. One prediction of this hypothesis is that ensemble amplitude-to-rate mappings will exhibit much more variability during learning than either before or after. The combination of WTA dynamics together with heterogeneous amplitude-to-rate mappings across an ensemble provides a specific and testable mechanism through which the beta rhythm could accomplish this goal of dynamic coordination.
Independent of possible functional roles played by the amplitude-to-rate mapping, phase-to-rate mappings may shift the relative probabilities of precisely-timed spike sequences. Simulation studies show that polychronous groups – sets of cells where activity propagates due to precise spike timing relations – can serve as the building blocks for cognitive operations such as working memory 
, and exhibit activity-dependent growth and decay useful for learning and pattern recognition 
. Empirically, Havenith et al. 
showed that relative spike timing in visual cortex reflects properties such as stimulus orientation. Importantly, given inter-connected pools of neurons, synchronous propagation of activity is more stable than asynchronous propagation. In fact, propagating synfire chains yield stable and robust spiking precision in the millisecond range that supports the self-stabilization of synfire chain activity 
. That is, given the right starting conditions, initially weak synfire chains (with few active members or poor synchronization) can recruit additional members and reduce spike-timing variance across the group. However, slightly different initial conditions may force a synfire chain to cross a dynamical systems separatrix between attractors, forcing the synfire chain to quickly decay 
. Since phase-to-rate mappings can influence spike timing, strong phase-to-rate mappings can increase the likelihood of some synfire chains while rendering others less likely. Since beta amplitude appears to act as a gain control mechanism for the strength of the phase-to-rate mapping, the influence of beta on the probability of different spike sequences can be adjusted by changing beta power. However, shows that a fixed change in beta amplitude will have a differential response on different cells, with some strongly increasing their phase preference while others show only moderate changes. Thus, the mapping from beta amplitude to spike sequence probabilities is not a simple one, but depends on the diversity of CLC parameters that hold across the population. Finally, task-dependent remapping of the preferred phase (c.f. ) provides a mechanism for the selective and task-dependent control of synfire chain activation and propagation. That is, during a given task the relative probabilities of a set of (function-specific) multi-neuron spike sequences can be controlled via adjustments in beta amplitude, while switching to another task involves a remapping of CLC parameters in order to call a different set of spike sequences into action. Since motor cortical function involves both rate modulation as well as spike synchronization 
, a mechanism to selective control synchronization while leaving rate modulation unchanged may prove useful to a system controlling distributed networks.
While the amplitude- and phase-to-rate mappings appear most relevant to local computation within a given cortical area, the phase-difference-to-rate mapping may play a role in the regulation of long-range communication between areas. According to the communication through coherence (CTC) hypothesis, the effective gain between interacting areas is a function of the phase difference between them 
. It is difficult to see how the brain could implement CTC control systems without the use of neurons that detect phase differences between areas, on the one hand, as well as neurons than can evoke shifts in the relative phase between distal areas, on the other. Neurons that could serve as phase difference detectors and effectors appear to be fundamental elements required by any distributed system of oscillatory network control. Furthermore, hierarchical predictive coding models suggest that the gamma rhythm is indicative of bottom-up feed-forward processing, while the alpha and beta rhythms serve as signatures of top-down feedback influence 
. Distinct phase-difference-to-rate mappings that operate at these frequencies appear to be one way to control the relative balance of feedforward and feedback processing. The phase-difference to amplitude-envelope-correlation relationship shown in Figure S8D
appears to support the communication through coherence hypothesis, but further studies targeting the role of spiking neurons in long-range interactions are required to clarify their role in the oscillatory control of distributed networks.
Prior work studying neural dynamics in motor cortex has tended to focus on the correlation between spiking activity and “external” factors (e.g. movement velocity, environmental state, behavior-dependent sensory feedback, etc). In contrast, this study focused on “internal” factors that arise from spontaneous, ongoing brain activity – including beta amplitude and phase within an area, or the difference in beta phase between areas. Specifically, we showed that most neurons exhibited a sigmoid dependence on beta amplitude (considered alone; ), a cosine dependence on beta phase (considered alone; ), and that beta amplitude provided a quadratic gain control for the beta phase preference (). What is the relationship between these “external” and “internal” factors? provides an example of external and internal tuning for one example neuron, showing how input variables can be mapped to a predicted rate, which can then be compared to a measured rate. For example, shows how time-in-trial and target ID can be mapped to a predicted rate (color), while shows how this predicted rate compares to the rate that actually occurs. Similarly, show this for beta amplitude and phase; these figures show that the range of the predicted rate generated from the beta-to-rate mapping (rinternal
) is about half that of the range of the predicted rate generated from trial information (rexternal
). The sum of these terms (r
) often has a larger range than either rexternal
alone. However, this sum assumes that rexternal
are independent – an assumption that is not appropriate for many neurons. For example, while shows neurons where target direction appears independent of the phase-to-rate modulation depth, show examples where there is a clear interaction between internal and external factors. The focus of this study was to investigate the dependence of spiking on internal factors, and to determine if this dependence changes from one task to another. Determining the relation between internal and external factors will require further investigation.
Nonetheless, the majority of neurons show a dependence on “internal” beta-related factors that is not mediated by external factors such as direction tuning (Figure S3
). A related concern is that the observed changes in CLC are more directly linked to bottom-up demands related to the trial substages (e.g. hold vs. movement period) than to top-down modulation associated with the task context. Figure S5
addresses this concern by directly comparing endogenous and exogenous factors; the fact that cross-task, within-stage differences are larger than within-task, cross-stage differences indicates that task context is a factor in determining neuronal responses related to CLC. That is, it appears that cells are influenced both by bottom-up, exogenous input related to the processing demands of the different trial substages as well as by top-down, endogenous input related to the maintenance of task context and rule selection.
An interesting aspect of this analysis has been the observation of the strong heterogeneity of neuronal sensitivities to different types of input, considering external vs. internal factors or top-down vs. bottom-up aspects of the experimental demands, compared to the stability of the average population responses. For example, Figure S3A
shows the baseline firing rates for each neuron during BC and MC, and makes it clear that many neurons exhibit large task-dependent shifts in the baseline spike rate. The average spike rate over the population, however, is relatively unchanged (red and blue lines, Figure S3A
). That is, with a shift in task the neuronal ensemble seems to reassign firing rates around a constant population mean rate. Similarly, the amplitude-to-rate and phase-to-rate mappings computed using spikes from all neurons (average population mappings) do not show the strong task-dependent shifts seen in the mappings of individual neurons. Therefore, we would predict that electrophysiological measures that depend on average ensemble activity, such as coupling between beta and the broadband ECoG signal 
, will be less likely to exhibit strong task-dependent changes than will individual neurons.
Finally, the empirical findings reported here are consistent with the hypothesis that dynamic changes in coupling between multiple spatial and temporal scales provide a simple mechanism to bias functional network activity 
. In particular, coupling between single neurons and the motor beta rhythm exhibits several properties that appear positioned to influence local cortical computation – namely, the phase-regulation of relative spike timing on a scale of tens of milliseconds and the amplitude-regulation of winner-take-all dynamics within neuronal ensembles occurring on a scale of hundreds of milliseconds. Similarly, long-distance communication appears to be modulated by the relative phase difference between areas. The presence of neurons that are sensitive to these properties could provide a mechanistic route for this information about relative phase differences to be detected and actively used in the dynamic regulation of large-scale network activity. While future studies employing casual intervention will be required to fully test the functional role of different oscillatory rhythms, here we have shown that the mapping from beta activity to firing rate changes in a reversible, task-dependent way. Given that beta oscillations are generated by the coordinated population activity of hundreds of thousand cells involved in a distributed network that spans both hemispheres 
, the results presented here suggests that the relationship of multiscale coupling between single neurons and larger networks is flexible and can be dynamically remapped in order to support new functional roles.