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The authors have declared that no competing interests exist.

Conceived and designed the experiments: RTC KG JMC. Performed the experiments: KG JMC. Analyzed the data: RTC JMC. Contributed reagents/materials/analysis tools: RTC JMC. Wrote the paper: RTC JMC.

Understanding the principles governing the dynamic coordination of functional brain networks remains an important unmet goal within neuroscience. How do distributed ensembles of neurons transiently coordinate their activity across a variety of spatial and temporal scales? While a complete mechanistic account of this process remains elusive, evidence suggests that neuronal oscillations may play a key role in this process, with different rhythms influencing both local computation and long-range communication. To investigate this question, we recorded multiple single unit and local field potential (LFP) activity from microelectrode arrays implanted bilaterally in macaque motor areas. Monkeys performed a delayed center-out reach task either manually using their natural arm (Manual Control, MC) or under direct neural control through a brain-machine interface (Brain Control, BC). In accord with prior work, we found that the spiking activity of individual neurons is coupled to multiple aspects of the ongoing motor beta rhythm (10–45 Hz) during both MC and BC, with neurons exhibiting a diversity of coupling preferences. However, here we show that for identified single neurons, this beta-to-rate mapping can change in a reversible and task-dependent way. For example, as beta power increases, a given neuron may increase spiking during MC but decrease spiking during BC, or exhibit a reversible shift in the preferred phase of firing. The within-task stability of coupling, combined with the reversible cross-task changes in coupling, suggest that task-dependent changes in the beta-to-rate mapping play a role in the transient functional reorganization of neural ensembles. We characterize the range of task-dependent changes in the mapping from beta amplitude, phase, and inter-hemispheric phase differences to the spike rates of an ensemble of simultaneously-recorded neurons, and discuss the potential implications that dynamic remapping from oscillatory activity to spike rate and timing may hold for models of computation and communication in distributed functional brain networks.

How is the functional role of a particular neuron established within an ensemble? The concept of a neural tuning curve – the mapping from input variables such as movement direction to output firing rate – has proven useful in investigating neural function. However, prior work shows that tuning curves are not fixed but may be remapped as a function of task demands – presumably via high-level mechanisms of cognitive control. How is this accomplished? Brain rhythms may play a causal role in this process, but the coupling of single cells to network activity remains poorly understood. We investigated the coupling between rhythmic beta activity and spiking as macaques performed two different tasks. This coupling can be described in terms of a function that maps oscillatory amplitude and phase to instantaneous spike rate. Similarly to direction tuning, this “internal” tuning curve also exhibits task-dependent changes. We characterize these changes across a large ensemble of simultaneously-recorded cells, and consider some of the neuro-computational implications presented by cross-level coupling between single cells and large-scale networks. In particular, relative to the slow time-scale of behavior, the observed beta-to-rate mappings may prove useful for modulating winner-take-all dynamics on intermediate time-scales and relative spike timing on fast time-scales.

Our understanding of the biophysical mechanisms governing the dynamics of single neurons has increased dramatically over the past decades. In contrast, a principled understanding of the mechanisms governing ensembles of interacting neurons – from local cortical microcircuits to discrete functional areas to large-scale brain networks – remains elusive. Recent imaging advances have generated detailed structural maps that span from the micro-scale of local synaptic connectivity

Several groups have proposed that neuronal oscillations play a critical role in the dynamic coordination of multi-scale brain networks

CLC between spikes and internally-generated brain rhythms remains less well understood than the coupling between spikes and externally-associated factors such as visual orientation

However, compared to other rhythms such as theta and gamma, the cellular/network origins

Therefore, a primary purpose of this study was to provide a quantitative characterization of CLC between the motor beta rhythm (10–45 Hz) and a large ensemble of simultaneously-recorded neurons in primary motor cortex (M1) across two distinct but related behavioral tasks (

A) Schematic of the MC task, where monkeys use their right arm to perform a delayed center-out reaching task to move an on-screen cursor from a center cue to one of 8 peripheral targets. B) Schematic of the BC task, where monkeys use changes in the firing rates of a subset of recorded cells in order to move a cursor from the center to one of 8 targets (irrespective of physical movement). C) Timing of different trial sub-stages in the MC and BC tasks. Trials start with the appearance of central cue. A hold period (MC: 500 ms; BC: 100 ms) begins once the cursor enters the central cue. Upcoming target appears onscreen once cursor enters center. Go cue (central cue color change) indicates that monkeys can move the cursor to the designated target (via hand movement in MC or firing rate changes in BC), with a range of movement times. Holding the cursor within the target (MC: 400 ms; BC: 50 ms) triggers juice reward (500 ms), followed by the start of the next trial.

Here we present several findings. First, we provide a descriptive (phenomenological) model of the coupling between the instantaneous spike rate of a given cell and frequency-specific oscillatory activity. Importantly, this model accounts separately for the influence on rate of oscillatory amplitude, phase, and the interaction between amplitude and phase. Second, we show that this model describes the coupling for a large ensemble of cells, but that a wide range of model parameters holds across the population during a given task. In particular, some cells were more sensitive to amplitude than phase or vice versa, or had differential sensitivity to the interaction between amplitude and phase. Third, for a given cell the coupling of beta activity to spiking was stable across multiple sessions of a given task, but was often remapped when subjects switched to a different task. The parameter changes induced across the ensemble by this reversible remapping were reliable across multiple datasets. Fourth, it appears that this rhythm-to-rate mapping and task-dependent remapping have properties that would prove useful for the causal control of functional networks interactions. We conclude with a discussion of how these empirical results point to potential mechanisms for the control of neuro-computational processes. In sum, cross-level coupling between micro-scale spiking and meso- and macro-scale network activity appears to be a robust, flexible bridge linking together the different levels of brain organization required for effective perception, cognition, and action.

Two adult male rhesus monkeys (

Monkeys were trained to perform a delayed center-out reaching task using either their natural arm inside a Kinarm exoskeleton (BKIN Technologies, Kingston, Ontario) (Manual Control, MC), or under direct neural control through a brain-machine interface (BMI) and irrespective of overt physical movement (Brain Control, BC)

Analyses were done using MATLAB (Mathworks). Filtering to extract beta amplitudes and phases was performed by convolving signals with Gabor time-frequency basis functions (Gaussian envelope). A Gabor time-frequency atom is fully defined by three parameters; namely, the center time t_{0}, the center frequency v_{0}, and the duration parameter s_{0}. In the time domain, the Gabor atom g is given as g(t | t_{0}, v_{0}, s_{0}) = 2^{1/4} exp[( − 1/4) s_{0} − p(t − t_{0})^{2} exp[ − s_{0}] + 2p v_{0} (t − t_{0})]. Since there was no significant difference in the frequency corresponding to the power spectral peak (power of −46 10*log_{10}(µV^{2}/Hz) at a frequency of 28 Hz), a fixed center frequency v_{0} of 28 Hz centered on the observed PSD peak and a duration parameter s_{0} of −5.075 (frequency domain standard deviation of 3.57 Hz) were used to extract the “beta signal” this study. For the amplitude-to-rate, phase-to-rate, phase-difference-to-rate, and amplitude-to-weight mappings, a spatial average of all LFPs recorded from the 64 electrodes in one microelectrode array was generated and used as the raw input signal. Each 8 × 8 microelectrode array covers an area of 3.5 × 3.5 mm^{2}, and therefore this spatial average is similar in scale to a single ECoG macroelectrode. Two average signals s_{L} and s_{R}, were generated for left and right M1, respectively, prior to additional analyses. After concatenating separate recording blocks, the BC dataset had a duration of 410 minutes for monkey P (97 minutes for monkey R), while the MC dataset had a duration of 172 minutes for monkey P (58 minutes for monkey R). To compute event-related potentials (ERPs) and event-related time-frequency amplitude maps, the signal indices of go cue onsets were identified for BC and MC. For ERPs, trial epochs −1000 ms before to 10000 ms after go cue indices were extracted from signals s_{L} and s_{R} were averaged. For time-frequency analyses, first the signals s_{L} and s_{R} were filtered around a given center frequency as described above. 40 center frequencies spaced semi-logarithmically from 1 to 300 Hz were employed. Second, the amplitude of each filtered signal was normalized such that the mean amplitude across all data was 1. Third, trial epochs −1000 ms before to 10000 ms after go cue indices were extracted from the amplitude time series and averaged. RT-sorted single-trial analyses (e.g.,

A) Go-cue related ERP activity in right primary motor cortex (M1) during Brain Control (BC). The neurons driving the brain-machine interface are in right M1. B) Time-frequency plot showing frequency-specific changes in mean amplitude relative to the onset of the go cue (0 ms) in right M1 during BC, including amplitude changes in the theta (6 Hz), high beta (28 Hz), low gamma (36 Hz), and high gamma (>70 Hz) bands. Across all frequencies, color scale indicates increase (red) or decrease (blue) in amplitude relative to the mean of 1. Notice the strong drop in beta amplitude locked to the onset of the go cue. C) Smoothed single-trial traces of beta amplitude, sorted by movement duration (from go cue to cursor entering target). Vertical black line at −100 ms indicates when cursor entered the center cue, 0 ms is go cue onset. First curved black line indicates when cursor enters target, second indicates reward onset, and third indicates reward offset and beginning of the next trial. The sharp drop in beta amplitude during movement is followed by a beta amplitude increase during the reward delivery. D–F) As in A–C, during Manual Control (MC).

To generate the beta amplitude-to-rate mapping, first the time series of instantaneous amplitudes was extracted from one of the average signals (s_{L} or s_{R}) described above. Call this N × 1 vector of amplitudes x_{A}. Amplitudes were normalized such that the mean amplitude across all data was 1. Second, the spike times from one neuron were used to generate a N × 1 binary vector x_{S}, where x_{S}[t] equals 1 for spike times t and equals 0 otherwise. Third, a N × 2 matrix M_{1} = [x_{A} x_{S}] was formed. Fourth, this matrix M_{1} was truncated to allow reshaping of the array in a future step – given the number of amplitude bins to be used later (n_{b}), this matrix M_{1} was truncated to form the N_{t} × 2 matrix M_{2}, where N_{t} is the largest integer less than or equal to N for which n_{b} * P = N_{t} for some integer P. Fifth, the rows of the matrix M_{2} were sorted according to the amplitude values in the first column of M_{2}; M_{2} = sortrows(M_{2},1). Sixth, a 3-dimensional array of size P × n_{b} × 2 was created by reshaping the sorted matrix M_{2}: M_{3} = reshape(M_{2},[P n_{b} 2]). Seventh, the mean amplitude for each bin was computed: A = mean(M_{3}(:,:,1),1), where A is a 1 × n_{b} vector of amplitudes. Eighth, the average spike rate for each bin was computed: R = (S_{R}/P)*sum(M_{3}(:,:,2) = = 1), where S_{R} is the sampling rate and R is a 1 × n_{b} vector of spike rates. The rate values R over the amplitude support A describe the empirically-observed amplitude-to-rate mapping. Ninth, this histogram-based mapping was fit with 4-parameter sigmoidal function F_{S} using the MATLAB function lsqcurvefit.m, where F_{S}(a) = p_{1} + p_{2} tanh ((a − p_{3})/(2 p_{4})), where a<0 is beta amplitude and tanh is the hyperbolic tangent function. In order to assess task-related changes, amplitude-to-rate mappings were computed separately for the full BC and MC datasets. In order to assess within-task stability of the mappings, the BC dataset was split into two disjoint datasets, BC1 and BC2 consisting of odd and even trials, respectively, and the above procedure performed separately on each. Similarly, split-half reliability during MC was assessed using two disjoint datasets MC1 and MC2. The empirical-observed estimates of the beta phase-to-rate and phase-difference-to-rate mappings were produced in an identical way, where the x_{A} time series of step 1 was replaced with a N × 1 vector of instantaneous phases from one hemisphere (phase-to-rate) or a N × 1 vector of inter-hemispheric phase differences (phase-difference-to-rate). For the phase-to-rate mapping, a cosine-type function was used in fitting: F_{C}(q) = p_{1} + p_{2} cos (θ − p_{3}), where θ is beta phase, p_{2}>0, and q, p_{3} are in the interval [ − p, p). For the phase-difference-to-rate mapping, a von Mises-type function was used in fitting: F_{D}(j) = p_{1} + p_{2} exp[p_{3} cos (φ − p_{4})], where φ is the phase difference, p_{3}>0, and φ, p_{4} is in the interval [ − p, p). For target-specific and trial-stage-specific analyses, the data was presorted to extract only relevant time intervals.

To determine the amplitude-to-weight mapping that describes the multiplicative gain effect beta amplitude has on the phase-to-rate mapping, a procedure similar to that describe above was performed, but sorting datapoints jointly by amplitude and phase. That is, first the amplitude time series x_{A}, the phase time series x_{P}, and the spike time series x_{S} were combined into a matrix W = [x_{A} x_{P} x_{S}], such that each row represents the amplitude, phase, and spike status of one sample point. Second, the rows of W were sorted according to the values in the amplitude column and partitioned into n_{ab} (amplitude) bins, where each bin has the same number of sample points. Third, the data in each (amplitude) bin was further sorted into n_{pb} (phase) bins. Fourth, the spike rate for each (amplitude, phase) bin was computed, generating a n_{ab} × n_{pb} matrix of spike rates. Fifth, this matrix was used to constrain the fitting of the 7-parameter function describing the full beta-to-rate mapping: F_{B}(a,θ) = R_{BETA}(a,θ) = p_{1} + p_{2} tanh ((a − p_{3})/(2 p_{4})) + (p_{5} a + p_{6} a^{2}) cos (θ − p_{7}). Given the amplitude-to-rate mapping R_{AMP}(a) and the phase-to-rate mapping R_{PHASE}(θ) described above, the quadratic weight factor or amplitude-to-rate mapping w_{AMP}(a) = b_{1} a + b_{2} a^{2} can be extracted from the relation R_{BETA}(a,θ) = R_{AMP}(a) + w_{AMP}(a) R_{PHASE}(θ).

The modeling approach described above is inherently univariate and does not extend easily to multivariate approaches. For the multivariate analysis we used a procedure similar to that described in _{channel} LFP signals from a training dataset were filtered to generate a N_{channel} × N_{samples} complex-valued matrix, where for each entry the absolute value gives the beta amplitude and the argument gives the beta phase. Second, this matrix was used to fit the parameters describing the complex multivariate Gaussian distribution _{B} − μ_{B})^{H} R_{B}^{−1} (x_{B} − μ_{B})], where x is a N_{channel} × 1 vector of complex values, x_{B} = [x; conj(x)] is the 2N_{channel} × 1 (augmented) vector of complex values, μ_{B} is a 2N_{channel} × 1 vector of complex values representing the mean of x_{B}, R_{B} is the 2N_{channel} × 2N_{channel} (augmented) covariance matrix of x_{B}, b = 1/(π^{N} sqrt(det(R_{B}))) is a normalization term, conj(x) returns the complex conjugate of x, and the superscript H represents the conjugate transpose operation. Call this distribution, fit using all data, the baseline distribution p_{BASE}(x). Third, perform another distribution fitting using LFP data from spike times only; call this the spike-triggered distribution p_{ST}(x). Fourth, from a new training dataset of filtered LFP signals, extract the N_{channel} × 1 vector representing each sample point and compute the log-likelihood ratio L(x) = log[p_{ST}(x)/p_{BASE}(x)], generating a 1 × N_{samples} time series of log-likelihood ratio values. Call this time series L. Fifth, compute the L-to-rate mapping for this training dataset, as was described above for the amplitude-to-rate mapping. Sixth, find the best 4-parameter sigmoid fit F_{S}(L | p) for the L-to-rate mapping, where p is a 4 × 1 parameter vector (see sigmoid function definition above). Seventh, given a novel test dataset of filtered LFP signals, extract the N_{channel} × 1 vector representing each sample point and compute the predicted instantaneous spike rate estimate R_{EST}(x) = F_{S} [log[p_{ST}(x)/p_{BASE}(x)]]. Eighth, evaluate the prediction by computing the estimated-rate-to-measured-rate mapping, computed as was done to estimate the amplitude-to-rate mapping.

We first consider the mapping from beta amplitude alone to spike rate (amplitude-to-rate mapping), then from beta phase alone to spike rate (phase-to-rate mapping), before examining the joint influence of beta amplitude and phase on neuronal spiking. Both the amplitude-to-rate and phase-to-rate mappings exhibit task-dependent changes, as does the full beta-to-rate mapping, and we characterize the distribution of mapping parameters across the population. Next, we consider the dependence between spiking and the beta phase difference between left and right primary motor cortices; whereas amplitude and phase provide information about a single area, the phase difference provides macroscopic information about the relationship between areas. Finally, we consider the issue of spike dependence on meso-scale spatial patterns, and the effect that task-dependent changes have on the predictability of spiking.

Given the strong event-related changes in beta amplitude during both MC and BC (

Tuning curves characterize neural properties by conditioning spike rates on external factors such as movement direction or internal factors such as beta amplitude or phase. A–E show the external tuning properties for one neuron (sig045a), while F–J characterize the internal tuning properties for the same cell. A) Trial-related rate changes relative to baseline, collapsed across all targets. Go-cue onset is 0 ms. Four disjoint datasets are shown (BC1 & BC2, red; MC1 & MC2, blue). B) Target-specific rate changes (relative to baseline and trial-related activity) for the 8 BC targets. Solid lines show responses for BC1, dotted lines BC2. C) As in B, for MC. D) External tuning: joint display of trial- and target-related rate changes in BC1; color indicates spike rate. E) External tuning components (r_{baseline}, r_{trial}, and r_{target}) are learned from training data (BC1) and applied to novel test data (BC2) to predict instantaneous spike rate. F) Rate changes associated with different beta amplitudes. Beta amplitude normalized to a mean of 1. For this neuron, large beta amplitudes are associated with reduced firing and low amplitudes with increased spiking, but rate of change is task-dependent. G) As in F, conditioning spike rate on beta phase rather amplitude. H) Weight term governing the interaction between amplitude and phase (see _{baseline}, r_{amp}, r_{phase}, and w_{amp}) are learned from training data (BC1) and applied to novel test data (BC2) to predict instantaneous rate.

A) A diversity of amplitude-to-rate mappings hold across neurons during a given task; shown are 12 example neurons during BC. Dots indicate measured spike rates, lines show best-fit sigmoids. Increased beta amplitude associated with decreased rate in some neurons while others exhibit increased firing; vertical lines indicate cross-over points associated with change in firing-rate rank order within ensemble. B–G) Amplitude-to-rate mapping can change as function of task; six example neurons shown. H) Within-task CLC parameter stability assessed by computing amplitude-to-rate mapping for disjoint BC datasets; positive (negative) rate changes indicate that spike probability and amplitude are positively (negatively) correlated. I) As in H, for MC. J–K) Direct comparison of BC/MC datasets provides evidence for cross-task remapping; the amplitude-to-rate mapping for one task may not hold for different task. Similarity of J and K indicates reliable task-dependent remapping.

Across both the MC and BC tasks, a statistical dependence between spike rate and beta amplitude was observed for 86.7% of the cortical motor neurons examined (p<0.01 uncorrected randomized permutation test; c.f.

Interestingly, while there is variation in the exact crossover point for the population of amplitude-to-rate mappings – that is, the amplitude value where the sigmoid function intersects the baseline rate – the amplitude-to-rate mappings for many neurons intersect near the mean beta amplitude (

Thus, for an ensemble of neurons with similar baseline rates, the common intersection point of the amplitude-to-rate mappings near the mean beta amplitude results in two distinct spike density regimes for the ensemble. That is, when the instantaneous beta amplitude is above its mean value, then there is an associated rank ordering of the spike rates across the population of neurons (relative to the tonic baseline rate for each neuron). For example, given the seven example neurons shown in

The within-task diversity of amplitude-to-rate mappings observed across the ensemble of recorded neurons is complemented by a different type of diversity – within-neuron, cross-task diversity – that is associated with task switching. Interestingly, while the amplitude-to-rate mappings for a given neuron are relatively stable across multiple datasets as long as all recordings are acquired under the same task conditions (

As shown by

In addition to task-dependent remapping, the amplitude-to-rate mapping also exhibits changes as a function of trial-stage, as shown for three example neurons in

Importantly, both the within-task amplitude-to-rate mappings and the cross-task amplitude-to-rate remappings are stable across multiple data sets (

As with beta amplitude, cortical motor neurons exhibited a spike density dependence on beta phase. During BC (MC), the spike density of 87.3% (91.2%) of neurons exhibited cosine modulation when conditioned on beta phase (p<0.01, uncorrected randomized permutation test). That is, when considering beta phase alone (neglecting beta amplitude for now, but see beta-to-rate mapping below), the change in spike rate as a function of beta phase (phase-to-rate mapping) was well-fit by a 3-parameter cosine-type function (see

A–H) Eight example neurons that exhibit task-dependent remapping of beta phase-to-rate relationship; fits for all neurons shown in grey. Vertical lines indicate phase of maximal spiking for BC (red) and MC (blue). Preferred phase varies across neurons within a task, but all BC phases occur earlier than preferred MC phases. I) Preferred phase for BC vs. MC for all 53 neurons in right M1, exhibiting task-dependent shift to later phase for MC. J) Preferred beta phases map to times of peak spike probability relative to the ongoing beta rhythm; shown are peak times for all neurons in right M1, sorted relative to beta trough during BC (red). MC (blue) does not preserve the BC ensemble timing order. K) As in I, for neurons in left M1. L) As in J, for left M1.

For a given neuron, both the modulation depth (maximum rate minus baseline rate) and preferred beta phase (beta phase exhibiting the maximum spike rate) could change from one task to another. For example, some neurons show few cross-task changes (e.g.,

Given the peak in the LFP power spectrum at a center frequency of 28 Hz, the average beta cycle occurs over ∼36 ms. Therefore, we can convert a set of preferred phases into a set of most-probable spike times relative to a fixed point in the cycle of the ongoing beta rhythm. That is, if a neuron is going to spike only once in a given beta cycle, it is most likely to do so at its preferred beta phase, which corresponds to a fixed temporal lag relative to the peak of the beta waveform. The ordered set of these lags across the population imposes a probabilistic rank-ordering of spike times across the ensemble that spans ∼10 ms (sorted red dots in

Finally, the phase-to-rate mapping is strongly affected by the magnitude of beta amplitude (^{th} amplitude percentiles). The relative sizes of the phase-to-rate mappings shown in

Beta phase has a stronger impact on spike rate when beta amplitude is large, but gain modulation is not uniform across neurons. A–D show the phase-to-rate mappings for 9 example neurons, where instantaneous phases and spike times were pre-sorted into one of four bins based on beta amplitude (see

While the (within-task) phase-to-rate preferred beta angle is largely stable for most neurons across the full range of beta amplitudes (

Nonetheless, one consequence of this interaction between amplitude and the phase-to-rate modulation depth is that the spike timing preference relative to beta phase becomes stronger for higher beta power. For example, we saw above that neuron sig038a fires before neuron sig045a 61.0% percent of the time when looking at all beta cycles where each cell fires once. Sorting these individual cycles according to beta amplitude, however, reveals that sig038a spikes before sig045a only 56.2% of the time for cycles in the lowest decile beta amplitudes, compared to 72.0% of the time for the cycles in the highest decile of beta amplitudes. Similar effects are seen across the population, and thus variations in beta amplitude influence the probability of observing arbitrarily spike timing sequences within an ensemble.

As with the amplitude-to-rate mapping, the phase-to-rate mapping exhibits both task- and trial-stage-related changes (

Sorting trials based on the intended BC target prior to computing the phase-to-rate mapping reveals differences in baseline firing (due to direction tuning) as well as changes in the phase-to-rate modulation depth. A–L show 6 example neurons where the phase-to-rate modulation depth is positively correlated with the target-specific shift in baseline spike rate. Colors indicate phase-to-rate mappings computed from trials moving toward different targets. Shown are phase-to-rate mappings with target-specific baseline shifts included (A–D, I, J) or removed (E–H, K, L). For example, sig045b (A, E) fires the most for Target 8 (black) and the least for Target 4 (green), and also exhibits the largest phase-to-rate modulation depth for Target 8 and the least for Target 4 – that is, target-specific spike rates and phase-to-rate modulation depth are positively correlated (c.f. sig038a in B, F). In contrast, the phase-to-rate modulation depth is largely independent of target direction for sig043b and sig020a, despite the large target-specific shift in baseline spike rates. Finally, sig073b and sig043c provide examples of negative correlation between target-specific shifts in baseline spike rates and target-specific changes in phase-to-rate modulation depth. Correlations for these 6 examples are shown in M.

Because the spike density of a given neuron depends on the interaction between beta amplitude and phase, the most complete picture of the dependence between spike rates and the beta rhythm is given by the full beta-to-rate mapping (where the term ‘beta’ here implies both beta amplitude and beta phase). That is, the estimated spike rate R_{BETA}(a,θ) is a sum of two terms: an amplitude-only rate R_{AMP}(a) given by the amplitude-to-rate mapping, and another term that is the product of the phase-only rate R_{PHASE}(θ) and a weight factor that is a function of amplitude alone, w_{AMP}(a). Specifically,_{AMP}(a) is a sigmoidal function of amplitude, w_{AMP}(a) is a quadratic function of amplitude, and R_{PHASE}(θ) is a weighted cosine function of phase (see

However, while the beta rhythm exhibits strong event-related changes in power (c.f.,

A) Amplitude-to-rate mapping for one neuron (sig045a), computed for a range of center frequencies (1–100 Hz). Vertical axis gives filter center frequency, horizontal axis gives amplitude at that center frequency (normalized to a mean of 1 for all frequencies); color gives spike rate change relative to baseline. This neuron exhibits different responses for different frequencies; positive correlation of rate with theta and high gamma bands, but negative correlation with beta and low gamma. B) Same data as A, showing only four frequency bands at 6, 27, 34, and 90 Hz. Dots indicate measured rates, lines are best fit sigmoids. C) As in A, for the phase-to-rate mapping. Strongest response for this neuron is seen at 34 Hz for this neuron. D) As in B, for the phase-to-rate mapping. E) Range of rate change for sig045a as function of center frequency. Peaks of the amplitude- and phase-to-rate ranges are offset, with the amplitude-to-rate mapping strongest ∼27 Hz while the phase-to-rate mapping is strongest ∼34 Hz. F) As in A, for a finer frequency resolution from 20–40 Hz. G–J) As in F, for neurons sig062a, sig081b, sig031a, and sig029a. K) As in C, from 20 to 40 Hz for neuron sig045a. L–O) As in K, for neurons sig062a, sig081b, sig031a, and sig029a.

So far we have only considered the relation between (micro-scale) spiking of single neurons to (meso-scale) beta LFP activity averaged locally over several millimeters. That is, for a neuron in left M1, we examine the relation of its spike rate to the average beta activity recorded in left M1, while for neurons recorded in right M1 we examine the average field potential activity from right M1. The 8 × 8 electrode arrays used here cover 3.5 × 3.75 mm^{2}, such that the LFP signals recorded on opposite sides of the array are generated by distinct cell populations, and the spatial average of all 64 LFP signals from one array is a meso-scale signal similar in scale to the activity recorded from one electrocorticography (ECoG) electrode as employed for human neurosurgery. The results above show a clear dependence between micro- and meso-scale phenomena. However, the relation between micro-scale spiking activity and fully macro-scale phenomena – such as phase coupling between the left and right motor cortices – remains unclear. Are some neurons sensitive to the phase difference between left and right M1, above and beyond the influence that can be attributed to locally-generated field potential activity?

To address this question, we examined the relationship between (micro-scale) single-unit spiking and the (macro-scale) relative phase difference between the beta activity occurring in the left and right primary motor cortices. While this quantity neglects the beta amplitude in each area, it has the advantage of being statistically independent of the (absolute) beta phase local to the neuron. That is, knowing the instantaneous beta phase local to the neuron alone tells us nothing about the instantaneous beta phase in the other hemisphere; however, if we know the local (absolute) beta phase as well as the relative phase difference between the hemispheres, then we can calculate the distal (absolute) beta phase in the other hemisphere (

A) Joint probability density function of instantaneous phases of left and right primary motor cortex (M1). B) Same data as A, after change of variables to isolate the phase difference between left and right M1. Inter-hemispheric phase differences are statistically independent from local M1 phase. C) Example neuron sensitive to beta phase difference between left and right M1. During BC, lowest rate occurs near the most probable phase difference (peak of distribution in D), but the neuron increases spiking when the inter-hemispheric phase difference shifts to less probable values. Dots indicate measured rate, red line is best-fit von Mises type function (see

In contrast to the relative stability of the phase-difference distribution across different tasks, the phase-difference-to-rate mappings in

Over the population of recorded neurons, 68.4% exhibited significant variation in their spike rates as a function of the macro-scale, inter-hemispheric beta phase differences during BC. In order to facilitate comparisons among all recorded neurons, we computed the phase difference between left and right M1 using the average signal from all LFPs in each 8 × 8 electrode array (one array per area) – that is, we generated one time series of instantaneous phase differences against which we can examine the activity of all neurons. Interestingly, the phase-difference-to-rate mapping is often stronger when the LFP signal from an electrode proximal to the neuron is used (data not shown). However, using different pairs of LFPs for each neuron makes systematic comparisons across neurons more difficult. An alternative approach to multi-scale coupling – from macro- to meso- to micro-scale – is suggested by

The analyses above employ a strictly univariate approach – the univariate (meso-scale) signal representing the mean activity in M1 is generated by spatially averaging the individual LFPs from an 8 × 8 microelectrode array (3.5 mm × 3.5 mm; c.f.

A) Schematic of 8×8 microelectrode array implanted in left primary motor cortex (M1). Interelectrode separation is 500 microns on average. Color indicates groups with different numbers of electrodes, including 4 (blue), 16 (blue and green), 36 (blue, green, black), and 64 (blue, green, black, and red). B) Including more channels in a multivariate model improves prediction performance – the spike rate range increases as one considers groups of 4 (blue), 16 (green), 36 (black), or 64 (red) channels. Improvement of prediction performance suggests that distal electrodes contribute information independent of information from proximal electrodes. The predicted rate (x-axis) is shown in normalized units in order to emphasize the increase in range of the measured rate. C–H) Examples of 6 neurons where within-task predictability (red; train on BC data, test on novel BC data) is higher than cross-task predictability (blue; train on BC data, test on novel MC data). Dots indicate measured rates, lines give best linear fit. Within-task predictions are accurate for neurons across both BC and MC, implying that low cross-task predictive performance is due to task-dependent remapping rather than a lack of cross-level coupling in one of the tasks.

For example, ^{2}). This is despite the fact that fewer parameters are used per electrode to model the cross-level coupling than for the amplitude- and phase-to-rate mappings considered above. The fact that including spatial information (in the form of more distal electrodes) improves the spike-rate predictions for individual neurons suggests that cells may be sensitive to distinct spatiotemporal patterns of population or local network activity.

Because of the stability of the within-task mapping from beta activity to single unit spiking, when given training and (novel) test data collected under the same task conditions, the spike density of individual cells can be predicted well for a large subset of neurons (c.f.

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 (

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 _{1} and E_{2} (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 _{1} becomes active (_{2} to become active (

For a given neuron, the amplitude- and phase-to-rate mappings are produced by the combined synaptic input to that cell. But since information about the population rhythm is broadly accessible, neurons may use this information to dynamically organize relative activity within a functional ensemble. This activity includes winner-take-all interactions arising from recurrent local connectivity and relative spike timing among ordered sets of cells. A) Two excitatory cells (E_{1} and E_{2}, red) that connect to a common inhibitory cell (I, blue) – and which in turn provides inhibitory synaptic connections to E_{1} and E_{2} to form re-entrant or recurrent excitatory-inhibitory loops – can act as a simple winner-take-all (WTA) module. That is, given different levels of input to E_{1} and E_{2}, then either E_{1} or E_{2} (but not both) will produce tonic spike output. B–C) If two cells with different amplitude-to-rate mappings provide input to such a WTA module, then the WTA module will provide different output at low and high beta amplitudes. For example, given the purple (sig045a) and gold (sig062a) amplitude to rate mappings shown in _{1} generates spike output only at low amplitudes while E_{2} spikes at high amplitudes; E_{1} and E_{2} switch roles at the beta amplitude where the amplitude-to-rate sigmoids intersect. Critically, task-dependent remapping implies that this intersection point can shift to different values for each pair of input neurons. D) One second example trace of filtered LFP activity during BC showing beta amplitude (black) and phase (grey) variation over time. E) Amplitude-to-rate mappings for seven example neurons: sig015a (blue), sig029a (green), sig029b (red), sig031a (cyan), sig045a (purple), sig062a (gold), and sig081b (black). Baseline rate has been removed to emphasize rate changes associated with amplitude variation. F) Changes in spike rates (relative to baseline) over one second induced by the amplitude-to-rate mappings (color as in E). Colors are as in

Why would this be useful? First, recall the 12 cells shown in

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

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

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

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; _{internal}) is about half that of the range of the predicted rate generated from trial information (r_{external}). The sum of these terms (r = r_{external} + r_{internal}) often has a larger range than either r_{external} or r_{internal} alone. However, this sum assumes that r_{external} and r_{internal} are independent – an assumption that is not appropriate for many neurons. For example, while

Nonetheless, the majority of neurons show a dependence on “internal” beta-related factors that is not mediated by external factors such as direction tuning (

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,

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

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