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Edited by: Viktor Jirsa, Movement Science Institute CNRS, France

Reviewed by: G. B. Ermentrout, University of Pittsburgh, USA; Meng Hu, Drexel University, USA

*Correspondence: Stewart Heitmann, School of Psychiatry, The University of New South Wales, Sydney, NSW, Australia. e-mail:

This is an open-access article distributed under the terms of the

Adaptive changes in behavior require rapid changes in brain states yet the brain must also remain stable. We investigated two neural mechanisms for evoking rapid transitions between spatiotemporal synchronization patterns of beta oscillations (13–30 Hz) in motor cortex. Cortex was modeled as a sheet of neural oscillators that were spatially coupled using a center-surround connection topology. Manipulating the inhibitory surround was found to evoke reliable transitions between synchronous oscillation patterns and traveling waves. These transitions modulated the simulated local field potential in agreement with physiological observations in humans. Intermediate levels of surround inhibition were also found to produce bistable coupling topologies that supported both waves and synchrony. State-dependent perturbation between bistable states produced very rapid transitions but were less reliable. We surmise that motor cortex may thus employ state-dependent computation to achieve very rapid changes between bistable motor states when the demand for speed exceeds the demand for accuracy.

Spatiotemporal waves of electrical activity are a ubiquitous phenomena in the cortex yet the functional relevance of these activity patterns remains unknown (Wu et al.,

Waves arise naturally in oscillatory media and nearly all aspects of brain function exhibit some form of oscillatory neural activity (Buzsáaki,

There is abundant empirical evidence for the role of synchronized oscillations in motor cortex. Voluntary movement production coincides with task-specific increases in the long-range synchronization of beta bandwidth (13–30 Hz) oscillations common to the motor cortex, the pyramidal tract neurons which project from the motor cortex to the spinal motor neurons, as well as the contra-lateral muscles recruited by the movement (Baker et al.,

Networks of phase-coupled oscillators are ideal for modeling the synchronization of oscillatory neural activity in a simplified mathematical form (Ermentrout,

The Kuramoto oscillator (Kuramoto,

Building on Ermentrout and Kleinfeld (

The present cortical model adopts a center-surround style connection topology rather than sparse long-range inhibitory connections. Center-surround connection topologies are widely regarded as a neurobiologically realistic model of the lateral connectivity in cortex and are known to induce both synchrony and wave patterns in oscillatory neural networks (e.g., Ermentrout,

Center-surround coupling was modeled with a smooth coupling kernel

Systematic manipulation of parameter

The nature of the transition boundary between synchrony and waves was explored further by numerical continuation of stable solutions between

Transitions between synchronous and wave patterns thus correspond to transitions between high and low values of phase coherence. The observed transitions in coherence are shown in Figure

The extent of the observed bistable region in Figure

Yet another type of bistable spatial pattern was also observed, which we call

The existence of bistable wave and synchronous solutions implies that the cortex can support both states using a fixed coupling topology. In this regime, the requirement for a physiological mechanism to modulate the activity of the inhibitory surround would vanish since transitions between waves and synchrony could instead be achieved by direct perturbation of the oscillator phases between the co-existing attractor basins.

Random perturbation of the oscillator phases failed to elicit state transitions between waves and synchrony. In light of the growing evidence for state-dependent computation in the brain (Buonomano and Maass,

We conjectured that a perturbation of each oscillator phase away from the phase of the

A simple state-dependent perturbation scheme was proposed that repels the phase of each oscillator away from the mean field phase by the amount
_{x} is the phase of the oscillator at spatial position

The optimal value of the perturbation amplitude parameter _{low} < _{high}) of the perturbation amplitude where state transitions occurred at 50% probability or better—we refer to this range of

Achieving reliable perturbations from synchrony to waves (not shown) is also possible with this method but these perturbations require very large amplitudes (

To constrain the choice of parameters in the state-dependent perturbation method, we surveyed the parameter space seeking the optimal combination of

Figure _{low} = 1.6 (SE 0.049) and _{high} = 3.2 (SE 0.052). The midpoint of the transition zone

Similarly, Figure _{low} = 2.2 (SE 0.049) and _{high} = 2.6 (SE 0.049).

The survey of all such transition zones across a range of coupling topologies is shown in Figure

The validity of the present model was verified by comparing the time course of state transitions in the simulated local field potential with previously published magnetoencephalogram (MEG) data showing modulated beta oscillations (20–25 Hz) in human motor cortex during a repetitive finger-tapping task (Boonstra et al.,

Figure

Figure

Figure

We note that the fast time scale of the oscillatory rhythms in the model is naturally defined by the autonomous frequency of the oscillators (22.5 ± 1 Hz). However, the slower time scale, at which synchronization occurs, scales linearly with the magnitude of the coupling weights. Since, the coupling weights were arbitrarily normalized to unity, the absolute transition times between different synchronization states should not be over-interpreted. Nonetheless, the relative transition times of kernel-driven versus perturbation-driven transitions can still be meaningfully compared.

To compare the simulations results with physiological data as well as illustrate the difference in average transition times between kernel-driven and perturbation-driven transitions, we followed the methods of Boonstra et al. (

Figure

Figures

The isotropic form of the center-surround coupling topology investigated thus far does not constrain the spatial orientation of the emergent wave patterns, which instead show little spatial ordering and change from simulation to simulation. Having assumed that distinct motor actions are encoded by the morphologies of distinct spatial wave patterns we now introduce an anisotropic form of the center-surround coupling topology (illustrated in Figures _{0}, _{1}) to vary as a function of the angular orientation (α ∈ [0°, 360°]) of the coupling direction where parameter _{0} defines the inhibitory strength along the major axis and parameter _{1} defines the inhibitory strength along the minor (orthogonal) axis of the kernel.

_{0} = 0.52 and _{1} = 0.64 were chosen so that the peak power response along the minor axis is twice that of the major axis while keeping the mean value of _{0} and _{1} centered on the optimal

We sought the minimum degree of anisotropy in the kernel that reliably evoked spatially oriented waves without departing far from the optimal bistable coupling (_{0} = 0.52 and _{1} = 0.64) that produced a 2:1 ratio in the peak spectral power of the minor axis relative to the major axis, while satisfying the constraint that ½(_{0} + _{1}) = 0.58 (Figure

A linear stability analysis was undertaken to ascertain those conditions under which isotropic coupling supports stable waves and synchrony. The analysis was restricted to the case of planar waves,

The stability of the planar wave was determined by considering the growth in time of the spatial perturbation,
_{n} is its eigenvalue. The real part of λ_{n} is the growth rate of the perturbation and must be non-positive for all

Figure

The spatial stability is formalized by the so-called

Figure

Spatiotemporal synchronization patterns in cortex were modeled using an array of Kuramoto phase oscillators that were spatially coupled using a neurobiologically inspired center-surround coupling topology. Controlled switching between self-organized patterns of waves and synchrony was achieved by manipulating the strength of the inhibitory connections in the coupling topology. This controlled switching reproduced the characteristic fluctuations in the spectral power observed in the beta bandwidth (20–25 Hz) oscillations in human motor cortex during a finger tapping task (Boonstra et al.,

We propose that spatiotemporal wave solutions represent active states in motor cortex whereas synchrony represents inactive motor states. This proposal finds support on information theoretic grounds because spatial wave patterns afford greater information capacity than do spatially synchronous patterns. The general agreement between the simulation results and experimental observations of attenuated beta power during movement production (Sanes and Donoghue,

We suggest that switching spatially synchronous oscillatory neural activity to a specific spatiotemporal wave pattern may be governed by excitatory thalamocortical projections modulating the lateral inhibitory connections within the motor cortex. Voluntary movements are known to be initiated in cortex when these thalamocortical projections are dis-inhibited by the basal ganglia which is itself implicated in movement selection (see Alexander and Crutcher,

Bistable cortical processes have previously been implicated in spontaneous switching between two distinct modes of alpha activity (Freyer et al.,

The optimal bistable coupling topology (

Simulation confirmed that repeated perturbation trials are capable of producing sequential transitions between wave and ripple patterns and that these transitions converge significantly faster than those achieved by toggling the coupling topology between monostable dynamical regimes. However, transition speed comes at the cost of reliability since not all perturbation trials successfully induce a state transition.

The general shape of the coupling kernel in the bistable regime seems biologically plausible. For isotropic coupling, bistability is observed in numerical simulation for

Ermentrout and Kleinfeld (

Numerical simulation supports our functional assignments of waves encoding motor action and synchrony encoding motor rest. Controlled switching between self-organized patterns of waves and synchrony in the cortical model reproduces the general character of task-dependent fluctuations of beta bandwidth oscillations observed in the local field potential of human motor cortex. Such task-dependent fluctuations are widely interpreted as dynamic reorganizations of the phases of the neural oscillatory activity underlying the local field potential. This reorganization is typically envisaged as a shift between in-phase synchronization and desynchronized modes of operation. This view is exemplified by the cortical idling hypothesis, however, it lacks a theoretical account of the neural mechanism behind these modes of cortical synchronization. Nor does it explain how the cortex might encode information within the desynchronized state. The present model demonstrates that task-dependent fluctuations of the local field potential can be achieved by switching cortex between modes of spatial synchrony and spatiotemporal wave patterns without resort to desynchronization.

The present model also offers a theoretical neural mechanism by which information may be encoded within the morphology of the spatial synchronization patterns. Numerical simulation demonstrates that the spatial morphology of the wave patterns can be manipulated by using an anisotropic local coupling topology to potentially encode a variety of motor movement states within the same patch of motor cortex. We suggest the laterally spreading inter-neurons of the motor cortex may likewise be modulated by cortico-thalamic projections that selectively enable a target motor action as instructed by the basal ganglia. The motor cortex may also exploit bistable cortical topologies to evoke rapid transitions between motor rest and the target motor action. In this case, fast-onset instructed-delay movements may be primed in motor cortex as a bistable ripple pattern (representing the motor ready state) which is later perturbed into a full wave pattern causing the target movement to unfold rapidly. However, such rapid transition speed comes at the cost of reliability since perturbation does not always induce a successful transition from ripple to wave. Perturbation between bistable cortical states may therefore only be a feasible mechanism for achieving rapid changes in brain states when the demand for speed exceeds the demand for accuracy. Whether the motor control system actually adopts such a strategy is an open question.

Spatially-coupled phase oscillator models are appropriate models to study the spatiotemporal synchronization patterns of oscillatory cortical activity such as that observed in motor cortex during movement production and motor preparation. In such cases, the spatial extent of the lateral coupling topology determines the spatial scale of the emergent patterns thus experimental observation of such patterns in cortex depends crucially on the spatial resolution of the recording sites. We anticipate that cortical oscillatory activity which appears to be desynchronized at course spatial resolution will likely reveal fine spatiotemporal synchronization structure when observed at higher spatial resolutions. Recent findings that the dendritic tree of cortical pyramidal neurons can actively discriminate the speed and direction of sequences of synaptic input suggests that the spatial resolution of wave-like synchronization patterns may even occur at scales below that of the network (Branco et al.,

Motor cortex was modeled as a two-dimensional sheet of non-locally coupled Kuramoto (^{2}, ω(

The _{0}, _{1} ∈ [0,1] define the strength of the inhibitory surround along the major and minor (orthogonal) axes of the kernel, respectively (see Figures

The linear stability of planar wave solutions,

Stability of the wave solution was determined by considering the growth in time of some spatial perturbation ψ(_{y} − ψ_{x−y}] to be ignored. Substituting Equation (12) back into (11) yields the linearized growth rate of the perturbation as

Since, Equation (13) is a linear differential equation in ψ, we apply the well-known solution,
_{n} is its eigenvalue. The real part of the eigenvalue represents the growth rate of the perturbation. Substituting Equation (14) in (13) gives

Equation (16) thus describes the growth rate of a spatial perturbation (with wavenumber _{n}) < 0 for all _{n} versus

For computational efficiency, Equation (5) was reformulated as
^{*} denotes two-dimensional convolution. In all cases, the size of the oscillator array was fixed at (^{2}). The natural oscillator frequencies

Hysteresis curves (Figure

The instantaneous phase coherence

The PFP was defined as the sum of the cosine components of the oscillator phases. It serves as a gross approximation of the local field potential for an equivalent patch of cortex by regarding the cosine of oscillator phase as analogous to membrane voltage potential. It can also be expressed exclusively in terms of

Spectrograms of the time-varying PFP signal (Figure _{c} = 5, _{b} = 1) that was scaled from 5 to 45 Hz using regular 0.5 Hz increments. Sampling frequency was 1000 Hz.

The state-dependent perturbation applied to each oscillator phase was defined as

We separately estimated the optimal values of perturbation amplitude

The upper and lower transition boundaries were estimated separately by fitting a transition probability curve with the logistic form

We nominated the mid-line of the transition zone as the optimal _{opt} for achieving _{opt}. The SEs of the mid-lines were estimated by pooling the SEs of the upper and lower bounds of the transition zones, respectively (see Figure

We sought the minimal level of asymmetry needed in the anisotropic coupling kernel (Figures _{opt}) we manipulated the degree of discrepancy Δ _{0} = _{opt} − Δ _{1} = _{opt} + Δ

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Michael Breakspear and Stewart Heitmann acknowledge the financial support of ARC Thinking Systems Grant TS0669860 and Brain Sciences UNSW. Michael Breakspear also acknowledges the financial support of BrainNRG. We thank Tjeerd Boonstra for generously supplying the data for Figures