# A mean field model for movement induced changes in the beta rhythm

- 1.1k Downloads
- 2 Citations

## Abstract

In electrophysiological recordings of the brain, the transition from high amplitude to low amplitude signals are most likely caused by a change in the synchrony of underlying neuronal population firing patterns. Classic examples of such modulations are the strong stimulus-related oscillatory phenomena known as the movement related beta decrease (MRBD) and post-movement beta rebound (PMBR). A sharp decrease in neural oscillatory power is observed during movement (MRBD) followed by an increase above baseline on movement cessation (PMBR). MRBD and PMBR represent important neuroscientific phenomena which have been shown to have clinical relevance. Here, we present a parsimonious model for the dynamics of synchrony within a synaptically coupled spiking network that is able to replicate a human MEG power spectrogram showing the evolution from MRBD to PMBR. Importantly, the high-dimensional spiking model has an exact mean field description in terms of four ordinary differential equations that allows considerable insight to be obtained into the cause of the experimentally observed time-lag from movement termination to the onset of PMBR (∼ 0.5 s), as well as the subsequent long duration of PMBR (∼ 1 − 10 s). Our model represents the first to predict these commonly observed and robust phenomena and represents a key step in their understanding, in health and disease.

## Keywords

Post-movement beta rebound Movement related beta decrease Neural mass Synchrony Power spectra Magnetoencephalography MEG Electroencephalography EEG Mean field## 1 Introduction

The modelling of brain rhythms is now a well established and vibrant part of computational neuroscience. Ever since the first recordings of the human electroencephalogram (EEG) in 1924 by Berger (1929) electrophysiological brain recordings have been shown to be dominated by oscillations (rhythmic activity in cell assemblies) across a wide range of temporal scales and scientists have sought to develop large scale models to describe the five main frequency bands of delta (1 − 4 Hz), theta (4 − 8 Hz), alpha (8 − 13 Hz), beta (13 − 30 Hz) and gamma (30 − 200 Hz). Moreover, it has long been known, since the early works of Jasper and Andrews (1936, 1938), that different brain rhythms can be localised to specific areas of the brain, and that these rhythms can be functionally distinct. For example, they showed that the beta rhythm present in the vicinity of the central sulcus was not affected by the presentation of a weak visual stimulus which suppressed the alpha rhythm, recorded from the occipital lobe. Given the challenge of modelling such complex behaviour it is perhaps no surprise that the long and industrious history of brain modelling has delivered more than one tool for this job. For issues that relate to spike times and their synchrony we can appeal to conductance based modelling, large scale network simulations and theories for understanding coupled oscillators, as recently surveyed in Ashwin et al. (2016). For questions that relate to understanding the coarse grained activity of either synaptic currents, mean membrane potentials or population firing rates, it is more natural to appeal to neural mass models, as reviewed in Coombes (2010). Indeed the latter have proven especially fruitful in providing large scale descriptions of how neural activity evolves over both space as well as time (Coombes et al. 2014; Pinotsis et al. 2014). However, these two approaches are dangerously close to creating a dichotomy so that there is no ideal computational modelling framework for understanding the role of spike-timing in generating localised brain rhythms.

A case in point that challenges the modelling tools currently available to us is the work of Jasper and Penfield (1949) who showed that beta rhythms generated from the motor cortex are suppressed during voluntary movement. This phenomenon is known as movement related beta decrease (MRBD). It wasn’t until some years later that the post-movement beta rebound (PMBR) (a temporary rise in amplitude of beta oscillations following movement cessation) was discovered (Riehle and Vaadia 2004; Pfurtscheller et al. 1996; Jurkiewicz et al. 2006). MRBD usually lasts for approximately 0.5 seconds and PMBR can last for up to several seconds. MRBD and PMBR are extremely robust, with clear amplitude changes in individual subjects and trials (Pfurtscheller and Lopes da Silva 1999). Interestingly, similar effects have been seen in studies where the subject is asked to think about moving, without carrying out the movement (Schnitzler et al. 1997; Pfurtscheller et al. 2005). These beta band modulations are believed to be caused by changes of synchrony within a relatively localised region of motor cortex (Stancák and Pfurtscheller 1995). Hence, MRBD is regarded as a special case of event-related desynchronisation (ERD) and PMBR a special case of event-related synchronisation (ERS).

Multiple papers have employed a large number of carefully controlled paradigms, in humans and animals to further investigate beta rebound phenomena and their modulations by tasks (see Cheyne 2013; Kilavik et al. 2013 for reviews). However, despite the robust nature of the beta task induced decrease and post stimulus rebound, the effect itself is relatively poorly understood and, at the time of writing, there has been, to our knowledge, no computational model capable of describing the beta rebound. In general, high amplitude beta oscillations are thought to reflect inhibition (Cassim et al. 2001; Gaetz et al. 2011), a hypothesis supported by quantifiable relationships between beta amplitude and local concentrations of the inhibitory neurotransmitter gamma aminobutyric acid (GABA) (Gaetz et al. 2011; Hall et al. 2011; Jensen et al. 2005; Muthukumaraswamy et al. 2013). This means that the observed MRBD might reflect an increase in processing during movement planning and execution and the PMBR might reflect active inhibition of neuronal networks post movement (Alegre et al. 2008; Solis-Escalante et al. 2012). An alternative, but not mutually exclusive hypothesis which has been proposed by Donner and Siegel (2011) (also outlined in Liddle et al. 2016) is that the beta signal, in part, represents long range integration across multiple brain regions (see also Liddle et al. 2013). Indeed this is a hypothesis supported by some evidence suggesting that large scale distributed network connectivity is mediated by beta oscillations (Brookes et al. 2011; Hall et al. 2014; Hipp et al. 2012).

To describe beta rebound we are faced with modelling a mesoscopic brain scale and in particular the changes of synchrony within a population of say 10^{6−7} excitatory pyramidal cells and their associated inhibitory interneurons. A neural mass model would be ideal for this scale, if the question of interest related to rate rather than spike, which suggests instead a simulation of a spiking neural network model. Unfortunately the latter can be notoriously hard to gain insight from for very large numbers of neurons. Ideally we would have access to a statistical neurodynamics providing a bridge between the two levels of description. This is an open mathematical problem. However, recent progress in obtaining a mean field reduction for a very specific choice of large scale spiking model has been made, and is ideally suited as a basis for breaking the dichotomy noted above. The single neuron model of choice being either a *θ*-neuron (So et al. 2014; Luke et al. 2013) or a (formally equivalent) quadratic integrate-and-fire (QIF) neuron (Pazó and Montbrió 1009), and the coupling being global and mediated by pulses (namely instantaneous synapses). Given the dense connections of connections in cortex on small scales (Klinshov et al. 2014) the global coupling assumption is not so restrictive for our purposes, though the assumption of fast synapses should be relaxed to incorporate more realistic post synaptic responses. This is precisely the issue we address here to develop a model capable of explaining MRBD and PMBR.

In what follows, Section 2 gives a recapitulation of cortical rebound, illustrated with newly acquired magnetoencephalography (MEG) data, along with some recently published results. A candidate large scale computational model is described in Section 3, utilising realistic synaptic conductance changes. The model is cast in both voltage and phase variable so that it can be understood both as a QIF network, and also as a phase-oscillator network so that a connection to Kuramoto type networks can be made. Importantly we develop an exact low dimensional mean field description capable of capturing the behaviour of a globally coupled network in the limit of a large number of neurons. The macroscopic variables of interest are now firing rate and mean membrane potential (for the voltage description) or the Kuramoto measure of synchrony (for the phase description). Importantly we show in Section 4 that the response of the mean field model to stimulation leads to spectrograms with all of the key features observed in MRBD and PMBR. Finally in Section 5 we emphasise the benefits of this new type of neural mass model, capable of tracking not only changes in firing rate but also coherence within a population, in describing cortical rebound, as well as discuss natural extensions to our initial single population approach.

## 2 MRBD and PMBR: a recapitulation

The above indicates that stimulus related beta power loss and post stimulus rebound are generally observable effects, seen in many cortical areas, during both sensory and cognitive tasks. Further, the reduction of rebound in disease has been robustly demonstrated. Thus, the generation of new mathematical models from which we can accurately predict task induced beta band dynamics are of interest.

## 3 A mean field model for spiking networks

There are a now a plethora of single neuron models for describing the spiking dynamics of cortical cells, many of which are extensions of the basic Hodgkin-Huxley model to incorporate nonlinear ionic currents that allow low frequency firing in response to constant current injection. Importantly mathematical neuroscience has identified a number of mechanisms that can generate ‘f-I’ curves with this property, with perhaps the most well known being the saddle-node on an invariant circle (SNIC) bifurcation (Ermentrout and Kopell 1986). This has led to the formulation of the elegant *θ*-neuron model (or Ermentrout-Kopell canonical model) which can mimic the firing and response properties of a cortical cell with a purely one dimensional dynamical system evolving on a circle. In a certain limit this is formally equivalent to the quadratic integrate-and-fire (QIF) model, also designed explicitly for understanding the generation of low firing rates in cortex (Latham et al. 2000). Given the simplicity of these models they are a natural candidate for cortical network studies, not only because they are computationally cheap, but because there is more chance to develop a statistical neurodynamics for such models than their biophysically complicated conductance based counterparts. Indeed mean field models for globally pulse-coupled networks have recently been developed by Luke et al. (2013) for *θ*-neuron models, and by Montbrió et al. (2015) for QIF models. To make these models more relevant to the interpretation of brain imaging signals, and in particular EEG and MEG, it is vital to augment the networks with more realistic forms of synaptic interaction and to move away from the overly restrictive assumption of fast pulsatile synaptic currents.

*N*QIF neurons each with a voltage

*v*

_{ i }, for

*i*= 1, … ,

*N*, evolving according according to the following set of coupled ordinary differential equations (ODEs):

*C*is a capacitance,

*η*

_{ i }is a constant current drive,

*κ*is a proportionality constant, which from now on (without loss of generality) will be set to unity, and

*I*

_{ i }is the synaptic current,

*g*(

*t*) represents a common time-dependent synaptic drive, which we shall take to arise through global coupling. This acts to push the voltage toward the synaptic reversal potential

*v*

_{syn}. If the synaptic current is positive (negative) we say that the synapse is excitatory (inhibitory). The QIF network has discontinuous trajectories since whenever

*v*

_{ i }reaches a threshold value

*v*

_{th}it is

*reset*to the value

*v*

_{reset}. This firing condition is also used to define an implicit set of firing times according to \(v_{i}({T_{i}^{m}})=v_{\text {th}}\), where \({T_{i}^{m}}\) denotes the

*m*th firing time of the

*i*th neuron. These in turn can be used to generate a set of conductance changes for the

*i*th neuron that we write in the form \({\sum }_{m \in \mathbb {Z}} s (t-{T_{i}^{m}})\), where

*s*(

*t*) is a fixed temporal filter. For a globally coupled network, with strength of synapse

*k*/

*N*, the total synaptic conductance change at each neuron is then

*s*(

*t*) =

*δ*(

*t*), where

*δ*is a Dirac-delta function. For a more realistic form describing a normalised post synaptic potential (PSP) with an exponential decay we may set

*s*(

*t*) =

*α*e

^{−α t }Θ(

*t*), whilst for a more general PSP with both a rise and fall time we would set \(s(t)=(1/\alpha _{1} - 1/\alpha _{2})^{-1} [\alpha _{1} \mathrm {e}^{-\alpha _{1} t} - \alpha _{2} \mathrm {e}^{-\alpha _{2} t}]{\Theta }(t)\). Here Θ(

*t*) is a Heaviside step function included to enforce causality, and the parameters

*α*,

*α*

_{1,2}are decay rates. Exploiting the fact that in all these cases

*s*(

*t*) is the Green’s function of a linear differential operator

*Q*then we may also write

*g*as the solution to the ODE system

*Q*to the choice of

*s*see Table 1. For the rest of this paper we shall work with the choice

*s*(

*t*) =

*α*

^{2}

*t*e

^{−α t }Θ, describing the so-called

*α*-function. This can be obtained from the difference of exponentials form described above in the limit

*α*

_{1,2}→

*α*, so that the corresponding differential operator

*Q*is

*s*(

*t*) to generate a synaptic current according to Eq. (2). From this one may in principle use Maxwell’s equations to determine the magnetic field that would underly an MEG-like signal. However, for simplicity we shall take the average network current to be a proxy for this physiological signal. This is given explicitly by

*g*(

*t*)(

*v*

_{syn}−

*V*(

*t*)), where

Synaptic filtering: Examples of differential operators and their corresponding temporal filters

| |
---|---|

1 | |

\(\displaystyle \left (1 \,+\, \frac {1}{\alpha } \frac {\mathrm {d} }{\mathrm {d} t} \right )\) | \(\displaystyle \alpha \mathrm {e}^{-\alpha t} {\Theta }(t)\) |

\(\displaystyle \left (1 \,+\, \frac {1}{\alpha _{1}} \frac {\mathrm {d} }{\mathrm {d} t} \right ) \left (1 \,+\, \frac {1}{\alpha _{2}} \frac {\mathrm {d} }{\mathrm {d} t} \right )\) | \(\displaystyle \left (\frac {1}{\alpha _{1}} \,-\, \frac {1}{\alpha _{2}} \right )^{-1} \left [\alpha _{1} \mathrm {e}^{-\alpha _{1} t} \,-\, \alpha _{2} \mathrm {e}^{-\alpha _{2} t} \right ]{\Theta }(t)\) |

*k*= 0) it is a simple matter to show that, when

*v*

_{th}→

*∞*and

*v*

_{reset}→−

*∞*, the frequency of oscillation is given by \(2\sqrt {\eta _{i}}/C\). For further simplicity we shall work with these choices for the values of

*v*

_{th}and

*v*

_{reset}. From now on we shall choose

*η*

_{ i }to be random variables drawn from a Lorentzian distribution

*L*(

*η*) with

*η*

_{0}is the centre of the distribution and Δ the width at half maximum. For simplicity we will consider the average frequency of oscillations \(\omega _{0} =2\sqrt {\eta _{0}}/C\) as a single parameter, distributed with a width at half maximum \({\Delta }\omega =2\sqrt {\Delta }/C\). In the coupled network, and if the frequencies of the individual neurons are similar enough, then one may expect some degree of phase locking (ranging from synchrony to asynchrony), itself controlled in part by the time-to-peak, 1/

*α*, of the synaptic filter.

*η*

_{ i }. In this figure we also track the evolution of two macroscopic order parameters. These are respectively the average membrane potential, given by Eq. (6), and the instantaneous mean firing rate

*r*:

*N*both the order parameters

*V*and

*r*show a smooth temporal variation. In the case of synchrony we would expect these mean field signals to show a periodic temporal variation, essentially following a trajectory reminiscent of a single QIF neuron receiving periodic drive, whilst for an asynchronously firing population these mean field signals would be constant in time (modulo finite size fluctuations). To quantify the degree of coherence (or phase-locking) within an oscillatory population it is convenient to use a Kuramoto order parameter. First though it is necessary to recast the model in terms of

*phase*variables.

*θ*-neuron it is natural to introduce the phase variable

*θ*

_{ i }∈ [−

*π*,

*π*) according to

*v*

_{ i }= tan(

*θ*

_{ i }/2) (so that \(\cos \theta _{i} = (1-{v_{i}^{2}})/(1+{v_{i}^{2}})\) and \(\sin \theta _{i} = 2v_{i}/(1+{v_{i}^{2}})\)). In this case we arrive at the

*θ*-neuron network

*P*(

*θ*) =

*δ*(

*θ*−

*π*) and is periodically extended such that

*P*(

*θ*) =

*P*(

*θ*+ 2

*π*), and we have used the result that \(\delta (t-{T_{j}^{m}}) = \delta (\theta _{j}(t)- \pi ) |\dot {\theta }_{j}({T_{j}^{m}})|\). The network defined by Eqs. (9) and (10) describes a set of

*N*phase variables interacting via spike triggered currents every time that

*θ*

_{ j }passes through

*π*. We will only consider the case that

*θ*

_{ j }increases through

*π*(so that spikes are only generated on the upswing of the corresponding voltage variable). In the absence of synaptic coupling an isolated

*θ*-neuron supports a pair of equilibria

*θ*

_{±}, with

*θ*

_{+}< 0 and

*θ*

_{−}> 0 for

*η*

_{ i }< 0, and no equilibria for

*η*

_{ i }> 0. In the former case the equilibria at

*θ*

_{+}is stable and the one at

*θ*

_{−}unstable. In neurophysiological terms, the unstable fixed point at

*θ*

_{−}is a threshold for the neuron model. Any initial conditions with

*θ*∈ (

*θ*

_{+},

*θ*

_{−}) will be attracted to the stable equilibrium, while initial data with

*θ*>

*θ*

_{−}will make a large excursion around the circle before returning to the rest state. For

*η*

_{ i }> 0 the

*θ*-neuron oscillates with frequency \(2\sqrt {\eta _{i}}/C\). When

*η*

_{ i }= 0 the

*θ*-neuron is poised at a saddle-node on an invariant circle (SNIC) bifurcation.

*θ*-neuron network is more straight forward to simulate as the model has continuous trajectories on an

*N*-torus (and there is no need to handle the discontinuous reset conditions). The Kuramoto order parameter is then defined as

*R*provides a measure of the degree of coherence within the network and Ψ is the average phase. If a population is perfectly synchronised then

*R*= 1 and similarly if the system is perfectly asyncronous then

*R*= 0. In Fig. 6 we show a sequence of snapshots of the Kuramoto order parameter for the dynamics shown in Fig. 5, as well as the time evolution of the degree of coherence.

*r*,

*V*) vary smoothly with time (for large

*N*) then so does the pair (

*R*,Ψ). Indeed in the large

*N*limit Luke et al. (2013), making use of the Ott-Antonsen (OA) ansatz (Ott and Antonsen 2008), have shown that a globally pulse-coupled network of

*θ*-neurons has an exact mean field description. The OA ansatz allows the reduction of globally coupled phase oscillators in the infinite size limit to an explicit finite set of nonlinear ODEs. These describe the macroscopic evolution of the system, in terms of the Kuramoto order parameter for synchronisation, as long as the distribution of phases is at most single peaked. In essence the ansatz is well suited to describing systems which dynamically evolve between an incoherent asynchronous state and a partially synchronised state, which is often the case in systems with interactions that are prescribed by harmonic functions, such as found in Eq. (9). To illustrate the type of network evolution that can be generated with different values of synchrony, see Fig. 7. Here we show some plots of the phase distribution for different values of the network coherence as well as the average network current that would be produced.

*Z*. Importantly the OA ansatz can also be used to obtain a mean field model in the presence of non-pulsatile synaptic, extending the approaches in Luke et al. (2013) and Montbrió et al. (2015). In Appendix B we show that this yields the mean field model described by the fourth order ODE system:

*H*(

*Z*) is a global state dependent drive to the population given by

*H*as function of

*W*using (13) from which we find

*H*(

*Z*) as the firing rate of the population that drives the global synaptic current. Figure 8 shows

*H*as a function of

*Z*. As expected

*H*takes its highest value when

*Z*≃e

^{ i π }, corresponding to high synchrony where all of the neurons fire and reset at the same time.

*r*,

*V*) in the reduced mean field model are plotted in Fig. 10. Unsurprisingly they behave similarly to the corresponding order parameters for the large scale simulations plotted in Fig. 5. Likewise, the mean field representation of (

*R*,Ψ), plotted in Fig. 11, agree extremely well with those shown in Fig. 6. For a further discussion of the bifurcation structure of this model see Coombes and Byrne (2017).

## 4 A mechanistic interpretation of movement induced changes in the beta rhythm

In Section 2 we demonstrated how an externally cued thumb movement caused a ∼ 0.5 s decrease in beta band power followed by a ∼ 2 − 4 s increase in beta band power, typifying MRBD and PMBR, respectively. The median nerve stimulation lasts ∼ 50 ms, however the evoked response lasts significantly longer. Upon examining the time-frequency spectrograms in Fig. 1 we observed an increase in low frequency activity at *t* = 0, which appears to last for ∼ 0.3 − 0.4 s, corresponding to the transduced median nerve stimulation and corresponding movement. We base the design of the external drive on this transduced signal.

We model the transduced signal as a temporally filtered drive *A* = *A*(*t*) that is received by every neuron in the model. In this case the dynamics of *Z* obey (14) under the replacement *η* _{0} → *η* _{0} + *A*, with *Q* _{ D } *A*(*t*) = Ω(*t*), where *Q* _{ D } is the differential operator obtained from *Q* in Eq. 5 under the replacement *α* → *α* _{ D }, and Ω(*t*) is a rectangular pulse, Ω(*t*) = πΘ(*t*)Θ(*τ* − *t*), where π can be interpreted as the strength of the drive. Note that the pulse is not applied until after transients have dropped off. As the evoked response in the experimental data last for ∼ 0.3 − 0.4 s we set *τ* = 0.4 s.

*Z*=

*R*e

^{ iΨ}, as well as a time series for the within population synchrony

*R*, in response to the drive described above. The colours correspond to the different time periods; before drive (blue), during drive (red), after drive (green). The system oscillates in partial synchrony with

*R*oscillating between ∼ 0.05 − 0.6 in the absence of drive. Once the drive is switched on the amplitude of these oscillations decreases and hence the power is also reduced, corresponding to MRBD. Note that the frequency also increases during this period. After the drive is switched off the level of coherence is increased as

*Z*is drawn towards the edge of the unit disk before spiralling back to the original limit cycle, corresponding to PMBR. Importantly the system does not rebound until

*t*≃ 0.5 s as seen in the real data. It should be noted that the stimulus corresponds to ∼ 80% of this time to rebound, however as the evoked response is present in ∼ 60 − 80% of the ∼ 0.5 s of MRBD in the real data we believe that this is a good fit.

*t*≃ 1.5 s, indicating a PMBR of roughly 1 s, which is not as long as the PMBR seen in our experimental data. An increase in power can be seen at around 26 Hz at

*t*≃ 0 s, corresponding to the increase in frequency during the drive on period. This high beta activity can be interpreted as the processing of the motor input.

Interestingly, we see a direct correlation between synchrony and the synaptic current. The time series in Fig. 12 (bottom) shows a peak in synchrony at *t* ≃ 0.5 s, just as the time series in Fig. 13 (left) shows a sharp increase in the amplitude of the synaptic current at *t* ≃ 0.5 s. This increase in amplitude can also be seen in the spectrogram (right). The strength of the drive π dictates the extent of the rebound. However it also prescribes the frequency of the oscillations amid the period when the drive is switched on. Therefore it is important to find the balance, where we have a prominent PMBR but also a physically realistic frequency during the interval when the stimulus is switched on. Although the increase in power at a higher frequency cannot be seen in our experimental data (Fig. 1), these time-frequency spectrograms were calculated for a small area of the motor cortex, it can be seen in the results obtained in Robson et al. (2015) (Fig. 2). It is widely believed that an increase in high beta and gamma activity is present in motor preparation and execution, in a more frontal region of the motor cortex.

The parameters were chosen such that the system oscillated at beta frequency and a significant MRBD and PMBR could be observed. The model is robust and can reproduce MRBD and PMBR for a wide region of parameter space.

## 5 Discussion

We have presented a mechanistic model that exhibits both MRBD and PMBR. This low dimensional model is derived from a corresponding high dimensional spiking network model and maintains a faithful representation of synaptic currents. In the reduced model these currents are driven by a firing rate that is itself a function of the complex Kuramoto order parameter. This makes a significant departure from the usual phenomenological neural mass description of neuronal population dynamics for which the firing rate is usually only a function of synaptic activity or mean membrane potential. Importantly the transient response of the reduced model is sufficiently rich to capture the emergent time scales of both MRBD and PMBR, when it is stimulated whilst operating in the beta frequency range. Although the length of PMBR observed in the reduced model was shorter than that seen in the experimental data, it is still within the documented 1 − 10 s range. Given that model responses are linked to changes in within-population coherence, this gives further support to the notion that beta band amplitude changes, and in particular those in MRBD and PMBR, are in fact due to changes in synchrony. More generally the model parameters can be altered so that the population oscillates at other frequencies, and hence, used to explain other ERD/ERS phenomena in the brain.

One natural extension of the model is to small networks, with coupling through the mean-field variables of a set of local populations, to describe systems with a mixture of excitation and inhibition. This would lead to a richer set of structures within the phase space for the network and provide further mechanisms for controlling emergent time-scales (say as orbits can be made to approach saddle structures, leading to a slow down in dynamics), which may help lengthen the PMBR, see Coombes and Byrne (2017) for a discussion of the bifurcation structure for a two population model. An interesting study would be to couple two identical populations, corresponding to left and right motor cortex areas, and drive one of the populations to inspect the bilateral response as seen in experimental data (Robson et al. 2015; Liddle et al. 2016). It is also possible that noise may play a constructive role, and the OA ansatz for a mean-field reduction can also be performed in this case (Lai and Porter 2905). The introduction of noise may also result in a lengthening of the PMBR.

Perhaps of more interest in using these next generation neural models to replace existing neural mass descriptions, is the development of reductive techniques to handle other nonlinear integrate-and-fire models, such as those of Izhikevich type (Izhikevich 2003). However, this is a substantial mathematical challenge since the OA reduction technique that we have employed here will break down. However, it is possible to extend the work presented here to include gap junction coupling (Laing 2015). There is now little doubt that gap junctions play a substantial role in the generation of neural rhythms, both functional (Hormuzdi et al. 2004; Bennet and Zukin 2004) and pathological (Velazquez and Carlen 2000; Dudek 2002). It is also interesting to consider the spatially extended version of this model, we report on this elsewhere (Byrne et al. 2017).

Another possible extension would be to include a variety of different synaptic receptor. We have assumed that PMBR and MRBD are mediated by the same type of synaptic receptor. However, Hall et al. (2011) suggest that MRBD is a GABA-A mediated process, whilst PMBR appears to be generated by a non-GABA-A receptor mediated process. A further model that distinguishes between receptors, may offer important insights into motor processes, and can be readily accomplished within the framework that we have presented here.

## Notes

### Acknowledgements

SC was supported by the European Commission through the FP7 Marie Curie Initial Training Network 289146, NETT: Neural Engineering Transformative Technologies. MJB was funded by a Medical Research Council (MRC) New Investigator Research Grant (MR/M006301/1). We also acknowledge Medical Research Council Partnership Grant (MR/K005464/1).

### Compliance with Ethical Standards

### Conflict of interests

The authors declare that they have no conflict of interest.

### Ethical approval

All procedures performed in studies involving human participants were in accordance with the ethical standards of the University of Nottingham Medical School Ethics Committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

## References

- Alegre, M., Alvarez-Gerriko, I., Valencia, M., Iriarte, J., & Artieda, J. (2008). Oscillatory changes related to the forced termination of a movement.
*Clinical Neurophysiology: Official Journal of the International Federation of Clinical Neurophysiology*,*119*(2), 290–300.CrossRefGoogle Scholar - Ashwin, P., Coombes, S., & Nicks, R. (2016). Mathematical frameworks for oscillatory network dynamics in neuroscience.
*Journal of Mathematical Neuroscience*,*6*(2), 1–92.Google Scholar - Bennet, M.V.L., & Zukin, R.S. (2004). Electrical coupling and neuronal synchronization in the mammalian brain.
*Neuron*,*41*, 495–511.CrossRefGoogle Scholar - Berger, H. (1929). Über das Elektroenzenkephalogram des Menschen.
*Archiv für Psychiatrie und Nervenkrankheiten*,*87*, 527–70.CrossRefGoogle Scholar - Brookes, M.J., Vrba, J., Robinson, S.E., Stevenson, C.M., Peters, A.M., Barnes, G.R., Hillebrand, A., & Morris, P.G. (2008). Optimising experimental design for MEG beamformer imaging.
*NeuroImage*,*39*(4), 1788–802.CrossRefPubMedGoogle Scholar - Brookes, M.J., Wood, J.R., Stevenson, C.M., Zumer, J.M., White, T.P., Liddle, P.F., & Morris, P.G. (2011). Changes in brain network activity during working memory tasks: a magnetoencephalography study.
*NeuroImage*,*55*(4), 1804–15.CrossRefPubMedGoogle Scholar - Brookes, M.J., Liddle, E.B., Hale, J.R., Woolrich, M.W., Luckhoo, H., Liddle, P.F., & Morris, P.G. (2012). Task induced modulation of neural oscillations in electrophysiological brain networks.
*NeuroImage*,*63*(4), 1918–30.CrossRefPubMedGoogle Scholar - Byrne, A., Avitabile, D., & Coombes, S. (2017). A next generation neural field model: the evolution of synchrony within patterns and waves.
*Physical Review E*. In prep.Google Scholar - Donner, T.H., & Siegel, M. (2011). A framework for local cortical oscillation patterns.
*Trends in Cognitive Sciences*,*15*(5), 191–199.Google Scholar - Cassim, F., Monaca, C.A.C., Szurhaj, W., Bourriez, J.-L., Defebvre, L., Derambure, P., & Guieu, J.-D. (2001). Does post-movement beta synchronization reflect an idling motor cortex?
*Neuroreport*,*12*(17), 3859–3863.CrossRefPubMedGoogle Scholar - Cheyne, D.O. (2013). MEG studies of sensorimotor rhythms: a review.
*Experimental Neurology*,*245*, 27–39.CrossRefPubMedGoogle Scholar - Coombes, S. (2010). Large-scale neural dynamics: simple and complex.
*NeuroImage*,*52*, 731–739.CrossRefPubMedGoogle Scholar - Coombes, S., & Byrne, A. (2017).
*Lecture notes nonlinear dynamics in computational neuroscience: from physics and biology to ICT, chapter next generation neural mass models*. PoliTO Springer.Google Scholar - Coombes, S., beim Graben, P., Potthast, R., & Wright, J. (Eds.) (2014).
*Neural fields, theory and applications*. Berlin: Springer.Google Scholar - Dudek, F.E. (2002). Gap junctions and fast oscillations: a role in seizures and epileptogenesis?
*Epilepsy Currents*,*2*, 133–136.CrossRefPubMedPubMedCentralGoogle Scholar - Ermentrout, G.B., & Kopell, N. (1986). Parabolic bursting in an excitable system coupled with a slow oscillation.
*SIAM Journal on Applied Mathematics*,*46*(2), 233–253.CrossRefGoogle Scholar - Gaetz, W., Macdonald, M., Cheyne, D., & Snead, O.C. (2010). Neuromagnetic imaging of movement-related cortical oscillations in children and adults: age predicts post-movement beta rebound.
*NeuroImage*,*51*(2), 792–807.CrossRefPubMedGoogle Scholar - Gaetz, W., Edgar, J.C., Wang, D.J., & Roberts, T.P.L. (2011). Relating MEG measured motor cortical oscillations to resting
*γ*-aminobutyric acid (GABA) concentration.*NeuroImage*,*55*(2), 616–21.CrossRefPubMedPubMedCentralGoogle Scholar - Gross, J., Kujala, J., Hamalainen, M., Timmermann, L., Schnitzler, A., & Salmelin, R. (2001). Dynamic imaging of coherent sources Studying neural interactions in the human brain.
*Proceedings of the National Academy of Sciences of the United States of America*,*98*(2), 694–9.CrossRefPubMedPubMedCentralGoogle Scholar - Hall, S.D., Stanford, I.M., Yamawaki, N., McAllister, C.J., Rönnqvist, K.C., Woodhall, G.L., & Furlong, P.L. (2011). The role of GABAergic modulation in motor function related neuronal network activity.
*NeuroImage*,*56*, 1506–1510.CrossRefPubMedGoogle Scholar - Hall, S.D., Prokic, E.J., McAllister, C., Ronnqvist, K.C., Williams, A.C., Yamawaki, N., Witton, C., Woodhall, G.L., & Stanford, I.M. (2014). Gaba-mediated changes in inter-hemispheric beta frequency activity in early-stage parkinson’s disease.
*Neuroscience*,*281*, 68–76.CrossRefPubMedPubMedCentralGoogle Scholar - Hillebrand, A., Singh, K.D., Holliday, I.E., Furlong, P.L., & Barnes, G.R. (2005). A new approach to neuroimaging with magnetoencephalography.
*Human Brain Mapping*,*25*(2), 199–211.CrossRefPubMedGoogle Scholar - Hipp, J.F., Hawellek, D.J., Corbetta, M., Siegel, M., & Engel, A.K. (2012). Large-scale cortical correlation structure of spontaneous oscillatory activity.
*Nature Neuroscience*,*15*(6), 884–90.CrossRefPubMedGoogle Scholar - Hormuzdi, S.G., Filippov, M.A., Mitropoulou, G., Monyer, H., & Bruzzone, R. (2004). Electrical synapses: a dynamic signaling system that shapes the activity of neuronal networks.
*Biochimica et Biophysica Acta*,*1662*, 113–137.CrossRefPubMedGoogle Scholar - Huang, M.X., Mosher, J.C., & Leahy, R.M. (1999). A sensor-weighted overlapping-sphere head model and exhaustive head model comparison for MEG.
*Physics in Medicine and Biology*,*44*(2), 423–40.CrossRefPubMedGoogle Scholar - Izhikevich, E.M. (2003). Simple model of spiking neurons.
*IEEE Transactions On Neural Networks*,*14*, 1569–1572.CrossRefPubMedGoogle Scholar - Jasper, H.H., & Andrews, H.L. (1936). Human brain rhythms. I. Recording techniques and preliminary results.
*Journal of General Physiology*,*14*, 98–126.Google Scholar - Jasper, H.H., & Andrews, H.L. (1938). Brain potentials and voluntary muscle activity in man.
*Journal of Neurophysiology*,*1*(2), 87– 100.Google Scholar - Jasper, H.H., & Penfield, W. (1949). Electrocorticograms in man: Effect of voluntary movement upon the electrical activity of the precentral gyrus.
*Archiv für Psychiatrie und Nervenkrankheiten*,*183*(1-2), 163–174.CrossRefGoogle Scholar - Jensen, O., Goel, P., Kopell, N., Pohja, M., Hari, R., & Ermentrout, B. (2005). On the human sensorimotor-cortex beta rhythm: sources and modeling.
*NeuroImage*,*26*(2), 347–355.CrossRefPubMedGoogle Scholar - Jurkiewicz, M.T., Gaetz, W.C., Bostan, A.C., & Cheyne, D. (2006). Post-movement beta rebound is generated in motor cortex: evidence from neuromagnetic recordings.
*NeuroImage*,*32*(3), 1281–9.CrossRefPubMedGoogle Scholar - Kilavik, B.r.E., Zaepffel, M., Brovelli, A., MacKay, W.A., & Riehle, A. (2013). The ups and downs of
*β*oscillations in sensorimotor cortex.*Experimental Neurology*,*245*, 15–26.CrossRefPubMedGoogle Scholar - Klinshov, V.V., Teramae, J.N., Nekorkin, V.I., & Fukai, T. (2014). Dense neuron clustering explains connectivity statistics in cortical microcircuits.
*PloS One*,*9*, e94292.CrossRefPubMedPubMedCentralGoogle Scholar - Lai, Y.M., & Porter, M.A. (2905). Noise-induced synchronization, desynchronization, and clustering in globally coupled nonidentical oscillators.
*Physical Review E*,*88*(01), 2013.Google Scholar - Laing, C.R. (2015). Exact neural fields incorporating gap junctions.
*SIAM Journal on Applied Dynamical Systems*, page to appear.Google Scholar - Latham, P.E., Richmond, B.J., Nelson, P.G., & Nirenberg, S. (2000). Intrinsic dynamics in neuronal networks. I.
*Theory Journal of Neurophysiology*,*83*, 808–827.PubMedGoogle Scholar - Liddle, E.B., Price, D., Palaniyappan, L., Brookes, M.J., Robson, S.E., Hall, E.L., Morris, P.G., & Liddle, P.F. (2016). Abnormal salience signaling in schizophrenia: the role of integrative beta oscillations.
*Human Brain Mapping*,*37*(4), 1361–74.CrossRefPubMedPubMedCentralGoogle Scholar - Luke, T.B., Barreto, E., & So, P. (2013). Complete classification of the macroscopic behaviour of a heterogeneous network of theta neurons.
*Neural Computation*,*25*, 3207–3234.CrossRefPubMedGoogle Scholar - Mary, A., Bourguignon, M., Wens, V., Op de Beeck, M., Leproult, R., De Tiège, X., & P. Peigneux. (2015). Aging reduces experience-induced sensorimotor plasticity. A magnetoencephalographic study.
*NeuroImage*,*104*, 59–68.CrossRefPubMedGoogle Scholar - Montbrió, E., Pazó, D., & Roxin, A. (2015). Macroscopic description for networks of spiking neurons.
*Physica Review X*,*5*(02), 1028.Google Scholar - Muthukumaraswamy, S.D., Myers, J.F.M., Wilson, S.J., Nutt, D.J., Hamandi, K., Lingford-Hughes, A., & Singh, K.D. (2013). Elevating endogenous GABA levels with GAT-1 blockade modulates evoked but not induced responses in human visual cortex.
*Neuropsychopharmacology: Official Publication of the American College of Neuropsychopharmacology*,*38*(6), 1105–12.CrossRefGoogle Scholar - Ott, E., & Antonsen, T.M. (2008). Low dimensional behavior of large systems of globally coupled oscillators.
*Chaos*,*18*, 037113.CrossRefPubMedGoogle Scholar - Pazó, D., & Montbrió, E. (1009). Low-dimensional dynamics of populations of pulse-coupled oscillators.
*Physical Review X*,*4*(01), 2014.Google Scholar - Pfurtscheller, G., & Lopes da Silva, F.H. (1999). Event-related EEG/MEG synchronization and desynchronization: basic principles.
*Clinical Neurophysiology*,*110*(11), 1842–1857.CrossRefPubMedGoogle Scholar - Pfurtscheller, G., & Solis-Escalante, T. (2009). Could the beta rebound in the EEG be suitable to realize a “brain switch”?
*Clinical Neurophysiology*,*120*(1), 24–29.CrossRefPubMedGoogle Scholar - Pfurtscheller, G., Stancák, A., & Neuper, C. (1996). Event-related synchronization (ERS) in the alpha band–an electrophysiological correlate of cortical idling: a review.
*International Journal of Psychophysiology*,*24*(1-2), 39–46.CrossRefPubMedGoogle Scholar - Pfurtscheller, G., Neuper, C., Brunner, C., & da Silva, F.L. (2005). Beta rebound after different types of motor imagery in man.
*Neuroscience Letters*,*378*(3), 156–9.CrossRefPubMedGoogle Scholar - Pinotsis, D., Robinson, P., beim Graben, P., & Friston, K. (2014). Neural masses and fields: Modelling the dynamics of brain activity.
*Frontiers in Computational Neuroscience 8*(149).Google Scholar - Riehle, A., & Vaadia, E. (2004).
*Motor cortex in voluntary movements: a distributed system for distributed functions*. CRC Press. ISBN 0203503589.Google Scholar - Robinson, S.E., & Vrba, J. (1998). Functional neuroimaging by synthetic aperture magnetometry (SAM). In Yoshimoto, T., Kotani, M., Kuriki, S., Karibe, H., & Nakasato, N. (Eds.),
*Recent advances in biomagnetism*(pp. 302–305): Tohoku University Press.Google Scholar - Robson, S.E., Brookes, M.J., Hall, E.L., Palaniyappan, L., Kumar, J., Skelton, M., Christodoulou, N.G., Qureshi, A., Jan, F., Katshu, M.Z., Liddle, E.B., Liddle, P.F., & Morris, P.G. (2015). Abnormal visuomotor processing in schizophrenia.
*NeuroImage: Clinical*.Google Scholar - Sarvas, J. (1987). Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem.
*Physics in Medicine and Biology*,*32*, 11–22.CrossRefPubMedGoogle Scholar - Schnitzler, A., Salenius, S., Salmelin, R., Jousmäki, V., & Hari, R. (1997). Involvement of primary motor cortex in motor imagery: a neuromagnetic study.
*NeuroImage*,*6*(3), 201–8.CrossRefPubMedGoogle Scholar - So, P., Luke, T.B., & Barretto, E. (2014). Networks of theta neurons with time-varying excitability Macroscopic chaos, multistability, and final-state uncertainty.
*Physica D*,*267*, 16–26.CrossRefGoogle Scholar - Solis-Escalante, T., Müller-Putz, G.R., Pfurtscheller, G., & Neuper, C. (2012). Cue-induced beta rebound during withholding of overt and covert foot movement.
*Clinical Neurophysiology: Official Journal of the International Federation of Clinical Neurophysiology*,*123*(6), 1182–90.CrossRefGoogle Scholar - Stancák, A., & Pfurtscheller, G. (1995). Desynchronization and recovery of beta rhythms during brisk and slow self-paced finger movements in man.
*Neuroscience Letters*,*196*, 21–24.CrossRefPubMedGoogle Scholar - Stevenson, C.M., Brookes, M.J., & Morris, P.G (2011).
*β*-band correlates of the fMRI BOLD response.*Human Brain Mapping*,*32*(2), 182–97.CrossRefPubMedGoogle Scholar - Timmermann, L., & Florin, E. (2012). Parkinson’s disease and pathological oscillatory activity: is the beta band the bad guy? - New lessons learned from low-frequency deep brain stimulation.
*Experimental Neurology*,*233*(1), 123–5.CrossRefPubMedGoogle Scholar - van Drongelen, W., Yuchtman, M., Van Veen, B.D., & van Huffelen, A.C. (1996). A spatial filtering technique to detect and localize multiple sources in the brain.
*Brain Topography*,*9*(1), 39– 49.CrossRefGoogle Scholar - van Veen, B.D., van Drongelen, W., Yuchtman, M., & Suzuki, A. (1997). Localization of brain electrical activity via linearly constrained minimum variance spatial filtering.
*IEEE Transactions on Bio-medical Engineering*,*44*(9), 867–80.CrossRefPubMedGoogle Scholar - Velazquez, J.L.P., & Carlen, P.L. (2000). Gap junctions, synchrony and seizures.
*Trends in Neurosciences*,*23*, 68–74.CrossRefGoogle Scholar - Vrba, J., & Robinson, S.E. (2001). Signal processing in magnetoencephalography.
*Methods (San Diego California)*,*25*(2), 249–71.CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.