Abstract
We derive a twolayer multiplex Kuramoto model from WilsonCowan type physiological equations that describe neural activity on a network of interconnected cortical regions. This is mathematically possible due to the existence of a unique, stable limit cycle, weak coupling and inhibitory synaptic time delays. We study the phase diagram of this model numerically as a function of the interregional connection strength that is related to cerebral blood flow and a phase shift parameter that is associated with synaptic GABA concentrations. We find three macroscopic phases of cortical activity: background activity (unsynchronized oscillations), epileptiform activity (highly synchronized oscillations) and restingstate activity (synchronized clusters/chaotic behaviour). Previous network models could hitherto not explain the existence of all three phases. We further observe a shift of the average oscillation frequency towards lower values together with the appearance of coherent slow oscillations at the transition from restingstate to epileptiform activity. This observation is fully in line with experimental data and could explain the influence of GABAergic drugs both on gamma oscillations and epileptic states. Compared to previous models for gamma oscillations and restingstate activity, the multiplex Kuramoto model not only provides a unifying framework, but also has a direct connection to measurable physiological parameters.
Introduction
Fast electrochemical processes taking place on a complicated cytoarchitectural network structure render the human brain a highly complex dynamical system. Brain activity, as measured directly via EEG or MEG, or indirectly by means of MRI recordings, reveals characteristic macroscopic patterns such as oscillations in various frequency bands^{1}, synchronization^{2,3,4}, or chaotic dynamics^{5}. Generally it is believed that macroscopic activity (involving neurons) is closely related to highlevel functions such as cognition, attention, memory or task execution. To understand the mechanisms of this correspondence, both the overall network structure of the brain and the local properties of neural populations have to be taken into account^{4,6}. Regarding the latter, neural inhibition seems to be essential for cortical processing^{7}.
Two physiological phenomena received much attention lately in terms of a mathematical understanding: restingstate activity and gamma oscillations. Restingstate activity is spontaneous, highly structured activity of the brain during rest and can be described in terms of networks of simultaneously active brain regions^{8,9}. Models of restingstate networks often rely on anatomical networks derived from histological or imaging data and on local interactions between populations of excitatory and inhibitory neurons^{10,11,12,13}. Oscillatory neural activity in the gamma range ( Hz) is potentially related to consciousness and the binding problem although its precise function remains unclear^{14}. To understand the origin of gamma oscillations, two mechanisms have been proposed^{15}. One describes interactions between inhibitory neurons together with an external driving force^{16,17}. The other mechanism is based on excitatoryinhibitory coupling with synaptic time delays^{18,19,20}. The relation of gamma oscillations and inhibition is experimentally well established. In mice^{21}, rats^{22} and humans^{3,23}, a decrease of GABAconcentrations (gammaaminobutyric acid is the main inhibitory neurotransmitter in mammals) is accompanied by a strong attenuation of the gamma frequency band and sometimes by epileptiform activity.
Many existing network models for restingstate activity and gamma oscillations are based on singleneuron local dynamics^{10,11,16,17,18,19}. Since experimentally observed restingstate networks comprise individual regions containing about to individual neurons, we believe that a local description in terms of WilsonCowan equations is an attractive alternative. The subject of multiplex networks received recent attention with applications reaching from social and technological systems to economy and evolutionary games^{24,25}.
In this work we derive a simple twolayer multiplex model from classical physiological equations that is able to capture the main features of cortical activity such as oscillations, synchronization and chaotic dynamics. This model unifies the roles of neural network topology, synaptic time delays and excitation/inhibition. It provides a closed framework for simultaneously understanding the origin of restingstate activity and gamma oscillations.
Results
Derivation of the multiplex Kuramoto model
We consider cortical regions indexed by , see Fig. 1a. Each region is populated by ensembles of excitatory and inhibitory neurons (e.g. pyramidal cells and interneurons). We define the activity level of a region as the fraction of firing excitatory (inhibitory) neurons of the total number of excitatory (inhibitory) neurons in that region at a unit time interval and denote it by (). Neglecting for the moment interactions between different regions, we assume that individual cortical regions obey the WilsonCowantype dynamics^{20}
where is a sigmoidal response function, and are realvalued feedback parameters and and are positive synaptic coefficients (). and account for external inputs, e.g. from sensory organs. We now introduce interactions among regions of the network by replacing in Eq. (1) by
Here , , , are positive synaptic coefficients linking regions and . accounts for transmission delays at inhibitory synapses (not to be confused with axonal conduction delays). In physiology, can be altered by changing the synaptic concentration of GABA^{18,19,21,22}. In the present model, we assume that is proportional to the average synaptic GABA concentration in the brain. To derive a multiplex Kuramoto model (MKM) from Eqs. (1) and (2), we make the following three assumptions:
(i) Homogeneity. Cortical regions exhibit nearly identical dynamical behavior. We therefore assume the following parameters to be constant across regions,
for all , up to small perturbations, denoted by .
(ii) Stable local oscillations. We choose the parameters such that each uncoupled system Eq. (1), under the assumption given in Eq. (3), has a unique exponentially stable limit cycle . As a consequence, after a transient time solutions of Eq. (1) can be written as
where (t) is an arbitrary solution of Eq. (1) on ^{26,27}. accounts for specific initial values. Let denote the period of . We assume that the frequency lies in the physiological gamma range.
(iii) Weak coupling. Interactions between adjacent regions are weak and inhibitoryinhibitory interactions are very weak in the sense that,
for all . These assumptions are justified because the number of synaptic connections within a cortical region is much larger than between regions and excitatory neurons outnumber inhibitory neurons by approximately one order of magnitude^{28,29}.
Figure 1b summarizes the connectivity structure between regions and . Region receives excitatory (green arrows) and inhibitory (red arrows) inputs plus feedback (blue arrows), magnitudes are indicated by the arrow labels. For the sake of clarity, arrows representing inputs of magnitude , and are drawn in continuous, dashed and dotted style, respectively.
Under these assumptions, the system Eq. (1) with Eq. (2) is equivalent (see SI) to a twolayer MKM
Here describes the deviations from the uncoupled phases that are associated with solutions of the uncoupled system Eq. (4). Accordingly, describes the deviations from the uncoupled oscillation frequency . Time has been rescaled, see SI. is the adjacency matrix of the excitatoryexcitatory interaction network as defined in Eq. (5) and is a linear combination of the adjacency matrices and . accounts for the interaction between excitatory and inhibitory populations, see SI. and , are the corresponding average degrees. is a phase shift parameter related to the time delay via . is a global coupling constant that we assume to be proportional to the cerebral blood flow. This is reasonable because the latter is strongly correlated with the connection strengths of functional networks reconstructed in magnetic resonance imaging^{30}. , the socalled natural frequencies of the MKM, are the constant contribution to the frequency deviations . We take from a symmetric, unimodal random distribution , with mean . Since the 1parameter family of rotatingframe transformations , , leave Eq. (6) invariant for any , without loss of generality we assume, and . Note that for each solution with , there exists a solution with and . Physiological processes changing , and occur on a much slower timescale than neural activity.
It is known that weakly coupled, nearly identical limitcycle oscillators can be described in terms of phase variables^{27,31,32,33}. However, in terms of the new variables , interactions between cortical regions take place on two independent layers representing excitatoryexcitatory and excitatoryinhibitory coupling, respectively and the complicated connectivity structure of Fig. 1b reduces to a simple twolayer multiplex structure. Figure 1c shows the variable transformation from activity variables and , to the phase variable for any cortical region. In the unperturbed case, , the limit cycle is parametrized by . Since is exponentially stable, perturbations of activity dynamics lead to phase deviations . For we recover the Kuramoto model on a single network, see SI and^{34,35,36,37,38,39}.
Order parameters
We characterize solutions of Eq. (6) by the following order parameters:
Synchronization
We define the order parameter^{32,33,34,40}
It takes values between (no synchronization) and (full synchronization)^{27}. Let denote its time average .
Chaotic dynamics
The instantaneous largest Lyapunov exponent is given by
where measures the separation between a reference trajectory and a perturbed one . is the initial separation at and is the norm, see SI. For large times, approaches the “true” largest Lyapunov exponent, .
Average frequency deviation
We look at average frequency deviations across all regions,
once a stationary state is reached.
Numerical simulation of the model
Synchronization
We find that synchronization depends on the coupling strength and phase shift , Fig. 2a. For , we expect (see SI) a transition from an unsynchronized to a synchronized state at a critical value , which is confirmed by our simulations, Fig. 2a. With , stronger coupling is required for this transition to occur. Above a value of approximately , a global synchronized state ceases to exist.
Chaotic dynamics
Above the synchronization threshold, , synchronization and chaotic dynamics are mutually exclusive, see Fig. 2b. For small values of , there exists a small chaotic region () at the Kuramoto transition, in agreement with the wellknown results for , see SI. This region is expanding with increasing values of . At the boundary to the synchronized region, increasingly large values of are obtained. peaks at , for . In the unsynchronized region, , the dynamics is not chaotic, . For and , which constitutes the largest fraction of the chaotic region, the smallest values of Lyapunov exponents that we obtain are between and . Those values typically occur close to the border to the unsynchronized region, where is close to . For comparison, we note that at the classical Kuramoto transition ( and ), where chaotic behavior of the system is out of question^{35}, values of maximally 0.07 are encountered in our model setup. Figure 2c integrates both results (synchronization and chaotic dynamics) into a schematic phase diagram that clearly exhibits three phases.
Spectral properties
Figure 3a–c shows the stationary distributions of frequency deviations for selected values in the (,)plane. For , the distributions are practically identical for different values of , Fig. 3a. For , at , a synchronization peak appears close to frequency zero. With increasing , this peak moves towards increasingly negative values, until . Between and , the distribution is rapidly becoming broader and shifts towards positive values. After reaching a maximum at , it is finally centered around zero again, Fig. 3b. , is similar, however larger positive and negative values for occur, Fig. 3c.
Figure 2d shows the average frequency deviation as a function of and . As expected (see SI), we find frequency suppression associated with synchronization in the region of large and small , but also for large and intermediate . For fixed , maximal frequency suppression occurs at . For large and large (chaotic region) we find slightly positive .
Robustness issues
Homogeneity
The derivation of the MKM is based on three key assumptions, see Eqs. (3)–(5). If Eq. (3) is violated, i.e. the ensemble of uncoupled WilsonCowan oscillators is strongly heterogenous, several oscillation periods may occur . As a consequence, weak interactions become frequencymodulated^{27}: Two oscillators interact only if their frequencies and are similar, in the sense that , where and are small numbers.
Uniqueness of local oscillations
Regarding Eq. (4), discarding the uniqueness of the limit cycles would result in heterogenous coupling strengths , or .
Stability of local oscillations and weak coupling
In contrast, both the exponential stability of the limit cycles and the weak coupling assumption, Eq. (5), are strictly necessary for the derivation of the MKM, since they allow for a dimensional reduction from activity to phase deviation variables (see SI). If the dimensional reduction can not be carried through, the full system Eq. (1) with Eq. (2) has to be studied, whose properties are much harder to access.
Numerical simulation
We tested the model for robustness with respect to the particular choice of parameters. As suggested by various brain atlases and cortical parcellation schemes, a number of cortical regions seems reasonable^{41,42}. We tested up to and found no deviations from the presented qualitative picture. For the link density , we find that as long as it exceeds the percolation threshold, , differences in simulations are marginal. Finally, we observe that like in the original Kuramoto model^{32,33}, for different natural frequency distributions the qualitative behaviour remains practically unchanged as long as is unimodal and symmetric.
Discussion
In several variants of singlelayer Kuramoto models with a phase shift or a time delay, frequency suppression appears^{37,39,43,44}. In addition^{39}, mentions chaotic behavior. However, the existence of the phase diagram with the three distinct macroscopic phases can not be inferred from any of those models to the best of our knowledge.
In Ref. 45, several modes of synchronization are reported for a Kuramoto model on two interconnected networks with an internetwork time delay. While this computational model does not exhibit chaotic or unsynchronized phases, it suggests that a more complicated network topology can lead to a deeper structure within the synchronized phase in Kuramototype models.
Since the present work emphasizes the derivation of the MKM and the study of its stationary properties, we did not investigate the details of the synchronization transition. In this context we mention that the emergence of synchronization follows different paths in different types of networks^{46}. Further, if a correlation between natural frequencies and network properties is assumed, explosive synchronization and hysteretic effects may appear^{47}.
Summarizing, we can show mathematically that a set of weakly coupled WilsonCowan oscillators on a cortical network with a synaptic time delay between excitatory and inhibitory neural populations is identical to a simple Kuramototype phase model on a twolayer multiplex network. Numerical investigations of this model reveal the presence of three distinct macroscopic phases in the space of control parameters (associated with cerebral blood flow) and (associated with synaptic GABA concentration). For couplings , activities of individual cortical regions show independent oscillatory behavior (unsynchronized). Frequencies are distributed symmetrically around an average frequency that we assume to be located in the physiological gamma range. This dynamical state corresponds to “background activity” of the brain. For , two phases are possible: for small , the system becomes synchronized, which corresponds to “epileptic seizure activity” in physiology. For large , synchronized activity only appears in clusters; the system is chaotic in general. We identify this phase with “restingstate activity” in the brain. An important property of the present model is that the average oscillation frequency is shifted towards lower values when crossing the boundary to the synchronized phase. This could explain the experimental fact^{2,21,22,23} that a decrease of the GABA concentration in the restingstate both triggers the appearance of epileptiform slow waves and diminishes gamma activity in the brain.
Methods
Equation (6) is integrated with a standard 4thorder RungeKutta algorithm with time steps of size . The system size is , both layers are chosen to be ErdösRényi networks with . Natural frequencies are taken from a standard normal distribution, initial phase deviations from the interval . The first time steps are discarded to exclude transient effects. For the remaining time steps, , and are evaluated. All results are averaged over identical, independent runs with different realizations of the initial conditions.
Additional Information
How to cite this article: Sadilek, M. and Thurner, S. Physiologically motivated multiplex Kuramoto model describes phase diagram of cortical activity. Sci. Rep. 5, 10015; doi: 10.1038/srep10015 (2015).
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Acknowledgements
We acknowledge financial support from EC FP7 projects LASAGNE, agreement no. 318132 and MULTIPLEX, agreement no. 317532.
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Sadilek, M., Thurner, S. Physiologically motivated multiplex Kuramoto model describes phase diagram of cortical activity. Sci Rep 5, 10015 (2015). https://doi.org/10.1038/srep10015
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