Scale-freeness or partial synchronization in neural mass phase oscillator networks: Pick one of two?

Modeling and interpreting (partial) synchronous neural activity can be a challenge. We illustrate this by deriving the phase dynamics of two seminal neural mass models: the Wilson-Cowan firing rate model and the voltage-based Freeman model. We established that the phase dynamics of these models differed qualitatively due to an attractive coupling in the first and a repulsive coupling in the latter. Using empirical structural connectivity matrices, we determined that the two dynamics cover the functional connectivity observed in resting state activity. We further searched for two pivotal dynamical features that have been reported in many experimental studies: (1) a partial phase synchrony with a possibility of a transition towards either a desynchronized or a (fully) synchronized state; (2) long-term autocorrelations indicative of a scale-free temporal dynamics of phase synchronization. Only the Freeman phase model exhibited scale-free behavior. Its repulsive coupling, however, let the individual phases disperse and did not allow for a transition into a synchronized state. The Wilson-Cowan phase model, by contrast, could switch into a (partially) synchronized state, but it did not generate long-term correlations although being located close to the onset of synchronization, i.e. in its critical regime. That is, the phase-reduced models can display one of the two dynamical features, but not both.

Distinct criticality of phase and amplitude dynamics in the resting brain.

Converging research suggests that the resting brain operates at the cusp of dynamic instability, as signified by scale-free temporal correlations. We asked whether the scaling properties of these correlations differ between amplitude and phase fluctuations, which may reflect different aspects of cortical functioning. Using source-reconstructed magneto-encephalographic signals, we found power-law scaling for the collective amplitude and for phase synchronization, both capturing whole-brain activity. The temporal changes of the amplitude comprise slow, persistent memory processes, whereas phase synchronization exhibits less temporally structured and more complex correlations, indicating a fast and flexible coding. This distinct temporal scaling supports the idea of different roles of amplitude and phase fluctuations in cortical functioning.

Model selection for identifying power-law scaling.

Long-range temporal and spatial correlations have been reported in a remarkable number of studies. In particular power-law scaling in neural activity raised considerable interest. We here provide a straightforward algorithm not only to quantify power-law scaling but to test it against alternatives using (Bayesian) model comparison. Our algorithm builds on the well-established detrended fluctuation analysis (DFA). After removing trends of a signal, we determine its mean squared fluctuations in consecutive intervals. In contrast to DFA we use the values per interval to approximate the distribution of these mean squared fluctuations. This allows for estimating the corresponding log-likelihood as a function of interval size without presuming the fluctuations to be normally distributed, as is the case in conventional DFA. We demonstrate the validity and robustness of our algorithm using a variety of simulated signals, ranging from scale-free fluctuations with known Hurst exponents, via more conventional dynamical systems resembling exponentially correlated fluctuations, to a toy model of neural mass activity. We also illustrate its use for encephalographic signals. We further discuss confounding factors like the finite signal size. Our model comparison provides a proper means to identify power-law scaling including the range over which it is present.

Structure-function discrepancy: inhomogeneity and delays in synchronized neural networks.

The discrepancy between structural and functional connectivity in neural systems forms the challenge in understanding general brain functioning. To pinpoint a mapping between structure and function, we investigated the effects of (in)homogeneity in coupling structure and delays on synchronization behavior in networks of oscillatory neural masses by deriving the phase dynamics of these generic networks. For homogeneous delays, the structural coupling matrix is largely preserved in the coupling between phases, resulting in clustered stationary phase distributions. Accordingly, we found only a small number of synchronized groups in the network. Distributed delays, by contrast, introduce inhomogeneity in the phase coupling so that clustered stationary phase distributions no longer exist. The effect of distributed delays mimicked that of structural inhomogeneity. Hence, we argue that phase (de-)synchronization patterns caused by inhomogeneous coupling cannot be distinguished from those caused by distributed delays, at least not by the naked eye. The here-derived analytical expression for the effective coupling between phases as a function of structural coupling constitutes a direct relationship between structural and functional connectivity. Structural connectivity constrains synchronizability that may be modified by the delay distribution. This explains why structural and functional connectivity bear much resemblance albeit not a one-to-one correspondence. We illustrate this in the context of resting-state activity, using the anatomical connectivity structure reported by Hagmann and others.

Distinct fast and slow processes contribute to the selection of preferred step frequency during human walking.

Humans spontaneously select a step frequency that minimizes the energy expenditure of walking. This selection might be embedded within the neural circuits that generate gait so that the optimum is pre-programmed for a given walking speed. Or perhaps step frequency is directly optimized, based on sensed feedback of energy expenditure. Direct optimization is expected to be slow due to the compounded effect of delays and iteration, whereas a pre-programmed mechanism presumably allows for faster step frequency selection, albeit dependent on prior experience. To test for both pre-programmed selection and direct optimization, we applied perturbations to treadmill walking to elicit transient changes in step frequency. We found that human step frequency adjustments (n = 7) occurred with two components, the first dominating the response (66 ± 10% of total amplitude change; mean ± SD) and occurring quite quickly (1.44 ± 1.14 s to complete 95% of total change). The other component was of smaller amplitude (35 ± 10% of total change) and took tens of seconds (27.56 ± 16.18 s for 95% completion). The fast process appeared to be too fast for direct optimization and more indicative of a pre-programmed response. It also persisted even with unusual closed-loop perturbations that conflicted with prior experience and rendered the response energetically suboptimal. The slow process was more consistent with the timing expected for direct optimization. Our interpretation of these results is that humans may rely heavily on pre-programmed gaits to rapidly select their preferred step frequency and then gradually fine-tune that selection with direct optimization.