Self-similarity and vanishing diffusion in fluvial landscapes.

Complex topographies exhibit universal properties when fluvial erosion dominates landscape evolution over other geomorphological processes. Similarly, we show that the solutions of a minimalist landscape evolution model display invariant behavior as the impact of soil diffusion diminishes compared to fluvial erosion at the landscape scale, yielding complete self-similarity with respect to a dimensionless channelization index. Approaching its zero limit, soil diffusion becomes confined to a region of vanishing area and large concavity or convexity, corresponding to the locus of the ridge and valley network. We demonstrate these results using one dimensional analytical solutions and two dimensional numerical simulations, supported by real-world topographic observations. Our findings on the landscape self-similarity and the localized diffusion resemble the self-similarity of turbulent flows and the role of viscous dissipation. Topographic singularities in the vanishing diffusion limit are suggestive of shock waves and singularities observed in nonlinear complex systems.

High gamma coherence between task-responsive sensory-motor cortical regions in a motor reaction-time task.

Electrical activity at high gamma frequencies (70-170 Hz) is thought to reflect the activity of small cortical ensembles. For example, high gamma activity (often quantified by spectral power) can increase in sensory-motor cortex in response to sensory stimuli or movement. On the other hand, synchrony of neural activity between cortical areas (often quantified by coherence) has been hypothesized as an important mechanism for inter-areal communication, thereby serving functional roles in cognition and behavior. Currently, high gamma activity has primarily been studied as a local amplitude phenomenon. We investigated the synchronization of high gamma activity within sensory-motor cortex and the extent to which underlying high gamma activity can explain coherence during motor tasks. We characterized high gamma coherence in sensory-motor networks and the relationship between coherence and power by analyzing electrocorticography (ECoG) data from human subjects as they performed a motor response to sensory cues. We found greatly increased high gamma coherence during the motor response compared with the sensory cue. High gamma power poorly predicted high gamma coherence, but the two shared a similar time course. However, high gamma coherence persisted longer than high gamma power. The results of this study suggest that high gamma coherence is a physiologically distinct phenomenon during a sensory-motor task, the emergence of which may require active task participation.NEW & NOTEWORTHY Motor action after auditory stimulus elicits high gamma responses in sensory-motor and auditory cortex, respectively. We show that high gamma coherence reliably and greatly increased during motor response, but not after auditory stimulus. Underlying high gamma power could not explain high gamma coherence. Our results indicate that high gamma coherence is a physiologically distinct sensory-motor phenomenon that may serve as an indicator of increased synaptic communication on short timescales (∼1 s).

A minimalist model for coevolving supply and drainage networks.

Numerous complex systems, both natural and artificial, are characterized by the presence of intertwined supply and/or drainage networks. Here, we present a minimalist model of such coevolving networks in a spatially continuous domain, where the obtained networks can be interpreted as a part of either the counter-flowing drainage or co-flowing supply and drainage mechanisms. The model consists of three coupled, nonlinear partial differential equations that describe spatial density patterns of input and output materials by modifying a mediating scalar field, on which supply and drainage networks are carved. In the two-dimensional case, the scalar field can be viewed as the elevation of a hypothetical landscape, of which supply and drainage networks are ridges and valleys, respectively. In the three-dimensional case, the scalar field serves the role of a chemical signal, according to which vascularization of the supply and drainage networks occurs above a critical 'erosion' strength. The steady-state solutions are presented as a function of non-dimensional channelization indices for both materials. The spatial patterns of the emerging networks are classified within the branched and congested extreme regimes, within which the resulting networks are characterized based on the absolute as well as the relative values of two non-dimensional indices.

Variational analysis of landscape elevation and drainage networks.

Landscapes evolve towards surfaces with complex networks of channels and ridges in response to climatic and tectonic forcing. Here, we analyse variational principles giving rise to minimalist models of landscape evolution as a system of partial differential equations that capture the essential dynamics of sediment and water balances. Our results show that in the absence of diffusive soil transport the steady-state surface extremizes the average domain elevation. Depending on the exponent m of the specific drainage area in the erosion term, the critical surfaces are either minima (0 < m < 1) or maxima (m > 1), with m = 1 corresponding to a saddle point. We establish a connection between landscape evolution models and optimal channel networks and elucidate the role of diffusion in the governing variational principles.