This paper is the product of the wave modelling community and it tries to make a picture of the present situation in this branch of science, exploring the previous and the most recent results and looking ahead towards the solution of the problems we presently face. Both theory and applications are considered.The many faces of the subject imply separate discussions. This is reflected into the single sections, seven of them, each dealing with a specific topic, the whole providing a broad and solid overview of the present state of the art. After an introduction framing the problem and the approach we followed, we deal in sequence with the following subjects: (Section) 2, generation by wind; 3, non-linear interactions in deep water; 4, white-capping dissipation; 5, non-linear interactions in shallow water; 6, dissipation at the sea bottom; 7, wave propagation; 8, numerics. The two final sections, 9 and 10, summarize the present situation from a general point of view and try to look at the future developments. Keywords

The nonlinear, metastable dynamics of the brain are essential for large-scale integration of smaller components and for the rapid organization of neurons in support of behavior. Therefore, understanding the nonlinearity of the brain is paramount for understanding the relationship between brain dynamics and behavior. Explicit quantitative descriptions of the properties and consequences of nonlinear neural networks, however, are rare. Because the local field potential (LFP) reflects the total activity across a population of neurons, nonlinearites of the nervous system should be quantifiable by examining oscillatory structure. We used high-order spectral analysis of LFP recorded from the dorsal and intermediate regions of the rat hippocampus to show that the nonlinear character of the hippocampal theta rhythm is directly related to movement speed of the animal. In the time domain, nonlinearity is expressed as the development of skewness and asymmetry in the theta shape. In the spectral domain, nonlinear dynamics manifest as the development of a chain of harmonics statistically phase coupled to the theta oscillation. This evolution was modulated across hippocampal regions, being stronger in the dorsal CA1 relative to more intermediate areas. The intensity and timing of the spiking activity of pyramidal cells and interneurons was strongly correlated to theta nonlinearity. Because theta is known to propagate from dorsal to ventral regions of the hippocampus, these data suggest that the nonlinear character of theta decreases as it travels and supports a hypothesis that activity dissipates along the longitudinal axis of the hippocampus.

[1] The variation of seaward and shoreward infragravity energy fluxes across the shoaling and surf zones of a gently sloping sandy beach is estimated from field observations and related to forcing by groups of sea and swell, dissipation, and shoreline reflection. Data from collocated pressure and velocity sensors deployed between 1 and 6 m water depth are combined, using the assumption of cross-shore propagation, to decompose the infragravity wave field into shoreward and seaward propagating components. Seaward of the surf zone, shoreward propagating infragravity waves are amplified by nonlinear interactions with groups of sea and swell, and the shoreward infragravity energy flux increases in the onshore direction. In the surf zone, nonlinear phase coupling between infragravity waves and groups of sea and swell decreases, as does the shoreward infragravity energy flux, consistent with the cessation of nonlinear forcing and the increased importance of infragravity wave dissipation. Seaward propagating infragravity waves are not phase coupled to incident wave groups, and their energy levels suggest strong infragravity wave reflection near the shoreline. The cross-shore variation of the seaward energy flux is weaker than that of the shoreward flux, resulting in cross-shore variation of the squared infragravity reflection coefficient (ratio of seaward to shoreward energy flux) between about 0.4 and 1.5.

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Oscillations in the hippocampal local field potential at theta and gamma frequencies are prominent during awake behavior and have demonstrated several behavioral correlates. Both oscillations have been observed to increase in amplitude and frequency as a function of running speed. Previous investigations, however, have examined the relationship between speed and each of these oscillation bands separately. Based on energy cascade models where “…perturbations of slow frequencies cause a cascade of energy dissipation at all frequency scales” (Buzsaki G. Rhythms of the Brain, 2006), we hypothesized that cross-frequency interactions between theta and gamma should increase as a function of speed. We examined these relationships across multiple layers of the CA1 subregion, which correspond to synaptic zones receiving different afferents. Across layers, we found a reliable correlation between the power of theta and the power of gamma, indicative of an amplitude-amplitude relationship. Moreover, there was an increase in the coherence between the power of gamma and the phase of theta, demonstrating increased phase-amplitude coupling with speed. Finally, at higher velocities, phase entrainment between theta and gamma increases. These results have important implications and provide new insights regarding how theta and gamma are integrated for neuronal circuit dynamics, with coupling strength determined by the excitatory drive within the hippocampus. Specifically, rather than arguing that different frequencies can be attributed to different psychological processes, we contend that cognitive processes occur across multiple frequency bands simultaneously with organization occurring as a function of the amount of energy iteratively propagated through the brain. NEW & NOTEWORTHY Often, the theta and gamma oscillations in the hippocampus have been believed to be a consequence of two marginally overlapping phenomena. This perspective, however, runs counter to an alternative hypothesis in which a slow-frequency, high-amplitude oscillation provides energy that cascades into higher frequency, lower amplitude oscillations. We found that as running speed increases, all measures of cross-frequency theta-gamma coupling intensify, providing evidence in favor of the energy cascade hypothesis.