Direct Verification of the Kinetic Description of Wave Turbulence for Finite-Size Systems Dominated by Interactions among Groups of Six Waves.

The present work considers systems whose dynamics are governed by the nonlinear interactions among groups of 6 nonlinear waves, such as those described by the unforced quintic nonlinear Schrödinger equation. Specific parameter regimes in which ensemble-averaged dynamics of such systems with finite size are accurately described by a wave kinetic equation, as used in wave turbulence theory, are theoretically predicted. In addition, the underlying reasons that the wave kinetic equation may be a poor predictor of wave dynamics outside these regimes are also discussed.

Network mechanism for insect olfaction.

Early olfactory pathway responses to the presentation of an odor exhibit remarkably similar dynamical behavior across phyla from insects to mammals, and frequently involve transitions among quiescence, collective network oscillations, and asynchronous firing. We hypothesize that the time scales of fast excitation and fast and slow inhibition present in these networks may be the essential element underlying this similar behavior, and design an idealized, conductance-based integrate-and-fire model to verify this hypothesis via numerical simulations. To better understand the mathematical structure underlying the common dynamical behavior across species, we derive a firing-rate model and use it to extract a slow passage through a saddle-node-on-an-invariant-circle bifurcation structure.

A computational investigation of electrotonic coupling between pyramidal cells in the cortex.

The existence of electrical communication among pyramidal cells (PCs) in the adult cortex has been debated by neuroscientists for several decades. Gap junctions (GJs) among cortical interneurons have been well documented experimentally and their functional roles have been proposed by both computational neuroscientists and experimentalists alike. Experimental evidence for similar junctions among pyramidal cells in the cortex, however, has remained elusive due to the apparent rarity of these couplings among neurons.

Effective dispersion in the focusing nonlinear Schrödinger equation.

For waves described by the focusing nonlinear Schrödinger equation (FNLS), we present an effective dispersion relation (EDR) that arises dynamically from the interplay between the linear dispersion and the nonlinearity. The form of this EDR is parabolic for a robust family of "generic" FNLS waves and equals the linear dispersion relation less twice the total wave action of the wave in question multiplied by the square of the nonlinearity parameter. We derive an approximate form of this EDR explicitly in the limit of small nonlinearity and confirm it using the wave-number-frequency spectral (WFS) analysis, a Fourier-transform based method used for determining dispersion relations of observed waves.

A Role for Electrotonic Coupling Between Cortical Pyramidal Cells.

Many brain regions communicate information through synchronized network activity. Electrical coupling among the dendrites of interneurons in the cortex has been implicated in forming and sustaining such activity in the cortex. Evidence for the existence of electrical coupling among cortical pyramidal cells, however, has been largely absent.