Faculty Bibliography

Our objective is a realistic theory of the visual cortex that can explain the visual selectivity, dynamics, and the diversity of visual properties in cortical cell populations. To do this, we have studied a large-scale computational model of Macaque V1 (McLaughlin et al. 2000 PNAS) based on anatomy and physiology. The model consists of 4 hypercolumns of inhibitory and excitatory point neurons (a lattice of 128X128 neurons). The model's intra-cortical connectivity is nonspecific and isotropic. The spatial extent of cortical excitatory connections exceeds that of inhibitory connections, as indicated by anatomical data. Cells in the model are classified as Simple or Complex by the same index of linearity of spatial summation that has been used in physiology experiments. Previously we showed how Simple cells could exist in the model despite the non-linearity of the LGN input and of cortico-cortical excitation (Wielaard et al 2001 J. NS.)-through strong cortico-cortical inhibition, synaptic pooling of cell responses, and diversity in the arrangement of LGN On/Off subregions. Complex cells arise in the model by allowing for randomness in synaptic coupling strengths, which can increase the importance of network excitation, and randomness in the strength of LGN input. Also, there is a small population of neurons in the model that are intermediate, in linearity of summation, between Simple and Complex cells. Therefore, the model produces diversity and properties of cortical responses that are consistent with physiological observations in Macaque V1 (Ringach et al 2001 J. NS.)

We derived a numerical technique for the propagation of the electromagnetic field in a five-level reverse saturable absorber including the nonlinear Kerr effect and dispersion. The numerical method combines the split step beam propagation method and the Crank-Nicholson method. Using our numerical technique we observed new behavior, not previously observed nor predicted to our knowledge, including the temporal splitting caused by the dynamics of the carrier densities in a reverse saturable absorber and the enhancement of absorption due to the Kerr nonlinearity. Our numerical calculation enables the prediction of nonlinear absorption using material parameters such as the absorption cross-sections and decay rates. We can also investigate the interplay between the optical pulse properties such as the temporal pulse width, spatial radius, incident energy and the carrier dynamics and nonlinear absorption of the reverse saturable absorber

In this paper, we offer an explanation for how selectivity for orientation could be produced by a model with circuitry that is based on the anatomy of V1 cortex. It is a network model of layer 4Calpha in macaque primary visual cortex (area V1). The model consists of a large number of integrate-and-fire conductance-based point neurons, both excitatory and inhibitory, which represent dynamics in a small patch of 4Calpha-1 mm(2) in lateral area-which contains four orientation hypercolumns. The physiological properties and coupling architectures of the model are derived from experimental data for layer 4Calpha of macaque. Convergent feed-forward input from many neurons of the lateral geniculate nucleus sets up an orientation preference, in a pinwheel pattern with an orientation preference singularity in the center of the pattern. Recurrent cortical connections cause the network to sharpen its selectivity. The pattern of local lateral connections is taken as isotropic, with the spatial range of monosynaptic excitation exceeding that of inhibition. The model (i) obtains sharpening, diversity in selectivity, and dynamics of orientation selectivity, each in qualitative agreement with experiment; and (ii) predicts more sharpening near orientation preference singularities.

In this article we use one-dimensional nonlinear Schrodinger equations (NLS) to illustrate chaotic and turbulent behavior of nonlinear dispersive waves. It begins with a brief summary of properties of NLS with focusing and defocusing nonlinearities. In this summary we stress the role of the modulational instability in the formation of solitary waves and homoclinic orbits, and in the generation of temporal chaos and of spatiotemporal chaos for the nonlinear waves. Dispersive wave turbulence for a class of one-dimensional NLS equations is then described in detail-emphasizing distinctions between focusing and defocusing cases, the role of spatially localized, coherent structures, and their interaction with resonant waves in setting up the cycles of energy transfer in dispersive wave turbulence through direct and inverse cascades. In the article we underline that these simple NLS models provide precise and demanding tests for the closure theories of dispersive wave turbulence. In the conclusion we emphasize the importance of effective stochastic representations for the prediction of transport and other macroscopic behavior in such deterministic chaotic nonlinear wave systems. (C) 2000 American Institute of Physics. [S0022-2488(00)01606-6]

The nonlinear coupling of two scalar nonlinear Schrodinger (NLS) fields results in nonfocusing instabilities that exist independently of the well-known modulational instability of the focusing NLS equation. The focusing versus defocusing behavior of scalar NLS fields is a well-known model for the corresponding behavior of pulse transmission in optical fibers in the anomalous (focusing) versus normal (defocusing) dispersion regime [19], [20]. For fibers with birefringence (induced by an asymmetry in the cross section), the scalar NLS fields for two orthogonal polarization modes couple nonlinearly [26]. Experiments by Rothenberg [32], [33] have demonstrated a new type of modulational instability in a birefringent normal dispersion fiber, and he proposes this cross-phase coupling instability as a mechanism for the generation of ultrafast, terahertz optical oscillations. In this paper the nonfocusing plane wave instability in an integrable coupled nonlinear Schrodinger (CNLS) partial differential equation system is contrasted with the focusing instability from two perspectives: traditional linearized stability analysis and integrable methods based on periodic inverse spectral theory. The latter approach is a crucial first step toward a nonlinear, nonlocal understanding of this new optical instability analogous to that developed for the focusing modulational instability of the sine-Gordon equations by Ercolani, Forest, and McLaughlin [13], [14], [15], [17] and the scalar NLS equation by Tracy, Chen, and Lee [36], [37], Forest and Lee [:18], and McLaughlin, Li, and Overman [23], [24]

Bistable responses of Fabry-Perot cavities and all-optical arrays, in the presence of weak diffraction and strong diffusion, are studied both analytically and numerically. The model is a pair of nonlinear Schrodinger equations coupled through a diffusion equation. The numerical computations are based on a split-step method with three distinct characteristics. This bistable nonlinear system, with strong diffusion and weak diffraction, behaves very differently than dispersive bistability with Kerr nonlinearity. Nevertheless, focusing nonlinearity can improve its response characteristics significantly. For example, it is found that hysteresis loops are much wider when nonlinearity is self-focusing than when nonlinearity is self-defocusing. For self-defocusing nonlinearity, strong diffraction can close a hysteresis loop completely. Because of strong diffusion, refractive index distributions are smoothed versions of intensity distributions. This weakens the nonlinear behavior of the system. By approximating the refractive index by a constant, the model is reduced to a discrete map. This reduction incorporates diffraction into a nonlinear response function, allowing diffractive effects to be studied analytically, and shown to agree well with more extensive numerical simulations. Optical arrays are also studied numerically Because of weaker diffractive crosstalk and a wider "operation gap" between "on" bits and "off" bits, an array with focusing nonlinearity allows finer packing than with self-defocusing nonlinearity. Using the reduced map, optimal packing densities of optical arrays are estimated. (C) 2000 Elsevier Science B.V. All rights reserved

Reverse-saturable absorbers are of considerable interest for optical limiting. Using the electric dipole per turbation, we derived the rate equation for a five-level system describing reverse-saturable absorbers. Traditional theories for the propagating laser beam in these materials are expressed in terms of the optical intensity. However, with the introduction of high-power short-pulsed lasers, the propagation of light in these materials may be subject to nonlinear phenomena such as self-focusing and self-phase modulation. Furthermore, conventional theories treat the laser light as a continuous wave or as a very broad temporal pulse in which dispersive effects are neglected. In order to incorporate these other nonlinear or dispersive effects, and therefore determine their influence in reverse-saturable absorbers, we derived an equation for the propagation of the electromagnetic field, rather than the intensity, coupled to the rate equations for a five-level system. We also coupled our theory to experimentally measurable parameters for these materials and detailed the various physical approximations used to obtain the rate equations

Dispersive wave turbulence is studied numerically for a class of one-dimensional nonlinear wave equations. Both deterministic and random (white noise in time) forcings are studied. Four distinct stable spectra are observed-the direct and inverse cascades of weak turbulence (WT) theory, thermal equilibrium, and a fourth spectrum (MMT; Majda, McLaughlin, Tabak). Each spectrum can describe long-time behavior, and each can be only metastable (with quite diverse lifetimes)-depending on details of nonlinearity, forcing, and dissipation. Cases of a long-live MMT transient state dcaying to a state with WT spectra, and vice-versa, are displayed. In the case of freely decaying turbulence, without forcing, both cascades of weak turbulence are observed. These WT states constitute the clearest and most striking numerical observations of WT spectra to date-over four decades of energy, and three decades of spatial, scales. Numerical experiments that study details of the composition, coexistence, and transition between spectra are then discussed, including: (i) for deterministic forcing, sharp distinctions between focusing and defocusing nonlinearities, including the role of long wavelength instabilities, localized coherent structures, and chaotic behavior; (ii) the role of energy growth in time to monitor the selection of MMT or WT spectra; (iii) a second manifestation of the MMT spectrum as it describes a self-similar evolution of the wave, without temporal averaging; (iv) coherent structures and the evolution of the direct and inverse cascades; and (v) nonlocality (in k-space) in the transferral process.

Bistable responses of Fabry-Perot cavities and optical arrays in the presence of diffraction and diffusion are studied both analytically and numerically. The model is a pair of nonlinear Schrodinger equations coupled through a diffusion equation. The numerical computations are based on a split-step method, with three distinct characteristics. In these diffusion-dominated arrays with weak diffraction, this study demonstrates that focusing nonlinearity can improve the response characteristics significantly. The primary results of the study are that (1) for diffusion-dominated media a small amount of diffraction is sufficient to alter optical bistability significantly; (2) focusing nonlinearities enhance optical bistability in comparison with defocusing nonlinearities; (3) in diffusion-dominated media these focusing-defocusing effects are quite distinct from self-focusing behavior in Kerr media; (4) in the presence of diffraction the response of the array can be described analytically by a reduced map, whose derivation can be viewed as an extension of Firth's diffusive model to include weak diffraction; (5) this map is used to explain analytically certain qualitative features of bistability, as observed in the numerical experiments; and (6) optimal spacing predictions are made with a reduced map and verified with numerical simulations of small all-optical arrays. (C) 1999 Optical Society of America [S0740-3224(99)00107-1]

We establish the semiclassical limit of the one-dimensional defocusing cubic nonlinear Schrodinger (NLS) equation. Complete integrability is exploited to obtain a global characterization of the weak limits of the entire NLS hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows. Consequently, this also establishes the zero-dispersion limit of the modified Korteweg-de Vries equation that resides in that hierarchy. We have adapted and clarified the strategy introduced by Lax and Levermore to study the zero-dispersion limit of the Korteweg-de Vries equation, expanding it to treat entire integrable hierarchies and strengthening the limits obtained. A crucial role is played by the convexity of the underlying log-determinant with respect to the times associated with the commuting flows. (C) 1999 John Wiley & Sons, Inc