Recently a perturbative theory has been constructed, starting from the Feynman rules of the nonlinear sigma model at the tree level in the presence of an external vector source coupled to the flat connection and of a scalar source coupled to the nonlinear sigma model constraint (flat connection formalism). The construction is based on a local functional equation, which overcomes the problems du...

Recurrent interactions in the primary visual cortex make its output a complex nonlinear transform of its input. This transform serves preattentive visual segmentation, that is, autonomously processing visual inputs to give outputs that selectively emphasize certain features for segmentation. An analytical understanding of the nonlinear dynamics of the recurrent neural circuit is essential to ha...

This paper presents a unified view of stochastic optimal control theory as developed within the machine learning and control theory communities. In particular we show the mathematical connection between recent work on Path Integral (PI) and Kullback Leibler (KL) divergence stochastic optimal control theory with earlier work on risk sensitivity and the fundamental dualities between free energy a...

Abstract Recurrent interactions in the primary visual cortex makes its output a complex nonlinear transform of its input. This transform serves pre-attentive visual segmentation, i.e., autonomously processing visual inputs to give outputs that selectively emphasize certain features for segmentation. An analytical understanding of the nonlinear dynamics of the recurrent neural circuit is essenti...

Here, formation of continuous attractor dynamics in a nonlinear recurrent neural network is used to achieve a nonlinear speech denoising method, in order to implement robust phoneme recognition and information retrieval. Formation of attractor dynamics in recurrent neural network is first carried out by training the clean speech subspace as the continuous attractor. Then, it is used to recogniz...

Recurrent interactions in the primary visual cortex makes its output a complex nonlinear transform of its input. This transform serves pre-attentive visual segmentation, i.e., autonomously processing visual inputs to give outputs that selectively emphasize certain features for segmentation. An analytical understanding of the nonlinear dynamics of the recurrent neural circuit is essential to har...

Hasimoto [10] introduced the map from vortex filament solutions of Euler’s equations for incompressible fluids in the local induction approximation to solutions of the nonlinear Schrödinger equation and he showed vortex filament equation is equivalent nonlinear Schrödinger equation. After this discovering of Hasimoto, several authors [1, 5, 9, 12, 13, 14, 15, 17, 20, 21, 22, 23, 24] studied the...

We first prove the second order convergence of the Strang-type splitting scheme for the nonlinear Schrödinger equation. The proof does not require commutator estimates but crucially relies on an integral representation of the scheme. It reveals the connection between Strang-type splitting and the midpoint rule. We then show that the integral representation idea can also be used to study the sto...