Neural routing circuits for forming invariant representations of visual objects

This thesis presents a biologically plausible model of an attentional mechanism for forming position- and scale-invariant representations of objects in the visual world. The model relies on a set of control neurons to dynamically modify the synaptic strengths of intra-cortical connections so that information from a windowed region of primary visual cortex (Vl) is selectively routed to higher cortical areas. Local spatial relationships (i.e., topography) within the attentional window are preserved as information is routed through the cortex, thus enabling attended objects to be represented in higher cortical areas within an object-centered reference frame that is position and scale invariant. The representation in V1 is modeled as a multiscale stack of sample nodes with progressively lower resolution at higher eccentricities. Large changes in the size of the attentional window are accomplished by switching between different levels of the multiscale stack, while positional shifts and small changes in scale are accomplished by translating and rescaling the window within a single level of the stack. The control signals for setting the position and size of the attentional window are hypothesized to originate from neurons in the pulvinar and in the deep layers of visual cortex. The dynamics of these control neurons are governed by simple differential equations that can be realized by neurobiologically plausible circuits. In pre-attentive mode, the control neurons receive their input from a low-level "saliency map" representing potentially interesting regions of a scene. During the pattern recognition phase, control neurons are driven by the interaction between top-down (memory) and bottom-up (retinal input) sources. The model respects key neurophysiological, neuroanatomical, and psychophysical data relating to attention, and it makes a variety of experimentally testable predictions.

Enhancement of ultrafast four-wave mixing in a polar molecule medium

We investigate the ultrafast four-wave mixing (FWM) with two-color few-cycle ultrashort pulses propagating in a two-level polar molecule medium. It is found that the enhancement of FWM can be achieved even for low intensity pulses due to the effects of permanent dipole moments (PDM) in polar molecules. Moreover, the conversion efficiency of FWM can be controlled by the carrier-envelope phases (CEP) of two ultrashort pulses. (c) 2006 Optical Society of America

Ordinary mod p representations of the metaplectic cover of p-adic SL_2

We classify the genuine ordinary mod p representations of the metaplectic group SL(2,F)-tilde, where F is a p-adic field, and compute its genuine mod p spherical and Iwahori Hecke algebras. The motivation is an interest in a possible correspondence between genuine mod p representations of SL(2,F)-tilde and mod p representations of the dual group PGL(2,F), so we also compare the two Hecke algebras to the mod p spherical and Iwahori Hecke algebras of PGL(2,F). We show that the genuine mod p spherical Hecke algebra of SL(2,F)-tilde is isomorphic to the mod p spherical Hecke algebra of PGL(2,F), and that one can choose an isomorphism which is compatible with a natural, though partial, correspondence of unramified ordinary representations via the Hecke action on their spherical vectors. We then show that the genuine mod p Iwahori Hecke algebra of SL(2,F)-tilde is a subquotient of the mod p Iwahori Hecke algebra of PGL(2,F), but that the two algebras are not isomorphic. This is in contrast to the situation in characteristic 0, where by work of Savin one can recover the local Shimura correspondence for representations generated by their Iwahori fixed vectors from an isomorphism of Iwahori Hecke algebras.

Coherent population accumulations of multiphoton transitions induced by an ultrashort pulse train in polar molecules

Coherent population accumulations of multiphoton transitions induced by an ultrashort pulse train in a two-level polar molecule are investigated theoretically by solving the density-matrix equations without invoking any of the standard approximations. It is shown due to the effects of permanent dipole moments, that the population accumulation of multiphoton transitions can be obtained in the polar molecule. Moreover, the population accumulations depend crucially on the relative phase between two sequential pulses, and the period in which the maximum population accumulation occurs is 2 pi/N in N-photon transitions.

Carrier-envelope phase dependence of few-cycle ultrashort laser pulse propagation in a polar molecule medium

We theoretically investigate carrier-envelope phase dependence of few-cycle ultrashort laser pulse propagation in a polar molecule medium. Our results show that a soliton pulse can be generated during the two-photon resonant propagation of few-cycle pulse in the polar molecule medium. Moreover, the main features of the soliton pulse, such as pulse duration and intensity, depend crucially on the carrier-envelope phase of the incident pulse, which could be utilized to determine the carrier-envelope phase of a few-cycle ultrashort laser pulse from a mode-locked oscillator.

Neural and computational representations of decision variables

These studies explore how, where, and when representations of variables critical to decision-making are represented in the brain. In order to produce a decision, humans must first determine the relevant stimuli, actions, and possible outcomes before applying an algorithm that will select an action from those available. When choosing amongst alternative stimuli, the framework of value-based decision-making proposes that values are assigned to the stimuli and that these values are then compared in an abstract “value space” in order to produce a decision. Despite much progress, in particular regarding the pinpointing of ventromedial prefrontal cortex (vmPFC) as a region that encodes the value, many basic questions remain. In Chapter 2, I show that distributed BOLD signaling in vmPFC represents the value of stimuli under consideration in a manner that is independent of the type of stimulus it is. Thus the open question of whether value is represented in abstraction, a key tenet of value-based decision-making, is confirmed. However, I also show that stimulus-dependent value representations are also present in the brain during decision-making and suggest a potential neural pathway for stimulus-to-value transformations that integrates these two results.

More broadly speaking, there is both neural and behavioral evidence that two distinct control systems are at work during action selection. These two systems compose the “goal-directed system”, which selects actions based on an internal model of the environment, and the “habitual” system, which generates responses based on antecedent stimuli only. Computational characterizations of these two systems imply that they have different informational requirements in terms of input stimuli, actions, and possible outcomes. Associative learning theory predicts that the habitual system should utilize stimulus and action information only, while goal-directed behavior requires that outcomes as well as stimuli and actions be processed. In Chapter 3, I test whether areas of the brain hypothesized to be involved in habitual versus goal-directed control represent the corresponding theorized variables.

The question of whether one or both of these neural systems drives Pavlovian conditioning is less well-studied. Chapter 4 describes an experiment in which subjects were scanned while engaged in a Pavlovian task with a simple non-trivial structure. After comparing a variety of model-based and model-free learning algorithms (thought to underpin goal-directed and habitual decision-making, respectively), it was found that subjects’ reaction times were better explained by a model-based system. In addition, neural signaling of precision, a variable based on a representation of a world model, was found in the amygdala. These data indicate that the influence of model-based representations of the environment can extend even to the most basic learning processes.

Knowledge of the state of hidden variables in an environment is required for optimal inference regarding the abstract decision structure of a given environment and therefore can be crucial to decision-making in a wide range of situations. Inferring the state of an abstract variable requires the generation and manipulation of an internal representation of beliefs over the values of the hidden variable. In Chapter 5, I describe behavioral and neural results regarding the learning strategies employed by human subjects in a hierarchical state-estimation task. In particular, a comprehensive model fit and comparison process pointed to the use of "belief thresholding". This implies that subjects tended to eliminate low-probability hypotheses regarding the state of the environment from their internal model and ceased to update the corresponding variables. Thus, in concert with incremental Bayesian learning, humans explicitly manipulate their internal model of the generative process during hierarchical inference consistent with a serial hypothesis testing strategy.