Reduction and reconstruction of stochastic differential equations via symmetries

An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance

An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts

Even a cursory inspection of the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties reveals striking structures. These patterns correspond to webs of elliptic-K3 fibrations whose mirror images are also elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fibers. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, explains much of the structure of the observed patterns.Comment: 30 pages, 15 colour figure

Measuring latency distribution of transcallosal fibers using transcranial magnetic stimulation

Background: Neuroimaging technology is being developed to enable non-invasive mapping of the latency distribution of cortical projection pathways in white matter, and correlative clinical neurophysiological techniques would be valuable for mutual verification. Interhemispheric interaction through the corpus callosum can be measured with interhemispheric facilitation and inhibition using transcranial magnetic stimulation. Objective: To develop a method for determining the latency distribution of the transcallosal fibers with transcranial magnetic stimulation. Methods: We measured the precise time courses of interhemispheric facilitation and inhibition with a conditioning-test paired-pulse magnetic stimulation paradigm. The conditioning stimulus was applied to the right primary motor cortex and the test stimulus was applied to the left primary motor cortex. The interstimulus interval was set at 0.1 ms resolution. The proportions of transcallosal fibers with different conduction velocities were calculated by measuring the changes in magnitudes of interhemispheric facilitation and inhibition with interstimulus interval. Results: Both interhemispheric facilitation and inhibition increased with increment in interstimulus interval. The magnitude of interhemispheric facilitation was correlated with that of interhemispheric inhibition. The latency distribution of transcallosal fibers measured with interhemispheric facilitation was also correlated with that measured with interhemispheric inhibition. Conclusions: The data can be interpreted as latency distribution of transcallosal fibers. Interhemispheric interaction measured with transcranial magnetic stimulation is a promising technique to determine the latency distribution of the transcallosal fibers. Similar techniques could be developed for other cortical pathways

Reduced Caudate and Nucleus Accumbens Response to Rewards in Unmedicated Subjects with Major Depressive Disorder

Objective: Major depressive disorder is characterized by impaired reward processing, possibly due to dysfunction in the basal ganglia. However, few neuroimaging studies of depression have distinguished between anticipatory and consummatory phases of reward processing. Using functional MRI (fMRI) and a task that dissociates anticipatory and consummatory phases of reward processing, the authors tested the hypothesis that individuals with major depression would show reduced reward-related responses in basal ganglia structures. Method: A monetary incentive delay task was presented to 30 unmedicated individuals with major depressive disorder and 31 healthy comparison subjects during fMRI scanning. Whole-brain analyses focused on neural responses to reward-predicting cues and rewarding outcomes (i.e., monetary gains). Secondary analyses focused on the relationship between anhedonic symptoms and basal ganglia volumes. Results: Relative to comparison subjects, participants with major depression showed significantly weaker responses to gains in the left nucleus accumbens and the caudate bilaterally. Group differences in these regions were specific to rewarding outcomes and did not generalize to neutral or negative outcomes, although relatively reduced responses to monetary penalties in the major depression group emerged in other caudate regions. By contrast, evidence for group differences during reward anticipation was weaker, although participants with major depression showed reduced activation to reward cues in a small sector of the left posterior putamen. In the major depression group, anhedonic symptoms and depression severity were associated with reduced caudate volume bilaterally. Conclusions: These results suggest that basal ganglia dysfunction in major depression may affect the consummatory phase of reward processing. Additionally, morphometric results suggest that anhedonia in major depression is related to caudate volume.Psycholog

Reduced Caudate and Nucleus Accumbens Response to Rewards in Unmedicated Subjects with Major Depressive Disorder

Objective: Major depressive disorder is characterized by impaired reward processing, possibly due to dysfunction in the basal ganglia. However, few neuroimaging studies of depression have distinguished between anticipatory and consummatory phases of reward processing. Using functional MRI (fMRI) and a task that dissociates anticipatory and consummatory phases of reward processing, the authors tested the hypothesis that individuals with major depression would show reduced reward-related responses in basal ganglia structures. Method: A monetary incentive delay task was presented to 30 unmedicated individuals with major depressive disorder and 31 healthy comparison subjects during fMRI scanning. Whole-brain analyses focused on neural responses to reward-predicting cues and rewarding outcomes (i.e., monetary gains). Secondary analyses focused on the relationship between anhedonic symptoms and basal ganglia volumes. Results: Relative to comparison subjects, participants with major depression showed significantly weaker responses to gains in the left nucleus accumbens and the caudate bilaterally. Group differences in these regions were specific to rewarding outcomes and did not generalize to neutral or negative outcomes, although relatively reduced responses to monetary penalties in the major depression group emerged in other caudate regions. By contrast, evidence for group differences during reward anticipation was weaker, although participants with major depression showed reduced activation to reward cues in a small sector of the left posterior putamen. In the major depression group, anhedonic symptoms and depression severity were associated with reduced caudate volume bilaterally. Conclusions: These results suggest that basal ganglia dysfunction in major depression may affect the consummatory phase of reward processing. Additionally, morphometric results suggest that anhedonia in major depression is related to caudate volume.Psycholog

PREFACE: DROPS, BUBBLES, AND THIN FILMS, SPECIAL ISSUE HONORING PROFESSOR OLEG KABOV

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