Symmetries and itineracy in nonlinear systems with many degrees of freedom

Tsuda examines the potential contribution of nonlinear dynamical systems, with many degrees of freedom, to understanding brain function. We offer suggestions concerning symmetry and transients to strengthen the physiological motivation and theoretical consistency of this novel research direction: Symmetry plays a fundamental role, theoretically and in relation to real brains. We also highlight a distinction between chaotic "transience" and "itineracy"

A dendritic mechanism for decoding traveling waves: Principles and applications to motor cortex

Traveling waves of neuronal oscillations have been observed in many cortical regions, including the motor and sensory cortex. Such waves are often modulated in a task-dependent fashion although their precise functional role remains a matter of debate. Here we conjecture that the cortex can utilize the direction and wavelength of traveling waves to encode information. We present a novel neural mechanism by which such information may be decoded by the spatial arrangement of receptors within the dendritic receptor field. In particular, we show how the density distributions of excitatory and inhibitory receptors can combine to act as a spatial filter of wave patterns. The proposed dendritic mechanism ensures that the neuron selectively responds to specific wave patterns, thus constituting a neural basis of pattern decoding. We validate this proposal in the descending motor system, where we model the large receptor fields of the pyramidal tract neurons — the principle outputs of the motor cortex — decoding motor commands encoded in the direction of traveling wave patterns in motor cortex. We use an existing model of field oscillations in motor cortex to investigate how the topology of the pyramidal cell receptor field acts to tune the cells responses to specific oscillatory wave patterns, even when those patterns are highly degraded. The model replicates key findings of the descending motor system during simple motor tasks, including variable interspike intervals and weak corticospinal coherence. By additionally showing how the nature of the wave patterns can be controlled by modulating the topology of local intra-cortical connections, we hence propose a novel integrated neuronal model of encoding and decoding motor commands

The Foveal Confluence in Human Visual Cortex

The human visual system devotes a significant proportion of its resources to a very small part of the visual field, the fovea. Foveal vision is crucial for natural behavior and many tasks in daily life such as reading or fine motor control. Despite its significant size, this part of cortex is rarely investigated and the limited data have resulted in competing models of the layout of the foveal confluence in primate species. Specifically, how V2 and V3 converge at the central fovea is the subject of debate in primates and has remained “terra incognita” in humans. Using high-resolution fMRI (1.2 × 1.2 × 1.2 mm3) and carefully designed visual stimuli, we sought to accurately map the human foveal confluence and hence disambiguate the competing theories. We find that V1, V2, and V3 are separable right into the center of the foveal confluence, and V1 ends as a rounded wedge with an affine mapping of the foveal singularity. The adjacent V2 and, in contrast to current concepts from macaque monkey, also V3 maps form continuous bands (∼5 mm wide) around the tip of V1. This mapping results in a highly anisotropic representation of the visual field in these areas. Unexpectedly, for the centermost 0.75°, the cortical representations for both V2 and V3 are larger than that of V1, indicating that more neuronal processing power is dedicated to second-level analysis in this small but important part of the visual field

Clinical Applications of Stochastic Dynamic Models of the Brain, Part I: A Primer

Biological phenomena arise through interactions between an organism's intrinsic dynamics and stochastic forces-random fluctuations due to external inputs, thermal energy, or other exogenous influences. Dynamic processes in the brain derive from neurophysiology and anatomical connectivity; stochastic effects arise through sensory fluctuations, brainstem discharges, and random microscopic states such as thermal noise. The dynamic evolution of systems composed of both dynamic and random effects can be studied with stochastic dynamic models (SDMs). This article, Part I of a two-part series, offers a primer of SDMs and their application to large-scale neural systems in health and disease. The companion article, Part II, reviews the application of SDMs to brain disorders. SDMs generate a distribution of dynamic states, which (we argue) represent ideal candidates for modeling how the brain represents states of the world. When augmented with variational methods for model inversion, SDMs represent a powerful means of inferring neuronal dynamics from functional neuroimaging data in health and disease. Together with deeper theoretical considerations, this work suggests that SDMs will play a unique and influential role in computational psychiatry, unifying empirical observations with models of perception and behavior

Clinical Applications of Stochastic Dynamic Models of the Brain, Part II: A Review

Brain activity derives from intrinsic dynamics (due to neurophysiology and anatomical connectivity) in concert with stochastic effects that arise from sensory fluctuations, brainstem discharges, and random microscopic states such as thermal noise. The dynamic evolution of systems composed of both dynamic and random fluctuations can be studied with stochastic dynamic models (SDMs). This article, Part II of a two-part series, reviews applications of SDMs to large-scale neural systems in health and disease. Stochastic models have already elucidated a number of pathophysiological phenomena, such as epilepsy and hypoxic ischemic encephalopathy, although their use in biological psychiatry remains rather nascent. Emerging research in this field includes phenomenological models of mood fluctuations in bipolar disorder and biophysical models of functional imaging data in psychotic and affective disorders. Together with deeper theoretical considerations, this work suggests that SDMs will play a unique and influential role in computational psychiatry, unifying empirical observations with models of perception and behavior

Subjective estimates of uncertainty during gambling and impulsivity after subthalamic deep brain stimulation for Parkinson\u2019s disease

Subthalamic deep brain stimulation (DBS) for Parkinson\u2019s disease (PD) may modulate chronometric and instrumental aspects of choice behaviour, including motor inhibition, decisional slowing, and value sensitivity. However, it is not well known whether subthalamic DBS affects more complex aspects of decision-making, such as the influence of subjective estimates of uncertainty on choices. In this study, 38 participants with PD played a virtual casino prior to subthalamic DBS (whilst \u2018on\u2019 medication) and again, 3-months postoperatively (whilst \u2018on\u2019 stimulation). At the group level, there was a small but statistically significant decrease in impulsivity postoperatively, as quantified by the Barratt Impulsiveness Scale (BIS). The gambling behaviour of participants (bet increases, slot machine switches and double or nothing gambles) was associated with this self-reported measure of impulsivity. However, there was a large variance in outcome amongst participants, and we were interested in whether individual differences in subjective estimates of uncertainty (specifically, volatility) were related to differences in pre- and postoperative impulsivity. To examine these individual differences, we fit a computational model (the Hierarchical Gaussian Filter, HGF), to choices made during slot machine game play as well as a simpler reinforcement learning model based on the Rescorla-Wagner formalism. The HGF was superior in accounting for the behaviour of our participants, suggesting that participants incorporated beliefs about environmental uncertainty when updating their beliefs about gambling outcome and translating these beliefs into action. A specific aspect of subjective uncertainty, the participant\u2019s estimate of the tendency of the slot machine\u2019s winning probability to change (volatility), increased subsequent to DBS. Additionally, the decision temperature of the response model decreased post-operatively, implying greater stochasticity in the belief-to-choice mapping of participants. Model parameter estimates were significantly associated with impulsivity; specifically, increased uncertainty was related to increased postoperative impulsivity. Moreover, changes in these parameter estimates were significantly associated with the maximum post-operative change in impulsivity over a six month follow up period. Our findings suggest that impulsivity in PD patients may be influenced by subjective estimates of uncertainty (environmental volatility) and implicate a role for the subthalamic nucleus in the modulation of outcome certainty. Furthermore, our work outlines a possible approach to characterising those persons who become more impulsive after subthalamic DBS, an intervention in which non-motor outcomes can be highly variable

Encoding via conjugate symmetries of slow oscillations for globally coupled oscillators

Peter Ashwin and Jon Borresen, Physical Review E, Vol. 70, p. 026203 (2004). "Copyright © 2004 by the American Physical Society."We study properties of the dynamics underlying slow cluster oscillations in two systems of five globally coupled oscillators. These slow oscillations are due to the appearance of structurally stable heteroclinic connections between cluster states in the noise-free dynamics. In the presence of low levels of noise they give rise to long periods of residence near cluster states interspersed with sudden transitions between them. Moreover, these transitions may occur between cluster states of the same symmetry, or between cluster states with conjugate symmetries given by some rearrangement of the oscillators. We consider the system of coupled phase oscillators studied by Hansel et al. [Phys. Rev. E 48, 3470 (1993)] in which one can observe slow, noise-driven oscillations that occur between two families of two cluster periodic states; in the noise-free case there is a robust attracting heteroclinic cycle connecting these families. The two families consist of symmetric images of two inequivalent periodic orbits that have the same symmetry. For N=5 oscillators, one of the periodic orbits has one unstable direction and the other has two unstable directions. Examining the behavior on the unstable manifold for the two unstable directions, we observe that the dimensionality of the manifold can give rise to switching between conjugate symmetry orbits. By applying small perturbations to the system we can easily steer it between a number of different marginally stable attractors. Finally, we show that similar behavior occurs in a system of phase-energy oscillators that are a natural extension of the phase model to two dimensional oscillators. We suggest that switching between conjugate symmetries is a very efficient method of encoding information into a globally coupled system of oscillators and may therefore be a good and simple model for the neural encoding of information