What can a mean-field model tell us about the dynamics of the cortex?

In this chapter we examine the dynamical behavior of a spatially homogeneous two-dimensional model of the cortex that incorporates membrane potential, synaptic flux rates and long- and short-range synaptic input, in two spatial dimensions, using parameter sets broadly realistic of humans and rats. When synaptic dynamics are included, the steady states may not be stable. The bifurcation structure for the spatially symmetric case is explored, identifying the positions of saddle–node and sub- and supercritical Hopf instabilities. We go beyond consideration of small-amplitude perturbations to look at nonlinear dynamics. Spatially-symmetric (breathing mode) limit cycles are described, as well as the response to spatially-localized impulses. When close to Hopf and saddle–node bifurcations, such impulses can cause traveling waves with similarities to the slow oscillation of slow-wave sleep. Spiral waves can also be induced. We compare model dynamics with the known behavior of the cortex during natural and anesthetic-induced sleep, commenting on the physiological significance of the limit cycles and impulse responses

Critical Points and Traveling Wave in Locomotion: Experimental Evidence and Some Theoretical Considerations

The central pattern generator (CPG) architecture for rhythm generation remains partly elusive. We compare cat and frog locomotion results, where the component unrelated to pattern formation appears as a temporal grid, and traveling wave respectively. Frog spinal cord microstimulation with N-methyl-D-Aspartate (NMDA), a CPG activator, produced a limited set of force directions, sometimes tonic, but more often alternating between directions similar to the tonic forces. The tonic forces were topographically organized, and sites evoking rhythms with different force subsets were located close to the constituent tonic force regions. Thus CPGs consist of topographically organized modules. Modularity was also identified as a limited set of muscle synergies whose combinations reconstructed the EMGs. The cat CPG was investigated using proprioceptive inputs during fictive locomotion. Critical points identified both as abrupt transitions in the effect of phasic perturbations, and burst shape transitions, had biomechanical correlates in intact locomotion. During tonic proprioceptive perturbations, discrete shifts between these critical points explained the burst durations changes, and amplitude changes occurred at one of these points. Besides confirming CPG modularity, these results suggest a fixed temporal grid of anchoring points, to shift modules onsets and offsets. Frog locomotion, reconstructed with the NMDA synergies, showed a partially overlapping synergy activation sequence. Using the early synergy output evoked by NMDA at different spinal sites, revealed a rostrocaudal topographic organization, where each synergy is preferentially evoked from a few, albeit overlapping, cord regions. Comparing the locomotor synergy sequence with this topography suggests that a rostrocaudal traveling wave would activate the synergies in the proper sequence for locomotion. This output was reproduced in a two-layer model using this topography and a traveling wave. Together our results suggest two CPG components: modules, i.e., synergies; and temporal patterning, seen as a temporal grid in the cat, and a traveling wave in the frog. Animal and limb navigation have similarities. Research relating grid cells to the theta rhythm and on segmentation during navigation may relate to our temporal grid and traveling wave results. Winfree’s mathematical work, combining critical phases and a traveling wave, also appears important. We conclude suggesting tracing, and imaging experiments to investigate our CPG model

A theoretical and experimental study of the mechanism of axoplasmic convection in nerve fibers driven by peristaltic surface waves

In the study of cell biology, investigators have found that substances which are produced within the cell nucleus are sometimes found throughout the cell at points distant from the site of production. In the case of nerve cells (neurons), this is particularly dramatic because of the unusual elongated geometry of these cells. A neuron possesses a cylindrical tubular extension called an axon or axis cylinder which is characterized by a large length-to-diameter ratio (103-106). The existence of a continuous proximo-distal flow of axoplasm within these cylindrical axons has now been demonstrated by numerous investigators. In this study, engineering techniques are employed to explore the role of microperistalsis as a possible driving mechanism for this axoplasmic flow. An experimental technique for injecting axons (5-10 microns-diameter) with micropipettes under visual microscopic control has been perfected. A new technique for microcapillary tube viscometric measurements applicable to micro samples of biological materials is presented. Using these techniques, a flow curve has been obtained for the axoplasmic substance. The results of these experiments indicate that axoplasin behaves as a highly viscous, pseudo-plastic material. No evidence of significant time-dependent thixotropic or viscoelastic effects was apparent. A theoretical analysis of the peristaltic pumping of pseudoplastic fluids at low Reynolds numbers by means of an infinite train of sinusoidal peristaltic waves is. presented. Results are shown as a series of pump characteristic curves involving the geometrical properties of tlie wave and the flow properties of the pseudo-plastic fluid as parameters. Data obtained from experiments performed on a plane, two-dimensional model are used to confirm the theoretical results. Cinemicrographic evidence reported in the literature describing waves traveling over the surface of axons in culture is discussed. A study of the geometrical properties of the peristaltic waves taken from these motion picture data is presented. The viscometric data obtained from axoplasm are used to establish system resistance curves for axons idealized as uniform cylindrical tubes. These data are correlated with the theoretical pump characteristic curves to determine an expected flow rate. A comparison between the theoretical flow rates and the observed axonal flow rates gives quantitative support to the hypothesis that peristalsis is the mechanism for axoplasmic flow. In addition, it is shown that the peristaltic pumping of a pseudoplastic fluid depends only on the geometrical properties of the peristaltic waves and the flow behavior index of the fluid. For this case of axoplasm, this indicates that the theoretical flow speed of axoplasm is independent both of the consistency of the axoplasmic material and the diameter of the axon

Controlling Chimeras

Coupled phase oscillators model a variety of dynamical phenomena in nature and technological applications. Non-local coupling gives rise to chimera states which are characterized by a distinct part of phase-synchronized oscillators while the remaining ones move incoherently. Here, we apply the idea of control to chimera states: using gradient dynamics to exploit drift of a chimera, it will attain any desired target position. Through control, chimera states become functionally relevant; for example, the controlled position of localized synchrony may encode information and perform computations. Since functional aspects are crucial in (neuro-)biology and technology, the localized synchronization of a chimera state becomes accessible to develop novel applications. Based on gradient dynamics, our control strategy applies to any suitable observable and can be generalized to arbitrary dimensions. Thus, the applicability of chimera control goes beyond chimera states in non-locally coupled systems

Binding by Asynchrony: The Neuronal Phase Code

Neurons display continuous subthreshold oscillations and discrete action potentials (APs). When APs are phase-locked to the subthreshold oscillation, we hypothesize they represent two types of information: the presence/absence of a sensory feature and the phase of subthreshold oscillation. If subthreshold oscillation phases are neuron-specific, then the sources of APs can be recovered based on the AP times. If the spatial information about the stimulus is converted to AP phases, then APs from multiple neurons can be combined into a single axon and the spatial configuration reconstructed elsewhere. For the reconstruction to be successful, we introduce two assumptions: that a subthreshold oscillation field has a constant phase gradient and that coincidences between APs and intracellular subthreshold oscillations are neuron-specific as defined by the “interference principle.” Under these assumptions, a phase-coding model enables information transfer between structures and reproduces experimental phenomenons such as phase precession, grid cell architecture, and phase modulation of cortical spikes. This article reviews a recently proposed neuronal algorithm for information encoding and decoding from the phase of APs (Nadasdy, 2009). The focus is given to the principles common across different systems instead of emphasizing system specific differences

Fractal Cognitive Triad: The Theoretical Connection between Subjective Experience and Neural Oscillations

It has long been appreciated that the brain is oscillatory1. Early measurements of brain electrophysiology revealed rhythmic synchronization unifying large swaths of the brain. The study of neural oscillation has enveloped cognitive neuroscience and neural systems. The traditional belief that oscillations are epiphenomenal of neuron spiking is being challenged by intracellular oscillations and the theoretical backing that oscillatory activity is fundamental to physics. Subjective experience oscillates at three particular frequency bands in a cognitive triad: perception at 5 Hz (exogenous), action at 2 Hz (endogenous), and attention at 0.1 Hz (cognitive). This triad functions as a means of information flow across scales of magnitude in a biological fractal. The Homunculus Solution is proposed in which mental experience occurs at fixed scales of biology. The mind is composed of minds, perceived as "the voices in your head." Each voice has voices inside its head to increasingly microscopic scales, forming an interactive fractal of subjective experience