Interacting Turing-Hopf Instabilities Drive Symmetry-Breaking Transitions in a Mean-Field Model of the Cortex: A Mechanism for the Slow Oscillation

Electrical recordings of brain activity during the transition from wake to anesthetic coma show temporal and spectral alterations that are correlated with gross changes in the underlying brain state. Entry into anesthetic unconsciousness is signposted by the emergence of large, slow oscillations of electrical activity (≲1  Hz) similar to the slow waves observed in natural sleep. Here we present a two-dimensional mean-field model of the cortex in which slow spatiotemporal oscillations arise spontaneously through a Turing (spatial) symmetry-breaking bifurcation that is modulated by a Hopf (temporal) instability. In our model, populations of neurons are densely interlinked by chemical synapses, and by interneuronal gap junctions represented as an inhibitory diffusive coupling. To demonstrate cortical behavior over a wide range of distinct brain states, we explore model dynamics in the vicinity of a general-anesthetic-induced transition from “wake” to “coma.” In this region, the system is poised at a codimension-2 point where competing Turing and Hopf instabilities coexist. We model anesthesia as a moderate reduction in inhibitory diffusion, paired with an increase in inhibitory postsynaptic response, producing a coma state that is characterized by emergent low-frequency oscillations whose dynamics is chaotic in time and space. The effect of long-range axonal white-matter connectivity is probed with the inclusion of a single idealized point-to-point connection. We find that the additional excitation from the long-range connection can provoke seizurelike bursts of cortical activity when inhibitory diffusion is weak, but has little impact on an active cortex. Our proposed dynamic mechanism for the origin of anesthetic slow waves complements—and contrasts with—conventional explanations that require cyclic modulation of ion-channel conductances. We postulate that a similar bifurcation mechanism might underpin the slow waves of natural sleep and comment on the possible consequences of chaotic dynamics for memory processing and learning

Modelling general anaesthesia as a first-order phase transition in the cortex

Since 1997 we have been developing a theoretical foundation for general anaesthesia. We have been able to demonstrate that the abrupt change in brain state broughton by anaesthetic drugs can be characterized as a first-order phase transition in the population-average membrane voltage of the cortical neurons. The theory predicts that, as the critical point of phase-change into unconsciousness is approached, the electrical fluctuations in cortical activity will grow strongly in amplitude while slowing in frequency, becoming more correlated both in time and in space. Thus the bio-electrical change of brain-state has deep similarities with thermodynamic phase changes of classical physics. The theory further predicts the existence of a second critical point, hysteretically separated from the first, corresponding to the return path from comatose unconsciousness back to normal responsiveness. There is a steadily accumulating body of clinical evidence in support of all of the phasetransition predictions: low-frequency power surge in EEG activity; increased correlation time and correlation length in EEG fluctuations; hysteresis separation, with respect to drug concentration, between the point of induction and the point of emergence

Phase transitions in single neurons and neural populations: Critical slowing, anesthesia, and sleep cycles

The firing of an action potential by a biological neuron represents a dramatic transition from small-scale linear stochastics (subthreshold voltage fluctuations) to gross-scale nonlinear dynamics (birth of a 1-ms voltage spike). In populations of neurons we see similar, but slower, switch-like there-and-back transitions between low-firing background states and high-firing activated states. These state transitions are controlled by varying levels of input current (single neuron), varying amounts of GABAergic drug (anesthesia), or varying concentrations of neuromodulators and neurotransmitters (natural sleep), and all occur within a milieu of unrelenting biological noise. By tracking the altering responsiveness of the excitable membrane to noisy stimulus, we can infer how close the neuronal system (single unit or entire population) is to switching threshold. We can quantify this “nearness to switching” in terms of the altering eigenvalue structure: the dominant eigenvalue approaches zero, leading to a growth in correlated, low-frequency power, with exaggerated responsiveness to small perturbations, the responses becoming larger and slower as the neural population approaches its critical point–-this is critical slowing. In this chapter we discuss phase-transition predictions for both single-neuron and neural-population models, comparing theory with laboratory and clinical measurement

Cortical patterns and gamma genesis are modulated by reversal potentials and gap-junction diffusion

In this chapter we describe a continuum model for the cortex that includes both axon-to-dendrite chemical synapses and direct neuron-to-neuron gap-junction diffusive synapses. The effectiveness of chemical synapses is determined by the voltage of the receiving dendrite V relative to its Nernst reversal potential Vrev. Here we explore two alternative strategies for incorporating dendritic reversal potentials, and uncover surprising differences in their stability properties and model dynamics. In the “slow-soma” variant, the (Vrev - V) weighting is applied after the input flux has been integrated at the dendrite, while for “fast-soma”, the weighting is applied directly to the input flux, prior to dendritic integration. For the slow-soma case, we find that–-provided the inhibitory diffusion (via gap-junctions) is sufficiently strong–-the cortex generates stationary Turing patterns of cortical activity. In contrast, the fast-soma destabilizes in favor of standing-wave spatial structures that oscillate at low-gamma frequency ( 30-Hz); these spatial patterns broaden and weaken as diffusive coupling increases, and disappear altogether at moderate levels of diffusion. We speculate that the slow- and fast-soma models might correspond respectively to the idling and active modes of the cortex, with slow-soma patterns providing the default background state, and emergence of gamma oscillations in the fast-soma case signaling the transition into the cognitive state

Instabilities of the cortex during natural sleep

The electrical signals generated by the human cortex during sleep have been widely studied over the last 50 years. The electroencephalogram (EEG) observed during natural sleep exhibits structures with frequencies from 0.5 Hz to over 50 Hz and complicated waveforms such as spindles and K-complexes. Understanding has been enhanced by comprehensive intra-cellular measurements from the cortex and thalamus such as those performed by Steriade et al [1] and Sanchez-Vives and McCormick [2]. Models of the cerebal cortex have been developed in order to explain many of the features observed. These can be classified in terms of individual neuron models or collective models. Since we wish to compare predictions with gross features of the human EEG, we choose a collective model, where we average over a population of neurons in macrocolumns. A number of models of this form have been developed recently; that developed at Waikato draws from a number of different sources to describe the temporal and spatial dynamics of the system

Three dimensional numerical modelling of full-scale hydraulic structures

This study presents the three-dimensional (3D) hydraulic modelling of free surface flows over complex full-scale hydraulic structures. The work outlined therein forms part of a larger ongoing project which focuses on the assessment of the capabilities of different 3D computational fluid dynamics (CFD) techniques to reproduce hydraulic flows over real scale spillway structures and on the comparison with physical scale models. The aim of the first part of the study, presented in this work, is to evaluate a range of 3D free surface methods with a particular focus on the Eulerian mesh-based Volume of Fluid (VOF) technique. A range of 2D and 3D free surface approaches are initially investigated and validated using an experimental case with a simple geometry. The commercial solver Ansys Fluent and the CFD open source platform OpenFOAM are used to implement the VOF model and the DualSPHysics code is used to conduct simulations using the Lagrangian meshless particle-based Smoothed Particle Hydrodynamics (SPH) method. The hydraulic flow over a real hydraulic structure is subsequently modelled, applying the evaluated model implementations. The scheme consists of a newly constructed flood storage reservoir with a labyrinth weir and extended spillway. Different hydraulic conditions is modelled using a 1:25 physical scale hydraulic model of the prototype which is used to validate the numerical models. In order to remove numerical model uncertainties and provide insight into scale effects, numerical simulations are applied first to the physical scale hydraulic model and then to the full-scale prototype. Results show the model is capable of accurately predicting the interface features as well as the velocity and water depths measured in the physical model. It is observed that full-scale predictions present approximately a 17% increase in velocity and a 20% decrease in water depth compared to the equivalent scaled predictions

Dissection, in vivo imaging and analysis of the mouse epitrochleoanconeus muscle

Analysis of rodent muscles affords an opportunity to glean key insights into neuromuscular development and the detrimental impact of disease-causing genetic mutations. Muscles of the distal leg, for instance the gastrocnemius and tibialis anterior, are commonly used in such studies with mice and rats. However, thin and flat muscles, which can be dissected, processed and imaged without major disruption to muscle fibres and nerve-muscle contacts, are more suitable for accurate and detailed analyses of the peripheral motor nervous system. One such wholemount muscle is the predominantly fast twitch epitrochleoanconeus (ETA), which is located in the upper forelimb, innervated by the radial nerve, and contains relatively large and uniformly flat neuromuscular junctions (NMJs). To facilitate incorporation of the ETA into the experimental toolkit of the neuromuscular disease field, here, we describe a simple method for its rapid isolation (<5 min), supported by high-resolution videos and step-by-step images. Furthermore, we outline how the ETA can be imaged in live, anaesthetised mice, to enable examination of dynamic cellular processes occurring at the NMJ and within intramuscular axons, including transport of organelles, such as mitochondria and signalling endosomes. Finally, we present reference data on wild-type ETA fibre-type composition in young adult, male C57BL6/J mice. Comparative neuroanatomical studies of different muscles in rodent models of disease can generate critical insights into pathogenesis and pathology; dissection of the wholemount ETA provides the possibility to diversify the repertoire of muscles analysed for this endeavour

A continuum model for the dynamics of the phase transition from slow-wave sleep to REM sleep

Previous studies have shown that activated cortical states (awake and rapid eye-movement (REM) sleep), are associated with increased cholinergic input into the cerebral cortex. However, the mechanisms that underlie the detailed dynamics of the cortical transition from slow-wave to REM sleep have not been quantitatively modeled. How does the sequence of abrupt changes in the cortical dynamics (as detected in the electrocorticogram) result from the more gradual change in subcortical cholinergic input? We compare the output from a continuum model of cortical neuronal dynamics with experimentally-derived rat electrocorticogram data. The output from the computer model was consistent with experimental observations. In slow-wave sleep, 0.5–2-Hz oscillations arise from the cortex jumping between “up” and “down” states on the stationary-state manifold. As cholinergic input increases, the upper state undergoes a bifurcation to an 8-Hz oscillation. The coexistence of both oscillations is similar to that found in the intermediate stage of sleep of the rat. Further cholinergic input moves the trajectory to a point where the lower part of the manifold in not available, and thus the slow oscillation abruptly ceases (REM sleep). The model provides a natural basis to explain neuromodulator-induced changes in cortical activity, and indicates that a cortical phase change, rather than a brainstem “flip-flop”, may describe the transition from slow-wave sleep to REM

Experimental and numerical modelling of aerated flows over stepped spillways

Stepped spillways are a popular design choice for reservoir overflows due to the high rates of energy dissipation and air entrainment compared to smooth spillways. Air entrainment is important in spillway flows as it affects the pressures acting on the spillway surface, which in adverse conditions can damage the spillway. Air entrainment also causes flow bulking which increases the depth of flow. This study presents free surface and pressure data for aerated flows over an experimental stepped spillway, with pressures measured at different positions across the width of the channel. Within the step cavities, recirculating vortices are observed in both the stream-wise and cross-stream directions, with the direction of circulation alternating at each subsequent step. These 3D effects cause the pressures acting on the step edges to vary across the width of the channel. The Volume of Fluid (VOF) and Eulerian multiphase numerical models are used to predict flows over the spillway. The Eulerian multiphase model shows high levels of air entrainment and is able to predict the position of the free surface to reasonable accuracy. The VOF model, conversely, does not show any air entrainment and therefore under predicts the position of the free surface. The accuracy to which each numerical model predicts pressures on the step faces varies depending on the measurement location. Both of the numerical models accurately simulate the direction of circulation of the 3D vortices within the step cavities. Simulations with varying channel widths, conducted using the VOF model, show that the pattern of 3D vortices repeats as the channel width is increased

Expanding the Toolkit for In Vivo Imaging of Axonal Transport

Axonal transport maintains neuronal homeostasis by enabling the bidirectional trafficking of diverse organelles and cargoes. Disruptions in axonal transport have devastating consequences for individual neurons and their networks, and contribute to a plethora of neurological disorders. As many of these conditions involve both cell autonomous and non-autonomous mechanisms, and often display a spectrum of pathology across neuronal subtypes, methods to accurately identify and analyze neuronal subsets are imperative. This paper details protocols to assess in vivo axonal transport of signaling endosomes and mitochondria in sciatic nerves of anesthetized mice. Stepwise instructions are provided to 1) distinguish motor from sensory neurons in vivo, in situ, and ex vivo by using mice that selectively express fluorescent proteins within cholinergic motor neurons; and 2) separately or concurrently assess in vivo axonal transport of signaling endosomes and mitochondria. These complementary intravital approaches facilitate the simultaneous imaging of different cargoes in distinct peripheral nerve axons to quantitatively monitor axonal transport in health and disease