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

Subthreshold dynamics of a single neuron from a Hamiltonian perspective

We use Hamilton's equations of classical mechanics to investigate the behavior of a cortical neuron on the approach to an action potential. We use a two-component dynamic model of a single neuron, due to Wilson, with added noise inputs. We derive a Lagrangian for the system, from which we construct Hamilton's equations. The conjugate momenta are found to be linear combinations of the noise input to the system. We use this approach to consider theoretically and computationally the most likely manner in which such a modeled neuron approaches a firing event. We find that the firing of a neuron is a result of a drop in inhibition, due to a temporary increase in negative bias of the mean noise input to the inhibitory control equation. Moreover, we demonstrate through theory and simulation that, on average, the bias in the noise increases in an exponential manner on the approach to an action potential. In the Hamiltonian description, an action potential can therefore be considered a result of the exponential growth of the conjugate momenta variables pulling the system away from its equilibrium state, into a nonlinear regime

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

Modelling the anaestheto-dynamic phase transition of the cerebral cortex

This thesis examines a stochastic model for the electrical behaviour of the cerebral cortex under the influence of a general anaesthetic agent. The modelling element is the macrocolumn, an organized assembly of ∼10⁵ cooperating neurons (85% excitatory, 15% inhibitory) within a small cylindrical volume (∼1 mm³) of the cortex. The state variables are hₑ and hᵢ, the mean-field average soma voltages for the populations of excitatory (e) and inhibitory (i) neurons comprising the macrocolumn. The random fluctuations of hₑ about its steady-state value are taken as the source of the scalp-measured EEG signal. The randomness enters by way of four independent white-noise inputs representing fluctuations in the four types (e-e, i-e, e-i, i-i) of subcortic alactivity. Our model is a spatial and temporal simplification of the original set of eight coupled partial differential equations (PDEs) due to Liley et al. [Neurocomputing 26-27, 795 (1999)] describing the electrical rhythms of the cortex. We assume (i) spatial homogeneity (i.e., the entire cortex can be represented by a single macrocolumn), and (ii) a separation of temporal scales in which all inputs to the soma “capacitor” are treated as fast variables that settle to steady state very much more rapidly than do the soma voltages themselves: this is the “adiabatic approximation.” These simplifications permit the eight-equation Liley set to be collapsed to a single pair of first-order PDEs in hₑ and hᵢ. We incorporate the effect of general anaesthetic as a lengthening of the duration of the inhibitory post-synaptic potential (PSP) (i.e., we are modelling the GABAergic class of anaesthetics), thus the effectiveness of the inhibitory firings increases monotonically with anaesthetic concentration. These simplified equations of motion for hₑ,ᵢ are transformed into Langevin (stochastic) equations by adding small white-noise fluctuations to each of the four subcortical spike-rate averages. In order to anchor the analysis, I first identify the t → ∞ steady-state values for the soma voltages. This is done by turning off all noise sources and setting the dhₑ/dt and dhᵢ/dt time derivatives to zero, then numerically locating the steady-state coordinates as a function of anaesthetic effect λ, the scale-factor for the lengthening of the inhibitory PSP. We find that, when plotted as a function of λ, the steady-state soma voltages map out a reverse-S trajectory consisting of a pair of stable branches—the upper (active, high-firing) branch, and the lower (quiescent, low-firing) branch—joined by an unstable mid-branch. Because the two stable phases are not contiguous, the model predicts that a transit from one phase to the other must be first-order discontinuous in soma voltage, and that the downward (induction) jump from active-awareness to unconscious-quiescence will be hysteretically separated from (i.e., will occur at a larger concentration of anaesthetic than) the upward (emergence) jump for the return of consciousness. By reenabling the noise terms, then linearizing the Langevin equations about one of the stable steady states, we obtain a two-dimensional Ornstein-Uhlenbeck (Brownian motion) system which can be analyzed using standard results from stochastic calculus. Accordingly, we calculate the covariance, time-correlation, and spectral matrices, and find the interesting predictions of vastly increased EEG fluctuation power, attended by simultaneous redistribution of spectral energy towards low frequencies with divergent increases in fluctuation correlation times (i.e., critical slowing down), as the macrocolumn transition points are approached. These predictions are qualitatively confirmed by clinical measurements reported by Kuizenga et al. [British Journal of Anaesthesia 80, 725 (1998)] of the so-called EEG biphasic effect. He used a slew-rate technique known as aperiodic analysis, and I demonstrate that this is approximately equivalent to a frequency-scaling of the power spectral density. Changes in the frequency distribution of spectral energy can be quantified using the notion of spectral entropy, a modern measure of spectral “whiteness.” We compare the spectral entropy predicted by the model against the clinical values reported recently by Viertiӧ-Oja et al. [Journal of Clinical Monitoring 16, 60 (2000)], and find excellent qualitative agreement for the induction of anaesthesia. To the best of my knowledge, the link between spectral entropy and correlation time has not previously been reported. For the special case of Lorentzian spectrum (arising from a 1-D OU process), I prove that spectral entropy is proportional to the negative logarithm of the correlation time, and uncover the formula which relates the discrete H₁ Shannon information to the continuous H₂ “histogram entropy,” giving an unbiased estimate of the underlying continuous spectral entropy Hω. The inverse entropy-correlation relationship suggests that, to the extent that anaesthetic induction can be modelled as a 1-D OU process, cortical state can be assessed either in the time domain via correlation time or, equivalently, in the frequency domain via spectral entropy. In order to investigate a thermodynamic analogy for the anaesthetic-driven (“anaestheto-dynamic”) phase transition of the cortex, we use the steady-state trajectories as an effective equation of state to uncouple the macrocolumn into a pair of (apparently) independent “pseudocolumns.” The stable steady states may now be pictured as local minima in a landscape of potential hills and valleys. After identifying a plausible temperature analogy, we compute the analogous entropy and predict discontinous entropy change—with attendant “heat capacity” anomalies—at transition. The Stullken dog experiments [Stullken et al., Anesthesiology 46, 28(1977)], measuring cerebral metabolic rate changes, seem to confirm these model predictions. The penultimate chapter examines the impact of incorporating NMDA, an important excitatory neurotransmitter, in the adiabatic model. This work predicts the existence of a new stable state for the cortex, midway between normal activity and quiescence. An induction attempt using a pure anti-NMDA anaesthetic agent (e.g., xenon or nitrous oxide) will take the patient to this mid-state, but no further. I find that for an NMDA-enabled macrocolumn, a GABA induction can produce a second biphasic power event, depending on the brain state at commencement. The latest clinical report from Kuizenga et al. [British Journal of Anaesthesia 86, 354 (2001)] provides apparent confirmation

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

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