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 ($\lesssim 1\mathrm{Hz}$) 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.

Popular Summary

The number of neurons that form the cerebral cortex (part of the gray matter) of a human brain is immense: some $8\times {10}^{10}$ on average. The spatiotemporal patterning in the activity of these neurons, as observed in brain imaging studies, is believed to be linked to cognitive functions such as learning, recognition, and speech. But the precise linkage between pattern and function remains elusive and is one of the most hotly debated topics in neuroscience today. The problem is being tackled at many structural levels (from networks of individual neurons to neuronal populations or continuum) and using many scientific approaches (such as electrical recordings and functional magnetic resonance imaging). In this paper, we put forward the idea that many aspects of the spatial patterning observed in imaging studies are the result of spontaneous pattern formation generated from the electrical synaptic coupling, also known as gap junctions, between neurons. Such communication channels have been observed experimentally.

We pursue this idea theoretically, in the context of the wake-to-coma transition induced by general anesthesia. Our model, many ingredients of which have been proposed and investigated before, describes the cortex as an excitable, continuum medium of interconnected neurons that is poised at a special point where a spatial (Turing) instability and a temporal (Hopf) instability can coexist. We suggest that normal functioning in the awake brain requires a delicate balance between two competing pressures: the need to form spatial patterns versus the need to generate global rhythms in the cortex. Small changes in model parameters, which reflect altered neurophysiological conditions, can instigate a rebalancing of these two instabilities. Extreme imbalance in either direction is almost certainly pathological: Frozen activity patterns prevent information transfer, while synchronous oscillations across the whole cortex correspond to seizure.

Our study of the model reveals some interesting findings. If the gap-junction-based inhibitory coupling between neurons is sufficiently strong, spatiotemporally patterned neural activity emerges spontaneously with a temporal rhythm that is controlled by the responsiveness of inhibitory chemical synapses. In particular, the model predicts that introduction of anesthetics to the awake brain—in the form of an increased influence of the inhibitory chemical synapses—leads to the emergence of a turbulent, chaotic pattern of slow-wave activity, the result of a subtle rebalancing between Turing and Hopf influences. Gap-junction-facilitated interneuronal communication, acting across the cortex, is crucial to the emergence of this pattern.

Significantly, the slow-wave oscillation is the defining feature of scalp-measured electroencephalography, both under general anesthesia and during the natural nonREM (nonrapid-eye-movement) sleep. Our proposed dynamic mechanism for the slow-wave oscillation contrasts with conventional explanations for anesthesia-induced slow oscillations that require cyclic modulation of ion-channel conductances in single neurons. We suggest that a Turing-Hopf interaction may also underpin the slow-wave stage of natural sleep.

Figure 1

Long-range two-point connections on a $120\times 120$ grid representing a $25\mathrm{\text{−}}\mathrm{cm}\times 25\mathrm{\text{−}}\mathrm{cm}$ square of flattened cortical tissue. The diagram shows direct axonal connections from coordinate $(x_1,y_1)=(20,60)$ to $(x_2,y_2)=(40,60)$ (black arrow), and back (gray arrow). Communication between connected points occurs after a time delay $\Delta t=\Delta r/v$, where $v$ is the axonal conduction speed and $\Delta r$ is the physical separation $(L/N_x)|x_1-x_2|$, with $L=25\mathrm{cm}$ and $N_x=120$. We apply periodic (toroidal) boundaries.

Figure 2

Manifold of steady-state firing rates $Q_e$ across the $(\lambda ,\Delta V^{\mathrm{rest}})$ anesthesia domain. The control parameter $\lambda$ sets the anesthetic effect; $\Delta V^{\mathrm{rest}}$ is an additive offset representing background cortical excitation ($\Delta V^{\mathrm{rest}}>0$) or suppression ($\Delta V^{\mathrm{rest}}<0$). The yellow curve marks the edge of the reentrant “fold”; the dashed black curve shows the projection of this edge onto the lower and upper surfaces, bounding the zone of multiple steady states. The red-green-blue curve shows the distribution of steady states for varying anesthetic inhibition at constant cortical excitation (see Fig. 3).

Figure 3

Vertical slice through the homogeneous equilibrium manifold of Fig. 2 showing the distribution of steady-state firing rates as a function of the anesthetic effect for constant excitation $\Delta V^{\mathrm{rest}}=1.5\mathrm{mV}$. We associate the upper, high-firing branch with the awake state, and the lower, low-firing branch with the “asleep” state corresponding to anesthetic-induced coma. Circled points indicate reference states at ${\lambda }_i=1.0$ and 1.018 on awake and coma branches, respectively.

Figure 4

Predicted dispersion curves for spatially homogeneous cortex for (a) top branch and (b) bottom-branch equilibria at ${\lambda }_i=1.0$ (see Fig. 3). Blue traces show the real part of the dominant eigenvalue, $\alpha =\mathrm{Re}(\Lambda )$, as a function of wave number $q$; dashed-red traces indicate oscillation frequency (in Hz) $f=\omega /2\pi =\mathrm{Im}(\Lambda )/2\pi$. Selected inhibitory diffusion values are $D_2/{\mathrm{cm}}^2=0.7$, 0.4, 0.1.

Figure 5

Effect of inhibitory diffusion on wake state. Grid simulations, initialized at the high-firing wake state of Fig. 3, show dynamical effect of stepped reductions in diffusion strength from $D_2=0.7{\mathrm{cm}}^2$ (top row) to $D_2=0$ (bottom). (a) Time-series extracts showing excitatory firing rates $Q_e(t)$ for the final 4 s of a 20-s run at $x=30$ ( black lines), $x=85$ ( gray lines) on the midline ($y=60$) of the $120\times 120$ cortical grid. (b) $Q_e(t,x)$ space-time strip charts showing $x$-axis activity along the $y=60$ midline strip for a full 20-s simulation. (c) $Q_e(y,x)$ bird’s-eye view of cortical activity at $t=20\mathrm{s}$; the color bar indicates the firing rate in spikes/s. Simulation settings: toroidal boundary conditions; integration time step $\Delta t=0.4\mathrm{ms}$. Stimulus: spatiotemporal white noise with scale factor $\alpha =4$; direct two-way connection joins $x_1=20$ to $x_2=60$ on the $y=60$ midline.

Figure 6

Fourier spectra and phase coherence maps for the wake strip charts of Fig. 5. (a) ${\tilde{Q}}_e(f,k)$ shows the 2D discrete Fourier transform of $Q_e(t,x)$ images of Fig. 5b; $f$ is the temporal frequency (Hz), and $k$ is the spatial frequency in $\mathrm{\text{cycles}}/L_x$, where $L_x=25\mathrm{cm}$ is the $x$-axis side length of the cortical grid. Inset line graphs in black compare the average temporal spectrum ${\tilde{Q}}_e(f)$ (horizontal format: summed over wave number $k$) against the average 1D spatial spectrum ${\tilde{Q}}_e(k)$ (vertical format: summed over frequency $f$). (b) Phase coherence maps $R(x,x^{\prime })$ showing the degree of firing-rate coherence for all time-series pairs $Q_e(t,x)$, $Q_e(t,x^{\prime })$ along the $y=60$ midline of a $120\times 120$ cortical grid; coherence maps are computed from Hilbert transforms of the final 2 s (from $t=18$ to 20 s) of Fig. 5b. The color bar shows the mapping between color and phase coherence.

Figure 7

Phase coherence trends for a two-point-connected cortex. Spatially averaged phase coherence measurements for (a) high-firing wake (${\lambda }_i=1.00$) and (b) low-firing coma (${\lambda }_i=1.018$) states of the noise-driven cortex for default two-point connectivity strength ($\mu =200$) are shown. Gray circles represent the outcomes of 20-s noise-stimulated numerical experiments at a given value of inhibitory diffusion in the range 0.0 to $1.0{\mathrm{cm}}^2$ in steps of 0.01 cm; simulations were repeated 20 times at each $D_2$ value for a total of 2020 data points per panel. For each experiment, we computed the $R(x^{\prime },x)$ coherence matrix for the final 2 s of the space-time record [e.g., see Figs. 5b, 6b], then extracted its upper-triangular matrix mean as an estimate of global phase coherence. Bold-black curves show the data median to guide the eye. Solid- (striped-) red bars indicate regions of a strongly (weakly) positive Lyapunov exponent as extracted from paired noise-free simulation runs.

Figure 8

Effect of inhibitory diffusion on the anesthetic-coma state. Grid simulations are given, initialized at the low-firing coma state of Fig. 3, showing the effect of stepped reductions in inhibitory diffusion strength from $D_2=0.8$ (top) to $D_2=0.1{\mathrm{cm}}^2$ (bottom). (a) Time-series extracts showing excitatory firing rates $Q_e(t)$ for the final 4 s of a 20-s run at $x=30$ (black lines) and $x=105$ (gray lines) on the $y=60$ midline of the $120\times 120$ cortical grid. (b) $Q_e(t,x)$ space-time charts showing $x$-axis activity along the $y=60$ midline for a full 20-s simulation. (c) $Q_e(y,x)$ bird’s-eye image of cortical activity at $t=20\mathrm{s}$; the color bar indicates the firing rate in spikes/s. Points $x_1=20$ and $x_2=60$ on the $y=60$ midline are directly connected; see Fig. 5 for the remaining simulation details.

Figure 9

Fourier spectra and phase coherence maps for the coma strip charts of Fig. 8. (a) ${\tilde{Q}}_e(f,k)$ shows the 2D discrete Fourier transform of $Q_e(t,x)$ images of Fig. 8b; $f$ is the temporal frequency (Hz), and $k$ is the spatial frequency ($\mathrm{\text{cycles}}/L_x$). Inset line graphs in black compare the average temporal spectrum ${\tilde{Q}}_e(f)$ (horizontal format) against the average 1D spatial spectrum ${\tilde{Q}}_e(k)$ (vertical format). (b) Phase coherence maps $R(x,x^{\prime })$ showing the degree of firing-rate coherence for time-series pairs $Q_e(t,x),Q_e(t,x^{\prime })$ (with $\{x,x^{\prime }\}\in \{1,2,\dots 120\}$) along the $y=60$ midline; coherence maps are computed from the final 2 s of Fig. 8b strip charts. The color bar shows the mapping between color and phase coherence.

Figure 10

Firing-rate time series for (a) chaotic anesthetic slow-wave oscillations ($D_2=0.4{\mathrm{cm}}^2$) and (b) bursting ($D_2=0.3{\mathrm{cm}}^2$) for 10 equally spaced points lying on the $y=60$ midline of the cortical grid. These graphs correspond, respectively, to horizontal samples $x=1+12i$ ($i=0,\dots ,9$) within rows 4 and 5 of the strip-chart images of Fig. 8b.

Figure 11

Spatially averaged $|{\tilde{Q}}_e(f){|}^2$ power spectra for anesthetic slow-wave (thick-black curve: $D_2=0.4{\mathrm{cm}}^2$) and bursting (gray curve: $D_2=0.3{\mathrm{cm}}^2$) time series illustrated in Figs. 10a, 10b, respectively. Power is expressed in dB relative to the value at 100 Hz. Bursting, but not slow-wave, shows pronounced resonances at $f_1=1.75\mathrm{Hz}$ and its harmonics $n{f}_1$, $n=2\dots 12$. The red line shows the best-fit trend for the power law $P\sim f^{-4.4}$. Frequency resolution for both spectra is 0.05 Hz.

Figure 12

Phase coherence trends for (a) wake (${\lambda }_i=1.00$) and (b) coma (${\lambda }_i=1.018$) for a homogeneous cortex devoid of two-point connections (i.e., $\mu =0$). Bold-black curves show the median trend through data points (gray circles) obtained from 2020 homogeneous cortex experiments; these $\mu =0$ bold-black trends are superimposed on the $\mu =200$ thin-orange median fits from Figs. 7a, 7b to illustrate the impact of suppressing the two-point axonal connection.