Gap junctions modulate seizures in a mean-field model of general anesthesia for the cortex

Abstract

During slow-wave sleep, general anesthesia, and generalized seizures, there is an absence of consciousness. These states are characterized by low-frequency large-amplitude traveling waves in scalp electroencephalogram. Therefore the oscillatory state might be an indication of failure to form coherent neuronal assemblies necessary for consciousness. A generalized seizure event is a pathological brain state that is the clearest manifestation of waves of synchronized neuronal activity. Since gap junctions provide a direct electrical connection between adjoining neurons, thus enhancing synchronous behavior, reducing gap-junction conductance should suppress seizures; however there is no clear experimental evidence for this. Here we report theoretical predictions for a physiologically-based cortical model that describes the general anesthetic phase transition from consciousness to coma, and includes both chemical synaptic and direct electrotonic synapses. The model dynamics exhibits both Hopf (temporal) and Turing (spatial) instabilities; the Hopf instability corresponds to the slow (≲8 Hz) oscillatory states similar to those seen in slow-wave sleep, general anesthesia, and seizures. We argue that a delicately balanced interplay between Hopf and Turing modes provides a canonical mechanism for the default non-cognitive rest state of the brain. We show that the Turing mode, set by gap-junction diffusion, is generally protective against entering oscillatory modes; and that weakening the Turing mode by reducing gap conduction can release an uncontrolled Hopf oscillation and hence an increased propensity for seizure and simultaneously an increased sensitivity to GABAergic anesthesia.

Appendix

Phase coherence calculations for Figs. 2 and 3

Consider a (real) time-series X(t). Its Hilbert transform is a complex time-series known as the analytic signal; by computing the four-quadrant arctangent of the ratio of its imaginary and real parts we obtain a time-series for ϕ(t), the instantaneous phase angle. The phase similarity between two time-series X(t) and Y(t) can be determined by examining the time-series of their phase differences $\Updelta \phi(t) = \phi_X(t) - \phi_Y(t).$ If X and Y are tightly phase-coupled, then plotting $e^{i\Updelta \phi},$ the difference signal mapped to the unit circle, will give a tightly clustered angular distribution of phasors, whereas if X and Y have a relative phase relation that is incoherent, the difference phasors will be randomly distributed around the complex circle.

We define the mean phase coherence R of the X and Y time-series as the length of the time-averaged phasor for the angular distribution of phase differences, $R = |\langle e^{i\Updelta \phi}\rangle|.$ If X and Y are tightly phase coupled, then R ≈ 1, and if they are uncoupled, R ≈ 0.

A Matlab implementation of the phase coherence algorithm reads as follows:

Let X ≡ Q _e (x ₀, t) and Y ≡ Q _e (x _k , t) be a pair of cortical firing-rate time-series belonging respectively to rows 0 and k of a given Fig. 2c strip-chart. The x-axis separation between these two rows is $\Updelta x = k\delta x,$ and the mean phase coherence, $R(\Updelta x),$ of their time-series gives a measure of the degree to which cortical activity is correlated for an electrode pair separated by distance $\Updelta x.$ We fix the x ₀ reference position to be close to the center of the grid, and vary the position of the second electrode x _k with k = 0,  ±1,  ±2, … corresponding to stepped increases in electrode separation ranging from $\Updelta x = 0$ to 12.5 cm. In this way we are able to construct the Fig. 2e graphs showing the variation of phase coherence with distance.

In these calculations, we took care to skip over first 1 s of recording to allow time for the initial transients to settle out. We used a 5-s recording window with 1-s overlap, and followed Mormann et al. (2000) in applying a Hanning window, retaining only the middle 80% of each segment to minimize edge distortions from the Hilbert transform. Because we expect phase symmetry for rows ±k symmetrically displaced either side of the x ₀ reference position, we used the standard deviation in their phase coherence values to estimate the error bars at separation $\Updelta x = k\delta x.$