Equilibrium and Nonequilibrium Phase Transitions in a Continuum Model of an Anesthetized Cortex

Abstract

In this chapter we investigate a range of dynamic behaviors accessible to a continuum model of the cerebral cortex placed close to the anesthetic phase transition. If the anesthetic transition from the high-firing (conscious) to the low-firing (comatose) state can be modeled as a jump between two equilibrium states of the cortex, then we can draw an analogy with the vapor-to-liquid phase transition of the van der Waals gas of classical thermodynamics. In this analogy, specific volume (inverse density) of the gas maps to cortical activity, with pressure and temperature being the analogs of anesthetic concentration and subcortical excitation. It is well known that at the thermodynamic critical point, large fluctuations in specific volume are observed; we find analogous critically-slowed fluctuations in cortical activity at its critical point. Unlike the van der Waals system, the cortical model can also exhibit nonequilibrium phase transitions in which the homogeneous equilibrium can destabilize in favor of slow global oscillations (Hopf temporal instability), stationary structures (Turing spatial instability), and chaotic spatiotemporal activity patterns (Hopf–Turing interactions). We comment on possible physiological and pathological interpretations for these dynamics. In particular, the turbulent state may correspond to the cortical slow oscillation between “up” and “down” states observed in nonREM sleep and clinical anesthesia.

Appendix

1.1 Model Equations

The cortex is modeled as a 2-D continuum of excitatory and inhibitory neurons, interconnected via resistive gap junctions and neurotransmitter -mediated chemical synapses . The spatially-averaged excitatory (inhibitory) soma potentials V _e (V _i ) obey partial differential equations,

$$\displaystyle\begin{array}{rcl} \tau _{e}\,\frac{\partial V _{e}} {\partial t} \: =\: V _{e}^{\text{rest}} + \Delta V _{ e}^{\text{rest}} - V _{ e} + \left [\rho _{e}\,\psi _{\mathit{ee}}\varPhi _{\mathit{ee}} +\rho _{i}\,\psi _{\mathit{ie}}\varPhi _{\mathit{ie}}\right ] + D_{1}\nabla ^{2}V _{ e}\,,& &{}\end{array}$$

(15.3)

$$\displaystyle\begin{array}{rcl} \tau _{i}\,\frac{\partial V _{i}} {\partial t} \: =\: V _{i}^{\text{rest}} - V _{ i} + \left [\rho _{e}\,\psi _{\mathit{ei}}\varPhi _{\mathit{ei}} +\rho _{i}\,\psi _{\mathit{ii}}\varPhi _{\mathit{ii}}\right ] + D_{2}\nabla ^{2}V _{ i}\,,& &{}\end{array}$$

(15.4)

with chemical-synaptic inputs […] enclosed in square brackets, and gap-junction inputs entering as diffusion terms $D\nabla ^{2}V _{e,i}$. Here, τ _b (b = e, i) is the soma time-constant; $V _{b}^{\text{rest}}$ is the soma resting voltage; ρ _b is the chemical synaptic strength with ρ _e  > 0 (EPSP) and ρ _i  < 0 (IPSP). These strengths are scaled by dimensionless reversal-potential functions ψ _ab (a = e, i),

$$\displaystyle\begin{array}{rcl} \psi _{ab}(t)\: =\: \frac{V _{a}^{\text{rev}} - V _{b}(t)} {V _{a}^{\text{rev}} - V _{b}^{\text{rest}}}\,,& & {}\\ \end{array}$$

that are normalized to unity when the neuron is at its resting voltage, and are zero when the membrane voltage reaches the relevant reversal potential (see Table 15.1 for values). The Φ _eb, ib functions in Eqs. (15.3, 15.4) are chemical-synaptic input fluxes obeying second-order differential equations,

$$\displaystyle\begin{array}{rcl} \left ( \frac{d} {\mathit{dt}} +\gamma _{e}\right )^{2}\varPhi _{ \mathit{eb}}\: =\:\gamma _{ e}^{2}\big[N_{\mathit{ eb}}^{\alpha }\,\phi _{ \mathit{eb}}^{\alpha }(t) + N_{\mathit{ eb}}^{\beta }\,Q_{ e}(t) +\phi _{ \mathit{eb}}^{\text{sc}}(t)\big],& &{}\end{array}$$

(15.5)

$$\displaystyle\begin{array}{rcl} \left ( \frac{d} {\mathit{dt}} +\gamma _{i}\right )^{2}\varPhi _{ \mathit{ib}}\: =\:\gamma _{ i}^{2}N_{\mathit{ ib}}^{\beta }\,Q_{ i}(t),& &{}\end{array}$$

(15.6)

with dendritic rate constants γ _e, i . The cortico-cortical and local connectivities N ^α and N ^β scale their respective incoming fluxes ϕ ^α, Q _e, i respectively; these fluxes are supplemented by an unstructured subcortical stimulation $\phi _{}^{\text{sc}}$ modeled as a small white-noise variation ξ(t) about a constant tone $\langle \phi _{}^{\text{sc}}\rangle$,

$$\displaystyle\begin{array}{rcl} \phi _{\mathit{eb}}^{\text{sc}}(t)\: =\:\langle \phi _{ \mathit{ eb}}^{\text{sc}}\rangle \: +\:\alpha \sqrt{\langle \phi _{\mathit{ eb}}^{\text{sc}}\rangle }\,\xi _{\mathit{eb}}(t),& & {}\\ \end{array}$$

where α is a dimensionless noise-amplitude scale-factor. The local fluxes Q _e, i in Eqs. (15.5, 15.6) are defined by a sigmoidal mapping from soma voltage to firing rate,

$$\displaystyle\begin{array}{rcl} Q_{e,i}(t)\: =\: \frac{Q_{e,i}^{\max }} {1 +\exp \left [-C\left (V _{e,i}(t) -\theta _{e,i}\right )/\sigma _{e,i}\right ]}\,,& & {}\\ \end{array}$$

with $C =\pi /\sqrt{3}$. Here, θ is the population-average threshold for firing, σ is its standard deviation, and Q ^max is the maximum firing rate.

The cortico-cortical flux ϕ ^α in Eq. (15.5) is generated by excitatory sources $Q_{e}(\mathbf{r},t)$, and obeys a 2-D damped wave equation [22],

$$\displaystyle\begin{array}{rcl} \left [\left ( \frac{\partial } {\partial t} + v\varLambda _{\mathit{eb}}\right )^{2} -\, v^{2}\nabla ^{2}\right ]\phi _{\mathit{ eb}}^{\alpha }(\mathbf{r},t)\: =\: (v\,\varLambda _{\mathit{ eb}})^{2}\,Q_{ e}(\mathbf{r},t)\,,& &{}\end{array}$$

(15.7)

where Λ _eb is the inverse-length scale for e → b axonal connections, and v is the axonal conduction speed.

The ∇² diffusion terms in Eqs. (15.3, 15.4) describe the voltage contribution arising from gap-junction currents between adjacent neurons. Gap-junction i-to-i coupling between inhibitory interneurons is substantially more abundant than e-to-e coupling between excitatory neurons [3], and in layer-1 of cortex, over 90 % of the neural density is inhibitory [35], suggesting the existence of a syncytium of interneuron-to-interneuron diffusive scaffolding that spans the cortex . In view of the relative dominance of i-to-i diffusion , we set the e-to-e diffusion strength D ₁ to be small fraction of inhibitory diffusion D ₂ (viz. $D_{1} = D_{2}/100$), with D ₂ being an adjustable parameter (see [29] for detailed derivation and estimation of D ₂ diffusive coupling).

1.2 Modeling Propofol Anesthesia

Inductive anesthetic agents, such as propofol , suppress neural activity by prolonging the opening of GABA (gamma−aminobutyric acid) channels on the postsynaptic neuron [7], allowing increased influx of chloride ions leading to hyperpolarization. We model propofol effect by simultaneously scaling both the inhibitory synaptic strength ρ _i (in Eqs. (15.3, 15.4)) and the dendritic rate-constant γ _i (Eq. (15.6)) by a dimensionless scale-factor λ that is set to unity in the absence of propofol, and which grows proportionately to propofol concentration,

$$\displaystyle\begin{array}{rcl} \rho _{i} \rightarrow \lambda \rho _{i}^{0},\qquad \gamma _{ i} \rightarrow \gamma _{i}^{0}/\lambda & & {}\\ \end{array}$$

where γ _i ⁰ and ρ _i ⁰ are the anesthetic-free default values. This scaling prolongs the duration of the inhibitory postsynaptic potential (IPSP) without altering its peak amplitude [14] so that the area of the IPSP response (representing total charge transfer) increases linearly with drug concentration. We note that at very high propofol concentrations—well above the clinically relevant range—the charge-transfer versus drug-concentration curve shows saturation effects [14], but the assumption of linearity is accurate at low concentrations, and has been used by Hutt and Longtin [11] in their anesthesia modeling.

1.3 Linear Stability Analysis for Homogenous Stationary States

Equations (15.3–15.7) define the cortical model in terms of two first-order (V _e, i soma voltages), and six second-order ($\varPhi _{\mathit{ee},\mathit{ei}};\,\varPhi _{\mathit{ie},\mathit{ii}};\,\phi _{\mathit{ee},\mathit{ei}}$ firing-rate fluxes) partial differential equations (DEs). If we disable the subcortical noise and take note of the parameter symmetries evident in Table 15.1 (viz., $N_{\mathit{ee}}^{\alpha } = N_{\mathit{ei}}^{\alpha }$; $N_{\mathit{ee}}^{\beta } = N_{\mathit{ei}}^{\beta }$; $N_{\mathit{ie}}^{\beta } = N_{\mathit{ii}}^{\beta }$), the cortical system reduces to a set of two first-order and three second-order DEs, equivalent to eight first-order equations. We locate the homogenous equilibrium states by eliminating all space- the time-derivatives in differential equations (15.3–15.7) $(\nabla ^{2} = 0;\partial /\partial t = \partial ^{2}/\partial t^{2} = 0)$, then solving (numerically) the resulting set of nonlinear coupled algebraic equations for the steady-state firing rates (Q _e , Q _i ) of the excitatory and inhibitory neural populations as a function of anesthetic effect λ and resting potential offset Δ V _e ^rest. The resulting distribution of homogeneous stationary states are displayed in Figs. 15.1a and 15.2a.

We define an eight-variable state vector $\mathbf{X}\,=\,\left [V _{e},V _{i},\varPhi _{\mathit{eb}},\dot{\varPhi }_{\mathit{eb}},\varPhi _{\mathit{ib}},\dot{\varPhi }_{\mathit{ib}},\phi _{\mathit{eb}},\dot{\phi }_{\mathit{eb}}\right ]^{\text{T}}$ with homogeneous equilibrium value $\mathbf{X}^{(0)}$. We examine the linear stability of this stationary state by imposing a small spatiotemporal disturbance $\delta \mathbf{X}$ about $\mathbf{X}^{(0)}$,

$$\displaystyle\begin{array}{rcl} \mathbf{X}(t,\mathbf{r}) =\mathbf{ X}^{(0)} +\delta \mathbf{ X}(t,\mathbf{r})& & {}\\ \end{array}$$

with $\delta \mathbf{X}$ being a plane-wave perturbation

$$\displaystyle\begin{array}{rcl} \delta \mathbf{X}(t,\mathbf{r}) =\delta \mathbf{ X}(t)\,e^{i\mathbf{q}\cdot \mathbf{r}} =\delta \mathbf{ X}(0)\,e^{\Lambda t}\,e^{i\mathbf{q}\cdot \mathbf{r}}\,,& & {}\\ \end{array}$$

where $\mathbf{q}$ is the wavevector with wavenumber $\vert \mathbf{q}\vert = q$, and $\Lambda$ is an eigenvalue whose real part gives the growth rate of the $\delta \mathbf{X}(0)$ initial perturbation: if $\text{Re}(\Lambda ) > 0$, an instability is predicted. Substituting $\mathbf{X} =\mathbf{ X}^{(0)} +\delta \mathbf{ X}$ into Eqs. (15.3–15.7) and retaining only linear terms results in the matrix equation,

$$\displaystyle\begin{array}{rcl} \frac{d} {\mathit{dt}}\delta \mathbf{X} = \mathbf{J}(q)\,\delta \mathbf{X}\,,& & {}\\ \end{array}$$

where J is an 8 × 8 Jacobian matrix in which the ∇² Laplacians for excitatory and inhibitory diffusion (Eqs. 15.3, 15.4), and wave propagation (Eq. 15.7) appear as − q ² terms. The eight eigenvalues owned by J describe the linearized dynamics of the homogeneous cortex. For each wavenumber q, we extract and plot the dominant eigenvalue—i.e., that eigenvalue whose real part is most positive (or least negative)—since this describes the most strongly growing (or most long-lived) mode at a given spatial frequency. The resulting $\Lambda$ vs q dispersion curves are shown in Figs. 15.3 and 15.5.

Although the linear dispersion curve provides valuable guidance regarding the onset of instability (i.e., when the real part of the dominating eigenvalue will approach zero), it cannot predict accurately the new dynamics that will emerge once the homogeneous steady state has lost stability and the nonlinear terms can no longer be ignored. This mismatch between linear dispersion prediction and actual simulation outcome is nicely illustrated in Fig. 15.5b which suggests a zero-frequency instability at zero wavenumber will interact with a low frequency wave instability, while the simulation of Fig. 15.7 shows that the actual outcome is a 1.6-Hz Hopf oscillation at q = 0. Similarly, Fig. 15.5d indicates a zero-frequency instability at q = 0 competing with a stationary Turing, but the Fig. 15.9 simulation shows destabilization in favor of strongly turbulent unsteady interactions between Hopf and Turing instabilities.