Effects of the anesthetic agent propofol on neural populations

Abstract

The neuronal mechanisms of general anesthesia are still poorly understood. Besides several characteristic features of anesthesia observed in experiments, a prominent effect is the bi-phasic change of power in the observed electroencephalogram (EEG), i.e. the initial increase and subsequent decrease of the EEG-power in several frequency bands while increasing the concentration of the anaesthetic agent. The present work aims to derive analytical conditions for this bi-phasic spectral behavior by the study of a neural population model. This model describes mathematically the effective membrane potential and involves excitatory and inhibitory synapses, excitatory and inhibitory cells, nonlocal spatial interactions and a finite axonal conduction speed. The work derives conditions for synaptic time constants based on experimental results and gives conditions on the resting state stability. Further the power spectrum of Local Field Potentials and EEG generated by the neural activity is derived analytically and allow for the detailed study of bi-spectral power changes. We find bi-phasic power changes both in monostable and bistable system regime, affirming the omnipresence of bi-spectral power changes in anesthesia. Further the work gives conditions for the strong increase of power in the δ-frequency band for large propofol concentrations as observed in experiments.

Appendix

Variables from section "Linear stability"

This section gives the polynomial constants of the characteristic Eq. 27:

$$ \begin{aligned} C(p,k)&=\gamma_e+\gamma_i+A_1-B_1\\ D(p,k)&=\omega_0^2+1-A_0+(\gamma_e+A_1)\gamma_i-B_1\gamma_e+B_0\\ E(p,k)&=(\omega_0^2+B_0)\gamma_e+(1-A_0)\gamma_i+\omega_0^2A_1\\ F(p,k)&=\omega_0^2(1-A_0)+B_0 \end{aligned} $$

(51)

with

$$ \begin{aligned} A_0(p,k)=&\delta_E(p)M_{0}(\sigma_ek), A_1(p,k)=\delta_E(p)\tau_eM_{1}(\sigma_ek)\\ B_0(p,k)=&\omega_0^2f(p)\delta_I(p)M_{0}(\sigma_ik), B_1(p,k)=\omega_0^2f(p)\delta_I(p)\tau_iM_{1}(\sigma_ik) \end{aligned} $$

and τ _e  = σ _e /v,τ _i  = σ _i /v. For p = 1, δ _E , δ _I  ≈ 0 and A ₀, B ₀, A ₁, B ₁ ≈ 0. Hence the pre-factors of the polynom (27) read

$$ \begin{aligned} C(p,k)&=\gamma_e+\gamma_i, \quad D(p,k)=\omega_0^2+1+\gamma_e\gamma_i\\ E(p,k)&=\omega_0^2\gamma_e+\gamma_i, \quad F(p,k)=\omega_0^2 \end{aligned} $$

The autocorrelation function

To obtain the effective membrane potential u(x,t) in section "The general power spectrum", we may write

$$ u(x,t)=\int_\Upomega dx^\prime \int_{-\infty}^\infty dt^\prime G(x-x^\prime,t-t^\prime)\Upgamma(x^\prime,t^\prime) $$

(52)

$$ =\int_{-\infty}^\infty dk \int_{-\infty}^\infty d\omega \tilde{G}(k,\omega)\tilde{\Upgamma}(k,\omega)e^{ikx-i\omega t}. $$

(53)

Here G(x,t) is the Greens’ function of the system, $\tilde{G}(k,\omega)$ denotes its Fourier transform and $\tilde{\Upgamma}(k,\omega)$ is the Fourier transform of the external stimulus Γ(x,t). Considering (53), then the correlation function reads

$$ \begin{aligned} C_{LFP}(x,\tau)=&\langle u(x,t)u(x,t-\tau)\rangle\\ =&{\frac{1}{(2\pi)^2}}\int_{-\infty}^\infty dk\int_{-\infty}^\infty dk^\prime \int_{-\infty}^\infty d\omega\int_{-\infty}^\infty d\omega^\prime\\ &\times\tilde{G}(k,\omega)\tilde{G}(k^\prime,\omega^\prime) \langle\tilde{\Upgamma}(k,\omega)\tilde{\Upgamma}(k^\prime,\omega^\prime)\rangle e^{i(kx+k^\prime x)-i\omega t-i\omega^\prime(t-\tau)}. \end{aligned} $$

(54)

Since the power spectrum p _LFP (x,ω) is defined by C _LFP (x,τ), we deduce from (54) that the power spectrum is determined by the Fourier transform of the Greens’ function $\tilde{G}(k,\omega)$ and the input correlation function in Fourier space $\langle \tilde{\Upgamma}(k,\omega)\tilde{\Upgamma}(k^\prime,\omega^\prime)\rangle.$

The Greens’ function

To compute the Greens function, we apply the Fourier transform in space to Eqs. 33 and 34, and obtain

$$ \begin{aligned} \tilde{u}_e(k,t)=&a_e\delta_E\int_{-\infty}^t d\tau h_e(t-\tau) \int_\Upomega dz K_e(z)\left(\tilde{u}_e\left(k,\tau-{\frac{|z|} {v}}\right)-\tilde{u}_i\left(k,\tau-{\frac{|z|}{v}}\right)\right)e^{-ikz}+\tilde{\Upgamma}(k,t)\\ \tilde{u}_i(x,t)=&a_i\delta_If\omega_0^2\int_{-\infty}^t d\tau h_i(t-\tau) \int_\Upomega dz K_i(z)\left(\tilde{u}_e\left(k,\tau-{\frac{|z|}{v}}\right) -\tilde{u}_i\left(k,\tau-{\frac{|z|}{v}}\right)\right)e^{-ikz}. \end{aligned} $$

Then it follows that

$$ \begin{aligned} \tilde{u}(k,t)=\,&\tilde{u}_e(k,t)-\tilde{u}_i(k,t)\\ =\,&a_e\delta_E\int_{-\infty}^t d\tau h_e(t-\tau) \int_\Upomega dz K_e(z)\left(\tilde{u}_e\left(k,\tau-{\frac{|z|}{v}}\right)-\tilde{u}_i\left(k,\tau -{\frac{|z|}{v}}\right)\right)e^{-ikz}\\ &-a_i\delta_If\omega_0^2\int_{-\infty}^t d\tau h_i(t-\tau) \int_\Upomega dz K_i(z)\left(\tilde{u}_e\left(k,\tau-{\frac{|z|}{v}}\right)-\tilde{u}_i\left(k,\tau-{\frac{|z|} {v}}\right)\right)e^{-ikz} +\tilde{\Upgamma}(k,t)\\ =&\int_{-\infty}^t d\tau\int_\Upomega dz H(z,t-\tau)\tilde{u}\left(k,\tau-{\frac{|z|}{v}}\right)e^{-ikz}+\tilde{\Upgamma}(k,t) \end{aligned} $$

(55)

with

$$ H(z,t)=a_e\delta_E h_e(t) K_e(z)-a_i\delta_If\omega_0^2 h_i(t) K_i(z). $$

In addition

$$ g(k,t)={\frac{1}{\sqrt{2\pi}}}\int_{-\infty}^{\infty} d\omega \tilde{G}(k,\omega)e^{-i\omega t} $$

(56)

is the spatial Fourier transform of G(x,t) and we write $\tilde{u}(k,t)$ using (52) as

$$ \tilde{u}(k,t)=\int_{-\infty}^{\infty} d\tau g(k,t-\tau)\tilde{\Upgamma}(k,\tau). $$

(57)

Further we recall the identity (see e.g. Atay and Hutt 2006)

$$ \tilde{u}\left(k,t-{\frac{|z|}{c}}\right) =\sum_{n=0}^\infty{\frac{1}{n!}}\left( -{\frac{|z|}{c}}\right)^n{\frac{\partial^n\tilde{u}(k,t)}{\partial t^n}} $$

(58)

and obtain from (55), (56), (57) and (58) after a Fourier transformation into frequency space

$$ \tilde{G}(k,\omega)={\frac{1}{\sqrt{2\pi}}}{\frac{1}{1-\sum_{n=0}^\infty{\mathcal L}_n(k,\omega) (-i\omega)^n}} $$

(59)

with

$$ {\mathcal L}_n(k,\omega)=\,{\frac{1}{n!}}\left(-{\frac{1}{v}}\right)^n\int_0^\infty dt \left(a_e\delta_E h_e(t) \tilde{K}_e^{n}(k)-a_i\delta_If\omega_0^2 h_i(t) \tilde{K}_i^{n}(k)\right)e^{i\omega t} $$

(60)

and the kernel Fourier moments (Atay and Hutt 2005)

$$ \tilde{K}^{n}(k)=\int_\Upomega dz K(z)|z|^ne^{-ikz}. $$

The external input

Considering the input (36), then the Fourier transform in space and time yields

$$ \tilde{\Upgamma}(k,\omega)={\frac{1}{\sqrt{2\pi}}} \bar{h}_e(\omega)\tilde{\xi}(k,\omega), $$

(61)

$$ \bar{h}_e(\omega)=\int_0^\infty dt h_e(t)e^{i\omega t} $$

(62)

with the Fourier transform of the external signal $\tilde{\xi}(k,\omega).$ Since the external fluctuations in Fourier space are uncorrelated,

$$ \langle \tilde{\xi}(k,\omega)\tilde{\xi}(k^\prime,\omega^\prime) \rangle=Q\delta({k+k^\prime}) \delta({\omega+\omega^\prime}), $$

(63)

we obtain finally

$$ \langle \tilde{\Upgamma}(k,\omega)\tilde{\Upgamma}(k^\prime,\omega^\prime)\rangle ={\frac{Q} {2\pi}}\bar{h}_e (\omega)\bar{h}_e(\omega^\prime)\delta(k+k^\prime)\delta({\omega+\omega^\prime}). $$

(64)

dX/dp > 0 in single stationary solutions

Considering Eq. 47,

$$ \begin{aligned} {\frac{dX(p)}{dp}}=\,&f^\prime-{\frac{\delta^\prime\rho^\prime(a_e-a_if) -\delta a_if^\prime}{S_mca_i}}{\frac{(1+\rho)^2}{1-\rho}}\\ &+{\frac{1-\delta(a_e-a_if)}{S_mca_i}}{\frac{(3-\rho)(1+\rho)} {(1-\rho)^2}}\rho^\prime \end{aligned} $$

(65)

with f′ = df/dp > 0, δ′ = ∂δ/∂ρ > and ρ′ = dρ/dp > 0. Further Fig. 3b illustrates the limits $\bar{V}_-\gg\Uptheta (\rho\approx 0)$ for p ≈ 1 and $\bar{V}_-\to\Uptheta (\rho \to 1)$ for p → ∞ and section "The resting state" shows that a _e  − a _i f < 0. Then (65) gives ${\frac{dX(p)}{dp}} > 0$ for all p.

The power spectrum for large frequencies

To compute p _EEG(ν), we consider Eq. 41 and compute the Greens’ function (59) in the long wavelength limit as

$$ \tilde{G}(0,\nu)\approx{\frac{1}{\sqrt{2\pi}}}{\frac{1}{1-{\mathcal L}_0(p,\nu)+i2\pi\nu{\mathcal L}_1(p,\nu)}} $$

(66)

with ${\mathcal L}_0(p,\nu), {\mathcal L}_1(p,\nu)$ defined as

$$ \begin{aligned} {\mathcal L}_0(p,\nu)=\,&{\mathcal L}_{0,r}(p,\nu)+i2\pi\nu{\mathcal L}_{0,i}(p,\nu)\\ {\mathcal L}_1(p,\nu)=\,&{\mathcal L}_{1,r}(p,\nu)+i2\pi\nu{\mathcal L}_{1,i}(p,\nu) \end{aligned} $$

with

$$ \begin{aligned} {\mathcal L}_{0,r}(p,\nu)=\,&A_e(\nu)\delta_E(p)-A_i(p,\nu)\delta_I(p)f(p)\omega_0^2(p)\\ {\mathcal L}_{0,i}(p,\nu)=\,&B_e(\nu)\delta_E(p)-B_i(p,\nu)\delta_I(p)f(p)\omega_0^2(p)\\ {\mathcal L}_{1,r}(p,\nu)=\,&A_e(\nu)a_e\delta_E(p)\tilde{K}_e^{1}(0)/v -A_i(p,\nu)a_i\delta_I(p)f(p)\omega_0^2(p)\tilde{K}_i^{1}/v\\ {\mathcal L}_{1,i}(p,\nu)=\,&B_e(\nu)a_e\delta_E(p)\tilde{K}_e^{1}(0)/v -B_i(p,\nu)a_i\delta_I(p)f(p)\omega_0^2(p)\tilde{K}_i^{1}/v \end{aligned} $$

and

$$ \begin{aligned} A_e(\nu)=\,&{\frac{1-(2\pi\nu)^2}{1+(2\pi\nu)^2(\gamma_e^2-2)+(2\pi\nu)^4}}\\ A_i(\nu)=\,&{\frac{\omega_0^2-(2\pi\nu)^2}{\omega_0^4+(2\pi\nu)^2(\gamma_i^2 -\omega_0^2)+(2\pi\nu)^4}}\\ B_e(\nu)=\,&{\frac{\gamma_e}{1+(2\pi\nu)^2(\gamma_e^2-2) +(2\pi\nu)^4}}\\ B_i(\nu)=\,&{\frac{\gamma_i}{\omega_0^4+(2\pi\nu)^2(\gamma_i^2-\omega_0^2) +(2\pi\nu)^4}}. \end{aligned} $$

Equation 66 assumes the approximation of a large but finite propagation speed v. Then inserting (66) into (41) yields the power spectrum

$$ P_{\rm EEG}(\nu)={\frac{Q}{2\pi}} {\frac{A_e^2(\nu)+B_e^2(\nu)}{(1-{\mathcal L}_{0,r})^2 +4\pi^2\nu^2(2(1-{\mathcal L}_{0,r}) {\mathcal L}_{1,i}+({\mathcal L}_{0,i} +{\mathcal L}_{1,r})^2)+{\mathcal L}_{1,i}^2}}. $$

(67)

The functions ${\mathcal L}_{1,r}, {\mathcal L}_{1,i}$ depend on the propagation speed and are small compared to ${\mathcal L}_{0,r}.$ Hence neglecting terms containing ${\mathcal L}_{1,r}, {\mathcal L}_{1,i},$ we find the conditions (50).