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How the cortico-thalamic feedback affects the EEG power spectrum over frontal and occipital regions during propofol-induced sedation

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Abstract

Increasing concentrations of the anaesthetic agent propofol initially induces sedation before achieving full general anaesthesia. During this state of anaesthesia, the observed specific changes in electroencephalographic (EEG) rhythms comprise increased activity in the δ− (0.5−4 Hz) and α− (8−13 Hz) frequency bands over the frontal region, but increased δ− and decreased α−activity over the occipital region. It is known that the cortex, the thalamus, and the thalamo-cortical feedback loop contribute to some degree to the propofol-induced changes in the EEG power spectrum. However the precise role of each structure to the dynamics of the EEG is unknown. In this paper we apply a thalamo-cortical neuronal population model to reproduce the power spectrum changes in EEG during propofol-induced anaesthesia sedation. The model reproduces the power spectrum features observed experimentally both in frontal and occipital electrodes. Moreover, a detailed analysis of the model indicates the importance of multiple resting states in brain activity. The work suggests that the α−activity originates from the cortico-thalamic relay interaction, whereas the emergence of δ−activity results from the full cortico-reticular-relay-cortical feedback loop with a prominent enforced thalamic reticular-relay interaction. This model suggests an important role for synaptic GABAergic receptors at relay neurons and, more generally, for the thalamus in the generation of both the δ− and the α− EEG patterns that are seen during propofol anaesthesia sedation.

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Acknowledgments

The authors acknowledge funding from the European Research Council for support under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 257253.

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Authors and Affiliations

  1. INRIA Grand Est - Nancy, Team NEUROSYS, 615 rue du Jardin Botanique, Villers-lès-Nancy, 54600, France

    Meysam Hashemi & Axel Hutt

  2. CNRS, Loria, UMR nō 7503, Vandoeuvre-lès-Nancy, 54500, France

    Meysam Hashemi & Axel Hutt

  3. Université de Lorraine, Loria, UMR nō 7503, Vandoeuvre-lès-Nancy, 54500, France

    Meysam Hashemi & Axel Hutt

  4. University of Auckland, Hamilton, New Zealand

    Jamie Sleigh

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  1. Meysam Hashemi
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  2. Axel Hutt
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  3. Jamie Sleigh
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Correspondence to Meysam Hashemi.

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The authors declare that they have no conflict of interest.

Appendices

Appendix A: Stationary states

Under the assumption of the spatial homogeneity, the mean excitatory and inhibitory postsynaptic potentials in the cortical pyramidal neurons (E), cortical inhibitory neurons (I), thalamo-cortical relay neurons (S) and thalamic reticular neurons (R) shown in Fig. 1 obey

$$\begin{array}{@{}rcl@{}} \hat {L}_{e} {V^{e}_{E}}(t)&=& K_{EE}S_{C}\left[{V^{e}_{E}}(t)-{V^{i}_{E}}(t)\right]\\ &&+ K_{ES}S_{T}\left[{V^{e}_{S}}(t-\tau)-{V^{i}_{S}}(t-\tau)\right],\\ \hat{L}_{i} {V^{i}_{E}}(t)&=& f_{C}(p) K_{EI} S_{C}\left[{V^{e}_{I}}(t)-{V^{i}_{I}}(t)\right] ,\\ \hat {L}_{e} {V^{e}_{I}}(t)&=& K_{IE}S_{C}\left[{V^{e}_{E}}(t)-{V^{i}_{E}}(t)\right] ,\\ \hat {L}_{i} {V^{i}_{I}}(t)&=& K_{II}S_{C}\left[{V^{e}_{I}}(t)-{V^{i}_{I}}(t)\right] ,\\ \hat{L}_{e} {V^{e}_{S}}(t)&=& K_{SE}S_{C}\left[{V^{e}_{E}}(t-\tau)-{V^{i}_{E}}(t-\tau)\right]+I(t) ,\\ \hat{L}_{i} {V^{i}_{S}}(t)&=& f_{T}(p) K_{SR} S_{T}\left[{V^{e}_{R}}(t)\right] ,\\ \hat{L}_{e} {V^{e}_{R}}(t)&=& K_{RE} S_{C}\left[{V^{e}_{E}}(t-\tau)-{V^{i}_{E}}(t-\tau)\right]\\ &&+K_{RS} S_{T}\left[{V^{e}_{S}}(t)-{V^{i}_{S}}(t)\right], \end{array} $$
(19)

In this model, the long-range propagation of the signals has been considered by the connection between cortex and thalamus associated with a constant time delay and the model follows closely Victor et al. (2011) and Drover et al. (2010) showing that long connections between cortico-cortical populations through white matter are not necessary to describe experimental EEG dynamics. In the case of constant external input I(t)=I 0, the spatially-homogeneous resting states of Eqs. (19) can be obtained by \(dV^{e,i}_{a}/dt=0\), for a∈{E,I,S,R} leading to

$$\begin{array}{@{}rcl@{}} V^{*e}_{E}&=& K_{EE} S_{C}\left[V^{*e}_{E}-V^{*i}_{E}\right]+K_{ES} S_{T}\left[V^{*e}_{S}-V^{*i}_{S}\right],\\ V^{*i}_{E}&=& f_{C}(p)K_{EI}S_{C}\left[V^{*e}_{I}-V^{*i}_{I}\right],\\ V^{*e}_{I}&=& K_{IE} S_{C}\left[V^{*e}_{E}-V^{*i}_{E}\right],\\ V^{*i}_{I}&=& K_{II}S_{C}\left[V^{*e}_{I}-V^{*i}_{I}\right],\\ V^{*e}_{S}&=& K_{SE} S_{C}\left[V^{*e}_{E}-V^{*i}_{E}\right] +I_{0},\\ V^{*i}_{S}&=& f_{T}(p) K_{SR}S_{T}\left[V^{*e}_{R}\right],\\ V^{*e}_{R}&=& {K_{RE} S_{C}\left[V^{*e}_{E}-V^{*i}_{E}\right]+K_{RS} S_{T}\left[V^{*e}_{S}-V^{*i}_{S}\right]}. \end{array} $$
(20)

The dynamic of the reduced model are described by

$$\begin{array}{@{}rcl@{}} \hat{L}_{e}{V^{e}_{E}}(t)&=& K_{ES} S_{T}\left[{V^{e}_{S}}(t-\tau)-{V^{i}_{S}}(t-\tau)\right] ,\\ \hat{L}_{e}{V^{e}_{S}}(t)&= &K_{SE} S_{C}\left[{V^{e}_{E}}(t-\tau)\right]+I(t) ,\\ \hat{L}_{i}{V^{i}_{S}}(t)&=& f_{T}(p) K_{SR} S_{T}\left[{V^{e}_{R}}(t)\right] ,\\ \hat{L}_{e}{V^{e}_{R}}(t)&=& K_{RE} S_{C}\left[{V^{e}_{E}}(t-\tau)\right]\\ &&+K_{RS} S_{T}\left[{V^{e}_{S}}(t)-{V^{i}_{S}}(t)\right], \end{array} $$
(21)

where the resting states of the system obey

$$\begin{array}{@{}rcl@{}} V^{*e}_{E}&=& K_{ES} S_{T}\left[V^{*e}_{S}-V^{*i}_{S}\right],\\ V^{*e}_{S}&=& K_{SE} S_{C}\left[V^{*e}_{E}\right] +I_{0},\\ V^{*i}_{S}&=& f_{T}(p)K_{SR} S_{T}\left[V^{*e}_{R}\right],\\ V^{*e}_{R}&=& {K_{RE} S_{C}\left[V^{*e}_{E}\right]+K_{RS} S_{T}\left[V^{*e}_{S}-V^{*i}_{S}\right]}. \end{array} $$
(22)

By inserting these equations into each other

$$\begin{array}{@{}rcl@{}} V^{*e}_{E}&=&K_{SE} S_{T} \left[ K_{SE} S_{C}\left[V^{*e}_{E}\right]+I_{0} \right.\\ &&\qquad\qquad\left.- f_{T}(p) K_{SR} S_{T}\left[H\left[V^{*e}_{E}\right]\right]\right], \end{array} $$
(23)

where \(H[V^{*e}_{E}] \equiv {K_{RE} S_{C}[V^{*e}_{E}]+\frac {K_{RS}}{K_{ES}} V^{*e}_{E}} \). Since all the resting states \(V^{*e,i}_{a}\) for a∈{E,R,S} can be written as an implicit function of \(V^{*e}_{E}\), the number of solutions of \(V^{*e}_{E}\), i.e., the number of roots of Eq. (23), is identical to the number of resting states (Robinson et al. 1997; Robinson et al. 1998; Robinson et al. 2004).

Appendix B: Theoretical power spectrum

The solution of Eq. (14) for t→∞ is

$$ \boldsymbol{Y}(t)={\int}_{-\infty}^{\infty} \boldsymbol{G}(t-t^{\prime}) \boldsymbol{\xi}(t^{\prime})dt^{\prime}, $$
(24)

with the matrix Greens function \(\boldsymbol {G}\in {\mathbb R}^{N\times N}\), that has dimension N = 7. Substituting Eq. (24) into Eq. (14) leads to

$$ \hat{\boldsymbol{L}} (\partial/\partial t,p) \boldsymbol{G}(t) = \boldsymbol {A }(p) \boldsymbol{G}(t)+\boldsymbol {B }(p)\boldsymbol{G}(t-\tau) + \boldsymbol{1} \delta(t) , $$
(25)

with the unitary matrix \(\boldsymbol {1} \in \mathbb {R}^{N \times N}\). Then the Fourier transform of the matrix Greens function

$$ \boldsymbol{\tilde{G}}(\nu,p)=\frac{1}{\sqrt{2 \pi}} \left[{\boldsymbol{L}}(\nu,p) -\boldsymbol{A}(p)-\boldsymbol{B}(p)e^{-2 \pi i\nu \tau} \right]^{-1}, $$
(26)

and the Wiener-Khinchine theorem defines the power spectral density matrix to

$$ \boldsymbol P(\nu)= 2 \kappa \sqrt{2\pi}\boldsymbol{\tilde{G}}(\nu,p) \boldsymbol{\tilde{G}}^{\top} (-\nu,p), $$

where the high index ⊤ denotes the transposed vector or matrix. Essentially we assume that the EEG is generated by the activity of pyramidal cortical cells. By virtue of the specific choice of external input to relay neurons, the power spectrum of the EEG just depends on one matrix component of the Greens function by

$$\begin{array}{@{}rcl@{}} P_{E}(\nu)&=&{2 \kappa} \sqrt{2\pi}{\tilde{G}}_{1,5}(\nu,p) {\tilde{G}}_{1,5}(-\nu,p)\\ &=&{2 \kappa}\sqrt{2\pi} \left\vert {\tilde{G}}_{1,5}(\nu,p)\right\vert^{2}. \end{array} $$
(27)

In detail, we find

$$\begin{array}{@{}rcl@{}} \boldsymbol{Y}(t)\! &=&\! \left( {V^{e}_{E}}(t)-V^{*e}_{E}, {V^{i}_{E}}(t)-V^{*i}_{E},{V^{e}_{I}}(t)-V^{*e}_{I},{V^{i}_{I}}(t)\right.\\ &&\left.\!-V^{*i}_{I},{V^{e}_{S}}(t)-V^{*e}_{S}, {V^{i}_{S}} (t)\! -\! V^{*i}_{S}, {V^{e}_{R}}(t)-V^{*e}_{R}\right)^{\top}, \end{array} $$

and the diagonal matrix \(\hat {\boldsymbol {L}} (\partial /\partial t,p)\) with the entries \(\hat {L}_{1,1} = \hat {L}_{3,3} = \hat {L}_{5,5} = \hat {L}_{7,7}=\hat {L}_{e}(\nu )\), \(\hat {L}_{2,2} =\hat {L}_{4,4}=\hat {L}_{6,6} = \hat {L}_{i}(\nu ,p)\) and

$$\begin{array}{@{}rcl@{}} \boldsymbol{ A}(p)&=& \left( {\begin{array}{ccccccc} K_{1}&-K_{1}&0&0&0&0&0 \\ 0&0&f_{C}(p)K_{3}&-f_{C}(p)K_{3}&0&0&0 \\ K_{4}&-K_{4}&0&0&0&0&0 \\ 0&0&K_{5}&-K_{5}&0&0 &0\\ 0&0&0&0&0&0&0 \\ 0&0&K_{5}&-K_{5}&0&0 & f_{T}(p)K_{7}\\ 0&0&0&0&K_{9}&-K_{9}&0 \end{array}} \right),\\ \boldsymbol{B}(p) &=& \left( {\begin{array}{ccccccc} 0&0&0&0&K_{2}&-K_{2}&0 \\ 0&0&0&0&0&0&0 \\ 0&0&0&0&0&0 &0\\ 0&0&0&0&0&0&0 \\ K_{6}&-K_{6}&0&0&0&0&0 \\ 0&0&0&0&0&0&0 \\ K_{8}&-K_{8}&0&0&0&0 &0 \end{array}} \right) \end{array} $$

with

$$\begin{array}{@{}rcl@{}} \hat {L}_{e}(\nu,p)&=&\left( 1+\frac{2 \pi i \nu}{\alpha_{e}} \right)\left( 1+\frac{2 \pi i \nu}{\beta_{e}} \right),\\ \hat{L}_{i}(\nu,p)&=&\left( 1+\frac{2 \pi i \nu}{\alpha_{i}} \right)\left( 1+\frac{2 \pi i \nu}{\beta_{i}} \right),\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} K_{1}&=&K_{EE}\frac{d S_{C}[V]}{d V} \mid_{V=\left( V^{*e}_{E}-V^{*i}_{E}\right)},\\ K_{2}&=&K_{ES}\frac{d S_{T}[ V]}{d V} \mid_{V=\left( V^{*e}_{S}-V^{*i}_{S}\right)},\\ K_{3}&=&K_{EI}\frac{d S_{C}[V]}{d V} \mid_{V=\left( V^{*e}_{I}-V^{*i}_{I}\right)},\\ K_{4}&=&K_{IE}\frac{d S_{C}[V]}{d V}\mid_{V=\left( V^{*e}_{E}-V^{*i}_{E}\right)},\\ K_{5}&=&K_{II}\frac{d S_{C}[V]}{d V}\mid_{V=\left( V^{*e}_{I}-V^{*i}_{I}\right)},\\ K_{6}&=&K_{SE}\frac{d S_{C}[V]}{d V}\mid_{V=\left( V^{*e}_{E}-V^{*i}_{E}\right)},\\ K_{7}&=&K_{SR} \frac{d S_{T}[V]}{d V} \mid_{V=V^{*e}_{R}},\\ K_{8}&=&K_{RE} \frac{d S_{C}[ V]}{d V}\mid_{V=\left( V^{*e}_{E}-V^{*i}_{E}\right)},\\ K_{9}&=&K_{RS}\frac{d S_{T}[V]}{d V} \mid_{V=\left( V^{*e}_{S}-V^{*i}_{S}\right)}. \end{array} $$

The constants K i , i=1,…,9 depend on the anaesthetic factor p since they are evaluated at the resting state depending on p itself. Hence

$$\begin{array}{@{}rcl@{}} \boldsymbol{\tilde{G}}(\nu,p)=\frac{1}{\sqrt{2 \pi}} \left[\begin{array}{ccccccc} \hat {L}_{e}(\nu)-K_{1}&K_{1}&0&0&-K_{2} e^{-2 \pi i\nu\tau}&K_{2} e^{-2 \pi i\nu\tau}&0 \\ 0&\hat {L}_{i}(\nu)&-f_{C}(p)K_{3}&f_{C}(p)K_{3}&0&0&0 \\ -K_{4}&K_{4}&\hat {L}_{e}(\nu)&0&0&0&0 \\ 0&0&-K_{5}&\hat {L}_{i}+K_{5}(\nu)&0&0&0\\ -K_{6}e^{-2 \pi i\nu\tau}&K_{6}e^{-2 \pi i\nu\tau}&0&0 &\hat {L}_{e}(\nu)&0&0\\ 0& 0&0& 0&0&\hat {L}_{i} (\nu,p)&-f_{T}(p)K_{7}\\ -K_{8}e^{-2 \pi i\nu\tau}&K_{8}e^{-2 \pi i\nu\tau}&0&0&-K_{9} &K_{9}&\hat {L}_{e}(\nu) \end{array}\right]^{-1} \end{array} $$
(28)

and finally

$$ P_{E}(\nu)={2 \kappa}\sqrt{2\pi} \left\vert {\tilde{G}}_{1,5}(\nu,p)\right\vert^{2}. $$
(29)

Appendix C: Contribution to power spectrum

Here we consider the reduced model and parametrize the contribution of PSPs to the power spectrum in the δ− and α− frequency bands. Substituting the ansatz Y(t)=e λt u with eigenfunction \(\mathbf {u}=[ {u^{e}_{E}}, {u^{e}_{S}}, {u^{i}_{S}}, {u^{e}_{R}}]^{\top } \) into Eq. (14) yields

$$\begin{array}{@{}rcl@{}} \left (\frac{\lambda}{\alpha_{e} }\! +\! 1 \right )\! \left (\frac{\lambda}{\beta_{e} } +1 \right )\! {u^{e}_{E}}\! &=&\! K_{2} ({u^{e}_{S}}-{u^{i}_{S}}) e^{-\lambda \tau}, \\ \left (\frac{\lambda}{\alpha_{e} }\! +\! 1 \right )\! \left (\frac{\lambda}{\beta_{e} } +1 \right )\! {u^{e}_{S}}\! &=&\! K_{6} {u^{e}_{E}} e^{-\lambda \tau} , \\ \left (\frac{\lambda}{\alpha_{i} }\! +\! 1 \right )\! \left (\frac{\lambda}{\beta_{i}} +1 \right )\! {u^{i}_{S}}\! &=&\! f_{T}(p) K_{7} {u^{e}_{R}}, \\ \left (\frac{\lambda}{\alpha_{e} }\! +\! 1 \right )\! \left (\frac{\lambda}{\beta_{e} }\! +\! 1 \right )\! {u^{e}_{R}}\! &=&\! K_{8} {u^{e}_{E}} e^{-\lambda \tau}\! +\! K_{9} \left( {u^{e}_{S}}\! -\! {u^{i}_{S}}\right). \end{array} $$
(30)

Now we can write all the elements of eigenfunction u in terms of the first element \({u^{e}_{E}}\) as follows

$$\begin{array}{@{}rcl@{}} {u^{e}_{E}} &=&u_{1}(\lambda) {u^{e}_{E}}, \\ {u^{e}_{S}} &=& \left( \frac {K_{6} e^{- \lambda \tau}} {\left( 1+ \frac{\lambda}{\alpha_{e}}\right)\left( 1+ \frac{\lambda}{\beta_{e}}\right)} \right) {u^{e}_{E}} \equiv u_{2}(\lambda) {u^{e}_{E}}, \\ {u^{i}_{S}} &=& \left( \frac{K_{6} e^{- \lambda \tau}} {\left( 1+ \frac{\lambda}{\alpha_{e}}\right)\left( 1+ \frac{\lambda}{\beta_{e}}\right)} - \frac{\left( 1+ \frac{\lambda}{\alpha_{e}}\right)\left( 1+ \frac{\lambda}{\beta_{e}}\right)} {K_{2} e^{- \lambda \tau}} \right)\\ {u^{e}_{E}} &\equiv& u_{3}(\lambda) {u^{e}_{E}}, \\ {u^{e}_{R}} &=& \frac{\left( 1+ \frac{\lambda}{\alpha_{i}}\right)\left( 1+ \frac{\lambda}{\beta_{i}}\right)} {f_{T}(p) K_{7} } \left( \frac{K_{6} e^{- \lambda \tau}} {\left( 1+ \frac{\lambda}{\alpha_{e}}\right)\left( 1+ \frac{\lambda}{\beta_{e}}\right)}\right.\\ &&\left. - \frac{\left( 1+\frac{\lambda}{\alpha_{e}}\right)\left( 1+ \frac{\lambda}{\beta_{e}}\right)} {K_{2} e^{- \lambda \tau}} \right) {u^{e}_{E}} \equiv u_{4}(\lambda) {u^{e}_{E}}. \end{array} $$
(31)

Then the associated normalized eigenfunction with known eigenvalue λ is \(\hat {\mathbf {u}} = C\left (1,u_{2}(\lambda ),u_{3}(\lambda ),u_{4}(\lambda )\right )^{\top }\), where \(C=\frac {1}{\sqrt {1+{u^{2}_{2}}+{u^{2}_{3}}+{u^{2}_{4}}}}\). All elements of vector Y(t)=[y 1(t),y 2(t),y 3(t),y 4(t)]⊤ can be written as \(y_{n}(t)=\hat u_{n} e^{\lambda t}+ \hat {u}_{n}^{\ast } e^{\lambda ^{\ast } t}\) for n=1,...,4. Here the superscript ∗ denotes the complex conjugate. Let λ=γ+2π i ν and \(\hat u_{n} =R_{n}+i I_{n}\), then y n (t) becomes

$$\begin{array}{@{}rcl@{}} y_{n}(t) &=& (R_{n}+iI_{n}) e^{(\gamma+2\pi i \nu)t}+(R_{n}-iI_{n}) e^{(\gamma-2\pi i \nu)t}\\ &=&2 e^{\gamma t}(R_{n}\cos(2\pi \nu t) -I_{n} \sin(2\pi \nu t)), \end{array} $$
(32)

and the contribution of excitatory and inhibitory currents to power in a certain frequency ν, in different populations can be defined by Eq. (18).

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Hashemi, M., Hutt, A. & Sleigh, J. How the cortico-thalamic feedback affects the EEG power spectrum over frontal and occipital regions during propofol-induced sedation. J Comput Neurosci 39, 155–179 (2015). https://doi.org/10.1007/s10827-015-0569-1

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