Electrorhythmogenesis and anaesthesia in a physiological mean field theory

doi:10.1016/j.neucom.2004.01.185

Neurocomputing

Volumes 58–60, June 2004, Pages 1197-1202

https://doi.org/10.1016/j.neucom.2004.01.185 Get rights and content

Abstract

We solve eight partial-differential, two-dimensional, nonlinear mean field equations, which describe the dynamics of large populations of cortical neurons. Linearized versions of these equations have been used to generate the strong resonances observed in the human EEG, in particular the α-rhythm (8– $13 Hz$ ), with physiologically plausible parameters. We extend these results here by numerically solving the full equations on a cortex of realistic size, which receives appropriately “colored” noise as extra-cortical input. A brief summary of the numerical methods is provided. As an outlook to future applications, we explain how the effects of GABA-enhancing general anaesthetics can be simulated and present first results.

Introduction

Rhythmic cortical activity has been observed with the electroencephalogram (EEG) for 74 years. The EEG has become a standard tool for examining brain function in a clinical setting [1]. One recent clinical application is monitoring the progress of anaesthesia by its effect on the EEG. Nevertheless, the underlying physiological mechanisms of even prominent EEG features, like the α-rhythm at 8– $13 Hz$ , remain poorly understood. We demonstrate here that the model of Liley et al. [3], [4] produces stable α-rhythms and qualitatively accounts for the effects of GABA-enhancing general anaesthetics. Physically, the EEG is generated by current flowing in response to synaptic activity in the apical dendrites of pyramidal neurons perpendicular to the cortical surface [6]. Electrodes sample the electric activity of this two-dimensional, cortical current dipole layer. On the scalp, they average over the potential of more than $1 cm^{2}$ of the cortex, or ∼10⁷ neurons. Thus the recorded electrode voltage will vary smoothly across the cortex, even though the activity of individual neurons may fluctuate strongly.

Similarly, the electric activity of macrocolumns, defined by the typical size 0.5– $3 mm$ of the axon collateral systems, varies smoothly over the cortex. One can define the average over the ∼10⁵ neurons of a macrocolumn as one “spatially averaged neuron”. A two-dimensional mean field model can describe the collective actions of such large ensembles of neurons, just like the macroscopic effects of large numbers of gas atoms can be described by thermodynamics. Liley et al. [3], [4] have suggested such a model and they and others [7] have shown that even simplified versions of it can reproduce typical EEG features. Here we go further by numerically solving the full model: $τ_{k} ∂ h_{k} (x →,t+ξ) ∂ t =−[h_{k} (x →,t+ξ)−h^{rest}_{k}]+ ∑ l ψ_{lk} (h_{k})I_{lk} (x →,t),$ $ψ_{lk} (h_{k})=[h^{eq}_{lk} −h_{k} (x →,t+ξ)]/|h^{eq}_{lk} −h^{rest}_{k} |,$ $∂ ∂ t +γ_{lk}^{2} I_{lk} (x →,t)= exp (1)γ_{lk} Γ_{lk} [N^{β}_{lk} S_{l} (h_{l})+Φ_{lk} (x →,t)+p_{lk} (x →,t)],$ $S_{k} (h_{k})=S_{k}^{max} 1+(1−r_{abs} S_{k}^{max}) exp − 2 h_{k} (x →,t+ξ)− μ ̄_{k} σ ̂_{k},$ $∂ ∂ t +vΛ_{ek}^{2} − 32 v^{2} ∇^{2} Φ_{ek} (x →,t)=v^{2} Λ_{ek}^{2} N^{α}_{ek} S_{e} (h_{e}),$ $Φ_{ik} ≡0,$ with k,l=e(excitatory),i(inhibitory) denoting the two distinct types of spatially averaged sub-populations. The subscripts lk means l→k, i.e., type l acting on type k. EEG voltage is expected to be linearly related to h_e, the spatially averaged excitatory soma membrane potential [3], [6].

In this model the mean soma membrane potentials h_k of Eq. (1) relax to their resting values h_k^rest with characteristic time constants τ_k, in the absence of any synaptic input. The factor ψ_lk of Eq. (2) takes into account the reversal (Nernst) potentials h_lk^eq associated with excitation and inhibition. Although this is basic membrane physiology, it is a new feature in mean-field models and crucial for avoiding unphysiological solutions. The postsynaptic activations I_lk in Eq. (3) are taken to have the impulse response $Γ_{lk} γ_{lk} t exp (1−γ_{lk} t)$ , based on “fast” excitatory (AMPA/kainate) and inhibitory (GABA_A) neurotransmitter kinetics, respectively. Presynaptic activity in Eq. (3) has three sources: short-range via intra-cortical connections N_lk^β with mean firing rates S_l(h_l), long range cortico–cortical activity Φ_lk, and extra-cortical input p_lk. Since the α-rhythm is attenuated or blocked by sensory stimuli, alerting, mental concentration, or drowsiness, one can speculate that it is the natural rhythm of the awake cortex in the absence of structured extra-cortical input. This motivates using noise for p_lk, as specified below. Mean firing rates are given by the sigmoid of Eq. (4). Finally, one can derive Eq. (5) by assuming homogeneous, isotropic connectivity functions that fall off $∼Λ_{ek}^{2} exp (−Λ_{ek} Δ x)$ . The mean cortico–cortical conduction velocity enters via the conduction time delay Δx/v and the Laplacian represents a first order approximation. It is assumed that cortico–cortical fibers are exclusively excitatory, see Eq. (6).

All 36 parameters in , , , , , can be related to physiological or anatomical data. However, available experimental constraints are generally weak, see Ref. [3] for further discussion. In the following, we will set ξ and r_abs to zero. Our canonical choice for the other parameters is: (h_k^rest, h_ek^eq, h_ik^eq, $μ ̄_{k}$ , $σ ̂_{k}$ , Γ_ek, $Γ_{ik}) = (−70,45,−90,−50,5,0.18,0.37) mV,(N_{ek}^{α}$ , N_ek^β, $N_{ik}^{β}) = (2000,3034,536)$ , (γ_ek, γ_ik, $S_{k}^{max}) =(300,65,500) s^{−1}$ , and Λ_ek=0.4/cm for both k=e and k=i; $v=300 cm / s$ , and $(τ_{e},τ_{i})=(0.1,0.02) s$ . Shaping noise for the extra-cortical input p_lk is computationally expensive. Since thalamocortical connections to pyramidal cells in the cortex are mainly excitatory, we fill only p_ee and set the other p_lk=0. The noise is distributed normally (mean $5000 s^{−1}$ , variance $1000 s^{−1}$ ), but its variation over grid and time steps is low-pass filtered (cutoffs: $k_{c} =2π/5 mm$ , $f_{c} =75 Hz$ ) to avoid unphysiologically rapid changes. Note that in Ref. [7] all p_lk were filled with unshaped white noise.

Section snippets

Implementation and results

The model is solved numerically by discretizing it in time and space. , , , are transformed to a set of first-order differential equations and solved by forward Euler iterations. Eq. (5) is iterated directly with three-point time derivatives and a five-point Laplacian. A full size cortex is simulated by employing a 512×512 toroidal space grid with a grid spacing of $1 mm$ . Time steps of $50 μs$ are sufficient to obtain stable and convergent solutions at this grid spacing up to the highest tested

Conclusions

The full mean field theory of Liley et al. [3], [4] has been solved numerically on a “cortex” of realistic size for the first time. Like in previous simplified analyses, EEG resonances like the α-rhythm can be generated with physiologically plausible parameters. The impact of anaesthesia has been simulated by changing one parameter systematically and the experimentally observed “biphasic” response was qualitatively reproduced without further tuning. Comparisons with experimental EEG data and

Ingo Bojak is a Postdoctoral Research Fellow within the Centre for Intelligent Systems and Complex Processes. He graduated in 2000 with a PhD in Theoretical High Energy Physics from the University of Dortmund, Germany. After two more years in physics as Postdoctoral Researcher at the Centre for the Subatomic Structure of Matter in Adelaide, Australia, he has now switched to theoretical neuroscience with a focus on EEG phenomenology.

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David T.J. Liley is a Senior Lecturer in Medical Biophysics. He is a graduate of medicine and obtained a PhD in brain dynamics in 1996 from the University of Auckland, New Zealand. His current research interests are focused on the theoretical elucidation of the dynamical relationship between hemodynamic and electrical measures of cortical activity.

Peter J. Cadusch is a Professorial Research Fellow within the School of Biophysical Sciences and Electrical Engineering. He graduated in 1977 with a PhD in Physics from the University of Melbourne, Australia.

Kan Cheng graduated with a DPhil in Mathematics from the University of Oxford in 2000. He now works in the private telecommunications industry.

¹: Supported by Grant DP0209218 from the Australian Research Council.

²: Supported by Grant EPPNSW067/2002 from the VPAC.

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