The fact that the brain produces electrical signals, or brain waves, has been known for over 150 years, and the first recording in humans using EEG was made almost 100 years ago. The pioneering work of Cajal in the late 19th century established the neuron doctrine that the nervous system is made up of discrete individual cells (neurons), which is one of the central tenets of modern neuroscience. Neurons are known to generate electrical signals as a result of their ability to maintain a voltage difference across their membranes that generates an electrochemical pulse known as an action potential that can travel rapidly along the axon. Consequently, the majority of approaches to characterizing brain dynamical behavior are based on the assumption that signal propagation along well-known anatomically defined pathways, such as major neural fiber bundles, tracts, or groups of axons, should be sufficient to deduce the dynamical characteristics of brain activity at different spatiotemporal scales. Characterizing brain networks is important for understanding many aspects of brain function, from neural processes underlying cognition to aberrant brain electrical activity, such as seen in epileptic seizures.

However, this view cannot explain the entire picture of observed brain activity propagation. Recently spatiotemporally organized, circular wave-like patterns of electrophysiological activity (traveling waves) were described at the macroscopic (scalp EEG, MEEG) and mesoscopic scale (invasive EEG), in animal models and humans, and during cognitive tasks and sleep (Muller et al., 2018; Zhang et al., 2018; Zanos et al., 2015). These findings represent a formidable challenge for current network theories to explain such a remarkable synchronization across a multitude of different local networks.

In the following sections, we introduce the basic physical problem, present the rationale behind the ubiquitous ‘quasi-static approximations’, and detail why it is a poor model for brain activity, and then outline our recently developed more general universal theory of brain waves and how it ultimately leads to a solution of the inverse problem for EEG data.

The general equations governing the propagation of EM waves are called Maxwell’s equations (Maxwell, 1865) and in an inhomogeneous and anisotropic medium take the form(1a)∇⋅D=ρ(1b)∇⋅H=0(1c)∇×E=−∂H∂t(1d)∇×H=Jt

where D=εE is the electric displacement field, where the (scalar) permittivity ε takes into account the polarization of the dielectric material in the electric field E, H is the magnetic field intensity, and the total current density Jt is given by the sum of the free current density Jf and bound current density Jb:(2)Jtotal=Jfree+Jbound=JConductive+JDisplacement

where(3)JC=σ⋅E,JD=∂D∂t

are the conductive current and displacement current, respectively.

This is the problem setup. To understand the electric fields in the brain, one needs to solve Maxwell’s equations. At this point, the standard procedure (e.g. Nunez et al., 1997) is to simply ignore the temporal variations in both the magnetic field and the electric field by setting ∂H∂t=0 and ∂E∂t=0. This simplified Equation 1c to ∇×E=0 and Equation 1d to ∇×H=JC since eliminating the time dependence of the electric field eliminates the displacement current: JD=∂D∂t=ε∂E∂t=0. This is the so-called quasi-static approximation (Gaugain et al., 2023; Rapetti and Rousseaux, 2014) ubiquitous in EEG analysis methods.

What are the justifications for these simplifications? It turns out that in biological tissues, the inductive effects are small or negligible (Plonsey and Heppner, 1967), so that eliminating the time dependence of the magnetic field is indeed justified. This is important as the simplified form of Equation 1c implies, for simple vector relations, that the electric field can be written in terms of a field potential ϕ since(4)∇×E=0⇒E=−∇ϕ

Solving for the electric field E is then equivalent to solving for ϕ. And we can ignore any magnetic field effects.

However, the assumption that the electric field does not vary with time is not justified in biological materials (Plonsey and Heppner, 1967), and, therefore, the expression in Equation 1d is correct as is - the displacement current must be retained. Maxwell’s equations in the brain thus take on a somewhat odd configuration in that they are, in the standard physics parlance, magnetostatic but not electrostatic.

It is somewhat ironic that the introduction of the displacement current, which was in some sense Maxwell’s greatest insight and the final piece of the puzzle in solving the equations of electromagnetism, turns out to be the key to the puzzle of brain activity, where it had once again been ignored.

Because of the ubiquity of the quasi-static approximation, it is worth pausing here to consider its consequences, since in our view they have led to confusion in the understanding of brain electrical activity and the problem of EEG reconstruction.E=−∇ϕ∇⋅E=ρ/ε}⇒∇2ϕ=ρε

which is Poisson’s equation and relates the electric field potential ϕ to ‘sources’ ρ. The most striking aspect of this solution, though it was the obvious endpoint by construction, is that there is no time dependence in the solution. One would have guessed this to be a giant red flag for the description of brain electrical activity, but the persistence of this approach has nevertheless been tenacious. Consequently, from the perspective of EEG reconstruction, the problem is framed in terms of ‘source reconstruction’. We note that these equations are also called the ‘quasi-static volume conduction’ equations, and, therefore, this problem is often referred to as the ‘volume conduction’ problem.

There is also another massive source of confusion that is often used to justify the quasi-static approximation. The logic goes something like this. We know that the brain has, for example, alpha waves which for the sake of simplicity we will assume the frequency to be a typical value of ω=10Hz. For EM waves in a medium of permittivity ε, the wavelength of these waves is related to the velocity v of the waves as(5)λ=vωwherev=cε

in which c=3×108m/s is the speed of light. For a typical tissue permittivity ε=100, the wave velocity is v=3×107m/s so that the wavelength is(6)λ=vω=3×10710=3×106m=3000km

Because the wavelength is so much greater than the spatial dimensions of the head, there can be no appreciable phase difference anywhere in the head, and EM wave propagation effects can be ignored. Indeed, this is true - there are effectively no EM wave propagation effects in the brain. But that argument is not a justification for the assumption of a time-independent electric field. Indeed, the logic is backward. One must first solve Maxwell’s equations under the proper conditions, then eliminate contributions that appear insignificant.

Indeed, if one simply assumes the absence of free charges, Maxwell’s equations Equation 1c and Equation 1d in a medium of permittivity ε and permeability μ combine to give(7)∇2E−1v2∂2E∂2t=0⇒E(r,t)=E0ei(k⋅r−ωt)

the solution to which are complex plane waves, which in turn implies the dispersion relation ω=v|k|. As we will show below, the dispersion relation derived from the correct version of Maxwell’s equations is quite different and provides a key insight into the interesting characteristics of brain waves.

The fact that observed alpha waves have velocities many orders of magnitude slower than EM alpha waves should be an obvious clue that something else is going on. This much is recognized in that they are ascribed to ill-defined concepts such as ‘neuronal oscillations’. But a consequence of that should be a reexamination of Maxwell’s equations in light of these experimental observations. We will do that in the next section and demonstrate that these ‘slow’ waves are not mysterious at all, but a direct consequence of the displacement current and the inhomogeneity and anisotropy of the tissues. They are not EM waves, but surface waves.

In this section, we provide a brief outline of the more detailed theoretical description in Galinsky and Frank, 2020a; Galinsky and Frank, 2020b. From the above discussion, the proper form of Maxwell’s equations to solve, from Equation 1a and d and Equation 3, is∇⋅D=ρ∇×H=σ⋅Jc+∂D∂t}⇒∂ρ∂t+∇⋅Jc

where the RHS is a statement of charge continuity. These equations along with Equation 4 and Equation 3 give the charge continuity equation in terms of the quantity of interest, the electric field potential ϕ:(8)∂∂t(∇2ϕ)=−∇⋅Σ⋅∇ϕ

where Σ={σij/ε} is the scaled conductivity tensor. Thus, the inclusion of the displacement current has produced a wave equation.

A simple linear wave analysis, i.e., substitution of ϕ∼exp⁡[−i(k⋅r−Ωt)], where k is the wave number, r is the coordinate, Ω is the frequency, and t is the time, gives the following complex dispersion relation, now written in tensor form where i,j={x,y,z} and repeated indices are summed:(9)D(Ω,k)=−iΩki2−Σijkikj−i∂iΣijkj=0,

which is composed of the real and imaginary components:(10)γ≡I[Ω]=Σijkikjk2ω≡R[Ω]=−∂iΣijkjk2

Several interesting features of this relation are worth noting. Because it is complex, it will result in both an oscillatory component (proportional to the frequency ω) and a decaying component (proportional to the decay rate γ). Both γ and ω are functions of the tissue parameters through Σij so there is a direct connection between tissue properties and the wave dynamics. The tissue properties are encoded in the tensor Σij so spatial variations due either to inhomogeneity or anisotropy will also influence wave propagation. But perhaps the most interesting feature of this dispersion relation is that ω∼1/k, and so quite different than the dispersion relation for EM waves. This has significant consequences for the nature of brain electrodynamics, as shown below.

These results bring us to a central important point. For typical low-frequency (≲10Hz) average values of white (WM) and gray matter (GM) conductivity and permittivity (i.e. from Gabriel et al., 1996a; Gabriel et al., 1996b), the decay rates give strong wave damping, and no waves would be observed. For example, typical values for GM and WM are εGM=4.07⋅107ε0, εWM=2.76⋅107ε0, σGM=2.75⋅10−2 S/m, σWM=2.77⋅10−2 S/m, where ε0=8.854187817⋅10−12 F/m is the vacuum permittivity so the damping rate γ is in the range of 75–115 s^–1, which would give strong wave damping. This leads immediately to the question of the effects of the anisotropy, which is encoded in the scaled conductivity tensor Σ. A full discussion of the effects of anisotropy is provided in Galinsky and Frank, 2020a; Galinsky and Frank, 2020b, but, here, we review the key novel and important finding of the general theory: the existence of previously unrecognized (at least theoretically) waves transverse to the fiber direction.

To see this, we take a very simple idealized tissue model: fibers are packed in a half space aligned in z direction, and their number decreases in x direction in a relatively thin layer at the boundary. We assume that small cross fiber currents can be characterized by a small parameter ϵ and represent the conductivity tensor as(11)Σ=(ϵυϵυϵυϵυϵυ0ϵυ0υ).

where υ≡υ(x). For the υ(x) dependence, we will assume that the conductivity is changing only through a relatively narrow layer at the boundary, and the conductivity gradient is directed along x axis.(12a)([∂t+a]∂z2+b∂z)ϕ∥=0,∼ϵ0(damped oscillator)(12b)(∂t∂y2+b∂y)ϕ⊥=0,∼ϵ1(wave equation)

where a=υ(x)|x0 and b=∂xυ(x)|x0 are considered constant evaluated at the boundary x0, and ϵ0 and ϵ1 denote the zeroth and the first orders of ϵ power. We emphasize that this approximation for a and b is specifically allowed because we are considering a thin boundary layer problem.

The first equation, Equation 12a, describes a potential along the fiber direction and is a damped oscillator equation that has a decaying solution. But the second equation, Equation 12b, describes a potential perpendicular to the fiber direction and does not include a damping term; hence, it describes a pure wave-like solution that propagates in the thin layer transverse to the main fiber direction. Thus, although this wave-like solution ϕ⊥ has a smaller amplitude than along the fiber action potential ϕ∥, it can nevertheless have a much longer lifetime. Such waves are called weakly evanescent transverse cortical waves, or WETCOW for short.

Though these produce many interesting effects, two aspects are most important for the current application. First, the decay rates γ for brain tissue are sufficiently small that waves can persist for time significantly longer than ‘spiking’. The persistence of the stable waves can be characterized by the ratio of the decay rate to the frequency, which from simple geometric considerations from Equation 10 for the longest waves (with the smallest amount of damping) withγω=Σijkikj∂iΣijkj≈∼0.02−0.04.

Anisotropy (Σ⊥<Σ∥) will reduce this estimate even further (see Galinsky and Frank, 2020a; Galinsky and Frank, 2020b, for more details). In other words, anisotropy can result in decay rates that can vary from these maximum (homogeneous) values above, all the way down to 0, based on the direction of propagation, increasingly supporting the existence of transverse waves. Without taking anisotropy into account, i.e., assuming the mean tissue values above as is done in the ‘standard model’, the decay is so rapid that transverse weakly evanescent waves are not supported.

Second, the inverse relationship between the frequency and wavelength in the dispersion relation (Equation 9) means that waves can extend throughout the entire volume of the brain. One can also recognize immediately from Equation 10 the existence of significant phase variations across the brain (characterized by both phase and group velocity) proportional to tensor products ∇⋅Σ and ∇⋅Σ⋅k that characterize wave propagation normal to the conductivity gradient and thus normal to the fiber orientation. This contradicts the long-standing belief that there are no significant phase variations across the head. There are, but they are not due to EM waves, but WETCOW waves.

The existence of these waves has profound implications for the understanding of brain electrical activity and communications and has been shown to explain a wide range of observed collective brain behaviors, including spiking in the extracellular space (Galinsky and Frank, 2020a; Galinsky and Frank, 2020b), rapid signal synchronization (Galinsky and Frank, 2021) that provides a mechanism for learning and memory (Galinsky and Frank, 2023b), and neuronal avalanches (Galinsky and Frank, 2023a; Galinsky and Frank, 2023c). And of course, they require rethinking what is meant by a brain ‘network’, since signal propagation must now be considered not only along fiber pathways, but between structures that may not even be neuronally directionally connected. But for the present purposes, they imply the existence of waves of electrical activity throughout the brain.

The WETCOW theory predicts the existence of waves satisfying Maxwell’s equations in the brain where the morphology and tissue characteristics have been properly taken into account. The EEG inverse problem therefore involves estimating the electric field potential ϕ that satisfies Maxwell’s equations constructed with the tissue properties of a particular brain, satisfying boundary conditions determined by the morphology of the brain, and consistent with measurements made in an array of electrodes on the surface of the brain. Our method for solving the inverse EEG problem can be summarized as follows. Given a standard EEG dataset from N electrodes and a high-resolution anatomical (HRA) MRI dataset with high contrast between GM and WM, the solution to the inverse EEG problem can be formulated as an approximation for the volumetric distribution of electrostatic potential inside the complex inhomogeneous and anisotropic tissues and complicated morphology of the MRI domain (Galinsky et al., 2018).

The solution to the EEG inverse problem entails solving Equation 8 for the electric field potential ϕ. Taking the temporal Fourier transform (i.e. replacing ∂/∂t→−Iω, I2=−1, where ω is the frequency), the electrostatic potential satisfies the equation in the Fourier (i.e. frequency) domain, and using the notation ∂i=∂/∂qi, qi={x,y,z}, can be written in tensor form as(13)(Σij−Iωεδij)∂i∂jϕω=[Iω(∂iε)δij−(∂iΣij)]∂jϕω+Fω,

where δ is the Dirac delta function, and a summation is assumed over repeated indices. This can be expressed in the form L^ϕω=R^ϕω+F^ω in terms of the operators L^≡∂i∂i, a frequency-dependent source term F^ω and the operatorR^≡σ+Iωεσ2+ω2ε2[Iω(∂iε)δij−(∂iΣij)−(Σij−σδij)∂i]∂j

where Σ={Σij} is a local tissue conductivity tensor and σ=TrΣ/3=Σii/3 is an isotropic local conductivity. Terms in square brackets show that the parts of R^ϕω can be interpreted in terms of different tissue characteristics and may be important for understanding the origin of sources of the electro-/magnetostatic signal detected by the EEG sensors. The first term (ω(∂iε)(∂iϕω)) corresponds to areas with sudden change in permittivity, e.g., the WM/GM interface. The second term ((∂iΣij)(∂jϕω)) corresponds to regions where the conductivity gradient is the strongest, i.e., the GM/CSF (cerebral spinal fluid) boundary. Finally, the last term (Σij∂i∂jϕω−σ∂i∂iϕω) includes areas with the strongest conductivity anisotropies, e.g., input from major WM tracts. The frequency- and position-dependent internal sources F^ω can be used to incorporate various nonlinear processes, including multiple frequency effects of the efficient synchronization/desynchronization by brain waves or effects of their critical dynamics. This term is ignored in the current paper because they are higher-order terms that complicate the processing (and interpretation) but do not substantially change the main results. They will be considered in future work.

The inverse problem can be solved by constructing an approximate solution for the potential ϕ across an entire brain volume iteratively as L^ϕω(k)=R^ϕω(k−1) and ϕ~ω(K)=αK∑k=0Kϕω(k)(Galinsky et al., 2018), where a single iteration forward solution is found using a Fourier-space pseudo-spectral approach (Gottlieb and Orszag, 1977). The volumetric frequency-dependent potential ϕ~ω(K) is the central quantity of interest, and it can be calculated over arbitrary frequency ranges ω=ω1…ω2, such as the standard frequency bands of interest in EEG. These potentials can then be converted to the time domain ϕ~(t,x) from which space-time modes can be determined using our entropy field decomposition (EFD) method for analysis for complex nonlinear systems (Frank and Galinsky, 2016a; Frank and Galinsky, 2016b) (see Appendix 1: The entropy field decomposition). Alternatively, as in this work, the estimated potentials ϕ~(t,x) can be used in the joint estimation scheme presented in Galinsky and Frank, 2019, as an additional modality QijE in the intermodality coupling matrix Qij (see Appendix 1: Multi-modality EFD (JESTER)). The potential depends upon the electrical properties of the tissue permittivity, permeability, and conductivity. These parameters can be estimated from the HRA MRI data. Using joint estimation with entropy regularization (JESTER), data from MRI can be used to define the complex brain tissue morphology and constrain the tissue-specific values of Σ and ε. This procedure of inverting the WETCOW brain wave model constrained by MRI-defined tissue properties is called SPatially resolved EEG Constrained with Tissue properties by Regularized Entropy (SPECTRE).

An approximate pseudo-spectral solution for the potential ϕ was constructed across an entire brain volume using either MRI Montreal Neurological Institute (MNI) 2 mm resolution (91 × 109 × 91 voxel dimensions), 1 mm MNI resolution (182 × 218 × 182 voxel dimensions), or 0.7 mm resolution (207 × 256 × 215 voxel dimensions). For the current study, we only used the anatomical data for estimation and assignment of different tissue types, and no diffusion MRI (dMRI) data was used. To register between different modalities, including MNI, HRA, fMRI, etc., and to transform the tissue assignment into an appropriate space, we used the symplectomorphic registration (SYMREG) registration method (Galinsky and Frank, 2019).

The pseudo-spectral computational approach used in SPECTRE has some important advantages over the finite/boundary element approaches typically used for electrostatic modeling of brain activity (Gramfort et al., 2010; Kybic et al., 2005; von Ellenrieder et al., 2009; Gutiérrez and Nehorai, 2008; Schimpf et al., 2002; Ermer et al., 2001; Mosher et al., 1999). It does not use surface meshes and so does not require limiting the location of activity sites to a small number of surfaces with fixed number of static dipole sources constrained to the surfaces. And the distribution of both electrostatic and geometric properties of the media (conductivity, permittivity, anisotropy, inhomogeneity - derived from the MRI data) is incorporated at every location throughout the volume. It is thus able to find a time-dependent spatial distribution of the electrostatic potential at every space-time location of a multidimensional volume as a superposition of source inputs from every voxel of the same volume (Galinsky et al., 2018). These traits allow it to model wave-like signal propagation inside the volume and can detect and characterize significantly more complex dynamical behavior of the sources of the electrostatic activity recorded at the sensor locations than traditional methods.

It is notoriously difficult to get high-quality EEG data in simultaneous fMRI/EEG studies as the presence of the rapidly varying magnetic fields present in an fMRI acquisition distorts the EEG signal. However, one recent open-source simultaneous fMRI/EEG study of a well-controlled visual task (the periodic flashing checkerboard) on multiple subjects (Telesford et al., 2023, available from the Nathan Kline Institute) provides important data to address this question.

The fMRI procedure samples the data at relatively coarse temporal sampling and thus is most sensitive to low-frequency variations in blood oxygenation level-dependent (BOLD) activity. The most useful comparison of SPECTRE with fMRI is therefore in the lowest frequency band, 0−1Hz (for details on the fMRI acquisitions, see Visual paradigm data). The SPECTRE reconstruction in this frequency band is shown in Figure 1 and demonstrates the ability of SPECTRE to faithfully reconstruct the spatial distribution similar to fMRI. Importantly, this comparison was performed on data from a single subject, since brain activity patterns can vary significantly between individuals, and averaging over multiple subjects obscures specific spatial variations important for validation. In the top rows of Figure 1 is shown the fMRI EFD mode that automatically detects the activation in the primary visual cortex. In the middle row are shown the SPECTRE modes reconstructed using the 2 mm MNI anatomical atlas, chosen because it was closest in resolution (2mm3) to the fMRI data (∼3mm3). The very close correspondence between the spatial patterns is evident.

The bottom rows in Figure 1 clearly demonstrate one of the most compelling, and perhaps surprising, aspects of SPECTRE - its ability to reconstruct activation at spatial resolution significantly higher than fMRI. This is a consequence of the SPECTRE reconstruction being based on the solution of the propagation of EM waves through specific tissue morphologies and bioelectric properties, provided by arbitrary resolution anatomical MRI data. The finer the resolution of the MRI scans, the more details can be available for the reconstruction. This is, of course, dependent upon the number and distribution of the EEG sensors, but certainly holds for the standard array configurations used in this paper.

Although it is an almost universally believed notion that EEG and fMRI are complementary because EEG has excellent temporal resolution but poor spatial resolution, while fMRI has poor temporal resolution but good spatial resolution, in fact, SPECTRE EEG reconstructions can achieve much higher intrinsic temporal and spatial resolution. Moreover, because there are no spatial distortions in SPECTRE, this mitigates one of the aspects of fMRI that most confounds spatial localization through signal loss and nonlinear geometric distortions. This is shown in Figure 2.

The slice-by-slice correlation coefficient between the activation patterns estimated by SPECTRE and fMRI is shown in Figure 3. Regions of very high correlation, most notable in the inferior brain regions, indicate the similarity in activation patterns detected between the two completely different neuroimaging methods (SPECTRE and fMRI). The correlations are not as strong in the superior regions of the brain, possibly due to the increased distortions in that region in this fMRI dataset. Even with perfect activation detection by both methods, the correlations would not be perfect (i.e. 1) as the two methods are measuring different physical processes. However, the smooth variations are indicative of nonrandom correlations between two vastly different imaging modalities.

It should be noted that the ‘simple’ periodic flickering checkerboard stimulus not only activates the primary visual cortex but activates other visual and supplementary fields as well, as is evident from the activity patterns in Figure 1. A simple stimulus does not imply a simple activation pattern. This notion was a primary motivation for our development of the EFD method for fMRI (Frank and Galinsky, 2016b). The activation mode reconstructions for both the fMRI and SPECTRE data are based on the EFD, which detects complex nonlinear interacting spatial-temporal modes of activity (Frank and Galinsky, 2016b). Thus, although the task is a ‘simple’ visual stimulation, our analysis is not expected to simply detect activity in only the visual cortex, as would be produced by a more standard regression approach (Telesford et al., 2023), but in a more complex set of brain networks. Indeed, multiple EFD modes are produced, though we have only shown the one incorporating the primary visual cortex. As we have argued previously (Frank and Galinsky, 2016b), EFD analysis is more sensitive than simple regression techniques to the complex brain activation patterns predicted by neuroscience, and less sensitive to erroneous identification of noise or nonindependent modes than the independent component analysis (ICA) (Frank and Galinsky, 2016b). Indeed, one of our observations from both the fMRI and EEG data used in this study (Telesford et al., 2023) is the appearance of PFC activations associated with visual stimulation, which has been suggestive of conscious visual perception (Libedinsky and Livingstone, 2011; Paneri and Gregoriou, 2017). Addressing this question is beyond the scope of the current paper.

Simultaneous EEG/fMRI were collected from subjects within a standard clinical 3T MRI scanner (see Attention paradigm data for details). The stimuli and paradigm are described in detail in Grinband et al., 2017. Briefly, bimodal stimuli consisting of short (∼1s) streams of simple tones (600 and 1000 Hz) alternating at 10 Hz were delivered concurrently with phase-reversing (6 Hz) checkerboard patterns presented at fixation. Participants were instructed to selectively attend to either the visual or auditory aspect of the bimodal stimulus and respond when the stream of stimuli in the attended modality ends.

SPECTRE processing was performed in the alpha band. The appearance of visual stimuli elicited a reduction of ongoing alpha (7–14 Hz) activity (‘event-related desynchronization’ [ERD]) over occipital cortex, believed to occur when cortical regions are brought ‘online’ for information processing (Klimesch, 2012). As in previous studies, e.g., Foxe and Snyder, 2011, attended visual stimuli elicited increased (more negative) amplitude of the alpha ERD compared to unattended stimuli (Figure 4A and B). In contrast, unattended, compared to attended, visual stimuli elicited a greater reduction in ongoing spectral activity within the 5–15 Hz frequency range over bilateral middle frontal cortex (Figure 4C and D). We estimated the neural sources of these attention-related modulations of oscillatory activity across the 8–12 Hz frequency band, which encompassed both the occipital and frontal activities (Figure 4E). Their anatomical localization was remarkably consistent across several individuals (Figure 5).

A direct comparison of the activation maps derived from both fMRI and EEG using SPECTRE for a single study within two subjects (i.e. without any average over studies or subjects) is shown in Figure 6. The comparison is made by choosing specific regions of interest defined in the MNI atlas (occipital cortex and cerebellum) and correlating the activation maps derived from EFD for fMRI and SPECTRE from EEG. Comparison of the similarity of activated regions in individual subjects is generally a nontrivial problem. This is particularly true in the current case where the spatial distortions in fMRI (and lack of them in SPECTRE) make measures such as mean-squared error difficult to interpret. Therefore, the computation of the correlation coefficient over a predefined atlas ROI is a reasonable conservative measure of statistical significance.

As an example of the type of whole-brain electric field activation maps that are possible with SPECTRE is shown in the montage of orthogonal slices from a 2 mm reconstruction, Figure 7, from one of the subjects of this same attention study.

Direct comparison of activation maps from two participants in the bimodal (auditory + visual) stimulation paradigm described for Figures 3 and 4 is shown in Figure 6. In each subject, two brain regions - the cerebellum and the occipital pole (top and bottom rows, respectively) - were delineated based on the MNI atlas, and EFD activation maps were correlated across these entire regions. Correlation coefficients were as follows: for Subject A, cerebellum = 0.74, occipital pole = 0.70; for Subject B, cerebellum = 0.70, occipital pole = 0.84. Correlations were computed only for regions exhibiting activation levels above 0.1. In contrast to fMRI, the SPECTRE technique identified robust activations in bilateral middle and inferior frontal cortex (indicated by yellow arrows) and middle temporal cortex (red arrows). It also discerned activations along the superior temporal cortex, including areas encompassing the primary auditory cortex (green arrows). We emphasize that the difference in activation maps provided by SPECTRE and fMRI is not expected to be identical, as EEG and fMRI are not measuring the same physical quantities. Indeed, fMRI is measuring rather poor proxies of the brain electrical fields. Therefore, it would be remarkable if these two methods did not have significant differences. Figure 4 provides the best example, where our EEG method showing deactivation (blue in E) is consistent with what can be considered a gold standard for EEG - the direct surface recordings near the scalp where the deactivation in the frontal lobes (C, D in blue) corresponds. It should also be noted that this is not simply a visual task, but an attention task, for which these activation patterns are well known, and thus provides yet another form of validation.

Therefore, the high correlation coefficients between the maps, Figure 6, are therefore indicative of the consistency between the fMRI and SPECTRE results in the ROI. Note that this does not imply similarity over the entire region shown. Indeed, the SPECTRE results show enhanced sensitivity to activation in regions not seen in the fMRI.

While comparison with fMRI can validate the correct detection of activated brain regions and networks, as shown in the previous section, it cannot inform the question of correct detection of electrical signals, since fMRI is based on a completely different contrast mechanism related to blood oxygenation. A direct validation of SPECTRE’s ability to faithfully reconstruct deep EM activity is, to our knowledge, only achievable with one type of data: iEEG recordings such as those used in medically refractory epilepsy patients for seizure onset localization where the electrodes are known to be adjacent to the site of electrical activity (Ramantani et al., 2016; van Mierlo et al., 2020). We analyzed an iEEG recording of a seizure localized in the left medial temporal region acquired at Northwell Health, NY. All implanted electrodes are shown in Figure 8 (top row) with each yellow dot depicting one recording contact. Comparing the SPECTRE reconstruction using all of the sensor data with one using only a subset of the data comprised of only the sensors on the surface of the brain (red dots in Figure 8, top row) allows the quantitative assessment of how closely the results from a set of surface electrodes correspond to those produced by intracranial measurements recording signal very close to the sources. The results are shown for the alpha frequency band in Figure 8 and reveal a very close correspondence between the SPECTRE mode reconstruction. Results for Subjects 2–4 are shown in Appendix 2—figures 1–3 and show similar agreement.

A traditional approach for validating estimation methods is to compare results against a ‘ground truth’ derived from numerically simulated signals. This is a standard procedure in source reconstruction methods, where simulated point dipole sources are embedded within a brain model (e.g. a high-resolution MRI scan). The forward problem is then solved to generate the dipole fields at the brain surface, and the resulting simulated signals are used to estimate the original source locations. While this approach works well for dipole-based models of brain activity, it is not directly applicable to the more realistic WETCOW model of brain waves.

However, an alternative validation strategy is possible by comparing SPECTRE data to the most reliable ground-truth data available: iEEG. One of the most striking predictions of the WETCOW theory is the presence of coherent, sustained cortical wave loops - a phenomenon demonstrated through numerical simulations in a realistic brain model derived from HRA data (Figure 10, top). This prediction provides a natural benchmark for validating the SPECTRE method.

An application of SPECTRE to a WETCOW analysis of iEEG recording of several epileptic seizure onsets in insular posterior opercular and in hippocampus areas that provides our first experimental evidence of the existence of our hypothesized cortical loops is shown in Figure 10 (bottom right). SPECTRE reconstruction confirms the existence of cortical wave loops in Figure 10 (bottom right), consistent with the numerical simulations shown in Figure 10 (top). The intracranial leads are shown in Figure 10 (bottom left).

A deep-surface-full comparison of modes for seizures datasets was run for all 5 iEEG subjects with 44 events total. The correlation plot shown in Figure 9 demonstrates that the correlations are very high. We ran three t-tests on Fisher’s Z-transformed correlation values (full-deep/full-surface, full-deep/deep-surface, full-surface/deep-surface), and the t-tests show that full-deep/full-surface correlations are very similar (null hypothesis is not rejected, p=0.2368, t(43)=1.19, p>0.001, 95% CI [–0.0370, 0.1457], SD = 0.3004), but the full-deep/deep-surface and full-surface/deep-surface t-tests show statistically significant differences (p=1e−8, t(43)=7.05, p>0.001, 95% CI [0.1827, 0.3291], SD = 0.2408, and p=8e−6, t(43)=5.06, p>0.001, 95% CI [0.1213, 0.2818], SD = 0.2640). These results support the claim that the SPECTRE reconstruction of the spatial distribution of deep electrical activity from the surface measurements accurately reflects the true spatial localization of the deep electric fields.

Having validated the SPECTRE method directly with simultaneous fMRI/EEG, iEEG, and an attention paradigm, we investigated the ability of SPECTRE to faithfully reconstruct the well-known neural ‘reward circuit’ that is one of the most important in understanding human cognition, emotion, and behavior (Schultz, 2015; Haber, 2017; Banich and Floresco, 2019) and is of great clinical significance in the understanding of addiction (Koob and Roberts, 1999; Gardner, 2011), mood disorders (Naranjo et al., 2001; Russo and Nestler, 2013), and a variety of other conditions (Lewis et al., 2021; Figure 10).

We demonstrate that SPECTRE using standard EEG data can accurately map human reward pathways akin to results previously only seen via fMRI. Indeed, fMRI results have highlighted a reward system within the brain that includes midbrain dopamine-producing regions (the substantia nigra pars compacta, the ventral tegmental area), the ventral striatum, and multiple regions within the human PFC (McClure et al., 2004). Other research using fMRI and source localization of EEG data suggests that the anterior cingulate cortex also plays a key role in reward processing (Holroyd and Coles, 2008). In a unifying theory, it has been proposed that all the aforementioned regions work together as a neural system for the optimization of reward-driven behavioral change (i.e. reinforcement learning; Holroyd and Coles, 2002).

This is of particular clinical significance because addictive behaviors have long been known to be subserved by specific brain regions operating in concert as the reward circuit (Koob and Volkow, 2010; Leshner, 1997; Tapert et al., 2003; Tyree and de Lecea, 2017; Kallen et al., 2023). The reward circuit is involved in processing rewarding stimuli of any sort, and, in drug addiction, substances of abuse (e.g. amphetamine) increase dopamine release in a protracted and less regulated manner as compared to typical stimuli, resulting in synaptic plasticity and altered functioning of this circuit over time.

For our analysis, we used a large gambling task dataset that includes 500 participants available for download from https://osf.io/65x4v/. The details of the dataset and an extensive analysis using standard EEG analysis methodologies are presented in Williams et al., 2021. The relevant information from this study is presented in Appendix 2: Reward Circuit Data.

For each subject trial n=10 power modes were calculated and summed to form the single space-time SPECTRE mode (see Mode reconstruction). Figure 11 shows three orthogonal slices of the difference in EFD power summed over all modes between conditions, averaged over all subjects. Activation in key regions of the reward circuit, including the frontal lobes, anterior cingulate gyrus, accumbens, and amygdala, is clearly evident. Strong negative activation (i.e. deactivation) is evident in several structures, including the supplementary motor cortex and the parietal operculum cortex. Activation is also apparent in the lingual gyrus and around the calcarine fissure and, as expected, in bilateral subcortical structures.

In Figure 12 is shown the power per brain regions as defined by the Harvard-Oxford 2 mm cortical (top) and subcortical (bottom) atlases. In the cortical regions (top), strong activation is apparent in the frontal cortex (medial, orbital, operculum), cingulate gyrus, paracingulate gyrus, and insular cortex. Activation in the accumbens is apparent from the data in the subcortical atlas Figure 12 (bottom). These activated regions are consistent with the known elements of the human brain reward circuitry.

Images of statistical significance (p<0.0001) are shown in Figure 13. It should be noted that the determination of statistical significance with SPECTRE by ‘traditional’ methods is potentially misleading as they will tend to underestimate activation significance. The estimation of the modes in SPECTRE employs EFD (Frank and Galinsky, 2016a; Frank and Galinsky, 2016b), which is a probabilistic formulation that by construction incorporates space-time neighborhood connectivity so that spatially and temporally coherent patterns (‘clusters’) are more probable. Traditional methods have the option for ‘clustering’ regions of activation post hoc into their general class of techniques called ‘bootstrapping’ or ‘permutation inference’. Cluster post-detection of an activation is incommensurate with our view of the estimation process, wherein the clustering in space-time is a key component indicator of high-probability regions of space-time. Spatially and temporally coherent patterns may be of low amplitude with apparent low significance by traditional means, but those intensities are within a mode that contains very high significance in cortical regions (e.g. Figure 13), which is predicted by the WETCOW model.

Mass univariate voxel-wise statistical analysis across the whole brain was performed using AFNI 3dttest++. The first level fixed effects were analyzed to produce contrast estimates computing the mean activation for each condition (the SPECTRE power modes obtained for pre- and poststimulus reward experiment). Statistical significance (t-statistic) between the SPECTRE power modes pre- and poststimulus reward experiment. Calculations of statistical significance (two-sample t-statistic) between the SPECTRE power modes pre- and poststimulus reward experiment were performed using the standard AFNI 3dttest++ algorithm. Significance threshold was p=1e−8, indicating strong statistical significance. The permutation/randomization multiple comparisons correction method was used to control the family-wise error rate and false discovery rate with AFNI’s 3dttest++ cluster-level thresholding through the -ClustSim option of the AFNI 3dttest++ algorithm. Images of statistical significance (p<0.0001) are shown in Figure 13. These results support the claim that SPECTRE can reliably reconstruct whole-brain electric field activity.

There is a long history of attempts to spatially localize EEG activity, and these are generally called ‘source localization’ or ‘source reconstruction’ methods (Pascual-Marqui et al., 1994; Hallez et al., 2005; Hallez et al., 2007; Dattola et al., 2020). These methods are fundamentally different from the SPECTRE approach as they are solving a different problem than SPECTRE (as described in A new physical theory of brain waves) that involves numerous stringent assumptions about brain electrical activity such as a fixed set of static dipole sources, an idealized geometric model of the head reduced to a few (typically three) shells, that spatially close points are more likely synchronized and the smoothness of the solution (see Taulu and Larson, 2021, and references therein).

These methods all implicitly assume the ‘quasi-static’ approximation to the EM field equation, which entails ignoring the time-dependent terms in Maxwell’s equations, which are dependent on tissue conductivity properties which are themselves frequency dependent. The resulting solutions are therefore static, have no frequency dependence, and are insensitive to the detailed spatially variable electrical properties of the tissues. However, as discussed in detail in the development of the WETCOW model (Galinsky and Frank, 2020b; Galinsky and Frank, 2021), these assumptions are incompatible with the basic physics of brain electrical activity. The SPECTRE approach is to employ the WETCOW model and solve the actual physical problem of the complete Maxwell’s equations in an inhomogeneous and anisotropic medium. It is specifically these dependencies that give rise to the previously undiscovered WETCOW waves that propagate preferentially along the gradients of local tissue inhomogeneity and anisotropy and thus propagate preferentially perpendicular to neuronal pathways. The WETCOW theory provides a comprehensive framework for characterizing the propagation of EM fields through the complex brain tissue microstructure and larger-scale morphology (e.g. cortical folding) and provides the dynamic solution to the electric potential field necessary to solve the EEG inverse problem.

The problem of spatially localizing the EEG signal involves estimating the most probable distribution of electric field amplitudes given an array of sensors. This is essentially a problem of correctly modeling the physics of how EM waves propagate through the complex environment of the convoluted brain tissue morphology and the anisotropic and inhomogeneous nature of brain tissue. The current state-of-the-art approach to this problem, called ‘source localization’, such as low-resolution electromagnetic tomography or LORETA algorithm with its many variations (Pascual-Marqui et al., 1994; Hallez et al., 2005; Hallez et al., 2007; Dattola et al., 2020), also called ‘EEG source imaging’ (Michel et al., 2004; Ojeda et al., 2021), involves using a predefined brain atlas, arbitrarily placing dipole sources on the surface, and calculating the contribution from these sources. Some methods propose using fMRI as a prior, which has the disadvantage of requiring fMRI acquisitions (Rosa et al., 2010; Schultze-Kraft et al., 2011; Jorge et al., 2014; Abreu et al., 2022). The current source localization methods are based on a static model for the electric field caused by a fixed set of predefined dipole sources (see Michel and Brunet, 2019, for a review of current methods). This model is inherently limited because in reality the brain’s electrical field variations are time dependent and generated by an essentially continuous distribution of sources through the entire brain. This description is the essence of the WETCOW theory (Galinsky and Frank, 2020b; Galinsky and Frank, 2021), which describes how highly coherent localized electric field phenomena, such as cortical wave loops and synchronized spiking, are produced by the complex nonlinear interactions of waves across multiple spatial and temporal scales.

The theoretical discussion above in Maxwell’s equations in the brain clarifies that the problem being solved by SPECTRE is the inversion of a dynamical wave equation model in contrast to the solution of Poisson’s equation for static sources being solved by traditional ‘source localization’ methods. Therefore, it makes little sense to compare the two methods, as they are attempting to solve two completely different problems. However, it perhaps serves some purpose to illustrate with a specific practical example a most basic computational problem encountered if one tries to even attempt a ‘source localization’ solution with the basis state even remotely resembling the setup used in SPECTRE.

In a typical application of the SPECTRE method, we use an MNI volumetric grid with 2 mm (902,629 voxels), 1 mm (7,221,032 voxels), or 0.73 mm (11,393,280 voxels). All voxels in our models are considered sites of EM activity consistent with the local intravoxel tissue characteristics (via Equation 13) rather than any assumed dipolar form used in source localization methods. This setup is facilitated by the pseudo-spectral computational approach used in SPECTRE detailed above in Numerical implementation. Our highest resolution (0.73 mm) SPECTRE processing can be completed on a modern workstation using 16–20 Gb of memory in a matter of minutes.

For comparison with a current state-of-the-art source localization method, we downloaded the currently available LoretaKey1 software (Dattola et al., 2020), which uses as a default a set of 6239 fixed dipoles. In order to make a fair comparison, we tried to use LoretaKey1 with a number of dipoles comparable to the number of voxels we use for our lower resolution (2mm voxel) reconstruction in this paper. We began with the equivalent 902,629 voxels used by SPECTRE in the 2mm reconstruction, but found that LoretaKey1 is not able to handle this size because of memory limitations (i.e. ‘out-of-memory’ crashes). We subsequently scaled down the number of dipoles by factors of 2, 4, 10, and 20 times. Only with around 45K dipoles were we able to make Loreta run. It ran for approximately 24 hr, but then again crashed due to out-of-memory problems. And it should be kept in mind that this is in the absence of even any attempt by Loreta to provide a solution to a time-dependent wave equation. On the contrary, our processing with 2 mm requires around 650 Mb of memory and takes only a matter of minutes to complete. At this point, it was decided that it was not possible for Loreta to provide a result that would usefully inform the efficacy of the SPECTRE method.