Multi-Modality Imaging and Modeling
of Dynamic Brain Function

J.S. George, D.M. Shmidt, C.C. Wood, G. Kenyon (P-21), B.J. Travis (EES-2)

The past few decades have witnessed extraordinary progress in the development of techniques for structural and functional imaging of the human brain. MRI is the premier technique for imaging the soft tissue anatomy of the human brain, but it has significant limitations for defining the geometry of the skull, which computerized tomography does very well. Functional MRI provides detailed pictures of spatial patterns of neural activation based on associated hemodynamic changes but does not capture the characteristic temporal dynamics of neurophysiological activation. MEG and EEG provide excellent temporal resolution of neural population dynamics but are limited in spatial resolution by the ambiguity and ill-posed nature of the source reconstruction problem. Electrophysiology, microanatomy, optical imaging, and other methods each provide important though limited insight into neural function and functional organization. Although the mix and relative importance of imaging technologies will evolve, the need to integrate information from multiple methods will remain.

Dynamic neuroimaging techniques allow measurement of neural population responses that reflect integrated activity of the underlying networks. Evolving analytical strategies allow increasingly reliable source localization and time-course estimation based on MEG and EEG, incorporating anatomy from MRI.^1,2 These tools allow us to observe the dynamic responses of neural populations; but to understand the nature and basis of network function, it is necessary to build models. Computational models of the physical systems and of the physiological processes that give rise to observable responses are essential to connect experimental measures with models of the distribution and dynamics of neural activation. Forward models of field or potential distributions at the head surface associated with primary currents within the brain are the basis for inverse procedures for MEG and EEG that attempt to estimate the number, location, extent, and time courses of regions of cortical activation. Simulation tools that capture the three-dimensional architecture and functional dynamics of neurons and of extended networks allow us to predict interesting dynamic responses and to optimize network models that account for experimental observations.

Advanced Forward Models

Source localization based on MEG and EEG has traditionally employed analytical or semi-analytical forward calculations based on simple geometries such as spherical shells, assuming that errors introduced by the forward calculation are small relative to the uncertainty associated with the inverse estimation procedure. For some applications, boundary element methods have been used, incorporating the gross geometry of the major conductivity boundaries of the head and employing a small number of tissue classes of simplified geometry.¹ However, as inverse methods get better, the simplifying assumptions inherent in these methods are less tenable.

Finite difference calculation.

We have adopted an alternative strategy based on a finite difference method (FDM) that can incorporate more detailed geometry based on MRI and estimates of volume conductivity provided by emerging imaging methods, including diffusion tensor (DT) MRI, current density MRI, and electrical impedance tomography. FDM does not require construction of a specialized computational mesh or explicit identification and topological checking of boundaries; the calculation is performed on a rectangular grid that is the most natural representation of the MRI data used to define the geometry. Various techniques improve the performance of the FDM on a rectangular grid, e.g., adaptive mesh refinement to control error in regions with high field variation and formulations that reduce errors caused by the staircase approximation of a curved boundary.

Because the FDM allows easy manipulation of geometrical details of the volume conductor, we have examined the effects of skull penetrations on observed field distributions. These studies demonstrated significant effects of the optic nerve and ear canal on both the magnitude and distribution of the potential field compared to a model that did not incorporate these shunts (Figure 1). These effects might significantly influence localization of sources in frontal or temporal lobes. We anticipate that it will be important to account for surgical penetrations of the cranium for neurosurgical patients in whom source localization studies are undertaken.

Conductivity estimation.

Although our computational formalism handles anisotropic conductivity, in the past such capability was of little consequence because there were no methods for the noninvasive estimation of tissue conductivity or anisotropy. However, our recent work has demonstrated the feasibility of estimating anisotropic conductivity based on DT-MRI [Figure 1(g)]. Tuch³ has shown that DT-MRI has a well-defined relationship to tissue conductivity. Diffusion and conductivity tensors share the same eigenvectors as a result of the common microgeometry. The relationship between the eigenvalues for diffusion and conductivity can be derived using an “effective medium” theory; we conclude that conductivity and diffusion are strongly and linearly related. Advances in measurement technology and analytical procedures may allow estimation of head conductivity using electrical impedance tomography coupled with models of tissue geometry from MRI.

The Neural Electromagnetic Inverse Problem

To estimate the dynamics of neural systems, an adequate model of the spatial distribution of the underlying sources is needed. Building this source model is the principal business of inverse procedures. Over the past decade, we and others have made significant advances in the development and implementation of inverse procedures for MEG and EEG. Increasingly, these methods employ information derived from other imaging modalities such as MRI to inform or constrain source localization procedures.

Bayesian Inference.

We have previously described a technique for Bayesian Inference that addresses the fundamental ambiguity of the inverse problem and the complex error surface associated with the model parameter space by explicitly sampling the posterior probability distribution.⁴ A Markov Chain Monte Carlo (MCMC) technique is used to conduct a series of numerical experiments and to see which stochastic solutions best account for the data (Figure 2). Source models accommodate an extended region of activation within a bounding volume defined by a few parameters. Because Bayesian methods explicitly employ prior knowledge to help solve the inverse problem, they provide a natural and formal method to integrate multiple forms of image data.

Parametric distributed source model.

In the 1999 paper describing Bayesian Inference,⁴ we employed a parametric source model consisting of a set of elemental currents, each aligned orthogonal to the local cortical surface and all contained with a bounding sphere centered on some cortical voxel. We recently have implemented a new technique to define the bounding volume for our activation model that produces regular two-dimensional patches across the cortex sheet. The patch is defined by a series of dilation operations (i.e., stepwise labeling of successive layers of contiguous voxels) about some seed voxel. This method produces source models based on patches of cortical activation, more consistent with our expectations, and allows useful constraints on the polarity of cortical currents.

Spatio-Temporal Bayesian Inference.

Our initial formulation of Bayesian Inference was applied to single instantaneous field maps. However, our experience has underscored the value of spatio-temporal modeling procedures that attempt to fit an extended sequence of field maps across time with a consistent ensemble of sources. This strategy produces a more parsimonious model and exploits the strong, local correlation within the time domain of integrated neural population activity. We have developed a scheme for Spatio-Temporal Bayesian Inference (STBI) in which each parametric model source has an associated time course. The MCMC algorithm is used to sample the posterior probability distribution of the time courses structured as vectors with an element corresponding to each time point of the sampled field distributions. By this strategy, we are able to estimate the form and variance of the time course associated with each probabilistic source.

Studies with simulated data.

Figure 3 outlines the results of a numerical study applying STBI to a simulated data set. In this example, we defined three small, distributed sources. Two of the sources have time courses that are highly correlated—a condition that creates difficulties for some spatio-temporal methods. STBI recovered all three of the sources and time courses with surprising fidelity and with very tight confidence intervals. In another study, we were able to resolve two sources separated by less than a centimeter near the posterior pole of occipital cortex on the basis of differences in orientation and time course. Our experience with simulated data suggests that much of the ambiguity of the MEG inverse problem is eliminated by STBI.

Coupling Experiments and Network Models

In spite of the advances in analytical methodology, noninvasive methods are unlikely to ever provide the spatial and temporal resolution required to monitor the activity of the entire ensemble of individual neurons in a local circuit within the brain. Network-modeling techniques offer the only viable strategy to truly understand dynamic measures of brain function in terms of the underlying synaptic and cellular dynamics and network connectivity. To do this properly, we must model at least some of the physiological and geometrical complexity of real neurons while accommodating very-large-scale networks.

Visual system model.

The mammalian visual system represents an ideal structure for employing cellular-level network models to relate dynamic measures of neural activity to underlying neural architecture and population dynamics. In the retina, cellular-network simulations can be related to measures of activity provided by individual microelectrodes or electrode arrays or by dynamic-optical-imaging techniques, thus providing a method to validate and optimize the first steps of the system simulation. We have already used such models to predict and account for experimentally observed dynamic responses.⁵

Large-scale, biologically realistic networks are modeled with the Sensory Enhanced Neural Simulation Engine (SENSE)—a general-purpose neural simulator originally developed by LANL investigators.⁶ SENSE can model systems containing as few as one neuron up to millions of geometrically and physiologically realistic neurons. SENSE has recently been implemented as a parallel code to increase the size and complexity of tractable models. With high-performance computers, extended systems such as the early visual pathways can be simulated. SENSE has been coupled to conjugate gradient-based inverse and optimization algorithms, allowing data fitting or optimization as a function of any SENSE variable or combination of variables.

Extensions for physical modeling.

SENSE captures the realistic three-dimensional spatial arrangements and connectivity patterns of neurons seen in neural tissue and the geometric, electrochemical, and synaptic properties of individual neuronal types. Evolution of the voltage of each neuron’s compartments is computed through a series of time steps. We have exploited this capability to compute the net current vector associated with activation of an individual neuron or population of neurons. Such a model can be coupled with the forward models previously described to predict patterns of response observable with MEG or EEG.

Optimization and applications of network models.

We have begun to compare model network responses to functional imaging data at the cellular-network level in order to disclose dynamic spatial and temporal patterns of activation that underlie interesting functional properties of neuronal networks. We continue to develop a network model of the retina that can be fit to experimental ensemble data produced with electrode arrays and optical imaging techniques in an effort to optimize neuronal and network properties. We have prototype models of much of the early visual system, and we will calibrate coupled-system model parameters against MEG and EEG data using efficient optimization algorithms to produce quantitative models of neural electromagnetic responses. This work is motivated by our role in a DOE-sponsored project to develop an electroneural prosthetic retinal implant.

Conclusion

Computational integration of multiple techniques for structural and functional neuroimaging provides a much more powerful strategy for dynamic measurement of brain function than the use of any individual method in isolation. Network models allow us to generate experimentally testable predictions of network behavior, such as modulation of dynamic activity and phase-locked oscillations within a population that should set up large signals detectable by MEG or EEG. Such responses are of increasing theoretical interest for understanding the computation by the brain.

Figure Captions

Figure 1. Forward calculations by the FDM.(a) Influence scheme showing resistive links between neighboring nodes. (b) Segmented MRI used to set conductivities for simulation. (c) Skull segmented from MRI before post processing. (d) Surface/cut plane rendering of potentials from Figure 1(c). (e and F) Two slices through computed potential volume, showing current leakage along the optic nerve penetration. (g) Computed conductivity tensors based on DT-MRI. Note the anisotropy of conductivity corresponding to white matter tracts.

Figure 2. Bayesian Inference probability maps of source locations derived from MEG studies of visual responses.

Figure 3. Source location and dynamics estimated by STBI. (a) Locations and (b) timecourses of simulated sources. (c) Locations and timecourses estimated by STBI.

References

1. J.S. George, C.J. Aine, J.C. Mosher, D.M. Ranken, H.A. Schlitt, C.C. Wood, J.D. Lewine, J.A. Sanders, and J.W. Belliveau, “Mapping function in the human brain with MEG, anatomical MRI, and functional MRI,” Journal of Clinical Neurophysiology 12(5), 406-431 (1995).
2. J.S. George, D.M. Schmidt, D.M. Rector, and C.C. Wood, “Dynamic functional neuroimaging integrating multiple modalities,” in Functional MRI: An Introduction to Methods (Oxford University Press, United Kingdom, 2001), pp. 353-382.
3. D.S. Tuch, V.J. Wedeen, A.M. Dale, J.S. George, and J.W. Belliveau, “Conductivity tensor mapping of the human brain using diffusion MRI,” in Proceedings of the National Academy of Science 98,11697-11701 (2001).
4. D.M. Schmidt, J.S. George, and C.C. Wood, “Bayesian Inference applied to the electromagnetic inverse problem,” Human Brain Mapping 7, 195-212 (1999).
5. G.T. Kenyon, B. Moore, J. Jeffs, G.S. Stephens, B.J. Travis, J.S. George, J. Theiler, and D.W. Marshak, “A model of high frequency oscillatory potentials in retinal ganglion cells,” Visual Neuroscience (in press).
6. S. Coghlan, M.V. Gremillion, and B.J. Travis, “NeuroBuilder: A user interface and network simulator for building neurobiological networks,” in Analysis and Modeling of Neural Systems I, F.H. Eeckman, Ed. (Kluwer Academic Publishers, Massachusetts, 1992), pp. 115-122.

Acknowledgment

This work was supported, in part, by the LANL LDRD program, the MIND Institute, and NIH.

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For more information, contact John George at jsg@lanl.gov.