▪ Abstract Electric fields can stimulate excitable tissue by a number of mechanisms. A uniform long, straight peripheral axon is activated by the gradient of the electric field that is oriented parallel to the fiber axis. Cortical neurons in the brain are excited when the electric field, which is applied along the axon-dendrite axis, reaches a particular threshold value. Cardiac tissue is thought to be depolarized in a uniform electric field by the curved trajectories of its fiber tracts. The bidomain model provides a coherent conceptual framework for analyzing and understanding these apparently disparate phenomena. Concepts such as the activating function and virtual anode and cathode, as well as anode and cathode break and make stimulation, are presented to help explain these excitation events in a unified manner. This modeling approach can also be used to describe the response of excitable tissues to electric fields that arise from charge redistribution (electrical stimulation) and from time-varying magnetic fields (magnetic stimulation) in a self-consistent manner. It has also proved useful to predict the behavior of excitable tissues, to test hypotheses about possible excitation mechanisms, to design novel electrophysiological experiments, and to interpret their findings.

Tumor treating fields (TTFields) are a non-invasive, anti-mitotic and approved treatment for recurrent glioblastoma multiforme (GBM) patients. In vitro studies have shown that inhibition of cell division in glioma is achieved when the applied alternating electric field has a frequency in the range of 200 kHz and an amplitude of 1-3 V cm(-1). Our aim is to calculate the electric field distribution in the brain during TTFields therapy and to investigate the dependence of these predictions on the heterogeneous, anisotropic dielectric properties used in the computational model. A realistic head model was developed by segmenting MR images and by incorporating anisotropic conductivity values for the brain tissues. The finite element method (FEM) was used to solve for the electric potential within a volume mesh that consisted of the head tissues, a virtual lesion with an active tumour shell surrounding a necrotic core, and the transducer arrays. The induced electric field distribution is highly non-uniform. Average field strength values are slightly higher in the tumour when incorporating anisotropy, by about 10% or less. A sensitivity analysis with respect to the conductivity and permittivity of head tissues shows a variation in field strength of less than 42% in brain parenchyma and in the tumour, for values within the ranges reported in the literature. Comparing results to a previously developed head model suggests significant inter-subject variability. This modelling study predicts that during treatment with TTFields the electric field in the tumour exceeds 1 V cm(-1), independent of modelling assumptions. In the future, computational models may be useful to optimize delivery of TTFields.

Diffusion tensor magnetic resonance imaging (DT-MRI) provides a statistical estimate of a symmetric, second-order diffusion tensor of water, D, in each voxel within an imaging volume. We propose a new normal distribution, p(D) /spl prop/ exp(-1/2 D : A : D), which describes the variability of D in an ideal DT-MRI experiment. The scalar invariant, D : A : D, is the contraction of a positive definite symmetric, fourth-order precision tensor, A, and D. A correspondence is established between D : A : D and the elastic strain energy density function in continuum mechanics-specifically between D and the second-order infinitesimal strain tensor, and between A and the fourth-order tensor of elastic coefficients. We show that A can be further classified according to different classical elastic symmetries (i.e., isotropy, transverse isotropy, orthotropy, planar symmetry, and anisotropy). When A is an isotropic fourth-order tensor, we derive an explicit analytic expression for p(D), and for the distribution of the three eigenvalues of D, p(/spl gamma//sub 1/, /spl gamma//sub 2/, /spl gamma//sub 3/), which are confirmed by Monte Carlo simulations. We show how A can be estimated from either real or synthetic DT-MRI data for any given experimental design. Here we propose a new criterion for an optimal experimental design: that A be an isotropic fourth-order tensor. This condition ensures that the statistical properties of D (and quantities derived from it) are rotationally invariant. We also investigate the degree of isotropy of several DT-MRI experimental designs. Finally, we show that the univariate and multivariate distributions are special cases of the more general tensor-variate normal distribution, and suggest how to generalize p(D) to treat normal random tensor variables that are of third- (or higher) order. We expect that this new distribution, p(D), should be useful in feature extraction; in developing a hypothesis testing framework for segmenting and classifying noisy, discrete tensor data; and in designing experiments to measure tensor quantities.

Temporal synchrony of signals arriving from different neurons or brain regions is essential for proper neural processing. Nevertheless, it is not well understood how such synchrony is achieved and maintained in a complex network of time-delayed neural interactions. Myelin plasticity, accomplished by oligodendrocytes (OLs), has been suggested as an efficient mechanism for controlling timing in brain communications through adaptive changes of axonal conduction velocity and consequently conduction time delays, or latencies; however, local rules and feedback mechanisms that OLs use to achieve synchronization are not known. We propose a mathematical model of oligodendrocyte-mediated myelin plasticity (OMP) in which OLs play an active role in providing such feedback. This is achieved without using arrival times at the synapse or modulatory signaling from astrocytes; instead, it relies on the presence of global and transient OL responses to local action potentials in the axons they myelinate. While inspired by OL morphology, we provide the theoretical underpinnings that motivated the model and explore its performance for a wide range of its parameters. Our results indicate that when the characteristic time of OL’s transient intracellular responses to neural spikes is between 10 and 40 ms and the firing rates in individual axons are relatively low (10 Hz), the OMP model efficiently synchronizes correlated and time-locked signals while latencies in axons carrying independent signals are unaffected. This suggests a novel form of selective synchronization in the CNS in which oligodendrocytes play an active role by modulating the conduction delays of correlated spike trains as they traverse to their targets.