Analytic modeling of neural tissue: II. Nonlinear membrane dynamics.

Computational modeling of neuroactivity plays a central role in our effort to understand brain dynamics in the advancements of neural engineering such as deep brain stimulation, neuroprosthetics, and magnetic resonance electrical impedance tomography. However, analytic solutions do not capture the fundamental nonlinear behavior of an action potential. What is needed is a method that is not constrained to only linearized models of neural tissue. Therefore, the objective of this study is to establish a robust, straightforward process for modeling neurodynamic phenomena, which preserves their nonlinear features. To address this, we turn to decomposition methods from homotopy analysis, which have emerged in recent decades as powerful tools for solving nonlinear differential equations. We solve the nonlinear ordinary differential equations of three landmark models of neural conduction-Ermentrout-Kopell, FitzHugh-Nagumo, and Hindmarsh-Rose models-using George Adomian's decomposition method. For each variable, we construct a power series solution equivalent to a generalized Taylor series expanded about a function. The first term of the decomposition series comes from the models' initial conditions. All subsequent terms are recursively determined from the first. We show rapid convergence, achieving a maximal error of < 1 0 - 12 with only eight terms. We extend the region of convergence with one-step analytic continuation so that our complete solutions are decomposition splines. We show that this process can yield solutions for single- and multi-variable models and can characterize a single action potential or complex bursting patterns. Finally, we show that the accuracy of this decomposition approach favorably compares to an established polynomial method, B-spline collocation. The strength of this method, besides its stability and ease of computation, is that, unlike perturbation, we make no changes to the models' equations; thus, our solutions are to the problems at hand, not simplified versions. This work validates decomposition as a viable technique for advanced neural engineering studies.

Imaging of current flow in the human head during transcranial electrical therapy.

BACKGROUND: It has been assumed that effects caused by tDCS or tACS neuromodulation are due to electric current flow within brain structures. However, to date, direct current density distributions in the brains of human subjects have not been measured. Instead computational models of tDCS or tACS have been used to predict electric current and field distributions for dosimetry and mechanism analysis purposes. OBJECTIVE/HYPOTHESIS: We present the first in vivo images of electric current density distributions within the brain in four subjects undergoing transcranial electrical stimulation. METHODS: Magnetic resonance electrical impedance tomography (MREIT) techniques encode current flow in phase images. In four human subjects, we used MREIT to measure magnetic flux density distributions caused by tACS currents, and then calculated current density distributions from these data. Computational models of magnetic flux and current distribution, constructed using contemporaneously collected T1-weighted structural MRI images, were co-registered to compare predicted and experimental results. RESULTS: We found consistency between experimental and simulated magnetic flux and current density distributions using transtemporal (T7-T8) and anterior-posterior (Fpz-Oz) electrode montages, and also differences that may indicate a need to improve models to better interpret experimental results. While human subject data agreed with computational model predictions in overall scale, differences may result from factors such as effective electrode surface area and conductivities assumed in models. CONCLUSIONS: We believe this method may be useful in improving reproducibility, assessing safety, and ultimately aiding understanding of mechanisms of action in electrical and magnetic neuromodulation modalities.

Biocompatible, high precision, wideband, improved Howland current source with lead-lag compensation.

The Howland current pump is a popular bioelectrical circuit, useful for delivering precise electrical currents. In applications requiring high precision delivery of alternating current to biological loads, the output impedance of the Howland is a critical figure of merit that limits the precision of the delivered current when the load changes. We explain the minimum operational amplifier requirements to meet a target precision over a wide bandwidth. We also discuss effective compensation strategies for achieving stability without sacrificing high frequency output impedance. A current source suitable for Electrical Impedance Tomography (EIT) was simulated using a SPICE model, and built to verify stable operation. This current source design had stable output impedance of 3.3 MΩ up to 200 kHz, which provides 80 dB precision for our EIT application. We conclude by noting the difficulty in measuring the output impedance, and advise verifying the plausibility of measurements against theoretical limitations.

Quantification of intraventricular hemorrhage with electrical impedance tomography using a spherical model.

We have developed a robust EEG-based current pattern which shows promise for the detection of intraventricular hemorrhage (IVH) in neonates. Our reconstructions to date are based on a layered spherical head model. In this study, the current pattern was used to gather data from three realistic-shaped neonatal head models and a physical phantom based on one of these models. We found that a sensitivity matrix calculated from a spherical model gave us satisfactory reconstructions in terms of both image quality and quantification. Incorporating correct geometry information into the forward model improved image quality. However, it did not improve quantification accuracy. The results indicate that using a spherical matrix may be a more practical choice for monitoring IVH volumes in neonates for whom patient-specific models are not available.

Can high-field MREIT be used to directly detect neural activity? Theoretical considerations.

We sought to determine the feasibility of directly studying neural tissue activity by analysis of differential phase shifts in MRI signals that occurred when trickle currents were applied to a bath containing active or resting neural tissue. We developed a finite element bidomain model of an aplysia abdominal ganglion in order to estimate the sensitivity of this contrast mechanism to changes in cell membrane conductance occurring during a gill-withdrawal reflex. We used our model to determine both current density and magnetic potential distributions within a sample chamber containing an isolated ganglion when it was illuminated with current injected synchronously with the MR imaging sequence and predicted the resulting changes in MRI phase images. This study provides the groundwork for attempts to image neural function using Magnetic Resonance Electrical Impedance Tomography (MREIT). We found that phase noise in a candidate 17.6 T MRI system should be sufficiently low to detect phase signal differences between active and resting membrane states at resolutions around 1 mm(3). We further delineate the broad dependencies of signal-to-noise ratio on activity frequency, current application time and active tissue fractions and outline strategies that can be used to lower phase noise below that presently observed in conventional MREIT techniques. We also propose the idea of using MREIT as an alternative means of studying neuromodulation.

A robust current pattern for the detection of intraventricular hemorrhage in neonates using electrical impedance tomography.

We compared two 16-electrode electrical impedance tomography (EIT) current patterns on their ability to reconstruct and quantify small amounts of bleeding inside a neonatal human head using both simulated and phantom data. The current patterns used were an adjacent injection RING pattern (with electrodes located equidistantly on the equator of a sphere) and an EEG current pattern based on the 10-20 EEG electrode layout. Structures mimicking electrically important structures in the infant skull were included in a spherical numerical forward model and their effects on reconstructions were determined. The EEG pattern was found to be a better topology to localize and quantify anomalies within lateral ventricular regions. The RING electrode pattern could not reconstruct anomaly location well, as it could not distinguish different axial positions. The quantification accuracy of the RING pattern was as good as the EEG pattern in noise-free environments. However, the EEG pattern showed better quantification ability than the RING pattern when noise was added. The performance of the EEG pattern improved further with respect to the RING pattern when a fontanel was included in forward models. Significantly better resolution and contrast of reconstructed anomalies was achieved when generated from a model containing such an opening and 50 dB added noise. The EEG method was further applied to reconstruct data from a realistic neonatal head model. Overall, acceptable reconstructions and quantification results were obtained using this model and the homogeneous spherical forward model.

Detection of intraventricular blood using EIT in a neonatal piglet model.

We have developed a sensitive EIT protocol for detection of intraventricular bleeding. A common model of human neonates is the neonatal piglet. We used our method to test the sensitivity of our method and device to small amounts of blood-like fluid injected near the left and right ventricles of a piglet cadaver. Comparing blood-like fluid detection in open an closed piglet skulls, we found that we could detect amounts of blood less than 0.5 ml, which is smaller than that required for our target of detecting grade II intraventricular hemorrhages in human neonates.

Normalization of a spatially variant image reconstruction problem in electrical impedance tomography using system blurring properties.

The electrical impedance tomography (EIT) image reconstruction problem is ill posed and spatially variant. Because of the problem's ill-posed nature, small amounts of measurement noise can corrupt reconstructed images. The problem must be regularized to reduce image artifacts. In this paper, we focus on the spatially variant characteristics of the problem. Correcting errors due to spatial variance should improve reconstruction accuracy. In this paper, we present methods to normalize the spatially variant image reconstruction problem by equalizing the point spread function (PSF). In order to equalize the PSF, we used the reconstruction blurring properties obtained from the sensitivity matrix. We compared three mathematical normalization schemes: pixel-wise scaling (PWS), weighted pseudo-inversion (WPI) and weighted minimum norm method (WMNM) to equalize images. The quantity index (QI), defined as the integral of pixel values of an EIT conductivity image, was considered in investigating spatial variance. The QI values along with reconstructed images are presented for cases of two-dimensional full array and hemiarray electrode topologies. We found that a spatially invariant QI could be obtained by applying normalization methods based on equalization of the PSF using conventional regularized reconstruction methods such as truncated singular value decomposition (TSVD) and WMNM. We found that WMNM normalization applied to WMNM regularized reconstruction was the best of the methods tested overall, for both hemiarray and full array electrode topologies.

Electrode configurations for detection of intraventricular haemorrhage in the premature neonate.

Intraventricular haemorrhage is a common cause of death in premature human infants. As preventative measures and treatments become available, a method for monitoring and detection is required. Electrical impedance tomography (EIT) is a viable monitoring method compared to modalities such as ultrasound, MRI or CT because of its low cost and contrast sensitivity to blood. However, its sensitivity to blood may be obscured by the low conductivity skull, high conductivity cerebrospinal fluid (CSF) and shape changes in the head and body. We estimated the sensitivity of three 16-electrode and impedance measurement configurations to bleeding using both idealized spherical and realistic geometry three-dimensional finite element models of the neonatal head. Sensitivity distribution responses to alterations in skull composition as well as introduction of conductivity anomalies were determined. Of the three patterns tested, a measurement scheme that employed electrodes at locations based on the 10-20 EEG layout, and impedance measurements involving current return over the anterior fontanelle produced superior distinguishabilities in regions near the lateral ventricles. This configuration also showed strongly improved sensitivities and selectivities when skull composition was varied to include the anterior fontanelle. A pattern using electrodes placed in a ring about the equator of the model had similar sensitivities but performed worse than the EEG layout in terms of selectivity. The third pattern performed worse than either the Ring or EEG-based patterns in terms of sensitivity. The overall performance of the EEG-based pattern on a spherical homogeneous model was maintained in a sensitivity matrix calculated using a homogeneous realistic geometry model.

Imaging and quantification of anomaly volume using an eight-electrode 'hemiarray' EIT reconstruction method.

Electrical impedance tomography (EIT) is particularly well-suited to applications where its portability, rapid acquisition speed and sensitivity give it a practical advantage over other monitoring or imaging systems. An EIT system's patient interface can potentially be adapted to match the target environment, and thereby increase its utility. It may thus be appropriate to use different electrode positions from those conventionally used in EIT in these cases. One application that may require this is the use of EIT on emergency medicine patients; in particular those who have suffered blunt abdominal trauma. In patients who have suffered major trauma, it is desirable to minimize the risk of spinal cord injury by avoiding lifting them. To adapt EIT to this requirement, we devised and evaluated a new electrode topology (the 'hemiarray') which comprises a set of eight electrodes placed only on the subject's anterior surface. Images were obtained using a two-dimensional sensitivity matrix and weighted singular value decomposition reconstruction. The hemiarray method's ability to quantify bleeding was evaluated by comparing its performance with conventional 2D reconstruction methods using data gathered from a saline phantom. We found that without applying corrections to reconstructed images it was possible to estimate blood volume in a two-dimensional hemiarray case with an uncertainty of around 27 ml. In an approximately 3D hemiarray case, volume prediction was possible with a maximum uncertainty of around 38 ml in the centre of the electrode plane. After application of a QI normalizing filter, average uncertainties in a two-dimensional hemiarray case were reduced to about 15 ml. Uncertainties in the approximate 3D case were reduced to about 30 ml.

Modeling skull electrical properties.

Accurate representations and measurements of skull electrical conductivity are essential in developing appropriate forward models for applications such as inverse EEG or Electrical Impedance Tomography of the head. Because of its layered structure, it is often assumed that skull is anisotropic, with an anisotropy ratio around 10. However, no detailed investigation of skull anisotropy has been performed. In this paper we investigate four-electrode measurements of conductivities and their relation to tissue anisotropy ratio (ratio of tangential to radial conductivity) in layered or anisotropic biological samples similar to bone. It is shown here that typical values for the thicknesses and radial conductivities of individual skull layers produce tissue with much smaller anisotropy ratios than 10. Moreover, we show that there are very significant differences between the field patterns formed in a three-layered isotropic structure plausible for bone, and those formed assuming that bone is homogeneous and anisotropic. We performed a measurement of conductivity using an electrode configuration sensitive to the distinction between three-layered and homogeneous anisotropic composition and found results consistent with the sample being three-layered. We recommend that the skull be more appropriately represented as three isotropic layers than as homogeneous and anisotropic.

Method for detecting the disconnection of an extracorporeal device using a patient's endogenous electrical voltages.

Tubing (especially venous) disconnections using pumped devices cause significant hemorrhage, and current monitoring techniques are imperfect because they rely on intraluminal pressure changes. We devised a passive detection method based on a patient's electrical voltages being transmitted via blood tubing to our alarm circuit. As the arterial and venous access sites are in close proximity, the signals are nearly identical during connection, and markedly different with disconnection. We built a prototype and tested it in vitro with saline and during hemodialysis treatments (n=7). The connection status is determined by examining the difference between endogenous voltages in the blood tubing from and to the patient, and when it exceeds a threshold an alarm condition is triggered. We tested for possible confounding by an electrical shunt through the dialyzer and determined that pathway had an impedance approximately three times (>350 kOmega) that of the tubing to the patient. As the roller blood-pump periodically occluded the tubing, the resultant intermittent very high impedance prevented that potential shunting problem and improved the sensitivity of our device. Disconnections were detectable at various bloodline sites (needles, sampling ports, drip chambers). Thus, the circuit's sensors can be placed remotely at the dialysis machine, with electrical continuity to blood made by inexpensive conductive elements at the tubing wall or drip chambers. Appropriate threshold and noise-eliminating circuitry, as well as alarm states that alert the staff and stop the blood pump, make our prototype a promising low-cost safety enhancement.

Detection and quantification of intraperitoneal fluid using electrical impedance tomography.

A prototype electrical impedance tomography system was evaluated prior to its use for the detection of intraperitoneal bleeding, with the assistance of patients undergoing continuous ambulatory peritoneal dialysis (CAPD). The system was sensitive enough to detect small amounts of dialysis fluid appearing in subtractive images over short time periods. Uniform sensitivity to blood appearing anywhere within the abdominal cavity was produced using a post-reconstructive filter that corrected for changes in apparent resistivity of anomalies with their radial position. The image parameter used as an indication of fluid quantity, the resistivity index, varied approximately linearly with the quantity of fluid added. A test of the system's response to the introduction of conductive fluid out of the electrode plane (when a blood-equivalent fluid was added to the stomach) found that the sensitivity of the system was about half that observed in the electrode plane. Breathing artifacts were found to upset quantitative monitoring of intraperitoneal bleeding, but only on time scales short compared with the fluid administration rate. Longer term breathing changes, such as those due to variations in the functional residual capacity of the lungs, should ultimately limit the sensitivity over long time periods.