Granger causality between multiple interdependent neurobiological time series: blockwise versus pairwise methods

Granger causality is becoming an important tool for determining causal relations between neurobiological time series. For multivariate data, there is often the need to examine causal relations between two blocks of time series, where each block could represent a brain region of interest. Two alternative methods are available. In the pairwise method, bivariate autoregressive models are fit to all pairwise combinations involving one time series from the first block and one from the second. The total Granger causality between the two blocks is then derived by summing pairwise causality values from each of these models. This approach is intuitive but computationally cumbersome. Theoretically, a more concise method can be derived, which we term the blockwise Granger causality method. In this method, a single multivariate model is fit to all the time series, and the causality between the two blocks is then computed from this model. We compare these two methods by applying them to cortical local field potential recordings from monkeys performing a sensorimotor task. The obtained results demonstrate consistency between the two methods and point to the significance potential of utilizing Granger causality analysis in understanding coupled neural systems.     

Interactions among toxins that inhibit N-type and P-type calcium channels

A number of peptide toxins from venoms of spiders and cone snails are high affinity ligands for voltage-gated calcium channels and are useful tools for studying calcium channel function and structure. Using whole-cell recordings from rat sympathetic ganglion and cerebellar Purkinje neurons, we studied toxins that target neuronal N-type (Ca(V)2.2) and P-type (Ca(V)2.1) calcium channels. We asked whether different toxins targeting the same channels bind to the same or different sites on the channel. Five toxins (omega-conotoxin-GVIA, omega-conotoxin MVIIC, omega-agatoxin-IIIA, omega-grammotoxin-SIA, and omega-agatoxin-IVA) were applied in pairwise combinations to either N- or P-type channels. Differences in the characteristics of inhibition, including voltage dependence, reversal kinetics, and fractional inhibition of current, were used to detect additive or mutually occlusive effects of toxins. Results suggest at least two distinct toxin binding sites on the N-type channel and three on the P-type channel. On N-type channels, results are consistent with blockade of the channel pore by omega-CgTx-GVIA, omega-Aga-IIIA, and omega-CTx-MVIIC, whereas grammotoxin likely binds to a separate region coupled to channel gating. omega-Aga-IIIA produces partial channel block by decreasing single-channel conductance. On P-type channels, omega-CTx-MVIIC and omega-Aga-IIIA both likely bind near the mouth of the pore. omega-Aga-IVA and grammotoxin each bind to distinct regions associated with channel gating that do not overlap with the binding region of pore blockers. For both N- and P-type channels, omega-CTx-MVIIC binding produces complete channel block, but is prevented by previous partial channel block by omega-Aga-IIIA, suggesting that omega-CTx-MVIIC binds closer to the external mouth of the pore than does omega-Aga-IIIA.     

Coherence and phase locking in the scalp EEG and between LORETA model sources, and microstates as putative mechanisms of brain temporo-spatial functional organization.

Brain electric mechanisms of temporary, functional binding between brain regions are studied using computation of scalp EEG coherence and phase locking, sensitive to time differences of few milliseconds. However, such results if computed from scalp data are ambiguous since electric sources are spatially oriented. Non-ambiguous results can be obtained using calculated time series of strength of intracerebral model sources. This is illustrated applying LORETA modeling to EEG during resting and meditation. During meditation, time series of LORETA model sources revealed a tendency to decreased left-right intracerebral coherence in the delta band, and to increased anterior-posterior intracerebral coherence in the theta band. An alternate conceptualization of functional binding is based on the observation that brain electric activity is discontinuous, i.e., that it occurs in chunks of up to about 100 ms duration that are detectable as quasi-stable scalp field configurations of brain electric activity, called microstates. Their functional significance is illustrated in spontaneous and event-related paradigms, where microstates associated with imagery- versus abstract-type mentation, or while reading positive versus negative emotion words showed clearly different regions of cortical activation in LORETA tomography. These data support the concept that complete brain functions of higher order such as a momentary thought might be incorporated in temporal chunks of processing in the range of tens to about 100 ms as quasi-stable brain states; during these time windows, subprocesses would be accepted as members of the ongoing chunk of processing.     

Voltage-dependent block by saxitoxin of sodium channels incorporated into planar lipid bilayers.

We have previously studied single, voltage-dependent, saxitoxin-(STX) blockable sodium channels from rat brain in planar lipid bilayers, and found that channel block by STX was voltage-dependent. Here we describe the effect of voltage on the degree of block and on the kinetics of the blocking reaction. From their voltage dependence and kinetics, it was possible to distinguish single-channel current fluctuations due to blocking and unblocking of the channels by STX from those caused by intrinsic channel gating. The use of batrachotoxin (BTX) to inhibit sodium-channel inactivation allowed recordings of stationary fluctuations over extended periods of time. In a range of membrane potentials where the channels were open greater than 98% of the time, STX block was voltage-dependent, provided sufficient time was allowed to reach a steady state. Hyperpolarizing potentials favored block. Both association (blocking) and dissociation (unblocking) rate constants were voltage-dependent. The equilibrium dissociation constants computed from the association and dissociation rate constants for STX block were about the same as those determined from the steady-state fractional reduction in current. The steepness of the voltage dependence was consistent with the divalent toxin sensing 30-40% of the transmembrane potential.     

Peculiarities of changes of EEG reactivity during performance of dual tasks in healthy subjects (voluntary postural control and calculation)

Complex EEG and stabilography investigation with separate and simultaneous performance of motor (voluntary postural control) and cognitive (calculation) tasks has been performed in 20 healthy subjects (22 +/- 0.7 yo.). Specific spatial and frequency reactive changes have been revealed during motor task performance. These included increase of coherence in alpha-band for long pair of channels in right hemisphere as well as in symmetric parietal-occipital regions in both hemispheres. Cognitive task performance has been accompanied by coherence increase for low bands (delta- and theta-) with higher activation in left hemisphere and frontal regions. In dual tasks where both components were performed worse comparing to control, performance led to reactive spatial and frequency changes of both--motor and cognitive--tasks, though these changes were less than during separate task performance. Decrease of coherence in alphal-band in frontal areas appeared as a zone of "conflict of interest - interferention". In dual tasks with better performance of each component comparing to control EEG coherence increased in each specific area as well as in areas of "conflict of interest".     

Ca2+ dependence of the Ca2+-selective TRPV6 channel

Microfluorimetry and patch-clamp experiments were performed on TRPV6-expressing HEK cells to determine whether this Ca(2+)-sensing Ca(2+) channel is constitutively active. Intact cells loaded with fura-2 had an elevated intracellular free Ca(2+) concentration ([Ca(2+)](i)), which decreased to the same level such as in non-transfected cells if external Ca(2+) was chelated by EGTA. Whole cell recordings from non-transfected HEK cells and cells expressing human TRPV6 revealed the presence of a basal inward current in both types of cells when the internal solution contained 0.1 mm EGTA and 100 nm [Ca(2+)](i) or if the cytosolic Ca(2+) buffering remained undisturbed in perforated patch-clamp experiments. If recombinantly expressed TRPV6 forms open channels, one would expect Ca(2+)-induced current inhibition, because TRPV6 is negatively regulated by internal Ca(2+). However, dialyzing solutions with high [Ca(2+)] such as 1 microm into TRPV6-expressing cells did not block the basal inward current, which was not different from the recordings from non-transfected cells. In contrast, dialyzing 0.5 mm EGTA into TRPV6-expressing cells readily activated Ca(2+) inward currents, which were undetectable in non-transfected cells. Interestingly, monovalent cations permeated the TRPV6 channels under conditions where no Ca(2+) permeation was detectable, indicating that divalent cations block TRPV6 channels from the extracellular side. Like human TRPV6, the truncated human TRPV6(Delta695-725), which lacks the C-terminal domain required for Ca(2+)-calmodulin binding, does not form constitutive active channels, whereas the human TRPV6(D542A), carrying a point mutation in the presumed pore region, does not function as a channel. In summary, no constitutive open TRPV6 channels were detected in patch-clamp experiments from transfected HEK cells. However, channel activity is highly regulated by intracellular and extracellular divalent cations.     

Technic in the Study of Vascular Systems in Plants

Methods have been improvised for studying the morphology of rosaceous fruits. One method is to dissect the buds and to immerse over night in 20% HCl; then in 5% NaOCl until bleached; after dehydrating, clearing and mounting in cedar oil, they are examined in dark field or with intense reflected light. Another method is to place the living stems, bearing buds, in a 0.01% aqueous solution of Niagara sky blue, forcing infiltration of the dye by means of pressure obtained in an ordinary pressure cooker thru the use of a bicycle pump; the buds are then dehydrated and embedded by ordinary methods. A series of photographs, if desired, may be taken of the cut surfaces of such a stained bud, after each section has been removed from the block.     

Technic in the Study of Vascular Systems in Plants

Methods have been improvised for studying the morphology of rosaceous fruits. One method is to dissect the buds and to immerse over night in 20% HCl; then in 5% NaOCl until bleached; after dehydrating, clearing and mounting in cedar oil, they are examined in dark field or with intense reflected light. Another method is to place the living stems, bearing buds, in a 0.01% aqueous solution of Niagara sky blue, forcing infiltration of the dye by means of pressure obtained in an ordinary pressure cooker thru the use of a bicycle pump; the buds are then dehydrated and embedded by ordinary methods. A series of photographs, if desired, may be taken of the cut surfaces of such a stained bud, after each section has been removed from the block.     

International Society of Chronobiology: Membership Information

Most variables of interest in laboratory medicine show predictable changes with several frequencies in the span of time investigated. The waveform of such nonsinusoidal rhythms can be well described by the use of multiple components rhythmometry, a method that allows fitting a linear model with several cosine functions. The method, originally described for analysis of longitudinal time series, is here extended to allow analysis of hybrid data (time series sampled from a group of subjects, each represented by an individual series). Given k individual series, we can fit the same linear model with m different frequencies (harmonics or not from one fundamental period) to each series. This fit will provide estimations for 2m + 1 parameters, namely, the amplitude and acrophase of each component, as well as the rhythm-adjusted mean. Assuming that the set of parameters obtained for each individual is a random sample from a multivariate normal population, the corresponding population parameter estimates can be based on the means of estimates obtained from individuals in the sample. Their confidence intervals depend on the variability among individual parameter estimates. The vari-ance-covariance matrix can then be estimated on the basis of the sample covariances. Confidence intervals for the rhythm-adjusted mean, as well as for the amplitude-acrophase pair, of each component can then be computed using the estimated covariance matrix. The p-values for testing the zero-amplitude assumption for each component, as well as for the global model, can finally be derived using those confidence intervals and the t and F distributions. The method, validated by a simulation study and illustrated by an example of modeling the circadian variation of heart rate, represents a new step in the development of statistical procedures in chronobiology.     

DNA contents of replication without DNA density labeling

A new method for determining the timing of DNA replication in specific regions of the mammalian genome without the use of DNA density labeling and DNA density centrifugation is described. The method is based on determination of average relative DNA copy numbers in specific genomic regions as cells progress through S phase, and "time of replication" for a specific region is described in terms of the cell's DNA content when the region is replicated. DNA is isolated from synchronized populations of G1 and S phase cells, it is slot-blotted at the same DNA concentration(s) for each population, and it is hybridized with 32P-labeled DNA probes that are specific to the regions of interest. Quantitation of the slot blot autoradiograms and flow cytometric analysis allows determination of (a) average relative DNA copy numbers for the regions of interest in synchronized cell populations, and (b) the average total DNA content in each population of synchronized cells. This information and the flow cytometry histograms are then used to calculate the cellular DNA content at which each region of interest is replicated. The results have a precision of less than or equal to +/- 10% of S phase for Chinese hamster (line CHO) rhodopsin, metallothionein II, the 5'-end of dihydrofolate reductase, the telomeric repeated sequence, pHuR-093 (also located near the centromeres in CHO chromosomes), and the c-Ki-ras family.     

A peptide derived from the Shaker B K+ channel produces short and long blocks of reconstituted Ca(2+)-dependent K+ channels.

A 20 amino acid synthetic peptide, corresponding to the amino-terminal region of the Shaker B (ShB) K+ channel and responsible for its fast inactivation, can block large conductance Ca(2+)-dependent K+ channels from rat brain and muscle. The ShB inactivation peptide produces two kinetically distinct blocking events in these channels. At lower concentrations, it produces short blocks, and at higher concentrations long-lived blocks also appear. The L7E mutant peptide produces only infrequent short blocks (no long-lived blocks) at a much higher concentration. Internal tetraethylammonium competes with the peptide for the short block, which is also relieved by K+ influx. These results suggest that the peptide induces the short block by binding within the pore of Ca(2+)-dependent K+ channels. The long block is not affected by increased K+ influx, indicating that the binding site mediating this block may be different from that involved in the short block. The short block of Ca(2+)-dependent K+ channels and the inactivation of Shaker exhibit similar characteristics with respect to blocking affinity and open pore blockade. This suggests a conserved binding region for the peptide in the pore regions of these very different classes of K+ channel.     

Dynamics of coupled thalamocortical modules

We develop a model of thalamocortical dynamics using a shared population of thalamic neurons to couple distant cortical regions. Behavior of the model is determined as a function of the connection strengths with shared and unshared populations in the thalamus, either within a relay nucleus or the reticular nucleus. When the coupling is via the reticular nucleus, we locate solutions of the model where distant cortical regions maintain the same activity level, and regions where one region maintains an elevated activity level, suppressing activity in the other. We locate and investigate a region where both types of solutions exist and are stable, yielding a mechanism for spontaneous changes in global activity patterns. Power spectra and coherence are computed, and marked differences in the coherence are found between the two kinds of modes. When, on the other hand, the coupling is via a shared relay nuclei, the features seen with the reticular coupling are absent. These considerations suggest a role for the reticular nucleus in modulating long distance cortical communication.     

Spatial coherence and stationarity of local field potentials in an isolated whole hippocampal preparation <Emphasis Type="Italic">in vitro</Emphasis>

Local field potential (LFP) multielectrode recordings of spontaneous rhythms in an isolated whole hippocampal preparation are characterized with respect to their spatial variability within the hippocampus, and their frequency properties. Using simulated data, we categorize potential relationships between frequency variation over time in LFP recordings and spatial variability between electrodes. We then use data recorded from the intact preparation to distinguish between our theoretical categories. We find that the LFP recordings have a close to spatially invariant frequency distribution (not phase) across the hippocampus, and differ in frequency only in a component that may be seen as physiological noise. From these facts, we conclude that the isolated hippocampal LFP recordings represent a single signal and may be regarded as a unitary circuitry. We additionally examine phase differences across our recording sites. We use our characterization of the hippocampal isolate’s properties to predict its spatial coherence in response to high frequency stimulation. We find that there is a finely tuned inverse relationship between temporal variability in the hippocampal isolate’s LFP recordings and their spatial coherence.     

A method for quantifying the anterior load–displacement behavior of the human knee in both the low and high stiffness regions

The anterior load–displacement behavior of the human knee with an intact ACL is characterized by a very low stiffness region initially and a high stiffness region that develops as anterior load is increased. Although this behavior has been well recognized for some time, a method for quantitatively describing the behavior in these two regions based on limits of motion at specific values of anterior/posterior force has not yet been developed. Thus, the purposes of this study were to describe and justify such a method for measuring the laxity and stiffness in both of these regions in the intact knee.

Unique to this study, low stiffness and high stiffness laxities were computed based on three limits of motion for seven cadaveric knees tested at flexion angles ranging from 0° to 90°. Defining the reference position of the tibia relative to the femur, one limit was the 0 N posterior limit which was determined using a specially designed load cycle to reduce uncertainty in establishing a reference position. Defining the upper bound of the load–displacement curve, a second limit was the 225 N anterior limit. A third intermediate limit was the 45 N anterior limit, which was the load that represented the transition from the low stiffness to the high stiffness region. Stiffnesses corresponding to each of the two regions were computed using regression analysis and also estimated based on the laxities. Comparison between the computed and estimated stiffnesses demonstrated that the stiffnesses in both the low and high stiffness regions can be estimated reasonably accurately based on the laxities. Therefore, the 0 N posterior limit and the two laxities are the three quantities needed to describe the load–displacement behavior of the normal knee.     

Focussed ion beam milling and scanning electron microscopy of brain tissue

This protocol describes how biological samples, like brain tissue, can be imaged in three dimensions using the focussed ion beam/scanning electron microscope (FIB/SEM). The samples are fixed with aldehydes, heavy metal stained using osmium tetroxide and uranyl acetate. They are then dehydrated with alcohol and infiltrated with resin, which is then hardened. Using a light microscope and ultramicrotome with glass knives, a small block containing the region interest close to the surface is made. The block is then placed inside the FIB/SEM, and the ion beam used to roughly mill a vertical face along one side of the block, close to this region. Using backscattered electrons to image the underlying structures, a smaller face is then milled with a finer ion beam and the surface scrutinised more closely to determine the exact area of the face to be imaged and milled. The parameters of the microscope are then set so that the face is repeatedly milled and imaged so that serial images are collected through a volume of the block. The image stack will typically contain isotropic voxels with dimenions as small a 4 nm in each direction. This image quality in any imaging plane enables the user to analyse cell ultrastructure at any viewing angle within the image stack.     

Robotics and Dynamic Image Analysis for Studies of Gene Expression in Plant Tissues

Gene expression in plant tissues is typically studied by destructive extraction of compounds from plant tissues for in vitro analyses. The methods presented here utilize the green fluorescent protein (gfp) gene for continual monitoring of gene expression in the same pieces of tissues, over time. The gfp gene was placed under regulatory control of different promoters and introduced into lima bean cotyledonary tissues via particle bombardment. Cotyledons were then placed on a robotic image collection system, which consisted of a fluorescence dissecting microscope with a digital camera and a 2-dimensional robotics platform custom-designed to allow secure attachment of culture dishes. Images were collected from cotyledonary tissues every hour for 100 hours to generate expression profiles for each promoter. Each collected series of 100 images was first subjected to manual image alignment using ImageReady to make certain that GFP-expressing foci were consistently retained within selected fields of analysis. Specific regions of the series measuring 300 x 400 pixels, were then selected for further analysis to provide GFP Intensity measurements using ImageJ software. Batch images were separated into the red, green and blue channels and GFP-expressing areas were identified using the threshold feature of ImageJ. After subtracting the background fluorescence (subtraction of gray values of non-expressing pixels from every pixel) in the respective red and green channels, GFP intensity was calculated by multiplying the mean grayscale value per pixel by the total number of GFP-expressing pixels in each channel, and then adding those values for both the red and green channels. GFP Intensity values were collected for all 100 time points to yield expression profiles. Variations in GFP expression profiles resulted from differences in factors such as promoter strength, presence of a silencing suppressor, or nature of the promoter. In addition to quantification of GFP intensity, the image series were also used to generate time-lapse animations using ImageReady. Time-lapse animations revealed that the clear majority of cells displayed a relatively rapid increase in GFP expression, followed by a slow decline. Some cells occasionally displayed a sudden loss of fluorescence, which may be associated with rapid cell death. Apparent transport of GFP across the membrane and cell wall to adjacent cells was also observed. Time lapse animations provided additional information that could not otherwise be obtained using GFP Intensity profiles or single time point image collections.     

Non-linear, non-invasive method for seizure anticipation in focal epilepsy

In this paper we discuss an approach, using methods of non-linear time series analysis applied to scalp electrode recordings, which is able to distinguish between epochs temporally distant from and just prior to, the onset of a seizure in patients with temporal lobe epilepsy. The method involves a comparison of recordings taken from electrodes adjacent to and remote from the site of ictal onset. In particular, we define a non-linear quantity which we call 'marginal predictability'. This quantity is computed using data from remote and from adjacent electrodes. We find that the difference between the marginal predictabilities computed for the remote and adjacent electrodes decreases several tens of minutes prior to seizure onset, compared to its value interictally.     

Fractal and Markov behavior in ion channel kinetics

Kinetic analysis of ion channel recordings attempts to distinguish the number and lifetimes of channel molecular states. Most kinetic analysis assumes that the lifetime of each state is independent of previous channel history, so that open and closed durations are Markov processes whose probability densities are sums of exponential decays. An alternative approach assumes that channel molecules have many configurtions with widely varying lifetimes. Rates of opening and closing then vary with the time scale of observation, leading to fractal kinetics. We have examined kinetic behavior in two types of channels from human and avian fibroblasts, using a maximum likehood method to test the dependence of rates on observational time scale. For both channels, openings showed mixed fractal and Markov behavior, while closings gave mainly fractal kinetics.     

Detecting determinism in short time series, with an application to the analysis of a stationary EEG recording

We have developed a new method for detecting determinism in a short time series and used this method to examine whether a stationary EEG is deterministic or stochastic. The method is based on the observation that the trajectory of a time series generated from a differentiable dynamical system behaves smoothly in an embedded phase space. The angles between two successive directional vectors in the trajectory reconstructed from a time series at a minimum embedding dimension were calculated as a function of time. We measured the irregularity of the angle variations obtained from the time series using second-order difference plots and central tendency measures, and compared these values with those from surrogate data. The ability of the proposed method to distinguish between chaotic and stochastic dynamics is demonstrated through a number of simulated time series, including data from Lorenz, R?ssler, and Van der Pol attractors, high-dimensional equations, and 1/f noise. We then applied this method to the analysis of stationary segments of EEG recordings consisting of 750 data points (6-s segments) from five normal subjects. The stationary EEG segments were not found to exhibit deterministic components. This method can be used to analyze determinism in short time series, such as those from physiological recordings, that can be modeled using differentiable dynamical processes.