This paper is a survey of topics related to Hermite interpolation. In the first part we present the standard analysis of the Hermite interpolation problem. Existence, uniqueness and error formula are ...included. Then some computational aspects are studied including Leibnitz' formula and devided differences for monomials. Moreover continuity and differentiation properties of divided differences are analyzed. Finally we represent Hermite polynomial with respect to different basis and give links between them.
Interictal fast oscillations between 100 and 500 Hz have been reported in signals recorded from implanted microelectrodes in epileptic patients and experimental rat models. Oscillations between 250 ...and 500 Hz, or fast ripples (FR), appeared related to the epileptic focus whereas ripples (80–200 Hz) were not. We report high-frequency oscillations recorded with intracranial macroelectrodes in seven patients with refractory focal epilepsy during slow-wave sleep. We characterize the relation of fast oscillations to the seizure focus and quantify their concordance with epileptiform transients, with which they are strongly associated. The patients were selected because interictal spikes were found within and outside the seizure onset zone. Visual inspection was used to identify and classify the ripples and FRs according to their relation to epileptiform spikes. Continuous-time wavelet analysis was used to compute their power. Ripples were present in all patients while FRs where found in five of the seven patients. Most ripples and FRs occurred at the same time as epileptiform transients. The rate of occurrence of ripples was higher within the seizure onset zone than outside in four of seven patients. The rate of FRs was much higher within the seizure onset zone than outside in four of the five patients with FRs (in these four patients, FRs were almost inexistent outside the seizure onset zone). The power of ripples and FRs tended to be higher in the electrodes where their rate was also higher. These results indicate that FRs were more restricted to the electrodes located within the seizure onset zone, especially to the hippocampus, than ripples. In only one patient, FRs were more frequent outside the seizure onset zone; this patient was the only one with cortical dysplasia and the electrode with a high rate of FRs was inside the lesion. This study demonstrates that interictal ripples and FRs can be recorded with depth macroelectrodes in patients. Most occur at the time of epileptiform spikes but some are isolated. Ripples do not show a clear differentiation between the seizure onset zone and remote areas, whereas FRs have a higher rate and higher power in the seizure onset zone. Our results also suggest a special capacity of the abnormal hippocampus to generate FRs, although they were also recorded in other structures.
Highlights • Scalp HFOs are evaluated in 32 epileptic patients with different spiking activity. • HFO occurrence reflects the spiking rate (high HFO frequency with frequent spikes). • HFOs are more ...frequent inside the irritative zone and inside the SOZ. • HFOs are less sensitive but more specific and accurate than spike for the SOZ.
We show that Halley’s basic sequence, resulting from accelerating the order of convergence of Newton’s method, is the most efficient way of doing so in terms of usage of certain derivatives. This ...fact could explain why this process of accelerating the convergence of Newton’s method is so frequently rediscovered. Then we present an algorithmic way of recognizing Halley’s family and we apply this algorithm to examples of rediscoveries.
In this paper, we introduce a nonlinear model which can be used as a regression model for modeling phenomena requiring a two-phase growth curves. The proposed model is defined as a continuous ...piecewise combination of two exponential functions with an unknown break point. Numerical results on real and simulated data sets are presented to illustrate the applicability and the utility of the proposed model on growth data.
OBJECTIVETo determine whether the maximum hemodynamic response to scalp interictal epileptic discharges (IEDs) corresponds to the region where IEDs originate and from where they propagate.
METHODSWe ...studied 19 patients who underwent first an EEG-fMRI showing responses in the gray matter, and then intracranial EEG (iEEG). We coregistered the hemodynamic responses to the iEEG electrode contacts and analyzed IEDs in the iEEG channel adjacent to a maximum response (labeled the main channel), in relation to IEDs in other channels during a widespread intracranial IED event. IEDs in the main channel were aligned at their peak, and IEDs in each channel were averaged time-locked to these instants. The beginning and peak of IEDs in the averaged trace were identified, blinded to the identity of the main channel. The latency of IEDs was computed between the earliest and all other channels.
RESULTSThe median latency of IEDs in the main channel was significantly smaller than in other channels for either the peak (15.5 vs 67.5 milliseconds, p = 0.00037) or the beginning (46.5 vs 118.4 milliseconds, p = 0.000048). The latency of IED was significantly correlated to the distance from the maximum hemodynamic response (p < 0.0001 for either the peak or the beginning).
CONCLUSIONIED adjacent to a maximum hemodynamic response, which often corresponds to the seizure onset zone, is more likely to precede IEDs in remote locations during a widespread intracranial discharge. Thus, EEG-fMRI is a unique noninvasive method to reveal the origin of IEDs, which we propose to label the spike onset zone.
Objective
Disturbed sleep is common in epilepsy. The direct influence of nocturnal epileptic activity on sleep fragmentation remains poorly understood. Stereo‐electroencephalography paired with ...polysomnography is the ideal tool to study this relationship. We investigated whether sleep‐related epileptic activity is associated with sleep disruption.
Methods
We visually marked sleep stages, arousals, seizures, and epileptic bursts in 36 patients with focal drug‐resistant epilepsy who underwent combined stereo‐electroencephalography/polysomnography during presurgical evaluation. Epileptic spikes were detected automatically. Spike and burst indices (n/sec/channel) were computed across four 3‐second time windows (baseline sleep, pre‐arousal, arousal, and post‐arousal). Sleep stage and anatomic localization were tested as modulating factors. We assessed the intra‐arousal dynamics of spikes and their relationship with the slow wave component of non‐rapid eye‐movement sleep (NR) arousals.
Results
The vast majority of sleep‐related seizures (82.4%; 76.5% asymptomatic) were followed by awakenings or arousals. The epileptic burst index increased significantly before arousals as compared to baseline and postarousal, irrespective of sleep stage or brain area. A similar pre‐arousal increase was observed for the spike index in NR stage 2 and rapid eye‐movement sleep. In addition, the spike index increased during the arousal itself in neocortical channels, and was strongly correlated with the slow wave component of NR arousals (r = 0.99, p < 0.0001).
Interpretation
Sleep fragmentation in focal drug‐resistant epilepsy is associated with ictal and interictal epileptic activity. The increase in interictal epileptic activity before arousals suggests its participation in sleep disruption. An additional increase in the spike rate during arousals may result from a sleep–wake boundary instability, suggesting a bidirectional relationship. ANN NEUROL 2020;88:907–920
What do intracerebral electrodes measure? von Ellenrieder, Nicolás; Khoo, Hui Ming; Dubeau, François ...
Clinical neurophysiology,
20/May , Letnik:
132, Številka:
5
Journal Article
Recenzirano
•The measurements of intracerebral macro-electrodes are not affected in an important way by the area of the contacts.•The width of the gap between the electrode and the functional cortex is the most ...important parameter defining the sensitivity.•Sensitivity of intracerebral channel decays with square of distance to small generators, but less rapidly for extended generators.
Gain insight and improve our interpretation of measurements from intracerebral electrodes. Determine if interpretation of intracerebral EEG is dependent on electrode characteristics.
We use intracerebral EEG measurements differing only in the recording electrodes (Dixi or homemade electrodes), and numerical simulations to determine the spatial sensitivity of intracerebral electrodes and its dependence on several parameters.
There is a difference in the high frequency (>20 Hz) power depending on the electrode type, which cannot be explained by the different contact sizes or distance between contacts. Simulations show that the width of the gap between electrode and brain and the extent of the generators have an effect on sensitivity, while other parameters are less important.
The sensitivity of intracerebral electrodes is not affected in an important way by the dimensions of the contacts, but depends on the extent of generators. The unusual insertion technique of homemade electrodes resulting in a large gap between functional brain and electrodes, explains the observed signal difference.
Numerical simulation is a useful tool in the choice or design of intracerebral electrodes, and increases our understanding of their measurements. The interpretation of intracerebral EEG is not affected by differences between typical commercially available electrodes.
In humans, the knowledge of intracranial correlates of spindles is mainly gathered from noninvasive neurophysiologic and functional imaging studies which provide an indirect estimate of neuronal ...intracranial activity. This potential limitation can be overcome by intracranial electroencephalography used in presurgical epilepsy evaluation. We investigated the intracranial correlates of scalp spindles using combined scalp and intracerebral depth electrodes covering the frontal, parietal and temporal neocortex, and the scalp and intracranial correlates of hippocampal and insula spindles in 35 pre-surgical epilepsy patients. Spindles in the scalp were accompanied by widespread cortical increases in sigma band energy (10–16Hz): the highest percentages were observed in the frontoparietal lateral and mesial cortex, whereas in temporal lateral and mesial structures only a low or no simultaneous increase was present. This intracranial involvement during scalp spindles showed no consistent pattern, and exhibited unexpectedly low synchrony across brain regions. Hippocampal spindles were shorter and spatially restricted with a low synchrony even within the temporal lobe. Similar results were found for the insula. We suggest that the generation of spindles is under a high local cortical influence contributing to the concept of sleep as a local phenomenon and challenging the notion of spindles as widespread synchronous oscillations.
•Spindles in the scalp are accompanied by widespread cortical spindle activity.•This activity is predominantly present in the frontoparietal lateral and mesial cortex.•The intracranial involvement during scalp spindles shows no consistent pattern.•The synchrony of spindles is unexpectedly low across different brain regions.•Hippocampal spindles were shorter and occurred mostly not at time of scalp spindles.
Focal cortical dysplasia (FCD) is often characterized by minor structural changes that may go unrecognized by standard radiological analysis. Visual assessment of morphological characteristics of FCD ...and sulci harbouring them is difficult due to the complexity of brain convolutions. Our purpose was to elucidate and quantify the spatial relationship between FCD lesions and brain sulci using automated sulcal extraction and morphometry. We studied 43 consecutive FCD patients using high-resolution MRI. Lesions were classified into small and large using qualitative (detection on initial clinical assessment of conventional MRI) and quantitative (volume) criteria. Sulci were identified and labelled automatically using an algorithm based on a congregation of neural networks. Segmented FCD lesions and sulci were then simultaneously visualized in 3D. We measured mean and maximum depth of sulci related to each FCD and of the corresponding sulci in 21 healthy controls. In addition, we calculated sulcal depth within the FCD neighbourhood. Twenty-one (21/43 = 49%) patients had small FCD lesions (volume range: 128–3093 mm3). Among them, 17 (81%) had been overlooked during initial radiological evaluation and were subsequently identified using image processing. Eighteen (18/21 = 86%) small FCD lesions were located at the bottom of a sulcus. Two others were related to the walls of two sulci and one was located at the crown of a gyrus. Mean and maximum depth of sulci related to the FCD was higher than that of the corresponding sulci in controls (P < 0.008). Sulcal depth within lesional neighbourhood had larger mean depth than that of the entire sulcus (P < 0.0002). Evidence that small FCD lesions are preferentially located at the bottom of an abnormally deep sulcus may be used to direct the search for developmental abnormalities, particularly in patients in whom large-scale MRI features are only mildly abnormal or absent.
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