We developed a modern numerical approach to the multivariate linear discrimination of Fisher from 1936 based upon singular value decomposition that is sufficiently stable to permit widespread application to spatiotemporal neuronal patterns. We demonstrate this approach on an old problem in neuroscience—whether seizures have distinct dynamical states as they evolve with time. A practical result was the first demonstration that human seizures have distinct initiation and termination dynamics, an important characterization as we seek to better understand how seizures start and stop. Our approach is broadly applicable to a wide variety of neuronal data, from multichannel EEG or MEG, to sequentially acquired optical imaging data or fMRI.
Models with a time delay often occur, since there is a naturally occurring delay in the transmission of information. A model with a delay can be noninvertible, which in turn leads to qualitative differences between the dynamical properties of a delay equation and the familiar case of an ordinary differential equation. We give specific conditions for the existence of noninvertible solutions in delay equations, and describe the consequences of noninvertibility.
Objective: To discriminate seizures from interictal dynamics based on multivariate synchrony measures, and to identify dynamics of a pre-seizure state.
Methods: A linear discriminator was constructed from two different measures of synchronization: cross-correlation and phase synchronization. We applied this discriminator to a sequence of seizures recorded from the intracranial EEG of a patient monitored over 6 days.
Results: Surprisingly, we found that this bivariate measure of synchronization was not a reliable seizure discriminator for 7 of 9 seizures. Furthermore, the method did not appear to reliably detect a pre-seizure state. An association between anti-convulsant dosage, frequency of clinical seizures, and discriminator performance was noted.
Conclusions: Using a bivariate measure of synchronization failed to reliably differentiate seizures from non-seizure periods in these data, nor did such methods show reliable detection of a synchronous pre-seizure state. The non-stationary variables of decreasing antiepileptic medication (without available serum concentration measurements), and concomitant increasing seizure frequency contributed to the difficulties in validating a seizure prediction tool on such data.
Significance: The finding that these seizures were not a simple reflection of increasing synchronization in the EEG has important implications. The non-stationary characteristics of human post-implantation intracranial EEG is an inherent limitation of pre-resection data sets.
Auditory-evoked potentials (AEPs) were triggered in real time as a function of ongoing electroencephalogram (EEG) phase. Phase triggering on-line or retrospective phase-selective averaging introduces phase artifacts such as spurious troughs or peaks, which mask mid-latency and affect the amplitude of late AEPs. We developed a method to control for phase artifacts by phase-selective averaging of trials, recorded without stimulation, and used this to uncover a previously unknown phase dependency of AEPs. Not only are such findings inconsistent with thestandardadditiveevokedpotentialmodel,butweidentified clear neural correlates at fixed latencies,whichare inconsistent with the recently proposed phase-resetting model. Our findings suggest that a new conceptualization is required to account for the interplay between the correlates of neural-evoked activity and modulation of ongoing EEG that together constitute evoked potentials.
Recent work has identified nonlinear deterministic structure in neuronal dynamics using periodic orbit theory. Troublesome in this work were the significant periods of time where no periodic orbits were extracted - "dynamically dark" regions. Tests for periodic orbit structure typically require that the underlying dynamics are differentiable. Since continuity of a mathematical function is a necessary but insufficient condition for differentiability, regions of observed differentiability should be fully contained within regions of continuity. We here vertify that this fundamental mathematical principle is reflected in observations from mammalian neuraonal activity. First, we introduce a null Jacobian transformation to verify the observation of differentiable dynamics when periodic orbits are extracted. Second, we show that a less restrictive test for deterministic structure requiring only continuity demonstrates widespread nonlinear deterministic structure only partially appreciated with previous approaches.
Nonlinear Mathematical Techniques can be used to Analyze the EEG Patterns of Epilepsy Patients enabling the Prediction of Seizures prior to the Onset of Symptoms
Experimental time-series of human H-reflexes were analyzed for the presence of fractal sturcture or deterministic chaos. Surrogate data sets consisting of stochastic time-series with preservation of selected properties of the experimental time-series were used as mathematical controls. Artifacts generated during the analysis of the experimental data are identified, and shown to be due to linear correlation in the original time-series. The method is simple and generally applicable to the non-linear analysis of time-series from any experimental system.
Many signals measured from the nervous system exhibit apparently random variability that is usually considered to be noise. The development of chaos theory has revealed that such random appearing variability may not, in fact, be random, but rather may be deterministic behavior that can reveal important information about the system's underlying mechanisms. We present some new methods for distinguishing determinism from randomness in experimental data, and we apply these methods to population neural responses recorded from hippocampal tissue slices.
An approach to discriminating deterministic versus stochastic dynamics from neuronal data is presented. Direct tests for determinism are emphasized, as well as using time series with clear physical correlates measured from small ensembles of neurons. Surrogate data are used to provide null hypothesis that the dyanmics in our data could be accounted for by linear stochastic systems. Algorithms are given in full, and the analysis fo an experimental example is given.
Long time series of monosynaptic la-afferent to alpha-motoneuron reflexes were recorded in the L7 or S1 ventral roots in the cat. Time series were collected before and after spinalization at T13 during constant amplitude stimulations of group la muscle afferents inthe triceps surae muscle nerves. Using autocorrelation to analyze the linear correlation in the time series demonstrated oscillations in the decerebrate state (4/4) that were eliminated after spinalization (5/5). Three tests for determinism were applied to these series: 1) local flow, 2) local dispersion, and 3) nonlinear prediction. These algorithms were validated with time series generated from known deterministic equations. For each experimental and theoretical time series used, matched time-series of stochastic surrogate data were generated to serve as mathematical and statistical controls. Two fo the time series collected in the decerebrate state (2/4) demonstrated evidence for deterministic structure. This structrue could not be accounted for by the autocorrelation in the data, and was abolished following spinalization. None of the time series collected in the spinalized state (0/5) demonstrated evidence of determinism. Although monosynaptic reflex variability is generally stochastic in the spinalized state, this simple driven system may display deterministic behavior in the decerebrate state.
Long time series of Schaffer collateral to CA1 yramidal cell presynaptic volleys (stratum radiatum) and population spikes (stratum pyramidale) were evoled (driven) in rat hippocampal slices. From the driven CA1 region in normal [K+] perfusate, both population spike amplitude and an input-output function consisting of population spike amplitude divided by the presynaptic volley amplitude were analyzed. Raising [K+] inthe perfusion medium to 8.5 mM, slices were induced to spontaneously burst fire in CA3 and long time series of inter-burst intervals were recorded. Three tests for determinism were applied to these series: a discrete adaption of a local flow approach, a local dispersion approach, and nonlinear prediction. Surrogate data were generated to serve as mathematical and statistical controls. All of the population spike (6/6) and input-output (6/6) time series form the normal [K+] driven circuity were stochastic by all three methods. Although most of the time series (5/6) fromthe autonomously bursting high [K+] state failed to demonstrated evidence of determinism, one (1/6) of these time series did demonstrate significant determinism. This single instance of predictability could not be accounted for by the linear correation in these data.
In order to compare methods for detecting determinism, we studied the ability of several recently developed algorithms to detect determinism in time series under various levels of additive noise. Theoretical chaotic time series were generated from the Henon and Lorenz equations with 0, 5, 25, 50, 75, and 100% colored or Gaussian white noise added. Three types of surrogate data were generated as controls in order to demonstrate that the data could be differentiated from a linear stochastic process.l The results for colored and Gaussian white noise were similar for each prediction method at a given level of noise. However, in the presence of noise, the tests for determinism differed in their ability to identify determinism in these time series.
To determine whether EEG spikes are predictable, time series of EEG spike intervals were generated from subdural and depth electrode recordings from four patients. The intervals between EEG apikes were hand edited to ensure high accuracy and eliminate false positive and negative spikes. Spike rates (per minute) were generated from longer time series, but for these data hand editing was usually not feasible. Linear and nonlinear models were fit to both types of data. One patient had no linear or nonlinear predictability, two had predictability that could be well accounted for with a linear stochastic model, and one had a degree of nonlinear predictability for both interval and rate data that no linear model could adequately account for.
To compare direct tests for detecting determinism in chaotic time series, data from Henon, Lorenz, and Mackey-Glass equations were contaminated with various levels of additive colored noise. These data were analyzed with a variety of recently developed tests for determinism, and the results compared.