Abstract
We address the issue of analyzing electroencephalogram (EEG) from seizure patients in order to test, model and determine the statistical properties that distinguish between EEG states (interictal, pre-ictal, ictal) by introducing a new class of time series analysis methods. In the present study: firstly, we employ statistical methods to determine the non-stationary behavior of focal interictal epileptiform series within very short time intervals; secondly, for such intervals that are deemed non-stationary we suggest the concept of Autoregressive Integrated Moving Average (ARIMA) process modelling, well known in time series analysis. We finally address the queries of causal relationships between epileptic states and between brain areas during epileptiform activity. We estimate the interaction between different EEG series (channels) in short time intervals by performing Granger-causality analysis and also estimate such interaction in long time intervals by employing Cointegration analysis, both analysis methods are well-known in econometrics. Here we find: first, that the causal relationship between neuronal assemblies can be identified according to the duration and the direction of their possible mutual influences; second, that although the estimated bidirectional causality in short time intervals yields that the neuronal ensembles positively affect each other, in long time intervals neither of them is affected (increasing amplitudes) from this relationship. Moreover, Cointegration analysis of the EEG series enables us to identify whether there is a causal link from the interictal state to ictal state.
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Notes
In time series models, a linear stochastic process has a unit root if 1 (one) is a root of the process's characteristic equation. The process will be non-stationary. If the other roots of the characteristic equation lie inside the unit circle, then the first difference of the process will be stationary. More generally, the ARMA(p,q) process with d unit roots may also be expressed as ARIMA(p,d,q) process and therefore when ARIMA(p,d,q) process is differentiated d times the stationary process ARMA(p,q) can be obtained. As a general signification, I(d) “intergated of order d” can also be used to indicate ARIMA(p,d,q). More detailed description can be found in Section 3.2.
In time series models, a linear stochastic process has a unit root if 1 (one) is a root of the process's characteristic equation. The process will be non-stationary and can be indicated as random walk. If the other roots of the characteristic equation lie inside the unit circle, then the first difference of the process will be stationary. We refer the reader to Section 3.2 for more detailed description.
Following from the literature of statistics, for the remainder of the paper we find it more adequate to use “short-run” in place of short-time intervals and “long-run” in place of relatively long-time intervals. “Short-run effects” is used in place of “the effects observed in short time intervals” and “Long-run effects” is used in place of “the effects observed in long-time intervals”. The choice of the extents on both the concepts of the “short-run” and “long-run” depend how greater is the data set as well as the objectives of the analysis. The detailed explanation can be found in Section 5.
The idea that the variables hypothesized to be linked by some “theoretical” relationship should not diverge from each other when the duration of stochastic process is sufficiently long is a fundamental one. In brain research we can view such variables may drift apart in the short time intervals or because the instantaneous effects, but if they were to diverge without bound, an equilibrium relationship among such variables could not be said to exist. The divergence from a stable equilibrium state must be stochastically bounded and, at some point, diminishing over time. “Cointegration” may be viewed as the statistical expression of the nature of such equilibrium relationships. The theoretical background of Cointegration can be found in Section 3.3.
For further details and related proofs refer to Banarjee et al. 1993, Ch.3.
Source Banarjee et al. (1993)
These definitions are adapted from Juselius (2006) Ch.4.
Such coefficients are formally correspond to the estimated components of the cointegrating vector which is introduced by Definition 2.
Please note that in this paper the term “shock” is synonomously used with the terms “innovation”, “error” and “noise”.
The regression results can be supplied upon to request.
The formal definition of “integrated processes of order d” is given in Section 3.2. More specifically, the integrated process of order 1 can be considered as a random walk which is a kind of non-stationary stochastic process having one root on complex unit circle.
(Phillips 1987b) presents asymptotic results for unit root and “near unit root” processes within a unified framework to explain the special properties of regressions estimated using borderline-stationary variables.
Our aim is to introduce a new methodolgy to examine the EEG data with the help of an example. Thus, there is no other special reason for us to choose mentioned patient’s data. We performed same analysis to other patients’ data and obtained similar results. For future studies, we consider that performing panel cointegration to all simultaneous series obtained from a patient, i.e. 19 simultaneous time series, can give more richer information about the nature of epilepsy.
The statistical outputs of these analyses can be supplied upon to request.
The regression results can be supplied upon to request
Both Engle and Granger (1987) and Johansen (1991a) give the related proofs which show that the asymptotic distribution of the estimators \( \left( {{{\hat \Gamma }_{11}},{{\hat \Gamma }_{12}},{{\hat \Gamma }_{21}},{{\hat \Gamma }_{22}}} \right) \) and \( \left( {\hat r,\hat s} \right) \) of \( \left( {{\Gamma_{11}},{\Gamma_{12}},{\Gamma_{21}},{\Gamma_{22}}} \right) \) and \( \left( {r,s} \right) \)respectively, are the same regardless of whether one uses \( \left( {{{\hat \beta }_1},{{\hat \beta }_2}} \right) \) of \( \left( {{\beta_1},{\beta_2}} \right) \).
References
Alarcon, G., Guy, C. N., Binnie, C. D., Walker, S. R., Owes, R. D. C., & Polkey, C. E. (1994). Intracerebral propagation of interictal activity in partial epilepsy: implications for source localization. Journal of Neurology, Neurosurgery and Psychiatry, 57, 435–449.
Avoli, M. (2001). Do interictal discharges promote or control seizures? Experimental evidence from an in vitro model of epileptiform discharge. Epilepsia, 3(42 Suppl), 2–4.
Bains, J. S., Longacher, J. M., & Staley, K. J. (1999). Reciprocal interactions between CA3 network activity and strength of recurrent collateral synapses. Nature Neuroscience, 2, 720–726.
Banarjee, A., Dolado, J. J, Galbraith, W. J., & Hendry D., (1993). Co-Integration,error correction, and the econometric analysis of non-stationary data. Oxford University Press.
Barbarosie, M., & Avoli, M. (1997). CA3-Driven hippocampal-entorhinal loop controls rather than sustains in vitro limbic seizures. Journal of Neuroscience, 17, 9308–9314.
Bazil, C. W., & Pedley, T. A. (1995). Neurophysiological effects of antiepileptic drugs. In R. H. Levy, R. H. Mattson & B. S. Meldrum (Eds.), Antiepileptic drugs (pp. 79–89). New York: Raven.
Blanco, S., Garcia, H., Quian Quiroga, R., Romanelli, L., & Rosso, O. A. (1995). Stationarity of the EEG series. IEEE Engineering in Medicine and Biology, 4, 395–399.
Chan, N. H., & Wei, C. Z. (1988). Limiting distribution of least-squares estimations of unstable autoregressive processes. Annals of Statistics, 16, 367–401.
Davidson, J. E. H., Hendry, D. F., Srba, F., & Yeo, S. (1978). Econometric modelling of the aggregate time series relationships between consumer’s expenditure and income in the United Kingdom. Economic Journal, 88, 661–692.
de Curtis, M., & Avanzini, G. (2001). Interictal spikes in focal epileptogenesis. Progress in Neurobiology, 63, 541–567.
Debanne, D., Thompson, S. M., & Gahwiler, B. H. (2006). A brief period of epileptiform activity strengthens excitatory synapses in the rat hippocampus in vitro. Epilepsia, 47, 247–256.
Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with unit root. Journal of the American Statistical Association, 74, 427–431.
Dikanev, T., Smirnov, D., Wennberg, R., Perez Velazquez, J. L., & Bezruchko, B. (2005). EEG nonstationarity during intracranially recorded seizures: statistical and dynamical analysis. Clinical Neurophysiology, 116(8), 1796–1807.
Duncan, J. S. (1987). Antiepileptic drugs and the EEG. Epilepsia, 28, 259–266.
Emerson, R., Turner, C. A., Pedley, T. A., Walczak, T. S., & Forgione, M. (1995). Propagation patterns of temporal spikes. Electroencephalography and Clinical Neurophysiology, 94, 338–348.
Engel, J., & Ackermann, R. F. (1980). Interictal spikes correlate with decreased, rather than increased epileptogenicity in amygdaloid kindled rats. Brain Research, 190, 543–548.
Engel, J., Rausch, R., Lieb, J. P., Kulh, D. E., & Crandall, P. H. (1981). Correlation of criteria used for localizing epileptic foci in patients considered for surgical therapy of epilepsy. Annals of Neurology, 9, 215–224.
Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: representation, estimation and testing. Econometrica, 55, 251–276.
Franaszczuk, P. J., & Bergey, G. K. (1999). An autoregressive method for the measurement of synchronization of interictal and ictal EEG signals. Biological Cybernetics, 81, 3–9.
Friedrich, R., & Uhl, C. (1996). Spatio-temporal analysis of human electroencephalograms: petit-mal epilepsy. Physica D, 98, 171–182.
Fuller, W. A. (1976). Introduction to statistical time series. New York: Wiley.
Gath, I., Feuerstein, C., Pham, D. T., & Rondouin, G. (1992). On the tracking of rapid dynamic changes in seizure EEG. IEEE Transactions on Biomedical Engineering, 39, 952–958.
Gotman, J. (1984). Relationships between triggered seizures, spontaneous seizures, and interictal spiking in the kindling model of epilepsy. Experimental Neurology, 84, 259–273.
Gotman, J. (1991). Relationships between interictal spiking and seizures: human and experimental evidence. Canadian Journal of Neurological Sciences, 18, 573–576.
Gotman, J., & Koffler, D. J. (1990). Interictal spiking increases after seizures but does not after decrease in medication. Electroencephalography and Clinical Neurophysiology, 75, 358–360.
Gotman, J., & Marciani, M. G. (1985). Electroencephalographic spiking activity, drug levels, and seizure occurrence in epileptic patients. Annals of Neurology, 17, 597–603.
Granger, C. W. J. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica, 37(3), 424–438.
Granger, C. W. J. (1981). Some properties of time series data and their use in econometric model specification. Journal of Econometrics, 16, 121–130.
Hall, A. (1994). Testing for a unit-root in time series with pretest databased model selection. Journal of Business and Economic Statistics, 12(4), 461–470.
Hamilton, J. D. (1994). Time series analysis. Princeton University Press.
Hendry, D. F., & Anderson, J. (1977). Testing dynamic specification in small simultaneous models: an application to a model of building society behavior in the United Kingdom. Ch. 8c. In M. D. Intrilligator (Ed.), Frontiers of quantitative economics (iii(a)th ed., pp. 361–383). Amsterdam: North-Holland.
Hernandez, J., Valdes, P. A., & Vila, P. (1996). EEG spike and wave modelled by a stochastic limit cycle. NeuroReport, 7, 2246–2250.
Iasemidis, L. D., & Sackellares, J. C. (1996). Chaos theory and epilepsy. The Neuroscientist, 2, 118–126.
Jefferys, J. G. R. (1989). Chronic epileptic foci in vitro in hippocampal slices from rats with the tetanus toxin epileptic syndrome. Journal of Neurophysiology, 62, 458–468.
Jensen, M. S., & Yaari, Y. (1988). The relationship between interictal and ictal paroxysms in an in vitro model of focal hippocampal epilepsy. Annals of Neurology, 24, 591–598.
Jing, H., & Takigawa, M. (2000). Comparison of human ictal, interictal and normal non-linear component analyses. Clinical Neurophysiology, 111(7), 1282–1292.
Jirsch, J. D., Urrestarazu, E., LeVan, P., Olivier, A., Dubeau, F., & Gotman, J. (2006). High-frequency oscillations during human focal seizures. Brain, 129(6), 1593–1608.
Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica, 59, 1551–1580.
Juselius, K. (2006). The cointgerated VAR model: Methodology and applications. New York: Oxford University Press.
Katz, A., Marks, D. A., McCarthy, G., & Spencer, S. S. (1991). Does interictal spiking change prior to seizures? Electroencephalography and Clinical Neurophysiology, 79, 153–156.
Kennel, M. B. (1997). Statistical test for dynamical nonstationarity in observed time series data. Physical Review E, 56, 316–321.
Layne, S. C., Meyer-Kress, G., & Holzfuss, J. (1986). Problems associated with dimensional analysis of electroencephalogram data. In G. Meyer-Kress (Ed.), Dimensions and entropies in chaotic systems (pp. 246–256). Berlin: Springer Verlag.
Lehnertz, K., & Elger, C. E. (1995). Spatio-temporal dynamics of the primary epileptogenic area in temporal lobe epilepsy characterized by neural complexity loss. Electroencephalography and Clinical Neurophysiology, 95, 108–117.
Librizzi, L., & de Curtis, M. (2003). Epileptiform ictal discharges are prevented by periodic interictal spiking in the olfactory cortex. Annals of Neurology, 53, 382–389.
Lüders, H. O., Engel, J., Jr., & Munari, C. (1993). General principles. In J. Engel Jr. (Ed.), Surgical treatment of the epilepsies (pp. 137–153). New York: Raven.
Manuca, R., Casdagli, C. D., & Savit, R. S. (1998). Nonstationarity in epileptic EEG data and implications for neural dynamics. Mathematical Biosciences, 147, 1–22.
Mayer, M. L., & Westbrook, G. L. (1987). Permeation and block of N-methyl-d-aspartic acid receptor channels by divalent cations in mouse cultured central neurones. Journal of Physiology, 394, 501–527.
Mormann, F., Andrzejak, R. G., Elger, C. E., & Lehnertz, K. (2007). Seizure prediction: the long and winding road. Brain, 130(2), 314–333.
Ojemann, G. A., & Engel, J., Jr. (1987). Acute and chronic intracranial recording and stimulation. In J. Engel Jr. (Ed.), Surgical treatment of epilepsies (pp. 263–288). New York: Raven.
Perez Velazquez, J. L., Cortez, M., Snead, O. C., III, & Wennberg, R. (2003). Dynamical regimes underlying epileptiform events: role of instabilities and bifurcations in brain activity. Physica D, 186, 205–220.
Phillips, P. C. B. (1987a). Time series regression with a unit root. Econometrica, 55, 277–301.
Phillips, P. C. B. (1987b). Towards a unified asymptotic theory of autoregression. Biometrika, 74, 535–548.
Phillips, P. C. B., & Ouliaris, S. (1990). Asymptotic properties of residual based tests for cointegration. Econometrica, 58, 165–193.
Phillips, P. C. B., & Perron, P. (1988). Testing for a unit root in time series regression. Biometrika, 75, 335–346.
Pijn, J. P., Neerven, J. V., Noest, A., & Lopes da Silva, F. H. (1991). Chaos or noise in EEG signals: dependence on state and brain site. Electroencephalography and Clinical Neurophysiology, 79, 371–381.
Pijn, J. P., Velis, D. N., van der Heyden, M. J., DeGoede, J., van Veerlen, C. W., & Lopes da Silva, F. H. (1997). Nonlinear dynamics of epileptic seizures on basis of intracranial EEG recordings. Brain Topography, 9, 249–270.
Ralston, B. L. (1958). The mechanism of transition of interictal spiking into ictal seizure discharges. Electroencephalography and Clinical Neurophysiology, 10, 217–232.
Rieke, C., Sternickel, K., Andrzejak, R. G., Elger, C. E., David, P., Lehnertz, K. (2002). Measuring nonstationarity by analyzing the loss of recurrence in dynamical systems. Phys Rev Lett, 88.
Said, S. E., & Dickey, D. A. (1984). Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika, 71, 599–607.
Sargan, J. D. (1964). Wages and prices in the United Kingdom: A study in econometric methodology. In P. E. Hart, G. Mills & J. K. Whitaker (Eds.), Econmetric analysis for economic plannin. London: Butterworth.
Schreiber, T. (1999). Interdisciplinary application of nonlinear time series methods. Physics Reports, 308(1), 1–64.
Staley, K. J., & Dudek, F. E. (2006). Interictal spikes and epileptogenesis. Epilepsy Currents, 6(6), 199–202.
Staley, K. J., Hellier, J. L., & Dudek, F. E. (2005). Do interictal spikes drive epileptogenesis? Neuroscientist, 11(4), 272–276.
Sundaram, M., Hogan, T., Hiscock, M., & Pillay, N. (1990). Factors affecting spike discharges in adults with epilepsy. Electroencephalography and Clinical Neurophysiology, 75, 358–360.
Swartzwelder, H. S., Lewis, D. V., Anderson, W. W., & Wilson, W. A. (1987). Seizure-like events in brain slices: suppression by interictal activity. Brain Research, 410, 362–366.
Talairach, J., & Bancaud, J. (1973). Stereotaxic approach to epilepsy. Progress in Neurological Surgery, 5, 297–356.
Weber, B., Lehnertz, K., Elger, C. E., & Wieser, H. G. (1998). Neuronal Complexity loss in the interictal EEG recorded with foramen ovale electrodes predicts side of primary epileptogenic area in the temporal lobe epilepsy: a replication study. Epilepsia, 39, 922–927.
Wilkus, R., Dodrill, C. B., & Troupin, A. S. (1978). Carbamazepine and the electroencephalogram of epileptics: a double blind study in comparison to phenytoin. Epilepsia, 19, 283–291.
Yu, D., Lu, W., & Harrison, R. G. (1998). Space time-index plots for probing dynamical nonstationarity. Physics Letters A, 250, 323–327.
Zoldi, S. M., Krystal, A., & Greenside, H. S. (2000). Stationarity and redundancy of multichannel EEG data recorded diring generalized tonic–clonic seizures. Brain Topography, 12, 187–200.
Acknowledgements
We thank Cigdem Özkara and her research team in Institute of Neurology, Cerrahpasa Faculty of Medicine, Istanbul University for their valuable remarks, and Ibrahim Oztura, Baris Baklan in Neurology Department of Dokuz Eylul University Hospital, Izmir for epileptic recordings and patient data. The statistical analysis were performed by Eviews 5.00.
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Özkaya, A., Korürek, M. Estimating short-run and long-run interaction mechanisms in interictal state. J Comput Neurosci 28, 177–192 (2010). https://doi.org/10.1007/s10827-009-0198-7
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DOI: https://doi.org/10.1007/s10827-009-0198-7