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A probabilistic method for determining cortical dynamics during seizures

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Abstract

This work presents a probabilistic method for inferring the parameter ranges in a biologically relevant mathematical model of the cortex most likely to be producing seizures observed in an electrocorticogram (ECoG) signal from a human subject. Additionally, this method produces a probabilistic pathway of the temporal evolution of physiological state in the cortex over the course of individual seizures, leveraging a model of the cortex that describes cortical physiology. We describe ways in which these methods and results offer insights into seizure etiology and have the potential to suggest new treatment options. To directly account for the stochastic and noisy nature of the mathematical model and the ECoG signal, we use a probabilistic Bayesian framework to map features of ECoG segments onto a distribution of likelihoods over physiologically-relevant parameter states. A Hidden Markov Model (HMM) is then introduced to incorporate the belief that cortical physiology has both temporal continuity and also a degree of reproducibility between individual seizures. By inspecting the ratio of likelihoods between HMMs run under two possible parameter regions, both of which produce seizures in the model, we determine which physiological parameter regions are more likely to be causing the observed seizures. We show that between individual seizures, there is consistency in these likelihood ratios between hypothesized regions, in the temporal pathways calculated, and in the separation of seizure from non-seizure time segment likelihood maps.

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Notes

  1. Note that other cortical models could be substituted into this methodology with few changes

  2. This metric is derived and described in Lopour and Szeri (2010)

  3. This particular parameter plane uses the sigmoid curve properties to change the excitatory behavior rather than P e e , a departure from previous literature (Kramer et al. 2005, 2007) that yields an alternative set of pathways for exploration

  4. These 75 runs were uniformly sampled within the state region and thus not from a single point in parameter space. Also, 4 overlapping epochs per 10 second run were extracted from each sample.

  5. In particular, the ksdensity function from MATLAB was used for pdf estimation

  6. Although there is transition matrix for each parameter space, these are chosen to have the same effect in coordinates normalized over the range of each parameter plane, in essence the standard deviation covers the equivalent of 4 neighboring states

  7. Note that this hypothesis ratio is only one way to evaluate and compare models. Often a more involved method called Bayesian model comparison is used to sweep across model parameters and assess model complexity in addition to comparing models (MacKay 2010). Here both hypothesized planes are of the same complexity so our comparison is valid. For certain types of future analysis one may wish to investigate models with differing numbers of parameters or complexity, in which Bayes factors may be appropriate

  8. Note that in addition, the intermediate component α of the alpha–beta algorithm for calculating the full HMM posterior marginal probability, can be used as a filtered version of the likelihood and run in real time

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Acknowledgments

This work was partially supported by an NSF Graduate Research Fellowship and in part by the National Science Foundation through the research grant CMMI 1031811. We would also like to thank Kevin Haas and Prashanth Selvaraj for their suggestions and insights.

Conflict of interests

The authors declare that they have no conflict of interest.

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Correspondence to Vera M. Dadok.

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Action Editor: Abraham Zvi Snyder

Appendices

Appendix A: Mathematical model of the cortex

The dimensionless version of the model developed in (Kramer et al. 2005) for seizures. In Tables 4, 5 and 6, the definitions of the variables and parameters of the model are described.

Table 4 Model variables
Table 5 Dimensionless model parameters
Table 6 Selected dimensional parameter definitions. Dimensional parameters are fully described in (Steyn-Ross et al. 2003)
$$\begin{array}{@{}rcl@{}} &&h_{m} = ({h_{o}^{e}})-\tilde{h}_{e}\tilde{I}_{m} \\ &&\frac{\partial\tilde{h}_{e}}{\partial\tilde{t}} = 1-\tilde{h}_{e}+ {\Gamma}_{e}({h_{e}^{0}}-\tilde{h_{e}})\tilde{I}_{ee} +{\Gamma}_{i}({h_{i}^{0}}-\tilde{h_{e}})\tilde{I}_{ie} \\ &&\frac{\partial\tilde{h}_{i}}{\partial\tilde{t}} = 1-\tilde{h}_{i}+ {\Gamma}_{e}({h_{e}^{0}}-\tilde{h_{i}})\tilde{I}_{ei} +{\Gamma}_{i}({h_{i}^{0}}-\tilde{h_{i}})\tilde{I}_{ii} \\ &&\left(\frac{1}{T_{e}}\frac{\partial}{\partial\tilde{t}}+1\right)^{2} \tilde{I}_{ee} = N_{e}^{\beta}\tilde{S}_{e}\left[\tilde{h}_{e}\right] +\tilde{\phi}_{e}+P_{ee}+\tilde{\Gamma}_{1} \\ &&\left(\frac{1}{T_{e}}\frac{\partial}{\partial\tilde{t}}+1\right)^{2} \tilde{I}_{ei} = N_{e}^{\beta}\tilde{S}_{e}\left[\tilde{h}_{e}\right] +\tilde{\phi}_{i}+P_{ei}+\tilde{\Gamma}_{2} \\ &&\left(\frac{1}{T_{i}}\frac{\partial}{\partial\tilde{t}}+1\right)^{2} \tilde{I}_{ie} = N_{i}^{\beta}\tilde{S}_{i}\left[\tilde{h}_{i}\right] +P_{ie}+\tilde{\Gamma}_{3} \\ &&\left(\frac{1}{T_{i}}\frac{\partial}{\partial\tilde{t}}+1\right)^{2} \tilde{I}_{ii} = N_{i}^{\beta}\tilde{S}_{i}\left[\tilde{h}_{i}\right] +P_{ii}+\tilde{\Gamma}_{4} \\ &&\left(\frac{1}{\lambda_{e}}\frac{\partial}{\partial\tilde{t}}+1\right)^{2} \tilde{\phi}_{e} = \frac{1}{{\lambda_{e}^{2}}} \frac{\partial^{2}\tilde{\phi}_{e}}{\partial\tilde{x}^{2}}+ \left(\frac{1}{\lambda_{e}}\frac{\partial}{\partial \tilde{t}}+1\right)N_{e}^{\alpha}\tilde{S}_{e}\left[\tilde{h}_{e}\right] \\ &&\left(\frac{1}{\lambda_{i}}\frac{\partial}{\partial\tilde{t}}+1\right)^{2} \tilde{\phi}_{i} = \frac{1}{{\lambda_{i}^{2}}}\frac{\partial^{2}\tilde{\phi}_{i}}{\partial\tilde{x}^{2}}+ \left(\frac{1}{\lambda_{i}}\frac{\partial}{\partial \tilde{t}}+1\right)N_{i}^{\alpha}\tilde{S}_{e}\left[\tilde{h}_{e}\right] \end{array} $$

The sigmoid transfer functions that transform the mean soma voltage to the dimensionless firing rates (\(\tilde {S}_{e}, \ \tilde {S}_{i}\)) are shown in Eqs. 16 and 17.

$$\begin{array}{@{}rcl@{}} \tilde{S}_{e}\left[\tilde{h}_{e}\right] &=& \frac{1}{1+\exp\left[-\tilde{g}_{e}(\tilde{h}_{e}-\tilde{\theta}_{e})\right]} \end{array} $$
(16)
$$\begin{array}{@{}rcl@{}} \tilde{S}_{i}\left[\tilde{h}_{i}\right] &=& \frac{1}{1+\exp\left[-\tilde{g}_{i}(\tilde{h}_{i}-\tilde{\theta}_{i})\right]} \end{array} $$
(17)

The stochastic input from subcortical sources is:

$$\begin{array}{@{}rcl@{}} \tilde{\varGamma}_{1}&=&\alpha_{ee}\sqrt{P_{ee}}\xi_{1}[\tilde{x},\tilde{t}] \\ \tilde{\varGamma}_{2}&=&\alpha_{ei}\sqrt{P_{ei}}\xi_{2}[\tilde{x},\tilde{t}] \\ \tilde{\varGamma}_{3}&=&\alpha_{ie}\sqrt{P_{ie}}\xi_{3}[\tilde{x},\tilde{t}] \\ \tilde{\varGamma}_{4}&=&\alpha_{ii}\sqrt{P_{ii}}\xi_{4}[\tilde{x},\tilde{t}] \end{array} $$

where ξ k are white Gaussian noise with zero mean, δ-function correlations and approximated numerically by:

$$ \xi_{k}[\tilde{x},\tilde{t}] = \frac{R(m,n)}{\sqrt{\Delta\tilde{x}{\Delta}{\tilde{t}}}} $$
(18)

and \([\tilde {x},\tilde {t}] = [m{\Delta }{\tilde {x}},\ \ n{\Delta }{\tilde {t}}]\)–integer coordinates on a grid spaced by \( [{\Delta }{\tilde {x}},\ \ {\Delta }{\tilde {t}}]\). This PDE was simulated in MATLAB with an Euler predictor-corrector method designed in (Lopour et al. 2011) and available as an online resource for that article.

Appendix B: Features and feature selection

The likelihood and HMM calculations described above assume a fixed set of appropriately chosen features. To find such a set, we start with a large set of dimensionless or normalized features (e.g. log power ratios, dimensionless quantities, etc.), and then use a feature selection algorithm to reduce this to a smaller set of relevant features. This large set included features in the time domain, frequency domain, wavelet features, measures of coherence, and correlation, among others.

Our assumptions at this point are that features can be classified in one of three categories: irrelevant features that do not reflect meaningful information about seizure dynamics, unmodeled features that are not captured by the mathematical model of the cortical tissue within these parameter planes, and relevant features that can be used in this algorithm for tracking.

1.1 Pruning

As a first step, to reduce the run time of the feature search algorithm, the feature set was pruned using a heuristic pruning technique.

Earlier, we made the assumption that the underlying cortical dynamics of seizures evolve through one of the two parameter planes \(\mathcal {H}_{j}\) hypothesized to have produced the seizures. Given this assumption, values of features from human subject seizures must be contained in the range of values produced by the cortical model under these hypothesized planes. Thus, we calculate for each feature over all human subject data the 5 % and 95 % quantiles, and compare these values to the minimum and maximum of the same feature calculated from all model simulated data. If the feature range identified by these quantiles in human subject data exceeds the range over all model simulations within a parameter plane, this feature is pruned and considered to be capturing noise or unmodeled dynamics. The 5 % and 95 % quantiles are used for human subject data rather than the full range to allow for artifacts and noise in the data.

1.2 Cost function

Next, we use a greedy feature selection algorithm to minimize a cost function, shown in Eq. (19) with components explained below. Roughly speaking, we penalize three different things: (1) classification error in the human subject data, to make sure seizure and non-seizure regions are separable in the data; (2) on-average inconsistent localization of seizure and non-seizure likelihoods in parameter planes; (3) number of features included in the feature set. As an additional step, due to the Naive Bayes assumptions of independence, at each greedy step, we only consider features that have an absolute value correlation coefficient less than 0.4 conditioned on the state.

$$ J(\mathbf{e}^{\boldsymbol{\tau}}_{sub}, \mathcal{L}(\cdot), N_{f}) =C_{Loc} + \lambda_{f} {N_{f}^{2}} $$
(19)

The first component C L o c function penalizes the amount of normalized average likelihood of human subject seizure epochs that maps into non-seizure regions of the parameter planes and likewise penalizes the fraction of average likelihood of non-seizure epochs that lies in the seizure-regions of the parameter planes. As this criterion involves both simulated data and human subject data, its goal is to pick features that are well-captured by this model and exclude features that are not. It also inherently penalizes classification errors in the human subject data as separability in the likelihood regions is an indication of separability in the human subject data alone.

This C L o c is a criterion based on the average likelihood regions of seizure and non-seizure subject data epochs mapped onto each hypothesis plane. Z j is a matrix over all states in the \(\mathcal {H}_{j}\) parameter plane that is 1 at states of the model plane that contain only seizures, 0 on states that contain only non-seizures, and the fraction of seizure epochs on states that contained both seizure and non-seizure epochs. \(\mathcal {N}_{j,sz}\) is the average likelihood of all seizure points normalized so that it sums to one, shown in Eq. (21). For speed, in these feature calculations, instead of a likelihood calculation based on a kernel estimate of the probability distribution of each feature, we used a simpler probability distribution function of an 8-bin histogram with add-one Laplace smoothing.

$$ C_{Loc} = \frac{1}{2}\sum\limits_{\mathcal{H}_{j}} \sum\limits_{s_{k}}\left((1-Z_{j})\mathcal{N}_{j,sz}+Z_{j}\mathcal{N}_{j,non-sz}\right) $$
(20)
$$ \mathcal{N}_{j,sz} = \frac{ \sum\limits_{t_{sz}} \mathcal{L}_{j}(\mathbf{e}^{t_{sz}}) }{ \sum\limits_{s_{k}} \sum\limits_{t_{sz}}\mathcal{L}_{j}(\mathbf{e}^{t_{sz}}) } $$
(21)

Finally, we add a regularization cost on the final number of features via λ f N f , where N f is the number of features and λ f =0.001.

The seizure regions were identified from the waveforms produced by the model over several stochastic runs. Waveforms with sustained oscillations of high amplitude waves were considered seizures. In any particular parameter state, multiple simulations produced multiple waveforms, resulting in states with fractional seizure scores.

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Dadok, V.M., Kirsch, H.E., Sleigh, J.W. et al. A probabilistic method for determining cortical dynamics during seizures. J Comput Neurosci 38, 559–575 (2015). https://doi.org/10.1007/s10827-015-0554-8

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