Exploring neural directed interactions with transfer entropy based on an adaptive kernel density estimator.

Zuo K; Bellanger JJ; Yang C; Shu H; Le Bouquin Jeannés R

doi:10.1109/embc.2013.6610507

Exploring neural directed interactions with transfer entropy based on an adaptive kernel density estimator.

Le Bouquin Jeannés R ²

Affiliations

1. LIST, Laboratory of Image Science and Technology [Nanjing] SouthEast University School of Computer Science and Engineering Si Pai Lou 2, Nanjing, 210096, CN
Authors
Zuo K^{1,

2}
Yang C^{1,

2}
Shu H^{1,

2}
(3 authors)
2. CRIBS, Centre de Recherche en Information Biomédicale sino-français INSERM : Laboratoire International AssociÉ Université de Rennes 1 9 Rue Jean Macé, 35700 Rennes, FR
Authors
Zuo K^{1,

2}
Bellanger JJ²
Yang C^{1,

2}
Shu H^{1,

2}
Le Bouquin Jeannés R²
(5 authors)

ORCIDs linked to this article

Le Bouquin Jeannés R | 0000-0002-4050-2895

Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Annual International Conference, 01 Jan 2013, 2013:4342-4345
https://doi.org/10.1109/embc.2013.6610507 PMID: 24110694 PMCID: PMC3898961

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Abstract

This paper aims at estimating causal relationships between signals to detect flow propagation in autoregressive and physiological models. The main challenge of the ongoing work is to discover whether neural activity in a given structure of the brain influences activity in another area during epileptic seizures. This question refers to the concept of effective connectivity in neuroscience, i.e. to the identification of information flows and oriented propagation graphs. Past efforts to determine effective connectivity rooted to Wiener causality definition adapted in a practical form by Granger with autoregressive models. A number of studies argue against such a linear approach when nonlinear dynamics are suspected in the relationship between signals. Consequently, nonlinear nonparametric approaches, such as transfer entropy (TE), have been introduced to overcome linear methods limitations and promoted in many studies dealing with electrophysiological signals. Until now, even though many TE estimators have been developed, further improvement can be expected. In this paper, we investigate a new strategy by introducing an adaptive kernel density estimator to improve TE estimation.

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Conf Proc IEEE Eng Med Biol Soc. Author manuscript; available in PMC 2014 Feb 3.

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Conf Proc IEEE Eng Med Biol Soc. 2013; 2013: 4342–4345.

https://doi.org/10.1109/EMBC.2013.6610507

PMCID: PMC3898961

HALMS: HALMS881459

PMID: 24110694

Exploring neural directed interactions with transfer entropy based on an adaptive kernel density estimator

Kai Zuo,^1,^2,^* Jean-Jacques Bellanger,^2,³ Chunfeng Yang,^1,² Huazhong Shu,^1,² and Regine Le Bouquin Jeannes^2,^3,^*

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Abstract

Keywords: bioelectric phenomena, brain models, causality, electroencephalography, entropy, medical signal detection, neurophysiology, nonlinear dynamical systems, stochastic processes

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I. Introduction

In neuroscience, recent works have been devoted to detecting effective connectivity [1] defined as a causal influence of the dynamics of a first system on the dynamics of a second one. In this context, two questions are commonly addressed: (i) how to choose a formal quantitative definition of effective connectivity and (ii) how to provide corresponding estimators defined as functions of signals recorded in both systems. Nowadays two approaches contrast. The first one does not rely on an underlying physiological model while the second one, namely dynamical causal modeling, does. In this contribution, we are only concerned with the first approach including linear and nonlinear methodologies, and we consider nonlinear nonparametric entropic characterization of this connectivity using the so-called transfer entropy (TE). When computed on a stationary bivariate time series (X, Y), this quantity measures the amount of information transferred from channel X (resp. Y) to channel Y (resp. X) and is denoted TE_x_→_y (resp. TE_y_→_x) hereafter. It was introduced by Schreiber [2] and defined as the Kullback-Leibler divergence between two different predictive probability distributions of Y_n. The first distribution is defined conditionally to amplitudes of X_n_′, n′ < n and Y_n_′, n′ < n, at time instants prior to n, and the second one is only defined conditionally to Y_n_′, n′ < n. A simple exchange of X and Y leads to the definition of TE_y_→_x. Formal definition is given by

{T E}_{x \to y} = \int_{R^{k + l + 1}} log (\frac{p_{Y_{n + 1} / Y_{n}^{k}, X_{n}^{l}} (y, y^{k}, x^{l})}{p_{Y_{n + 1} / Y_{n}^{k}} (y, y^{k})}) \times p_{Y_{n + 1}, Y_{n}^{k}, X_{n}^{l}} (y, y^{k}, x^{l}) d (y, y^{k}, x^{l})

(1)

with $U_{n}^{m} ≜ (U_{n}, \dots, U_{n - m + 1})$ , p_U (u) denotes the probability density of a random vector U at u, k and l are the predictor dimensions. Let us note that the definition of TE is qualitatively consistent with Wiener and Granger approaches, which only compare mean square prediction errors. In theory, TE_x_→_y is never negative and is equal to zero iif

p_{Y_{n + 1} / Y_{n}^{k}, X_{n}^{l}} (y, y^{k}, x^{l}) = p_{Y_{n + 1} / Y_{n}^{k}} (y, y^{k}), y \in R

(2)

The choice of k and l can impact drastically on theoretical TE value and, without a priori information on the hidden nonlinear dynamics generating (X, Y), this issue is not trivial and is not discussed in this paper. Given the theoretical index and an N point observation (X, Y)_n, n = 1..N, we have to determine an estimation procedure to compute ${\hat{T E}}_{x \to y}$ from

(y_{n + 1}, y_{n}^{k}, x_{n}^{l}), n = n_{0}, \dots, N - 1 (n_{0} = 1 + max (k, l))

(3)

If all probability densities are known, the trivial Monte Carlo estimator could be:

\frac{1}{N - n_{0}} \sum_{n = n_{0}}^{N - 1} log (\frac{p_{Y_{n + 1}, Y_{n}^{k}, X_{n}^{l}} (y_{n + 1}, y_{n}^{k}, x_{n}^{l}) p_{Y_{n}^{k}} (y_{n}^{k})}{p_{Y_{n + 1}, Y_{n}^{k}} (y_{n + 1}, y_{n}^{k}) p_{Y_{n}^{k}, X_{n}^{l}} (y_{n}^{k}, x_{n}^{l})})

(4)

Since the densities are unknown, a first method consists in replacing each density p_U by an estimation _U possibly obtained by a fixed size kernel estimation approach as proposed in [3]. A second method computing estimations $\hat{log (p_{U})}$ of log (p_U) from K Nearest Neighbors (KNN) selection was developed for mutual information estimation in [4] and applied in [5]. It is also possible to compute the estimation of a density p_U with adaptive size kernels. We propose this improvement to compute TE and compare our results on linear Gaussian models (i) with corresponding theoretical values, and (ii) with TE estimated using a fixed bandwidth kernel and/or KNN kernel as in [5]. Then, these methods are compared on a neurophysiological model [11].

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II. Methods and Materials

A. Kernel methods

In order to reconstruct a probability density p_U from N observed states u_n, the general form of a fixed kernel density estimator (FKDE) of bandwidth h is given by:

{\hat{p}}_{U} (u) = \frac{1}{N} \sum_{n = 1}^{N} \frac{1}{h} K (\frac{u - u_{n}}{h}) = \frac{1}{N} \sum_{n = 1}^{N} K_{h} (u - u_{n})

(5)

where K denotes a kernel function. For joint density probability estimated at (y, y^k, x^l), we write:

\begin{array}{l} {\hat{p}}_{YYX} (y, y^{k}, x^{l}) = \frac{1}{∣ Δ ∣} \sum_{m \in Δ} \frac{1}{h_{y}^{k + 1} h_{x}^{l}} K (\frac{y - y_{m + 1}}{h_{y}}) \\ \prod_{j = 1}^{k} K (\frac{y^{k} (j) - y_{m - j + 1}}{h_{y}}) \prod_{j = 1}^{l} K (\frac{x^{l} (j) - x_{m - j + 1}}{h_{x}}) \end{array}

(6)

where Δ = {m / max(k, l) < m ≤ N − 1, |m − n| > τ_c}, and τ_c is the decorrelation time defined as the minimum delay leading to a correlation coefficient equal to 0.1. Parameters h_x, h_y are the respective kernel bandwidths for signals x and y which are normalized. A fixed bandwidth (independent of m) is unable to deal satisfactorily with the tails of the distribution without over smoothing the main part of this distribution. To avoid this issue, two methods help in estimating a density f at a point x :

{\hat{f}}_{1} (x) = \frac{1}{N} \sum_{m = 1}^{N} K_{h_{x}} (x - x_{m}) where h_{x} \equiv h (x)

(7)

{\hat{f}}_{2} (x) = \frac{1}{N} \sum_{m = 1}^{N} K_{h_{m}} (x - x_{m}) where h_{m} \equiv h (x_{m})

(8)

The first method (7) uses a x dependent bandwidth, this bandwidth being unchanged for different points x_m. One example of this method is a KNN estimator [6]. The second method (8) uses a x_m dependent bandwidth, which does not depend on x, leading to the adaptive kernel density estimator (AKDE) [7] we adopted.

B. Adaptive kernel density estimator

AKDE is an improved alternative to FKDE. Given an initial bandwidth h and a first FKDE based estimation ₀, Abramson [8] adapted the bandwidth according to these initial quantities:

h_{m} \propto h_{0} / \sqrt{{\hat{f}}_{0} (x_{m})}

(9)

In [9], Hwang extended this procedure:

h_{m} = {({\hat{f}}_{0} (x_{m}) / g)}^{- r} h_{0}

(10)

where g is the geometric mean of (₀ (x_m))_m, i.e.

log g = \frac{1}{N} \sum_{m = 1}^{N} log {\hat{f}}_{0} (x_{m})

(11)

r is an user defined sensitivity parameter generally satisfying 0 < r < 1. For different dimensions, the value of r should change. However, it is difficult to choose the proper r for four different probability density estimations. We suggest to compute ${\hat{p}}_{YYX} (y_{n + 1}, y_{n}^{k}, x_{n}^{l})$ before its substitution in (4) using (10) and to leave the other densities unchanged. The three steps of the proposed algorithm are as follows:

Step 1: for n_h (10) values v_i of h, 0 < v₁ <..<v_{n_h} ≤3, compute fixed kernel density estimations at $(y_{n + 1}, y_{n}^{k}), (y_{n}^{k}, x_{n}^{l}), y_{n}^{k}$ and (y_n₊₁, $y_{n}^{k}, x_{n}^{l}$ ) ;
Step 2: compute h_m using (10) only to update ${\hat{p}}_{YYX} (y_{n + 1}, y_{n}^{k}, x_{n}^{l})$ leaving the other densities unchanged, and search the set V_h of values v_i leading to unimodality of $f_{v_{i}} : r \in {0 < r_{1}, \dots, r_{20} < 1} \to {\hat{T E}}_{x \to y} (h_{m} (v_{i}, r)) = f_{v_{i}} (r)$ for v_i {v₁,.., v_{n_h}}hence eliminating monotonic curves (no multimodal curve was observed);
Step 3: finally retain the maximum value ${\hat{T E}}_{x \to y} = {max}_{(h, r) \in V_{h} \times] 0; 1 [} (f_{h} (r))$ .

Experimentally, the selected value h_s in the last step is often close to the initial bandwidth h₀. Clearly, the computation time is increased with AKDE (multiplied by 20) for updating the density in step 2.

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III. Experimental Results

We tested our method with Gaussian kernels to compute TE_x_→ _y and TE_y _→_x on two kinds of signals. The first kind included two toy linear AutoRegressive (AR) models and the second one was a realistic EEG model. Predictor dimensions k and l were chosen equal to the corresponding AR models orders estimated by the generalized Bayesian Information Criterion as in [10]. For AR models, the decorrelation time was τ_c=20 and experiments were repeated 200 times on 1024-point signals to get averaged values.

A. Unidirectional linear model

For the first linear stochastic system (model 1), the following two signals were generated:

{\begin{cases} x (n) = 1.3435 x (n - 1) - 0.9025 x (n - 2) + e_{1} (n) \\ y (n) = 0.5 x (n - 3) - 0.4 y (n - 2) + e_{2} (n) \end{cases}

(12)

where e₁ and e₂ were independent white Gaussian noises with zero means and unit variances.

Fig. 1 displays TE computed with a fixed bandwidth h. Experimental transfer entropy ${\hat{T E}}_{x \to y}$ is smaller than the theoretical value which is equal to Granger Causality index divided by 2 (GC/2 was computed from the model coefficients) in the case of Gaussian signals (see Table I). Fig. 2 corresponds to TE values vs. r using AKDE. The flow propagation from signal x to signal y was correctly established whereas the estimated influence from signal y to signal x was not significant. Step 3 of the algorithm led to h_s = 0.45. Comparing Fig. 1 and Fig. 2, TE estimated using AKDE is much closer to the exact value (0.41). We also compared our estimator with Trentool toolbox [5] and concluded to its relevant behavior as seen in Table I which allows to compare the different estimators in terms of mean and standard deviation. It reveals visible improvement in ${\hat{T E}}_{x \to y}$ all other performance with Gaussian AKDE over estimators.

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Figure 1

Results of TE between two time series x and y (model 1) using a Gaussian kernel and a fixed bandwidth h

Solid line: ${\hat{T E}}_{x \to y}$ , dashed line: ${\hat{T E}}_{y \to x}$

Horizontal line: exact value TE_x_→_y=0.41

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Object name is halms881459f2.jpg

Figure 2

Results of TE between two time series x and y (model 1) using a Gaussian kernel and AKDE (10)

Solid line: ${\hat{T E}}_{x \to y}$ , dashed line: ${\hat{T E}}_{y \to x}$

Horizontal line: exact value TE_x_→_y=0.41

TABLE I

Mean values (and standard deviation in parentheses) of transfer entropy using the different estimators on model 1

Estimator	X → Y	Y → X
GC/2	0.4146	0
TE (fixed h)	0.3242 (0.0158)	0 (0)
TE (AKDE)	0.4063 (0.0179)	0.01267(0.0045)
Trentool	0.3484 (0.0115)	−0.0158 (0.0070)

B. Bidirectional linear model

For the second linear stochastic system (model 2), we generated the following signals:

{\begin{matrix} x (n) = 0.5 x (n - 1) + 0.3 y (n - 2) + e_{1} (n) \\ y (n) = 0.5 x (n - 3) - 0.4 y (n - 1) + e_{2} (n) \end{matrix}

(13)

where e₁ and e₂ were as in (12). Fig. 3 and Fig. 4 allow to compare TE values using either a fixed bandwidth (Fig. 3) or AKDE (Fig. 4, h_s = 0.45). For this model, the exact value of TE from signal x to signal y (resp. from signal y to signal x) given in Table II is represented by a solid grey line (resp. a dashed grey line) in Fig. 3 and and4.4. Focusing on Fig. 4, the bidirectional flow propagation was correctly detected using AKDE, the mean values of TE being close to the exact ones (see also Table II). This figure reveals that the bias of AKDE estimator is negligible. As for TE estimated with a fixed bandwidth (Fig. 3 and Table II), its values remain lower than the exact ones, similarly as those estimated with Trentool toolbox. For all estimators tested, the standard deviation is 5 to 10 times lower than the corresponding mean value.

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Figure 3

Results of TE between two time series x and y (model 2) using a Gaussian kernel and a fixed bandwidth h

Solid line: ${\hat{T E}}_{x \to y}$ , dashed line: ${\hat{T E}}_{y \to x}$ , Horizontal lines: exact values

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Object name is halms881459f4.jpg

Figure 4

Results of TE between two time series x and y (model 2) using a Gaussian kernel and AKDE (10)

Solid line: ${\hat{T E}}_{x \to y}$ , dashed line: ${\hat{T E}}_{y \to x}$ , Horizontal lines: exact values

TABLE II

Mean values (and standard deviation in parentheses) of transfer entropy using the different estimators on model 2

Estimator	X → Y	Y → X
GC/2	0.1511	0.0630
TE (fixed h)	0.1118 (0.0123)	0.0422 (0.0083)
TE (AKDE)	0.1457 (0.0133)	0.0689 (0.0087)
Trentool	0.1120 (0.0091)	0.0446 (0.0079)

C. Physiology based model

We simulated EEG signals with a nonlinear SDE (stochastic differential equation) model [11] of order 20 to represent the activities of two neuronal populations Pop_X and Pop_Y:

\begin{array}{l} d [X_{I}^{T} X_{E}^{T}] = g_{X} (X_{I}, X_{E}, θ) d t + {d W}_{X} \\ d [Y_{I}^{T} Y_{E}^{T}] = g_{Y} (Y_{I}, Y_{E}, θ, K_{X Y} Sig (X)) d t + {d W}_{Y} \\ X (t) = L (X_{I} (t), X_{E} (t)), Y (t) = L (Y_{I} (t), Y_{E} (t)) \end{array}

(14)

where the line vectors $X_{I}^{T} X_{E}^{T}$ and $Y_{I}^{T} Y_{E}^{T}$ are in ¹⁰ (T is the transpose operator) and denote dynamical state evolutions for Pop_X and Pop_Y aggregating components []_I and []_E modeling respectively interacting inhibitory and excitatory activities. Parameterized nonlinear functions (u, v) → g (u, v, θ) drive the state evolutions. Sig(.) is a nonlinear sigmoidal function. The coupling parameter K_XY allows effective connectivity adjustment from the first population to the second one. X and Y are computed with a linear function L and are interpretable as two intracranial EEG signals recorded from proximal field electrodes. The independent Brownian processes W_X and W_Y represent random surrounding populations influences. The parameters vector θ includes three scalar components A, B and G, allowing to modify the type of activity (normal/epileptic). The model was time discretized by Euler scheme to produce two discrete time outputs. As in [11], we fixed these parameters to 5, 3, 20 in Pop_X and to 3.5, 3.5, 84 in Pop_Y. This resulted in a narrow band fast activity around 25 Hz (similar to that observed at seizure onset) in each population. K_XY was set to 1500. Fifty blocks of 8-second length signals were simulated with a sampling rate of 256 Hz. In this experiment, as the influence from one physiological signal to another one may be largely delayed, and to get a not too large predictor dimension l, we first shifted signal Y as proposed in [1]. The delay corresponded to 33 sampling time instants and was determined from cross covariance maximization. The maximum order in the model (after shifting) was set to 2 and τ_c was set to 500. According to Fig. 5 and Table III, we conclude to the relevance of the new estimator compared to the “references” given by Granger causality index (GC/2) and Trentool toolbox. As a matter of fact, ${\hat{T E}}_{x \to y}$ and ${\hat{T E}}_{y \to x}$ are sensibly more contrasted (considering means and standard deviations) with Gaussian AKDE than with the two other methods (Table III). Moreover, when comparing the mean values of $\hat{T E}$ obtained for the different methods with this physiological model, a larger dispersion was observed than with previous linear models 1 and 2. The difference between the AKDE based estimator and GC/2 could be expected due to the nonrobustness of Granger index to nonlinearities. On the other hand, the difference between the AKDE based estimator and Trentool estimator (which even failed in detecting the flow direction) was unexpected.

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Figure 5

Results of TE between two delayed simulated EEG signals using a Gaussian kernel and AKDE (10)

Solid line: ${\hat{T E}}_{x \to y}$ , dashed line: ${\hat{T E}}_{y \to x}$

TABLE III

Mean values (and standard deviation in parentheses) of transfer entropy using the different estimators on the physiological model

Estimator	X → Y	Y → X
GC/2	0.011 (0.0193)	0.0028 (0.0009)
TE (AKDE)	0.2521 (0.1143)	0.1249 (0.0615)
Trentool	0.0049 (0.3182)	0.0091 (0.0083)

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IV. Conclusion

In this paper, we focused on information propagation between two observations using TE and introduced an adaptive kernel density estimator to improve fixed kernel TE estimator. Results on simulated AR models revealed a very low bias with AKDE approach and proved the relevance of this new method in detecting uni/bi-directional propagation flows. Using a fixed bandwidth or Trentool approach led to much more biased values. For physiological signals, even if we had no ground-truth, the causal effects were perfectly identified and allowed characterizing the driving system and the responding one. In the future, the AKDE method will be tested on real EEG signals and on more complex scenarios including stronger nonlinearities and/or multivariate observations. A validation phase including statistical hypothesis tests based on surrogate data will complete this work.

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References

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Exploring neural directed interactions with transfer entropy based on an adaptive kernel density estimator.

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Exploring neural directed interactions with transfer entropy based on an adaptive kernel density estimator

Kai Zuo

Jean-Jacques Bellanger

Chunfeng Yang

Huazhong Shu

Regine Le Bouquin Jeannes

Abstract

I. Introduction

II. Methods and Materials

A. Kernel methods

B. Adaptive kernel density estimator

III. Experimental Results

A. Unidirectional linear model

TABLE I

B. Bidirectional linear model

TABLE II

C. Physiology based model

TABLE III

IV. Conclusion

References

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Benchmarking Transfer Entropy Methods for the Study of Linear and Nonlinear Cardio-Respiratory Interactions.

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