A Blind Module Identification Approach for Predicting Effective Connectivity Within Brain Dynamical Subnetworks

Abstract

Model-based network discovery measures, such as the brain effective connectivity, require fitting of generative process models to measurements obtained from key areas across the network. For distributed dynamic phenomena, such as generalized seizures and slow-wave sleep, studying effective connectivity from real-time recordings is significantly complicated since (i) outputs from only a subnetwork can be practically measured, and (ii) exogenous subnetwork inputs are unobservable. Model fitting, therefore, constitutes a challenging blind module identification or model inversion problem for finding both the parameters and the many unknown inputs of the subnetwork. We herein propose a novel estimation framework for identifying nonlinear dynamic subnetworks in the case of slowly-varying, otherwise unknown local inputs. Starting with approximate predictions obtained using Cubature Kalman filtering, residuals of local output predictions are utilized to improve upon local input estimates. The algorithm performance is tested on both simulated and clinical EEG of induced seizures under electroconvulsive therapy (ECT). For the simulated network, the algorithm significantly boosted the estimation accuracy for inputs and connections from noisy EEG. For the clinical data, the algorithm predicted increased subnetwork inputs during the pre-stimulus anesthesia condition. Importantly, it predicted an increased frontocentral connectivity during the generalized seizure that is commensurate with electrode placement and that corroborates the clinical hypothesis of increased frontal focality of therapeutic ECT seizures. The proposed framework can be extended to account for several input configurations and can in principle be applied to study effective connectivity within brain subnetworks defined at the microscale (cortical lamina interaction) or at the macroscale (sensory integration).

Appendix

Neural Population Models

Early efforts by Wilson and Cowan showed that modeling of oscillations in electric potential recordings, such as EEG, can be reasonably linked to the average firing activity of interacting excitatory and inhibitory within a unit area of neural tissue. Later extensions showed lumped parameter representations of alpha rhythms (Rotterdam et al. 1982) and basic cortical column dynamics (Jansen and Rit 1995). The latter model was the basis for the neural population model developed by Wendling in 2001 that accounts rhythm generation as an interaction between the main excitatory pyramidal, the local fast inhibitory cells and the slow inhibitory dendritic targeting cells. In this model, different interaction strengths among the three populations can generate spontaneous normal activity, sporadic spiking and sustained large amplitude low-frequency spikes as seen in seizures (Shayegh et al. 2011).

The basic building block for a single neural population in all of the aforementioned models is a rate-to-current-to-rate transformation. In a given neural population, an intracellular current results from the summed activity of synaptic elements $H_e$, $H_i$ representing excitatory and inhibitory connection of the population. This current is then mapped to an output firing rate by a sigmoidal transformation. A synapse $H_l$, $l\in \{e,i\}$ is thought of as a second order linear system whose impulse response follows a double-exponential (or alpha) function $h_l(t)=A_la_lte^{-a_lt}$, where $a_l$ is the time constant and $A_l$ is an amplitude constant for that synapse (Fig. 16). Accordingly, for a given input rate $r_l(t)$ arriving at this synapse, the corresponding postsynaptic current $x_l(t)$ can be computed using a two-state linear system by introducing an intermediate state z(t) in

Fig. 16

Basic building blocks for a neuronal model. a excitatory and inhibitory blocks. b impulse responses of the various synapses

$$\begin{aligned} \dot{x_l}(t)= & {} z(t) \nonumber \\ \dot{z}(t)= & {} A_la_lr_l(t) -2a_lz(t)-a_l^2x_l(t) \end{aligned}$$

(45)

Finally, the population output is the firing rate which is produced after scaling and passing all synaptic currents $x(t)=\sum _l C_lx_l(t)$ through a static sigmoidal nonlinearity $r(t)=S\{x(t) \}$, $S\{x\}=\frac{2e}{1+e^{k(x_o-x)}}$.

As one of the earliest neural population models to simulate seizures, the Jansen model could reproduce basic sustained spike discharges but the fast EEG associated with low-voltage rapid discharges. To accommodate the latter, Wendling added to the slow inhibition, present in the Jansen model, a second class of inhibitory interneurons with faster dynamics, namely the somatic targeting fast interneurons (FIN) with fast GABAergic synapsis (Wendling et al. 2002).

Induced Seizure Neuronal models

The Wendling model became hence a popular tool to investigate spontaneous seizure genesis in pathological tissue. Recently, our group has demonstrated that the Wendling model can be readily modified to account for seizure induction in models of normal tissue by external stimuli such as those applied during ECT (Karameh et al. 2014).

Spontaneous Seizure Model

In the basic Wendling model (Wendling et al. 2002), a local area is represented by a primary pyramidal cell population that connects with three distinct interneuron populations that are common across cortical structures. As shown in Fig. 17, the pyramidal cell population (PYR) are reciprocally connected to (i) a local population of excitatory interneurons (EIN) that provide excitatory feedback to the pyramidal cells (ii) a population of fast inhibitory interneurons (FIN) that provides fast GABA-A somatic inhibition to PYR, and finally, (iii) a population of slow inhibitory interneurons (SIN) that provide slow GABA-A dendritic inhibition to the PYR as well as inhibit the FIN population. Starting with the above formulation of single synapses, the local area is represented by the following set of non-linear state-space equations:

$$\begin{aligned} \dot{x}_{p}(t)= & {} y_1(t) \end{aligned}$$

(46)

$$\begin{aligned} \dot{y}_1(t)= & {} AaS\{x_{e}(t)-x_{is}-x_{if}(t)\}-2ay_1(t)-a^2x_p(t) \nonumber \\ \dot{x}_{e}(t)= & {} y_2(t) \end{aligned}$$

(47)

$$\begin{aligned} \dot{y}_2(t)= & {} Aa\left[ u_0(t)+c_{ep}S\{c_{pe}x_p(t)\}\right] -2ay_2(t)-a^2x_{e}(t) \nonumber \\ \dot{x}_{is}(t)= & {} y_3(t) \end{aligned}$$

(48)

$$\begin{aligned} \dot{y}_3(t)= & {} Bbd_{sp}S\{c_{ps}x_p(t)\}-2by_3(t)-b^2x_{is}(t)\nonumber \\ \dot{x}_{if}(t)= & {} y_4(t) \end{aligned}$$

(49)

$$\begin{aligned} \dot{y}_4(t)= & {} Ggd_{fp}S\{c_{pf}x_p(t)-d_{sf}x_{ii}(t) \} )-2gy_4(t)-g^2 x_{if} \nonumber \\ \dot{x}_{ii}(t)= & {} y_5(t) \end{aligned}$$

(50)

$$\begin{aligned} \dot{y}_5(t)= & {} Bb S\{c_{ps}x_p(t)\}-2by_5(t)-b^2x_{ii}(t) \end{aligned}$$

(51)

$$\begin{aligned} z(t)= & {} K_{iv}\left( x_{e}(t)-x_{is}-x_{if}(t)\right) \end{aligned}$$

(52)

Fig. 17

Local Area model of a neuronal population that can be driven into an induced seizure by external inputs. Model is modified from (Wendling et al. 2002) to include the fast-to-slow inhibitory connection $d_{fs}$ (in red)

where $x_p(t)$ is the synaptic current output from pyramidal cell population to all local cells; $x_e$, $x_{is}$, and $x_{if}$ are, respectively, the excitatory, the slow GABA-A inhibitory input and the fast GABA-A inhibitory inputs to the pyramidal cells ); $x_{ii}$ is the slow inhibition from SIN to FIN population. The constants (A, a), (B, b) and (G, g) dictate the time profiles of postsynaptic currents associated with connections to EIN, FIN, SIN populations, respectively. $u_0(t)$ is an external input assumed to arrive as a firing rate to the local pyramidal population. The output of the model is the voltage trace z(t) which is proportional to the over all postsynaptic current in the pyramidal cell population and is assumed to be representative of the EEG traces of the overall local area. Finally, the constants $d_{m,n}$ and $c_{m,n}$ denote the excitatory and inhibitory connection strengths, respectively, originating from population m to population n and $(m,n) \in \{ \text{ p: } \text{ PYR, } \text{ e: } \text{ EIN, } \text{ f: } \text{ FIN, } \text{ s } \text{ SIN } \}$. For concise reference, the above continuous time state-space nonlinear system can written as

$$\begin{aligned} {\dot{{\mathbf x}}} (t)= & {} \tilde{f} \left( \mathbf {{x}} (t), u(t)\right) \nonumber \\ {z}(t)= & {} \tilde{h}\left( \mathbf {{x}} (t)\right) \end{aligned}$$

(53)

where $\mathbf {x} (t) \in {\mathbb R}^{10}$ are the firing states of this system, $\tilde{f}$ is the process nonlinear function, and $\tilde{h}$ is the measurement function. The above model produces, for a given set of fixed connections parameters, an oscillatory behavior that change little in character regardless of the applied input $u_o(t)$. That is, the dynamics of the nonlinear system exhibit single stable solutions for a given parameter set. For example, a change in the balance of slow-to-fast connection parameters can move the oscillations from normal to sporadic spikes and into sustained slow quasi-sinusoidal epileptic activity. Accordingly, the model could produce spontaneous seizures caused by pathology (functional imbalance) in the local population, but not induced seizures which are initiated by external inputs, such as those occurring under ECT.

Electrically-Induced Seizure Model

To accommodate the induction of seizures in normal neural population models, we have previously introduced a simple modification to the basic Wendling model that is physiological in nature and that allows for the normal activity to switch to seizure activity when subjected to large external stimuli without changing internal model parameters (Karameh et al. 2014), as shown in Fig. 18. The basic idea is rather simple: specifically, normal levels of dendritic inhibition act to balance the strength of excitatory afferents on the local pyramidal populations while reduce inhibition (or disinhibition) in this region will cause a runaway excitation. Several experimental lines of evidence suggest that impaired dendritic inhibition may play in important role in spontaneously occurring seizures (Wendling et al. 2005). Particularly, fast inhibition show significant firing increase (Curtis and Gnatkovsky 2009) and exhibit depolarization block (Cammarota et al. 2013) just prior to seizure onset, and the collapse of the inhibitory control lead to runaway pyramidal excitation (Trevelyan and Schevon 2013). The reciprocal connections between slow dendritic targeting interneurons and fast interneurons have also been reported in the middle (Beierlein et al. 2003) and upper cortical layers (Cruikshank et al. 2012; Markram et al. 2004), show preference in interneuron targeting (Pfeffer et al. 2013) with slow GABA-A inhibitory cells targeted by fast spiking cells (Staiger et al. 1997; Tamás et al. 2004; Palmer et al. 2012). We have therefore proposed the existence of an inhibitory pathway from the fast spiking FIN interneurons to the SIN populations that can under large external stimuli act to inhibit the SIN population, thereby creating brief dendritic disinhibition in the PYR population.

Accordingly, the Wendling model for a single area is modified so as to include a fast inhibitory to slow inhibitory neuron connection (Fig. 17, red line). The corresponding nonlinear state-space equations for the SIN input synapse becomes

$$\begin{aligned} \begin{aligned} \dot{x}_{is}(t)&= y_3(t) \\ \dot{y}_3(t)&= Bbd_{sp}S\{c_{ps}x_p(t)-d_{fs}x_{if}(t)\}-2by_3(t)-b^2x_{is}(t) \end{aligned} \end{aligned}$$

(54)

Fig. 18

Various EEG patterns produced by the multi-area model. a Background activity. b Sporadic spikes. c Sustained slow oscillations. d Seizures [for details, cf. (Karameh et al. 2014)]

When subjected to increasing levels of external stimulation, the modified Wendling (MW) model for a single area switched from a baseline normal background activity first into sporadic spikes and then into continuous low-frequency spikes similar to those observed in ECT induced seizures. The MW model was also utilized in a multi-area simulation where the interareal connections emanate from pyramidal cells of one area and contact both the pyramidal and FIN population of the other areas, as seen in Fig. 2). For a model of two interconnected areas and with only one area subjected to external stimulus, it was noted that the threshold for seizure induction in that area is now lowered and that the ensuing seizure also propagates to the other areas even though that area was not under the influence of the stimulus. Thus the MW model accounts for both the initiation and the propagation of electrically-induced seizures in normal tissue (see Karameh et al. 2014) for details).

Induced Seizure in a Network

For a total of M distinct EEG channels recordings, a network with M distinct blocks is constructed. Between any two channels, reciprocal connections are made via long-range excitatory synapses to the local pyramidal PYR population and its associated fast interneuron population FIN (as in Fig. 2). We also assume that long range connections from area q to area r incur a fixed propagation delay $\tau _{qr}$. This shows as an extra synaptic input in the corresponding equations (Eqs. 47, (49) of the local area Modified Wendling equations as follows

$$\begin{aligned} \dot{y}^{(r)}_2(t)= & {} Aa\left[ u_r(t)+c_{ep}S\left\{ c_{pe}x^{(r)}_p(t)+ \sum _{{\mathop {q\ne r}\limits ^{q=1}}}^M K_{qr}x^{(q)}_{p}(t-\tau _{qr})\right\} \right] \nonumber \\&-2ay^{(r)}_2(t)-a^2x^{(r)}_{e}(t) \end{aligned}$$

(55)

$$\begin{aligned} \dot{y}^{(r)}_4(t)= & {} Ggd_{fp}S\left\{ c_{pf}x^{(r)}_p(t)+ \sum _{{\mathop {q\ne r}\limits ^{q=1}}}^M L_{qr}x^{(q)}_{p}(t-\tau _{qr}) -d_{sf}x^{(r)}_{ii}(t)\right\} \nonumber \\&-2gy^{(r)}_4(t)-g^2 x^{(r)}_{if}(t) \end{aligned}$$

(56)

where $x^{(q)}_{p}(t)$ is the output from the pyramidal cells in area q. $K_{qr}$ and $L_{qr}$ are the connections from pyramidals in area q to the pyramidal and fast inhibitory cells in area r, respectively.

Kalman Filters for System Modeling

From a systems point of view, physiological recordings, such as EEG e(t) in Eq. 52, can often be thought of imperfect measurements or noisy observations related to an underlying dynamical system. The dynamical system response, in turn, can be decomposed into constituent internal states that are either physical in nature (e.g. neural firing rates $x_e(t)$ in Eq. 47) or mathematical entities that allow determining the time trajectory of the system output. In what follows, we will briefly outline Kalman filtering approaches to estimate the internal states of a system that fits the temporal dynamics of the observations.

State Estimation

We here consider the discrete-time state space description of a dynamical system that has the following form

$$\begin{aligned} \begin{aligned} \text {Process equation: } {\mathbf x}_{k}&= f({\mathbf x}_{k-1},{\mathbf u}_{k-1})+{\varvec{\mu }}_{k-1} \\ \text {Measurement equation: }{\mathbf z}_k&= h({\mathbf x}_{k},{\mathbf u}_{k})+{\varvec{\eta }}_{k} \end{aligned} \end{aligned}$$

(57)

where ${\mathbf x}_k \in \mathbb {R}^{n}$ is the n dimensional state vector at discrete time k; $f: \mathbb {R}^{n} \times \mathbb {R}^{n_u} \rightarrow \mathbb {R}^{n}$ is a known function that describes that states as a function of their history and of a control input ${\mathbf u}_k \in \mathbb {R}^{n_u}$; $h: \mathbb {R}^{n} \times \mathbb {R}^{n_u} \rightarrow \mathbb {R}^{n_z}$ is an known observation function that relates the measurements ${\mathbf z}_k \in \mathbb {R}^{n_z}$ to the hidden states of the system; finally, ${\varvec{\mu }}_{k}$ and ${\varvec{\eta }}_k$ are additive independent process and observation Gaussian-distributed noise sequences with zero means and covariances ${\mathbf Q}_{k}$ and ${\mathbf R}_k$ respectively. In the case when both h and f are known, finding the hidden states based on the noise observations is an estimation problem which will considered next. In the case when at least one of the functions h and f is partially unknown, finding the hidden states and the model unknowns is a model learning problem which will be considered after introducing the cubature Kalman filters.

Strictly speaking, however, a majority of the practical physiological systems are nonlinear (in the problem at hand, for example, the sigmoidal functions introduce nonlinearity). This implies the existence of a nonlinearity in either (or both) of the functions f and h. and hence a nonlinear transformation of the Gaussian priors in the above integrals, thus obfuscating the existence of simple general solutions. To overcome this problem, the extended Kalman filter (EKF) provides an approximate solution by linearizing the nonlinear dynamics $f(\mathbf x_{k-1}, \mathbf u_{k_1})$ locally around an estimated state by a linearized system, which is limited by the accuracy of this linearization. The unscented Kalman filter (UKF) provides an approximate solution to the integral(s) as a weighted summations of the nonlinearly transformed set of sigma points (so called unscented transformation). The UKF had earlier been a popular choice to solve these problems but suffered from numerical instabilities, which were mitigated by the Cubature Kalman filter (CKF), described next.

Developed by Arasaratnam and Haykin in 2009 (Arasaratnam and Haykin 2009), the Cubature Kalman Filter (CKF) approximated the integrals of the form (nonlinear $\times$ Gaussian) for n-dimensional vectors $\mathbf x\in \mathbb {R}^n$ using third degree spherical cubature rule as

$$\begin{aligned} \int f(\mathbf x)\times {\mathscr {N}}(\mathbf x;\mu ,\Sigma ) d\mathbf x \approx \frac{1}{2n} \sum _{i=1}^{2n} f(\mathbf X_i) \end{aligned}$$

(58)

where $\mathbf X_i$ are the set of 2n sigma points that symmetrically sample the distribution at locations $\mathbf X_i= \mu +\xi _i \sqrt{\Sigma }$, and the spread factor $\xi _i= \sqrt{n} e_i$, $i=1\dots n$ and $\xi _i= -\sqrt{n} e_i$, $i=n+1\dots 2n$ with $e_i$ as a unit vector for the $ith$ dimension. With the time and measurement update steps involve approximate computations of covariance matrices, numerical inaccuracies might still incorrectly produce matrices that are not positive semidefinite leading to algorithm divergence. This however does not occur in a variant of the CKF, the Square root Cubature Kalman filter (SCKF), which instead computes and propagates the square root of the covariance matrices. The SCKF will be used here and a brief description of the algorithm is given in the appendix.

Joint Estimation

The standard Kalman setup in Eq. (57) estimates the hidden states for known control inputs $\mathbf {u}_k$ and for known process and observation functions f(.) and h(.), respectively. In the case where the input is unknown, or the functions are partially unknown and parameterizable by an unknown parameter vector $\varvec{\theta }\in \mathbb R ^{n_\theta }$ (i.e. $f\left( \mathbf x_{k-1},\mathbf u_{k-1}\right) =f\left( \mathbf x_{k-1},\mathbf \theta _{k-1}, \mathbf u_{k-1}\right)$), Kalman filters allow to augment the state vector to search for $\mathbf {u_k}$ and $\varvec{\theta }$ as part of the hidden states by commonly assuming these to follow random walk processes, leading to the following overall state-space system representation

$$\begin{aligned} \mathbf {u}_{k}= & {} \mathbf {u}_{k-1}+\mathbf {w}_{k-1} \end{aligned}$$

(59)

$$\begin{aligned} \varvec{\theta }_{k}= & {} \varvec{\theta }_{k-1}+\varvec{v}_{k-1} \end{aligned}$$

(60)

$$\begin{aligned} \mathbf x_k= & {} f\left( \mathbf x_{k-1},\varvec{\theta }_{k-1},\mathbf {u}_{k-1} \right) +\varvec{\mu }_{k-1} \end{aligned}$$

(61)

$$\begin{aligned} {\mathbf z}_k= & {} h({\mathbf x}_{k},{\mathbf u}_{k})+{\varvec{\eta }}_{k} \end{aligned}$$

(62)

where $\mathbf {w}_{k-1}$ and $\varvec{v}_{k-1}$ are independent Gaussian-distributed sequences with zero mean and covariances $\varvec{\Lambda }_{W,k}$ and $\varvec{\Lambda }_{V,k}$, respectively. The equations above constitute joint-estimation problem for an augmented hidden state vector $\left[ \mathbf {u}^\top _k \varvec{\theta }^\top _k \mathbf x^\top _k \right] ^\top \in \mathbb R^{(n_u+n_\theta +n)}$ thereby allowing the SCKF to estimate time-varying control inputs and, particularly, time varying parameters which is a very desirable property in the case where the nonlinearity is not fixed in time.

The SCKF Algorithm

The main SCKF algorithm is presented next, where all of the steps can be deduced directly from the CKF except for the update of the posterior error covariance.

Time Update

• 1) Evaluate the cubature points $(i=1,2,{\ldots}m)$

  $$\begin{aligned} \mathbf {X}_{i,\,k-1|k-1} = \mathbf {S}_{k-1|k-1}\varvec{\xi }_i+\hat{\mathbf x}_{k-1} \end{aligned}$$

  (63)

  where $m=2n_x$

• 2) Evaluate the propagated cubature points $(i=1,2,{\ldots}m)$

  $$\begin{aligned} \mathbf {X}_{i,\,k|k-1}^* = f(\mathbf X_{i,\,k-1|k-1},\mathbf u_{k-1}) \end{aligned}$$

  (64)

• 3) Estimate the predicted state

  $$\begin{aligned} \hat{\mathbf x}_{k|k-1} = \frac{1}{m}\sum _{i=1}^m \mathbf X_{i,\,k|k-1}^* \end{aligned}$$

  (65)

• 4) Estimate the square-root factor of the predicted error covariance

  $$\begin{aligned} \mathbf S_{k|k-1} = \mathbf {Tria} ([\mathscr {X} _{k|k-1}^*\quad \mathbf S_{Q,\,k-1}]) \end{aligned}$$

  (66)

  where $\mathbf S_{Q,\,k-1}$ denotes a square-root factor of $\mathbf Q_{k-1}$ such that $\mathbf Q_{k-1}=\mathbf {S}_{Q,\,k-1} \mathbf {S}_{Q,\,k-1}^T$ and the weighted centered(prior mean is subtracted off) matrix

  $$\begin{aligned} \begin{aligned} \mathscr {X}_{k|k-1}^* = \frac{1}{\sqrt{m}}&[\mathbf {X}_{1,\,k|k-1}^*-\hat{\mathbf x}_{k|k-1} \quad \mathbf {X}_{2,\,k|k-1}^* -\hat{\mathbf x}_{k|k-1} \\&\ldots\mathbf {X}_{m,\,k|k-1}^*-\hat{\mathbf x}_{k|k-1}] \end{aligned} \end{aligned}$$

  (67)

Measurement Update

• 1) Evaluate the cubature points $(i=1,2,..m)$

  $$\begin{aligned} \mathbf X_{i,\,k|k-1} = \mathbf {S}_{k|k-1}\varvec{\xi }_i+\hat{\mathbf x}_{k|k-1} \end{aligned}$$

  (68)

• 2) Evaluate the propagated cubature points $(i=1,2,..m)$

  $$\begin{aligned} \mathbf {Z}_{i,\,k|k-1} = h(\mathbf X_{i,\,k|k-1},\mathbf u_k) \end{aligned}$$

  (69)

• 3) Estimate the predicted measurement

  $$\begin{aligned} \hat{\mathbf z}_{k} = \frac{1}{m}\sum _{i=1}^m \mathbf {Z}_{i,\,k|k-1} \end{aligned}$$

  (70)

• 4) Estimate the square-root of the innovation covariance matrix

  $$\begin{aligned} \mathbf {S}_{zz,\,k|k-1} = \mathbf {Tria} ([\mathscr {\mathbf Z}_{k|k-1}^*\quad \mathbf {S}_{R,\,k}]) \end{aligned}$$

  (71)

  where $\mathbf {S}_{R,\,k}$ denotes a square-root factor of $\mathbf {R}_k$ such that $\mathbf R_k=\mathbf {S}_{R,\,k} \mathbf {S}_{R,\,k}^T$ and the weighted centered matrix

  $$\begin{aligned} \mathscr {Z}_{k|k-1} = \frac{1}{\sqrt{m}}[\mathbf Z_{1,\,k|k-1}\hat{\mathbf z}_{k} \quad \mathbf Z_{2,\,k|k-1}-\hat{\mathbf z}_{k} ...\, \mathbf {Z}_{m,\,k|k-1}-\hat{\mathbf z}_{k}] \end{aligned}$$

  (72)

• 5) Estimate the cross-covariance matrix

  $$\begin{aligned} \mathbf {P}_{xz,\,k|k-1} = \mathscr {X}_{k|k-1}\mathscr {Z}_{k|k-1}^T \end{aligned}$$

  (73)

  where the weighted, centered matrix

  $$\begin{aligned} \begin{aligned} \mathscr {X}_{k|k-1} = \frac{1}{\sqrt{m}}&\left[ \mathbf {X}_{1,\,k|k-1}-\hat{\mathbf x}_{k|k-1} \quad \mathbf {X}_{2,\,k|k-1}-\hat{\mathbf x}_{k|k-1} \right. \\&...\, \left. \mathbf {X}_{m,\,k|k-1}-\hat{\mathbf x}_{k|k-1}\right] \end{aligned} \end{aligned}$$

  (74)

• 6) Estimate the Kalman gain

  $$\begin{aligned} \mathbf {G}_k = (\mathbf {P}_{xz,\,k|k-1}/\mathbf {S}_{zz,\,k|k-1}^T)/\mathbf {S}_{zz,\,k|k-1} \end{aligned}$$

  (75)

• 7) Estimate the updated state

  $$\begin{aligned} \hat{\mathbf x}_{k} = \hat{\mathbf {x}}_{k|k-1}+\mathbf {G}_k(\mathbf {z}_k-\hat{\mathbf {z}}_{k}) \end{aligned}$$

  (76)

• 8) Estimate the square-root factor of the corresponding error covariance

  $$\begin{aligned} \mathbf {S}_{k|k} = \mathbf {Tria} ([\mathscr {X}_{k|k-1} \,-\, \mathbf {G}_k \mathscr {Z}_{k|k-1} \quad \mathbf {G}_k\mathbf {S}_{R,\,k}]) \end{aligned}$$

  (77)

Generalization of the RA-SCKF Algorithm

It is noted that the developed procedure shares some properties with two-stage approaches common in linear system identification, namely the instrumental variables and the two-step least squares (Ljung 1999). These approaches are utilized whenever the residual (prediction error) of a linear regression (the first stage) is correlated with the regressors (including inputs) leading to biased parameters estimates. Starting with these biased estimates, the regressors for the second stage are adjusted to remove any correlation with the residual and hence reduce biases. By analogy, upon dissecting the error of the first stage SCKF into channel-specific components, inverting the linearized inout/output map to find a desired deviation signal, and then finding its linear predictability from the system input, the algorithm incorporates a gain corresponding to the input-correlated residual into the estimation structure. This allowing the second stage SCKF (the residual adjustment) to further reduce the bias in the estimates.

The presented mapping of the independent components is based on the assumption that all the channels are driven by distinct inputs. Here, a one-to-one mapping between the ICA-extracted components and the channel-specific errors was formed. In principle, this procedure can be modified to account for any segmentation or clustering of the channel-specific inputs to common or distinct sets of inputs. To find this segmentation, it is best to utilize a priori knowledge of common pathways that drive separate channels. Still, in the absence of such knowledge, it is also plausible to create different candidate mappings of the ICA-extracted components onto channel-specific errors. For example, a channel-specific error $e_c$ can be formed from every independent component j to each channel i (20, $\mathscr {P}$ is all combinations), along with the corresponding desired input deviation (Eq. 34) and the corresponding linear predictability (Eq. 37). A mapping from j to i is to be created only if such predictability exceeds a predefined threshold. The channel-specific error can then be constructed incrementally starting with components with the next highest predictability. A summary of the proposed algorithm is shown below. Subsequently, after all the mapping are formed, a residual-adjustment SCKF run is conducted for the candidate segmentation and its log-likelihood is computed. Out of all the segmentations, the winner is selected based on the largest log likelihood.

Supplementary Connectivity Tables

See (Table 3).

Table 3 Average absolute interconnection strengths for ictal activity under (a) normal and (b) reverse configurations for Subject 20 (self connections are fixed, and numbers in parentheses are percentage change from normal)