Ethics statement

The study was approved by the Riverside Research Ethics Committee (12/LO/2005) and all participants signed a written informed consent form prior to the study, according to the declaration of Helsinki (2013).

The generalised neuronal state equation

Causal models can be considered as the evolution of states x as a static function of themselves. For instance, for the i^th region: (1) where f is a nonlinear function, v are external inputs, and θ are model parameters–usually interpreted in terms of connectivity or rate constants. DCM extends the assumptions in Eq (1) by considering the ways in which states change with respect to time, so that: (2) which reduces to Eq (1) in the limit that inputs v vary slowly relative to the states x.

Extrapolating the logical progression from Eq (1) to Eq (2), we can define a generalised neuronal state equation in which states may vary, not just with respect to time as in Eq (2), but with respect to any arbitrary number of dimensions μ, such that: (3)

For example, if we consider the simple three-node network in Fig 1.

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Fig 1. Three-node network.

The structural connection between nodes 1 and 2 is given by s₁₂ and the structural connection between nodes 1 and 3 is given by s₁₃.

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We describe the evolution of the first node using Eq (3) as follows: (4)

We then retain time as the first dimension (μ₁ = t) on the left-hand side and assign the remaining two terms to the rate of change of states along the structural dimensions of the network, such that: (5) where σ is a (rate) constant of proportionality with dimensions T⁻¹; and e.g. describes the rate of change of the first node’s activity along the edge connecting the first and second nodes. Intuitively, this kind of extension can be thought of as generalising the dynamics of any one point in the brain to a partial differential equation that models the spatiotemporal dynamics (e.g., neuronal field models). However, here, instead of three spatial dimensions, we assign one dimension to each inbound edge for every node within the graph.

We write the general form of Eq (5) as follows for the i^th node in a network of N regions: (6) where k_ij is the adjacency matrix element connecting the i^th and j^th regions, the inclusion of which ensures that only nearest neighbours in the network are taken into account. Note that in the context of neural systems, the definition of nearest neighbours may also be taken to mean regions that are structurally connected via white matter tracts. This means that any given region or node can have more neighbours than if we were modelling the cortical sheet as a two-dimensional Markov field.

The structural gradients in Eq (6) can be discretized by expressing them in terms of the difference in state values (at a given time) at the i^th and j^th nodes, such that: (7) which, together with Eq (6), tells us that: (8) where we see that, in the limit of zero structural gradients (x_i = x_j), the second term on the right-hand side vanishes and Eq (8) reduces to the original DCM neuronal state equation in Eq (2). In other words, every node behaves in the same way and can be described with an ordinary differential equation. In summary, we obtain a third level of neuronal state equations building on the simple causal models in Eq (1), through the classical DCM in Eq (2), and ending with the generalised form in Eq (8), with each level reducing to the former in the appropriate limiting cases.

DCM and the Fokker-Planck equation

The second term on the right-hand side of Eq (8) can be re-written as follows: (9) where d_i is the degree of the i^th node; δ_ij is the Kronecker delta function, which equals unity when i = j, and otherwise equals zero; and l_ij is the graph Laplacian matrix element connecting the i^th and j^th nodes, where the graph Laplacian matrix L is defined as the difference between the degree matrix D and the adjacency matrix K, such that L = D−K.

Using Eq (9), we can write Eq (8) as follows: (10) and as the graph Laplacian is the discretized version of the Laplace operator [16], the second term on the right-hand side describes a diffusion process, hence lending an interpretation to σ as a diffusion coefficient. Therefore, Eq (10) takes the form of the discretized Fokker-Planck equation, with a drift term: f(x,v,θ) and a diffusion term: .

Linear stability analysis

In order to model neural time series, we assume first order interactions, which means Eq (8) can be written as: (11) where the matrix element p_ij reduces to the DCM intrinsic coupling matrix element a_ij in the limiting case of zero structural gradients (x_i = x_j); the matrix element q_ji reduces to the DCM extrinsic coupling matrix element c_ji in the limiting case of zero structural gradients (x_i = x_j); and ω⁽ⁱ⁾ are non-Markovian fluctuations in the i^th region’s activity [17]. These fluctuations model deviations from the linear flow of states under an adiabatic approximation. In other words, we assume a centre manifold for the dynamics, which are linear and assign fluctuations tangential to the manifold to ω⁽ⁱ⁾, which decay rapidly and return to the manifold under the centre manifold theorem [18].

The stability of a linear time invariant (LTI) system [19] is not dependent upon external driving inputs, which means that these do not affect system stability or resilience to perturbations. Therefore, if we wish to determine the long-term stability of a dynamical system, we need only to consider its Jacobian. Therefore, we can re-write Eq (11) by retaining only the first two (stability-relevant) terms, such that: (12) which we can write out explicitly for the three-node network in Fig 1 as follows: (13)

The Jacobian of this generalised DCM can thus be written as follows for a network comprising N regions: (14)

Diffusion and stability

In 1953, Turing showed that an initially stable dynamical system can be rendered unstable by virtue of a diffusion mechanism, allowing for the emergence of spatial inhomogeneities–now known as Turing patterns [20]. Here, we proceed via similar logic, except we begin with the opposite premise and ask the following question: can we push an initially unstable system (such as a neural system prone to seizures) into a stable regime by altering the system’s diffusivity?

To answer this, we multiply the diffusion coefficient σ by a constant α: (15)

By multiplying σ by some unknown quantity α in this way we can determine–in the subsequent linear stability analysis–whether this change should act in a way as to increase or decrease diffusivity in order to render an initially unstable system stable.

If the system in Eq (14) is initially unstable then at least one eigenvalue has a positive Real component. This is guaranteed to be the case when the trace (the sum of the eigenvalues) is positive: (16)

We then transform Eq (14) via Eq (15) to obtain: (17) which has been rendered stable due to the altered diffusion coefficient.

Stability implies that all the eigenvalues now have negative Real components, which in turn means that the trace must now be negative: (18) which, together with Eq (16), means that: (19)

We then know from Eq [16] that trJ>0. Furthermore, we know that diffusion coefficients are necessarily positive, given that they play the role of rate constants [21] and also that the adjacency matrix elements k are either 0 or 1. Therefore, the only way in which Eq (19) can be satisfied is if: (20) i.e. an initially unstable system can be rendered stable by virtue of a reduction in the diffusion coefficient.

Diffusivity can be altered via external driving inputs

In practice it is not possible to change the diffusion coefficient as in Eq (15), due to the fact that it is an intrinsic property of the dynamical system described by Eq (14). However, as the diffusion coefficient quantifies diffusivity via Fick’s law [22], we recover an important piece of information from Eq (20); namely, that if we want to push a system toward stability, we must act so as to decrease diffusivity.

If we look at the governing equation of motion [11], we note that the diffusion term comprises three factors: 1) the diffusion coefficient σ, which we noted above cannot be changed; 2) the adjacency matrix element k_ij which, similar to σ, is intrinsic to the system and is therefore also unchangeable, without resorting to severing connections e.g. via surgical intervention; and finally 3) the gradient (x_i−x_j), which is the only factor that can be decreased by applying external driving inputs in a way as to force x_j to mirror the activity profile of x_i as closely as possible–hence decreasing diffusivity.

Note that this discovery, i.e. that the gradients are the only alterable aspect of the equation of motion, follows directly from the generalisation of the DCM neural state equation. It should also be noted that manipulating the gradients in this way has no effect on the eigenvalues of the Jacobian and thus no effect on the intrinsic stability of the system. Instead, what we are proposing is that it should in practice be possible to induce a form of temporary ‘pseudo stability’ in strategic locations via externally forced gradient minimization.

Let us now consider the same three-node system shown in Fig 1 and assume that the bottom node is prone to instabilities (Fig 2A). In the case of epilepsy, we therefore propose a form of intervention in which we apply external driving input (Fig 2B) in such a way as to allow the two neighbouring nodes (Fig 2C) to mirror the activity in the region that is prone to seizures.

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Fig 2. Gradient reduction via stimulation.

A) The seizure-prone node. This displays the activity shown in the red graph therein. B) External driving input is applied in such a way as to allow the two neighbouring nodes to mirror the activity of the seizure-prone node in A). C) Extrinsic coupling to the two regions neighbouring A).

https://doi.org/10.1371/journal.pcbi.1008448.g002

In summary, we have derived a generalised DCM of neuronal activity that can generate empirical timeseries. We now proceed to estimate the parameters of the DCM using empirical timeseries from epilepsy patients. Equipped with these parameters, we can then evaluate the Jacobian and simulate the effects of an intervention that moves the ensemble or population dynamics implicit in Eq (11) from a regime of instability (i.e., seizure activity) to one of stability.

The seizure network

In all analyses presented here, we use a network comprising the following regions: frontal mid, frontal mid orbital, precuneus, and thalamus. This network is known to play an important role in generalised spike and wave (GSW) discharges [23] and, for simplicity, we will refer to it henceforth as the ‘seizure network’.

Participants and data acquisition

We analyse data recorded from 15 patients (6 male) with juvenile myoclonic epilepsy (JME) with a mean age of 24.5 years and no structural differences found between patients. In addition, we analyse data from 15 age-matched healthy controls (5 male) with a mean age of 25.2 years (see S1 Table). The data were acquired at the Institute of Psychiatry Psychology and Neuroscience (IoPPN), King’s College London. The patients did not have any neurological diagnoses other than epilepsy and had no history of drug or alcohol misuse. All participants underwent a resting state simultaneous EEG-fMRI and a diffusion tensor imaging (DTI) session, with 32 directions and b = 1500 s/mm². We do not obtain directional information regarding structural connectivity via DTI and subsequent adjacency matrices are therefore symmetric. A 3T scanner (MR750, GE Healthcare) was used to acquire 300 echo-planer images (3.3×3.3×3.3 mm, field of view 211 mm, repetition time 2.16 s, echo time 25 ms, flip angle 75°, 36 slices, slice thickness 2.5 mm).

Preprocessing

To preprocess the fMRI data, we use the statistical parametric mapping (SPM8, r613) software running on MATLAB (R2016b), together with the FIACH [24] package for R (3.2.2). First, we convert the data from DICOM to Nifti formats. We then delete the first four volumes of each session to avoid magnetic saturation effects. We subsequently re-align all images to the first remaining volume. To correct for possible artefacts, we apply the FIACH toolbox to the BOLD time series. We then normalise all data into standard MNI space with 2 mm isotropic voxels. All images are then spatially smoothed using a Gaussian filter of 8 mm fullwidth at half maximum. The BOLD signal is filtered between 0.04–0.07 Hz [25]. This frequency range is chosen to minimise the overlap between the BOLD signal and possible breathing and pulsation artefacts. We parcellate the brain into standard 90 automated anatomical labelling (AAL) regions (excluding the cerebellum) [26]. Finally, we apply principal component analysis (PCA) to the voxel time series within each region, from which we retain the first principal component to summarise the activity in each region [27].

Probabilistic tractography is used to preprocess DTI data using the iFOD2 algorithm within the MRtrix software [28]. Streamlines are filtered using SIFT [29], resulting in 10⁷ streamlines. We estimate a 90×90 structural connectivity matrix according to the number of streamlines connecting each pair of regions, normalised by their combined volumes. To reduce inter-subject variability, we then normalise each structural connectivity matrix relative to its maximum value. For each subject, we binarize the structural connectivity matrix according to 18 thresholds between 10% and 95% (in steps of 5%). Equipped with the structural measures k_ij and the summaries of regional activity in the seizure network, we estimate the latent states x_i and parameters of the DCM.

Bayesian model inversion

To create a DCM of observable timeseries, we can use Eq (11) as a state space model with fast (analytic) fluctuations ω⁽ⁱ⁾ and map the latent states x_i to observable quantities with additive observation noise. Eq (11) is used to model all the neural time series presented in this paper, using standard (variational) routines in the Statistical Parametric Mapping (SPM) software. Specifically, we use generalised or variational Bayesian filtering (Dynamic Expectation Maximisation) [30]. This enables us to first estimate the parameters of the model for different patients. We then use the average of these parameters as a ground truth to generate data which we can then manipulate on a known circuit with plausible parameter values. In other words, with the DCM framework we aim to: a) infer the latent states; b) estimate the parameters; and c) estimate the hyperparameters, i.e. the precision components of fluctuations on the states and observation noise. After recovering the posterior densities, hyperparameters, and variational free energies on an individual level for all 30 subjects, we then obtain two group-level models–patients and healthy controls.

In Fig 3 we provide a visual aid to the governing equation of motion [11] in order to describe the meaning of each term. Note that, at this stage, we are not performing any gradient manipulation. The stochastic differential Eq (11) furnishes a dynamic causal model, where fast fluctuations are assumed to be small. This means that if the underlying dynamics can be described by Eq (11), then the model parameters θ = (p, σ) can be recovered from observations of BOLD signals x (Fig 3A).

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Fig 3. Equation of motion.

A) BOLD signal intensity (i.), normalized between zero and unity, for an example time course (in seconds) in one patient (blue), together with the time series estimate (red) following Bayesian model inversion. This inversion provides posterior densities over the matrix elements p in the drift term, and the diffusion coefficient σ in the diffusion term, informed by a non-Markovian noise process ω. B) Example of an adjacency matrix from DTI data, from which we obtain matrix elements k in the diffusion term, following thresholding and binarization. C) The eight regions of the seizure network, in which the left and right frontal mid regions (red) give rise to unstable activity. Stimulation would then be applied to one or more of the remaining (black) regions in a way that mirrors the activity of the red nodes.

https://doi.org/10.1371/journal.pcbi.1008448.g003

As the parameter space associated with the eight-region seizure network is larger than can be accommodated by the time points available for each scan, we reduce the number of free parameters via functional connectivity priors [31,32]. This allows us to constrain the optimization such that we retain ~10× as many time points as free parameters. The adjacency matrix elements k_ij are not included as free parameters as they are known a priori from diffusion imaging (Fig 3B).) The system’s BOLD timecourses is measured in response to external stimulation v, which is supplied here in the form of EEG (Fig 3C).

Bayesian model reduction

Bayesian model reduction is a type of model comparison, in which we calculate the free energy estimation to log model evidence logp(d|m) for the reduced model in which σ = 0 and the reduced posterior density over the σ parameter.

The free energy combines both accuracy and complexity when scoring models: (21)

Here, logp(d|θ, m) is the log likelihood of the data d conditioned upon model states, parameters and hyperparameters θ, and model structure m. In this study we have two model structures: with diffusion and without diffusion; q(θ) is the approximate posterior density over θ; and p(θ|m) is the prior probability of θ given m, where the Kullback-Leibler divergence (KL) term allows for solutions q(θ) that are too complex to be penalised. In order to determine whether we are justified in including the diffusion term in the equation of motion [11], we first optimise the full model including a non-zero diffusion coefficient σ (i.e., including diffusion). We then use Bayesian model reduction [33,34] to estimate the evidence for the reduced model in which σ = 0 (i.e. excluding diffusion). The reduced model is specified by setting the prior variance over the diffusion coefficient σ to zero. The two models are compared by using the log Bayes Factor F(σ = 0)−F(σ≠0), which accounts for the relative evidence for non-diffusivity vs. diffusivity. Associated probabilities are then calculated by normalizing F, such that .

Diffusivity is higher in the patient group

We find that the patient group has higher diffusivity as compared with the control group across structural adjacency matrix thresholds used to define k_ij (see Fig 4).

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Fig 4. Diffusion coefficients.

The diffusion coefficient (σ) (following Bayesian model averaging) as a function of structural (DTI) adjacency matrix threshold (%) for patients and controls. The 90% Bayesian credible intervals are sufficiently small to be contained within the data points.

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The higher diffusivity in the patient group across all thresholds lends credence to the approach of suppressing seizure activity by decreasing diffusivity. We show the results in Fig 4 on an individual subject level in S1 Fig.

Stability is lower in the patient group: We find that the patient group is less stable across structural adjacency matrix thresholds, as compared with the control group (see Fig 5). The lower stability in the patient group justifies the approach of aiming to increase stability. All subsequent results shown in this paper are calculated using a threshold of 50% to define structural connectivity (i.e., k_ij). We show the results in Fig 5 on an individual subject level in S2 Fig.

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Fig 5. Stability.

The sum of the Real components of the eigenvalues of the Jacobian – measuring intrinsic stability of neural dynamics–as a function of structural (DTI) adjacency matrix threshold (%) for patients and controls following Bayesian model averaging. The 90% Bayesian credible intervals are sufficiently small to be contained within the data points.

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Accounting for diffusion improves models

We show, by using Bayesian model reduction (see Methods) in both patients and controls, that the variational free energies and associated probabilities are higher for the full models including diffusion, than in the reduced models excluding diffusion (Fig 6).

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Fig 6. Bayesian model reduction.

A) Control group. Approximate lower bound on log model evidence afforded by the free energy (F) following Bayesian model reduction for the reduced model without (w/o) diffusion and the full model with (w) diffusion. B) Probabilities derived from the log evidence in A). C) & D) Same layout as A) & B), but for the patient group.

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We show the results in Fig 6 on an individual subject level in S3 Fig.

This Bayesian model comparison provides evidence that the diffusion term in the equation of motion [11] gives a better description of the BOLD time series. Note that we do not increase the free energy automatically merely by adding another term. This would not be informative, as one can always improve model accuracy with more variables. Instead, the variational free energy describes a trade-off between accuracy and complexity. This means that the free energy can decrease if the complexity introduced by additional parameters is greater than the associated model accuracy increase. However, this is not the case here, in that we find that the increase in accuracy afforded by the diffusion term increases the variational free energy to a greater degree than the decrease caused by the heightened model complexity.

Pre-ictal perturbation

Following Bayesian model inversion and averaging in the patient group, we use the resulting parameters to create an in silico seizure network, in which we perturb the mid-frontal sources with pre-ictal activity taken from EEG measurements (Fig 7A and 7B).

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Fig 7. Pre-ictal perturbation.

A) EEG activity from the frontal lobe (FP1 & FP2). Ictal activity is shown by the yellow section and pre-ictal activity is shown by the red section. B) The seizure network shown in MNI space (left) with the mid-frontal regions (left & right) indicated by the red nodes. The mean pre-ictal signal intensity (i) (right), normalized between zero and unity, corresponding to the red sections in A), is used as the external driving input and is supplied to the red nodes, as indicated by the inward-pointing red arrows. C) The mean BOLD signal intensity (i), normalized between zero and unity, of the mid-frontal region to the stimulus in B), as indicated by the outward-pointing red arrows.

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We use the mean activity from EEG channels located close to the frontal region (FP1 & FP2), as these pick up a mixture of frontal source activities. Note that previous studies have indicated the vital role of the frontal lobe in JME [35,36]. Prior to GSW discharge events, we observe a change in the EEG channels exclusively in the frontal lobe region, with typical background activity seen in other channels. Following this stimulation, we see that the activity profiles from the mid-frontal regions climb in an uncontrolled manner, due to the associated eigenvalues with positive Real components (Fig 7C). Note, however, that which regions are stable or not varies across patients and that stability, or lack thereof, does not imply anything about focal pathology–given that we are dealing with generalized epilepsy.

Seizure suppression

Using the same setup as in Fig 7, we again run the forward model. However, this time–in addition to the pre-ictal driving stimulus–we supply additional external stimulation to all nodes except the mid-frontal region, in a way that mirrors the pre-ictal activity in Fig 7B. Note that, as stability is determined purely by the Jacobian of the system, this perturbation does not alter the stability in the regions. Rather, this pre-emptive intervention induces a temporary ‘pseudo stability’, in which the activity of an affected region can be driven back to baseline. We show that, in agreement with our theoretical predictions, reducing activity gradients in this way results in the response of the unstable mid-frontal region being suppressed (Fig 8).

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Fig 8. Simulating seizure suppression.

A) External stimulation signal intensity (i), normalized between zero and unity, applied to all nodes except for the mid frontal region for 1000 forward models ranging from zero stimulation (yellow) to a stimulation profile that perfectly mirrors the pre-ictal activity in Fig 7B (black). B) Response signal intensity (i), normalized between zero and unity, of the mid frontal region for the same 1000 forward models in A) with matching colours, i.e. the yellow response corresponds to zero stimulation and the black response corresponds to a stimulation profile that perfectly mirrors the pre-ictal activity in Fig 7B.

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We show an animated version of Fig 8 in S1 Movie.

We note from Fig 8 that the methodology we are proposing works best when the input exactly matches the signal, i.e. with minimized gradients. This can be seen by the gradual shift in the response as the stimulation becomes increasingly similar to the signal–from an initial divergence to baseline recovery. Note also that, although these simulations are based on fMRI data collected in patients with epilepsy, these patients did not experience seizures while in the scanner: the pre-ictal activity collected with EEG was recorded separately. The output of the model in Figs 7C and 8B should therefore not be viewed as mimicking a seizure-like BOLD response. Instead, what we show here is that, in response to a perturbation in the form of real pre-ictal activity (Figs 7B and 8A), an initially unstable system with a diverging response can be suppressed–a result that we suggest could be clinically advantageous in the treatment of epilepsy.

Minimum effective stimulation

In the previous section we stimulated all nodes except for the mid-frontal region to demonstrate proof of principle. However, for treatment purposes, it is clearly better to stimulate as few regions as possible, in order to remain minimally invasive. We therefore proceed by using each individual node in turn as the stimulation target. This allows us to determine the order of the regions, ranked in terms of the lowest stimulation strength required to suppress the mid-frontal response (Table 1).

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Table 1. Ranking order in terms of the stimulation strength required to suppress the BOLD response in the mid-frontal region to the same extent as in Fig 8B, by stimulating a single region in the seizure network.

Stimulation strengths are presented relative to the lowest value (thalamus left), which is assigned a value of unity. N/A values are assigned if it is not possible to suppress the BOLD response in the mid-frontal region by targeting the corresponding single region.

https://doi.org/10.1371/journal.pcbi.1008448.t001

Note that the mid-frontal region is not included in this ranking as it is being supplied with the driving input in the form of pre-ictal activity and our technique requires mirroring this activity in neighbouring nodes in the network, in order to decrease gradients and thus diffusivity. Using this in silico stimulation protocol, we find that the thalamus (left) requires the lowest stimulation strength (by a considerable margin) in order to suppress activity in the unstable mid-frontal region, relative to the other six regions in the seizure network.