Patient data

In order to initialise the connectivity of our models, we used data from 39 subjects, 19 of whom are suffering from left temporal lobe epilepsy. The subjects were selected from the dataset presented in [40, 41]. Written informed consent was obtained, signed by all participants, and conformed to local ethics requirements. The ethical review board of the medical faculty of Bonn gave IRB approval (032/08) and all experiments were performed in accordance with relevant guidelines and regulations. T1 weighted MRI scans and diffusion tensor imaging (DTI) data were obtained using a 3 Tesla scanner, a Siemens MAGNETOM TrioTim syngo (Erlangen, Germany). The T1 images were obtained using 1mm isovoxel, TR = 2500ms and TE = 3.5ms. The DTI data used 2mm isovoxel, TR = 10,000ms, TE = 91ms and 64 diffusion directions, b-factor 1000s mm−2 and 12 b0 images. In both caes FoV was 256mm.

To create the structural connectomes, FreeSurfer was used to obtain surface meshes of grey and white matter boundaries from the MRI data and to parcellate the brain into regions of interest (ROI) based on the Desikan atlas [42, 43]. This process identified 82 ROIs which spanned cortical and subcortical regions (Nucleus accumbens, Amygdala, Caudate, Hippocampus, Pallidum, Putamen and Thalamus). Streamline tractography was obtained from DTI images using the Fiber Assignment by Continuous Tracking (FACT) algorithm [44] through the Diffusion toolkit along with TrackVis [45]. First, we performed eddy-correction of the image by applying an affine transform of each diffusion volume to the b0 volume and rotating b-vectors using FSL toolbox (FSL, http://www.fmrib.ox.ac.uk/fsl/). After the diffusion tensor and its eigenvector was estimated for every voxel, we applied a deterministic tractography algorithm [44] initiating a single streamline from the centre of each voxel. Tracking was stopped when the angle change was too large (35 degree of angle threshold) or when tracking reached a voxel with a fractional anisotropy value of less than 0.2 [46].

The centre coordinates of each voxel were the start of a single streamline, the total number of streamlines never exceeded the number of seed voxels. The number of connecting streamlines were used to determine the connectivity matrix (S), as the streamline count has recently been confirmed to provide a realistic estimate of white matter pathway projection strength [47]. Distance matrices were also constructed using the mean fibre length of the streamlines connecting each pair of ROIs (Fig 1). The surface area of each ROI was found using FreeSurfer for cortical regions and for subcortical areas by computing the interface area to the white matter in T1 space [48].

Modified Wilson-Cowan model

Our model consists of a network of 82 coupled modified Wilson-Cowan oscillators, each representing a single brain region. In order to include divisive inhibition into our model, each W-C node consists of one excitatory and two inhibitory populations (Fig 2). The first inhibitory population represents interneurons firing at the dendrites of the postsynaptic neurons (subtractive inhibition) and the second inhibitory population represents interneurons firing directly at the soma of the postsynaptic neurons, delivering divisive inhibition. For the implementation of the model we followed the methodology and notation of [49]. All the notations that we use for the description of the model are summarised in Table 1.

Of course, the model described in [49] has been designed to simulate the connectivity of a cortical microcircuit and not the connectivity of sub-cortical regions. Still, a number of studies [50–52] have shown the presence of shunting inhibition (in addition to regular subtractive inhibition) in many of the subcortical areas we used in our study. Thus, we felt that the inclusion of both inhibitory populations in the nodes representing subcortical regions was justified.

According to this approach, the activity of each brain region is represented by a Wilson-Cowan model, governed by the following delayed differential equations (DDE’S): (1) (2) (3)

To account for the divisive inhibition a modified input-output function is required: (4)

For, j∈{e,i}, where e stands for excitatory and i stands for inhibitory. The inhibitory populations have the same input-output function and the same constants since they are assumed to respond to inputs in a similar way. However, the difference in the type of inhibition those neurons deliver to the excitatory population is due to their different targeting onto the postsynaptic neurons, that is, somatic vs dendritic.

The constant k_j, j∈{e,i} is given by: (5)

As is the case with the sigmoid function the constant k_j is the same for both inhibitory populations.

In our study, the constants of the sigmoid were set at θ_e = 4, θ_i = 3.7, a_e = 1.3, a_i = 2, following the values used at [49]. Moreover, the external inputs of the inhibitory populations were set to P_s = P_d = 1 while the input of the excitatory population was set to P_e = 2. Other values were considered for P_e ranging from 1.1 to 4 (the range where the system produces oscillations) with results similar to the ones presented here. Providing no input to the inhibitory populations (P_s = P_d = 0) results in a lack of long term stable oscillations and therefore we restricted the parameter value to P>0. A detailed description of all notation used is given in Table 1.

Connectivity and plasticity

The weights W_ij between network nodes representing brain regions were initialized according to the brain anatomy of each patient using the data described in the section ‘‘Patient data”. Specifically, given the matrix S of the streamline counts for an individual subject we followed the original study [40] and initialised the connectivity matrix M as: (6)

This connectivity matrix was the only element of our study taken from biological data, everything else refers to simulations (and not experimental results) using the model described in this section.

The connectomes of healthy and epileptic patients did not show any apparent differences and due to the small dataset and very high dimensionality of the data (82x82 matrices) training (and testing) a classifier, to examine the possibility of distinguishing between the two groups at this level, was deemed unfeasible.

During the simulation, the weights were updated every 10 milliseconds by the following learning rule: (7)

We chose this simple rule in order to apply a simple form of Hebbian plasticity [53, 54] (if high activity in a pre-synaptic node is followed by an increase of activity in the post-synaptic node, the connection weight increases, otherwise it decreases) in neuron populations. The learning rate was set at c = 0.1. Other values were considered, and similar results were obtained with the only difference being the speed of weight change. Still, the pattern of activity remained the same for all the values we examined as can be seen in Fig 3.

The weight matrix was normalised after each update—to avoid runaway plasticity as indicated by the findings of [55, 56]—by the following rule: (8)

For the internal weights of each node we used two different sets of initial values. The first set of values was chosen to represent the connectivity of a healthy brain region while the second set was chosen to represent an epileptogenic region. The values of the healthy region were decided after an extensive parameter search, starting at the values used by [49] and examining values between 8 and 21 (the range at which the system produces oscillations). The values we selected lead to high amplitude oscillations in all three populations during the first hours of the simulation. The amplitude of the oscillations gradually decreases and stabilizes after some hours. It must be noted that the final values were chosen to facilitate the dynamics of the system and may not correspond to the connectivity of a real biological system. Still, using different parameters usually resulted in oscillations of different amplitude and consequently slower stabilization periods, but as a general rule did not lead to radically different behaviour in the system.

After the values of the node representing a healthy region were established, the values of the nodes representing epileptogenic regions were derived by increasing the weights of excitatory connections and reducing the weights of the inhibitory connections. Those changes aimed at increasing the excitability of those nodes (increased excitatory and decreased inhibitory input) in order to simulate the dynamics associated with epilepsy. The difference in behaviour of the epileptogenic nodes was small but observable (oscillations of increased amplitude and occasional seizure-like activity when the input to their excitatory populations was increased), as with the original connection weights, choosing different values led to slightly different results (the more excitable a node is, the greater the effect of stimulation), but the main observations remained the same. The values chosen are presented in Fig 2.

The weights w₁,w₂,w₃,w₄,w₆ were updated every 10 milliseconds according to a modified version of the rule we used for the external connections with subsequent normalization after every update. (9) where Pre(t), Post(t) are the activities of the presynaptic and the postsynaptic populations, respectively. Several proposed mechanism of internal plasticity were considered, but due to the lack of a consensus about a general mechanism of inhibitory plasticity [53, 57]—especially in neural mass models—we chose to use this simple intuitive rule, similar to the rule we used for the external connections. The most commonly used learning rule for inhibitory plasticity, introduced in [58] could not be used in this model due to long term instability in the network’s dynamics.

For the normalization, we employed the same rule used for the global connectivity: (10)

Since there has been little research on how inhibitory to inhibitory plasticity could be implemented in a neural mass model, the weights w₅ and w₇ were kept stable. The learning rate was set at c = 0.05.

Finally, the delays were initialized for each patient, as the length of the fibres connecting two brain regions divided by the speed of spike propagation. For the calculation of the delays we assumed that activity propagates with the same speed in all connecting fibres, which was set at 7 m/s, following the convention used at [59, 60]. To calculate the distance between regions, we selected the fibre trajectory length—which we calculated using deterministic tracking of diffusion tensor imaging data—instead of the Euclidian distance in order for the delays to be more biologically realistic.

Stimulation

Each session of stimulation was modelled as a decrease of 50% (the stimulation is cathodal, due to better reported experimental results [21]) in the external input of three nodes representing the brain regions most commonly responsible for seizures in these patients (amygdala, hippocampus and parahippocampal gyrus), for a period of 30 minutes. Despite two of these brain regions being sub-cortical, the ability of transcranial stimulation to affect them has been demonstrated in past studies [61–63]. Stimulation in all cases started at t = 200s after the beginning of the simulation. This initial period was allowed for the oscillations of the system to stabilize before stimulation begins.

The choice of stimulation parameters was made in order for the model to correspond to a working protocol of TCS presented in [64]. Due to the computational constraints of such large simulations [65], we modelled only one session and an additional resting period of 24 hours.

Implementation and analysis of results

Simulations were run for three distinct groups of subjects, according to the global connectivity data and model used:

1. Healthy subjects: The global connectivity data were derived from the healthy individuals and the simulation was performed using a model where no epileptogenic (particularly excitable) nodes are present.
2. Epilepsy patients: The global connectivity data were derived from individuals suffering from left temporal lobe epilepsy and the simulation was performed using a model where the stimulated nodes were modelled as epileptogenic (highly excitable)
3. Control subjects: The global connectivity data were derived from individuals suffering from left temporal lobe epilepsy but the simulation was performed using the “healthy” model, where the stimulated nodes are not distinct in terms of excitability from all other nodes.

After the initial choice of connectivity and model parameters, two simulations–with and without stimulation- run in parallel for a period of 24 hours with snapshots of the weight matrices taken every 50 seconds. The large system of DDE’s (246 equations) was solved by using Matlab’s dde23 delayed differential equation solver (The code for the model can be found in the following repository: https://github.com/MGiannakakis/Epilepsy_Simulation)

The effect of the stimulation on the connectivity at every time step was measured in the following ways:

1. The global effect of the stimulation on the connectivity of the brain was measured as the difference (%) of the connectivity matrices M = (W_ij): (11) where W′_ij is the weight between nodes i and j at time t after stimulation and W_ij(t) is the weight between nodes i and j at time t without stimulation. This measure represents the effect stimulation has on the internode connections of the brain.
2. The effect of the stimulation on the internal connectivity of each node (local effect) was measured as the difference (%) of the internal weights in the stimulated and the non-stimulated versions:

(12) where i = 1,…,82 the brain node, is the k-th weight of the i-th node at time t in the stimulated version and is the i-th weight of the k-th node at time t in the non- stimulated version. These measures represent the effect of stimulation on the internal connectivity of each brain region.

Connectivity measure

In order to study the effect of stimulation on the nodes that received no direct stimulation, we examined several connectivity metrics that could explain such an effect. One of those metrics is the Jaccard index. The Jaccard index of two nodes measures the similarity in connectivity (the common neighbours) and is defined as: (13)

Where Γ(i) is the set of nodes connected to node i and |Γ(i)∩Γ(j)|,|Γ(i)∪Γ(j)| are the number of elements in the sets Γ(i)∩Γ(j) and Γ(i)∪Γ(j) respectively.

In our study, we defined the Jaccard index of a secondary node i to be: (14)

Where p,a,h are the stimulated nodes.

The network presents a larger global connectivity change at the end of the stimulation for epilepsy patient connectomes

The effect of stimulation on the inter-node connections in our model follows a similar pattern in all subjects. Specifically, during the period of stimulation, the global effect measure D(t) increases steadily (Fig 4), reaching a local maximum at the end of stimulation (t = 2000 s). A first difference between the three groups can be observed at this point since the value of D(t) at the end of the stimulation session (30 min) is on average significantly (p-value < 0.0001) greater for the epileptic subjects (2.9730% ± 0.7301) than the healthy subjects (1.9671% ± 0.3261) and the control subjects (1.7609% ± 0.5290). The similarity of the healthy and control groups in contrast to the epileptic group suggests that the increased excitability of the stimulated nodes and not the initial global connectivity is the main driver of the changes of the global effect measure. Indeed, the global connectivity on its own seems to make the healthy subjects more excitable, since the values of D(t) were slightly higher for the healthy than the control group (although the difference was not statistically significant).

After the end of stimulation session, the global effect D(t) keeps fluctuating for the remainder of the simulation with a clear increasing trend in the majority of subjects. The rate of this increase varies greatly from subjects to subject and it was calculated as the rate r = D(t₀)/D(t₁), where t₀ = 2000s is the end of the stimulation session and t₁ = 24h the end of the simulation. For all subjects the value d varies greatly (0.5846% ± 0.2751) and we can also observe a small difference (statistically insignificant) between the values of healthy subjects (0.5358% ± 0.2128), the similar values of control subjects (0.5372% ± 0.1609) and the slightly greater values of epileptic subjects (0.6328% ± 0.2533) which is not statistically significant (S1 Fig). Thus, the differences between the groups are attributable to different effect of stimulation and not the post-stimulation change in connectivity.

Finally, in order to examine the extent to which the initial global connectivity determines the development of the global difference measure D, we measured the correlation between D in control and epileptic subjects initialised with the same connectivity data and found it to be higher (0.7747 ± 0.1102), than the average correlation between random pairs of control and epileptic subjects (0.6199 ± 0.3213). This, suggests that although the scale of change is mainly determined by the excitability of the stimulated nodes, the exact global connectivity does (at least partially) determine the development of the global effect measure.

The inclusion of excitable nodes leads to a larger change in the local connectivity of the stimulated nodes during but not after stimulation

In the nodes that received direct stimulation (representing the amygdala, hippocampus and parahippocampal gyrus), the effect on the connectivity was most prominent during the period of stimulation, resulting in a constant increase of the local effect measure d_k(t) in all three nodes. Thus, the local measure invariably reaches a global maximum at the end of the stimulation session (t = 2000 s). As with the global connectivity, the effect on the epileptic subjects is greater than the effect on the other two groups (p-value < 0.0001 for all three nodes). Specifically, the average effect for all three nodes on a healthy subject is 0.4746% ± 0.0509, in a control subject is 0.3853% ± 0.0427 and on an epileptic subject is 1.0794% ± 0.0264.

A difference from global connectivity is that in this case the difference between healthy and control subjects is clearly significant (p-value < 0.0001). This suggests that the brain connectivity of epileptic patients conditions the epileptogenic regions to be less excitable than in healthy individuals. Of course, the internal connectivity that makes these regions highly excitable masks this effect as we observed from the metrics of the epileptic group. Still, this finding seems to suggest that the inter-regional connectivity of epileptic patients tends to limit the excitability of epileptogenic regions.

After the end of the stimulation session, the local measure d_k(t) changes similarly in the healthy/control groups but very differently in the epileptic group.

In the healthy/control subjects, the end of the stimulation session (30 min) is followed by a slow decrease in the value of the local effect d_k(t). Around 8 hours after the end of the stimulation session, the difference measure stabilizes at d_k(t)≈0.1%, for all three nodes (Fig 4), for a representative subject. The local effect measure d_k(t) of a node is considered to be stabilized at time t if the Coefficient of variation of the values of d_k(t) for the 5 minutes prior to t is less than 0.3. After that point, there may be some small oscillation in the value of d_k(t) but the change is minimal.There is much greater variation in the epileptic subjects, both between the nodes of the same subject as well as between equivalent nodes (representing the same brain region) of different subjects (Fig 5). Immediately after the end of the stimulation session and for a period lasting 5–6 hours, the local effect d_k(t) is sharply (more than in the healthy/control subjects) decreasing for all 3 nodes. With the exception of two subjects where there is a short increasing period in the values of the amygdala and the hippocampus, d_k(t) is strictly decreasing during this period for all three nodes of every subject. It should be noted that in almost all the epileptic subjects (18 out of 19), the connectivity of the node representing the parahippocampal gyrus is behaving differently than the connectivity of the nodes representing the other two stimulated regions. The local effect (measured by d_k(t)) on the parahippocampal gyrus node is diminishing faster than the equivalent measures of the other two regions, reaching values close to zero at the end of this first period.

For the remainder of the simulation, each subject presents different behaviour and the various stimulated nodes also present differences in each subject. In 10 of the subjects the local effect on the node representing the parahippocampal gyrus remains at the low levels it reached in the end of the decrease period (1–5/6 hours) with some minimal increases. In the remaining 9 subjects the local effect on that node starts increasing at some point between 8–12 hours after the end of stimulation and continues to increase for the remainder of the simulation reaching values comparable with those of the other two regions. The nodes representing other two stimulated regions (amygdala and hippocampus) behave almost identically in each subject. After the end of the first period of decrease the local effect measures of these areas stabilize in 10 of the subjects and decrease very slowly in 6 of the subjects for the remainder of the stimulation. In the remaining 3 subjects, the local effect measure increases for a period of 1.5–2 hours until it reaches values much higher than those of the other subjects (d_k(t)≈0.55), after that point the effect on those areas begins to slowly decrease.

At the end of the simulation, we can observe that the final values of d_k(t) for the epileptic subjects (0.1412% ± 0.0882) are slightly greater than those of the healthy subjects (0.1165% ± 0.0275) which in turn are slightly greater than those of the control subjects (0.1037% ± 0.0400) in the nodes that received stimulation. Still that differences are not statistically significant. This implies that the initial difference between healthy/control and epileptic subject does not lead to a long-term difference in the stimulation effects.

Some non-stimulated nodes show local connectivity changes after the end of the stimulation

The effects of stimulation can be seen not only on the internal connectivity of the nodes that are stimulated directly but also on the connectivity of other nodes that receive no direct stimulation (Fig 2).

Specifically, in all groups, the local effect d_k(t) of several nodes starts increasing and reaches a peak shortly after the end of the stimulation session. It should be noted that the change in those nodes does not absolutely coincide with the stimulation session, rather it happens shortly afterwards, possibly due to the time delays. Moreover, unlike the directly stimulated nodes where a difference can be observed between the healthy/control and epileptic model groups, no such difference can be observed in the values of those secondary regions.

After this initial increase, the local effect on all secondary nodes usually decreases and seems to stabilize after a period of about 8 hours. For the majority of subjects (29 out of 39) the values that the difference measures have at this point will be very close to the values they will have at the end of the stimulation. In most cases, the final value of the effect measures for those nodes are very close to the values of the other non-stimulated nodes that were not affected by the stimulation, but in some cases the final values for some of these secondary nodes (especially the entorhinal cortex) are much closer to–and in some cases higher than—the values of the stimulated nodes. Interestingly, in some epileptic subjects (5 out of 19) the local effect measure of some secondary nodes began to suddenly increase hours after the stimulation session when they were apparently stabilised for some time. This unpredictable behaviour suggests than even in the very simplified model used for this study, the dynamics of plasticity are not easily predictable for a timescale of hours. That is an indication that long-term effects that cannot be predicted from the initial response to stimulation.

Still, despite the fact that we do not know the exact cause of these post-stimulation changes, they seem to appear more frequently in some nodes than in others and several factors could explain why those nodes in particular were affected. Specifically, the brain regions that these nodes represent are characterized by increased connectivity with the stimulated regions as well as by a small Euclidian distance from the stimulated regions. Additionally, the effect the connections with the stimulated regions seemed to be greater than average (increased connection weights). Finally, the Jaccard index (common neighbours) of the affected nodes and the stimulated nodes was higher than in regions that were not affected. Moreover, the frequency of excitation among the six most commonly excited nodes (Fig 6) is correlated with the aforementioned metrics of the corresponding brain regions. For example, the node representing the entorhinal cortex that was affected in 17 of the subjects, scores higher in all the metrics (connectivity, Jaccard index, etc.) than the node representing the putamen which was excited in 2 of the subjects. A ranking of all the regions according to those metrics is given in Table 2. Moreover, the corresponding absolute values are presented in the supplementary information (S1 Table).

We wondered whether these observed secondary effects may have clinical significance. The patients from whom the data originated had received resective surgery (not stimulation) of the seizure causing brain regions (amygdalohippocampectomy) and the outcome of these surgeries was known for a number of them (17 subjects). We found that the increased effect in the secondary nodes that was observed in the epileptic group was weakly correlated with a worse outcome of resective surgery: Epileptic subjects who presented a long-lasting effect on secondary nodes after stimulation within our model, i.e. higher values of the local effect measures compared with other non-stimulated nodes at the end of the simulation, were on average less likely (3.225 ± 1.220 on the ILAE classification scale) than those who did not present such effects (2.011 ± 1.110) to benefit from surgery (p-value = 0.0484, Cohen’s d = 1.042). Still, given the important differences between surgery and stimulation as well as the small sample size, it is very possible that this finding is not meaningful and further research with larger samples and experimental data from patients who received brain stimulation is required to evaluate any potential clinical application of this framework.