In order to study the interhemispheric connectivity by TC, we considered two cortical areas, the right and the left SMA (each described by Equations 2.1–2.18), interconnected through long-range excitatory connections with a time delay. Figure 3 shows the model layout.

In TC modeling, we relied on the assumptions that all transcallosal fibers are excitatory at a neurochemical level (Bloom and Hynd, 2005) since the long-range connections originate exclusively from pyramidal population. However, both excitatory and inhibitory messages have been demonstrated to travel through the corpus callosum (Swayze, 1987; Bloom and Hynd, 2005); indeed, the functional effects of these connections depend on various factors, such as the receptors and the interneurons involved. In particular, a synaptic connection from pyramidal neurons in one area to pyramidal neurons in the other areas has an excitatory role on the target region, whereas a connection from pyramidal neurons in one area to inhibitory interneurons in the other area has an inhibitory effect.

Ursino et al. demonstrated that an excitatory connection to fast inhibitory interneurons is able to transmit the rhythms from one region to the other very efficaciously, while connections to slow inhibitory interneurons are less effective (Ursino et al., 2010). For these reasons, we modeled TC as projections from pyramidal cells to contralateral pyramidal neurons and to fast inhibitory interneurons.

To simulate connectivity, we assumed that the average spike density of pyramidal neurons of the presynaptic area (z_p) affects the target region via a connectivity constant to fast inhibitory interneurons (K_f) and to pyramidal neurons (K_p) with a time delay T.

This is achieved by modifying the input quantities u_p and u_f of the two cortexes.

We can write: (2.19)ui(t)=ni(t)+Ki · zp(t−T)i=p, f where n_i(t) represents the external input (see Section Driving Input).

In the model, we defined a global connectivity constant K as: (2.20)K=Kp+ Kf

We used the same value of K_p and K_f for both cortexes. To assign a relative value for these parameters, we made two approximations: (i) we considered only layers 2/3 and 5, provided that the origins of TC is mainly located in those layers (Koralek et al., 1990; Rouiller et al., 1991; Martínez-García et al., 1994; Karayannis et al., 2007; Molyneaux et al., 2007); (ii) we considered that the fibers from layers 2/3 and 5 generate more synapses to fast-spiking interneurons (about 70%) than to pyramidal neurons (about 30%). The contribution of synapses to the other interneurons was considered negligible (Dantzker and Callaway, 2000).

Accordingly, we assumed that 70% of synapses target the fast inhibitory interneurons (hence K_f = 0.7K), while the remaining 30% of synapses target the pyramidal ones (K_p = 0.3K). The choice of these percentages might seem decisive for model behavior. However, a preliminary sensitivity analysis showed that the overall behavior of the model does not change appreciably if this ration is changed even greatly. Indeed, only the edges of the working regions range, that will be described below, slightly changes.

It is known that the corpus callosum has a variable number of fibers and, depending on the considered theory (Theory of inhibition or Theory of Excitation), that number correlates negatively/positively with the information transfer and positively/negatively with the lateralization, respectively. To represent this variability, we tested the model with 101 K K-values equally distributed between 0 and 100; then K_p varies between 0 and 30 and K_f between 0 and 70.

The delay in the information transfer between the two cortexes, which depends on the complexity of the specific task, can vary in a range 10–300 ms. Considering the results of transcranial magnetic stimulation studies, transcallosal conduction times between motor cortexes of healthy subjects are typically about 13 ms (Liederman, 1998). So we used a time delay T equal as great as 13 ms.

Parameters sensitivity analysis showed that inputs to slow inhibitory and excitatory interneurons do not produce appreciable changes in the model dynamics (Ursino et al., 2010). Therefore, in the following, we will consider only external inputs to pyramidal neurons and to fast inhibitory interneurons.

The external input n_i(t) (i = p, f) is represented as a Gaussian white noise which accounts for all inputs not incorporated in the model, both excitation coming from the environment and the density of action potentials coming from other connected regions. It was modeled as a positive input to the pyramidal cells (hence excitatory) with mean m = 40 pps (pulses per second) and standard deviation σ = 1 and a positive input to the fast inhibitory interneurons (hence with an inhibitory effect) with m = 3 pps and σ = 1. The external inputs have the same m and σ for the two cortexes. The inputs are given for the entire simulation. The model parameters and the external input set the WP at a level similar to that used by other models in literature (Wendling et al., 2002; Zavaglia et al., 2006).

The driving input has a very important role for setting the population WP and then the potential cortex excitability. In particular, in agreement with the cortical activation model (Pfurtscheller, 2006), the baseline WP determines the modulation of the cortex, and so the induction of an ERD or an ERS.

To simulate the occurrence of the task, consisting in the persistent imagery of the right hand movement, we added a modulating input to the pyramidal population of the contralateral cortex (i.e., to the left SMA).

We simulated a typical motor imagery trial lasting 16 s: 4 s of baseline, 8 s of motor imagery and 4 s of rest (Suffczynski et al., 2001). The modulating input is modeled as a smoothed trapezoidal function, with the rise and fall times as long as 2 s and a maximum amplitude plateau as high as 100 pps lasting 4 s (Figure 4).

Outputs of the model are the average membrane potentials of the pyramidal populations, which simulate the local field potential recorded by superficial EEG over the left and the right SMA (in Figure 3 v_{out_L} and v_{out_R}, respectively). Moreover, during the simulations we also analyzed the average spike density z_p of the pyramidal neurons. This variable is related with the neuronal firing rate, then it contains a direct information on the interaction between the two cortexes. In particular, z_p can vary between 0 and 2 × e₀. In the following, we will call WP the value of z_p around which the cortex works. The simulated sample frequency is 100 Hz.

After introducing the modulating input and once fixed the WP, we computed the ERD/ERS as follows: (2.21)ERD/ERS(%)=P(t)−PBPB · 100 where P(t) is the power extracted of v_p(t) at each time-point over the 16 s, and P_B the mean power in baseline (during the first 4 s).

For each beta sub-bands, we considered 101 values of K to evaluate the role of the connection strength on the transmission of the signal from one cortex to the other. In this phase there is no modulating input on the cortexes.

The average value of z_p over time (16 s), for each of the two cortexes, is shown in Figure 6 as a function of K. LB, MB, and HB curves show the same behavior and include 3 regions:

The previous results can be explained by the following mechanisms: in the first two regions, the network start from an upper saturation point and the presence of TC move the WP of pyramidal neurons near a linear region of the sigmoid, increasing the interhemispheric communication. In particular, excitation from pyramidal neurons of the other hemisphere, via the inter-hemispheric connection K_f, causes excitation of fast inhibitory interneurons. Indeed, this population has a very important role since it is sensible to the input and produces the greater variations of the WP at the increase of K. Conversely, excitation to pyramidal neurons, through the connection K_p, has a less relevant role, since this population works close to the upper saturation. The strong increase in z_f, in turn, causes a significant inhibition of pyramidal neurons, with the global result to move the pyramidal WP toward a linear region of the sigmoid. Moreover, the decrease of pyramidal activity reduces the WPs of the slow inhibitory interneurons and of excitatory interneurons. Finally, it is worth while that a reduction in the WP (i.e., functioning in the linear region) is associated with larger fluctuations in the population activity induced by noise.

The behavior in the third zone reflects a competitive mechanism, similar to a winner takes all dynamics: only one region (due to a better influence of external noise) wins the competition causing the almost complete inhibition of the other. This mechanism reflects a strong lateralization of the brain.

Figure 7 shows a map of the v_{out_R} and v_{out_L} log-transformed PSD spectra of each beta sub-bands, computed for each connectivity constant K. In the figure, the three previously identified regions are well evident for each beta band. Specifically, when K is lower than 7, the maps show the main spectral content in the beta bands (light blue areas) for both the cortexes. Similar to the first region, values of K belonging to the second identified region (LB: 7 < K < 35, MB: 7 < K < 26 and HB: 7 < K < 16), provide PSD spectra with the main content in the beta bands but with a greater amplitude (red areas) for both right and left SMA. In the third region (LB: K > 35, MB: K > 26 and HB: K > 16), one of the cortex is completely inhibited (left SMA) and it shows the main spectral content in low frequency band (2–7 Hz) (light red and white areas), whereas the other cortex shows a behavior as in the “no-connectivity” condition with the main spectral content in beta bands (light blue and white areas).

To clarify this phenomenon Figure 8 shows the v_{out_R} and v_{out_L} PSD spectra of each beta sub-bands, computed using 3 connectivity constants belonging to the 3 above identified working regions.

The information transfer is well evident using the connectivity constant K = 4: the PSD spectra of the two cortexes are similar and have a power content greater that the “no-connectivity” condition (Figure 5).

Using K = 14 there is a very efficient transmission of the two rhythms: the two cortexes are indeed synchronized at the same frequency, and the PSD spectra show a peak of higher amplitude with respect to the K = 4 condition.

In the third region (K = 70), we found a different behavior between the two cortexes: one cortex works as in the “no-connectivity” condition, whereas the other is characterized by a low frequency content. This signifies that the strong connection strength form the first to the second cortex causes a slower rhythm (mainly in the theta range) but, due to the prevalence of inhibition, the average membrane potential in the second cortex lies below the threshold of the sigmoid relationship, i.e., the second region works in the low saturation region.

After analyzing the behavior of the system with the driving input alone, we added the modulating input to simulate the imagery task. Since simulation concerns the imagination of the right hand movement, we gave the input to the left SMA.

Figure 9 shows the ERD/ERS % maps (computed with Equation 2.21) as a function of the time and as function of K. In this phase, we focused on the first two previously identified K regions.

As shown in the figure, an increase in K causes a significant decrease in the average power (hence ERD) in the contralateral side, and a significant increase in power (hence ERS) in the ipsilateral side. The latter increases with K until a maximum is reached, than the effect decreases at larger values of K.

From Figure 9, we selected two K K-values, K_H and K_L, for each beta sub-bands, corresponding to a high and low amplitude of ERD/ERS, respectively, and tested the temporal pattern of ERD/ERS during the task (Figure 10). The selected values are 10 and 26 for LB band, 6 and 24 for MB band and 4 and 13 for HB band.

For each beta sub-bands the figure shows that, using K_H, an ERD occurs in the Left SMA, while an ERS occurs in the Right SMA, at the beginning of the imagery task. As the motor imagery task ends, the two powers return to their baseline values. Using K_L, the transmission becomes less efficient. Actually, the ERD follows the same behavior as for K_H, but with a lower amplitude. Conversely, the ERS is not generated.

To better explain the ERD/ERS generation, some exempla of v_{out_R}, v_{out_L}, z_{p_R} and z_{p_L} are showed as function of time in Figure 11. The figure shows the behavior of the network for low beta and the two K-values used in Figure 9 (K_H = 10 and K_L = 26). In particular, the ERD is generated for both K-values since the modulating input, during the motor imagery task, moves z_p of the target cortex (left cortex) toward upper saturation. The smaller fluctuations in z_p, in turn, are reflected in smaller fluctuations in the overall column, and so the amplitude of v_{out_L} decreases. For K_H, the TC produces a decrease of z_p in the ipsilateral cortex, (right), moving the WP in the linear region. This corresponds to large fluctuations (ERS). For K_L, z_p oscillation of the right cortex is already maximum and it cannot generate a further ERS.

Analyzing Figure 6, we distinguished 3 regions:

Behavior in the third region agrees with the inhibitory theory of metacontrol proposed by Banich. In fact, in metacontrol information presented to both hemispheres is completely managed by one dominant hemisphere (Banich, 1995). In other words, by inhibiting activity of the opposing hemisphere the other hemisphere becomes dominant for the processing of the stimulus information (Hellige et al., 1989). The model suggests a similar behavior in case of high values of K (i.e., in the third region described above). Here, model behavior implements a “winner takes all” dynamics, in which only one among the competitors (in this case one of the two areas of the cortex) wins the competition and completely dominates the task. Conversely, at intermediate vales of K, both hemispheres are involved in the process, and in this region we can observe the maximal synchronization between the two rhythms. Here, the model suggests that cooperation between the two hemispheres is the adopted strategy, in which binding of the rhythms allows the integration of the two pieces of information (left and right) into a single combined response.

In this section we will discuss how a modulating input is transmitted from the target cortex to the other. When K is equal to zero (no connection) the input is not transferred from one cortex to the other, than an ERD is generated on the target cortex but and ERS is not generated in the controlateral one. However, for low values of K the transfer is maximum (blue and red areas for ERD and ERS, respectively in Figure 8) and specifically there is a turning point after which, increasing the value of K, the transfer starts to decrease (light blue and light red areas for ERD and ERS, respectively in Figure 8). The condition of maximum transfer is a condition in which the two cortexes are near the transition from the concave region to the convex region. In this point the z_p amplitude and mean are not too high, so a modulating input is able to modulate the z_p in a significant way.

Our simulation results show that a modulating input produces a decrease of the power in beta band on the target cortex ERD and an increase of the power on the ipsilateral cortex ERS. This can be explained with the following mechanism. The input excites the target cortex, which inhibits more the ipsilateral cortex. The latter in turn inhibits less the target cortex. This process generates a loop to which the target cortex is increasingly excited and the ipsilateral one increasingly inhibited. As a consequence, the WP of the target cortex moves to a position closer to the upper saturation, i.e., a position where the amplitude of the rhythm decreases. In fact, here the spiking frequency of the neurons is less affected by the external noise. Conversely, the WP of the ipsilateral cortex moves down in the linear region of the sigmoid relationship, where the spiking frequency of neurons is maximally affected by noise, thus causing a power increase in the beta range. This mechanism allows the generation of the ERD (related with an excitation state, close to upper saturation) and ERS (related with a inhibition state, close to the central region) patterns.

Of course, this result strongly depends on model assumption on the initial WP of the two pyramidal populations, placed in the upper portion of the sigmoidal relationship, not too distant from saturation (Wendling et al., 2002; Zavaglia et al., 2006) and is consistent with the model of cortical activation (Pfurtscheller, 2006). According to this theory, when the cortical activation baseline level is high and the majority of neurons is occupied by synchronization processes, an increase of cortical activation, induced for example by an external excitatory input, can only induce an ERD. This is the behavior observed in the target cortex. In turn, the WP of the pyramidal population strongly depends on WP of the fast interneurons population, placed in the lower saturation portion of the sigmoid. These neurons are very sensitive and ready to spike as soon as they receive an input. As a consequence, excitation of fast inhibitory neurons in the ipsilateral cortex (induced by TC from the target other cortex) causes a decrease in the WP of their pyramidal neurons, thus inducing an increase in fluctuations. This is in accordance with the cortical activation model: when the cortical activation baseline level is high and the majority of neurons is occupied by synchronization processes, a decrease of cortical activation can only induce an ERS.

In this phase we have not considered the before identified third region, since in that region one of the cortexes takes the control of the process and the other is completely turned off.

In the context of the debate about excitatory or inhibitory theories of the corpus callosum, this model is collocated mainly in accordance with the inhibitory theory.

If there are no modulating inputs on the cortex, but only a common driving input, as K increases, the interaction between the two cortexes moves the WP toward a more linear region of the sigmoid. After a certain value of K, i.e., increasing the strength of the TC, the laterality increases because one hemisphere is inhibited and the other tasked with the performance of the specific cognitive process. This is in accordance with the finding of Fling et al., which reported a strong, positive relationship between the strength of interhemispheric inhibition and the microstructure of interhemispheric fibers that is specific to tracts connecting the primary motor cortexes. In particular, an increased fiber microstructure in young adults predicts interhemispheric inhibitory capacity (Fling et al., 2013).

With the introduction of the modulating input, the behavior remains of inhibitory type: the inhibition of one cortex leads to the excitation of the other cortex, and vice versa. This behavior, causing an opposite shift in the WP of the two areas, can explain the observed ERD in the target cortex and the observed ERS in the ipsilateral cortex, and enforces the inhibitory theory: the two hemispheres are in constant mutually inhibitory relationship with each other and this competition is mediated by the corpus callosum. Also, included in this theory there is the notion that the corpus callosum serves as an “inhibitory barrier” between hemispheres to prevent maladaptive cross talk between the hemispheres for which a given function is dominant (Kinsbourne, 1982; Bloom and Hynd, 2005; Welcome and Chiarello, 2008; Adam and Güntürkün, 2009).

As previously investigated by several researchers (Filippini et al., 2010; Lindenberg et al., 2012; Halder et al., 2013), the integrity and the number of connections of the deep white matter structure are predictors of SMR-based BCI performance, and there is a relation between the white matter architecture and SMR-BCI aptitude. In particular, the corpus callosum was one of the top five white matter regions which were found to be most discriminating in the low vs. high BCI-aptitude group comparison, showing significant correlations (p < 0.05) with individual BCI-performance. According to our findings, when corpus callosum had a too low number of fibers, we expected a lower ERD and ERS amplitude and consequently lower BCI performances. At the increasing of corpus callosum fibers number, the generation of ERD/ERS becomes optimal until reaching a too high number of fiber and only an ERD is produced.

Halder et al. proposed that the best strategy to improve BCI performance in low aptitude users is by conducting a long-term BCI training program consisting of multiple sessions, which not only targets to increase proficiency in BCI usage for communication and control, but also attempts to incorporate interventions to increase or stabilize the microstructural integrity of BCI-critical central white matter (Halder et al., 2013).

In this context and in the light of our findings, we propose a further strategy to be combined with the one just presented. It consists in the use of neuromodulation techniques during BCI training (Matsumoto et al., 2010; Wei et al., 2013; He et al., 2014). In particular, an appropriate stimulation could move the neural circuit in an ideal WP, which could improve the information transfer between the two cortexes and increase the ERD/ERS amplitudes.

The validation of a model requires predictive ability to be tested. As the proposed model is based on several assumptions separately validated in the previously discussed studies (Pfurtscheller and Neuper, 1997; Suffczynski et al., 2001; Wendling et al., 2002; Grabska-Barwińska and Żygierewicz, 2006; Zavaglia et al., 2006; Fling et al., 2013), the model should simulate accurately the phenomenon of beta ERD/ERS generation and its dependency on TC. A specific protocol for the validation of the proposed model should include the recording of both:

EEG signals from the SMA, during resting state condition and motor imagery tasks. The PSD and ERD/ERS should be extracted and fitted with model outputs to identify a K-value for each participant.

The analysis of the correlation between K and Function Anisotropy could provide the validation of the model, demonstrating the dependency of the ERD/ERS amplitude on TC and the phenomenon of Transcallosal Inhibition. This is planned in future works.