The authors have declared that no competing interests exist.
Conceived and designed the experiments: AB CW SL. Performed the experiments: AB CW. Analyzed the data: AB CW. Contributed reagents/materials/analysis tools: AB CW SL. Wrote the paper: AB CW SL.
Repetitive transcranial magnetic stimulation (rTMS) holds promise as a non-invasive therapy for the treatment of neurological disorders such as depression, schizophrenia, tinnitus, and epilepsy. Complex interdependencies between stimulus duration, frequency and intensity obscure the exact effects of rTMS stimulation on neural activity in the cortex, making evaluation of and comparison between rTMS studies difficult. To explain the influence of rTMS on neural activity (e.g. in the motor cortex), we use a neuronal network model. The results demonstrate that the model adequately explains experimentally observed short term effects of rTMS on the band power in common frequency bands used in electroencephalography (EEG). We show that the equivalent local field potential (eLFP) band power depends on stimulation intensity rather than on stimulation frequency. Additionally, our model resolves contradictions in experiments.
Concerning the repetitive application of transcranial magnetic stimulation (rTMS), it has been suggested that low stimulation frequencies (≤1 Hz) have lasting inhibitory effects at least on motor cortex areas
Furthermore, broad consensus has not yet been established regarding the dependency of the rTMS stimulation frequency on EEG band power. While some studies report no significant changes in the alpha band after 10 Hz rTMS stimulation
The influence of rTMS on the power in the delta band is also subject to experimental research. Activity increases in the delta band after the application of 10 Hz rTMS are stated
In order to understand the influence of stimulus frequency and stimulus intensities on EEG measurements, we provide a neuronal network model which allows for the simulation of rTMS and the observation of its effects after the end of the stimulation period.
Such networks have been intensively examined in the past few decades, resulting in different insights into network behaviour and a broad range of different models. While physical models focus mainly on the oscillatory behaviour of the model components (e.g. the Kuramoto oscillator)
Some of the existing models already integrate the effect of TMS on the network behaviour, but they differ in their degree of abstraction. Detailed models represent the behaviour of the neuronal entities and the TMS induced axial and transmembrane currents by sets of differential equations
We define a random graph based asymmetric Hopfield network with synchronous updates
Spiking neurons are coloured red, non-spiking neurons are coloured blue. Synaptic connections between the neurons are either excitatory (black) or inhibitory (orange). The functions Ii(t) and Ei(t) describe the input and output of each neuron. The threshold Θi triggers the generation of a spike and is set to −60 mV. The constant c is denoting the change in postsynaptic potential triggered by the arrival of an action potential (AP) at the synaptic cleft. The model output ρ(t) consists of the sum of the output Ei(t) of all neurons.
We state that the network size (1000, 2000, 4000, 8000 and 16000 neurons) has no significant influence on the band power. Instead, we identify the average number of synaptic connections maintained by each neuron as dominant parameter. Our numerical analysis of the network behaviour showed that an average node degree k = 120 could explain best the EEG measurements of the motor cortex areas subject to rTMS stimulation. We therefore assume that our model represents the effect of rTMS stimulation on the motor cortex.
The effects are summarized in
TIME | F(1,16) = 574,7 | p<5,8e-14 |
TMS-INTENSITY | F(3,48) = 3,3 | p<0.03 |
FREQUENCY-BAND | F(4,64) = 21626 | p<0,0000 |
TIME*TMS-INTENSITY | F(3,48) = 21,3 | p<6,7e-9 |
TIME*TMS-FREQUENCY | F(5,80) = 6,1 | p<0,000077 |
TIME*FREQUENCY-BAND | F(4,64) = 3152,5 | p<0,0000 |
TMS-INTENSITY*FREQUENCY-BAND | F(12,192) = 70,3 | p<0,0000 |
TMS-FREQUENCY*FREQUENCY-BAND | F(20,320) = 29,7 | p<0,0000 |
TIME*TMS-INTENSITY*TMS-FREQUENCY | F(15,240) = 1,9 | p<0,03 |
TIME*TMS-INTENSITY*FREQUENCY-BAND | F(12,192) = 93,8 | p<0,0000 |
TIME*TMS-FREQUENCY*FREQUENCY-BAND | F(20,320) = 40,1 | p<0,0000 |
TMS-INTENSITY*TMS-FREQUENCY*FREQUENCY-BAND | F(60,960) = 7,0 | p<0,0000 |
TIME*TMS-INTENSITY*TMS-FREQUENCY*FREQUENCY-BAND | F(60,960) = 7,7 | p<0,0000 |
In general, the stimulation frequency has no significant effect on the band power. But for a stimulation frequency of 0.5 Hz, we stated a selective influence on the delta, theta, alpha and gamma band.
The stimulation intensity has strong, but varying effects on the band power in the frequency bands. In the delta band, we observed a decreased band power for the stimulation intensities 4, 6, and 8 but an increase for a stimulus intensity of 10 compared to the mean band power before TMS (
Mean band power values are given in V/√Hz and represent an average over the runs.
Ignoring the stimulation frequency of 0.5 Hz, we observed a linear decrease of the alpha band power (Pearson product moment correlation: r = −1 p<0.05) and beta band power (r = −0.99, p<0.05) and a linear growth of the gamma band power (r = 0.99, p<0.05) while the power of the theta band (r = 0.6) did not show a statistically significant linear behaviour (
Data were averaged separately over all segments and separately for TMS-intensity over all TMS frequencies except 0.5 Hz for POST. Error bars indicate standard deviation of the factor POST.
Concerning the stimulation intensity, we observed significant changes of the band power (
Our model turns out to be beneficial for explaining the experimentally observed effects of rTMS mentioned in the related work section for the following reasons. First, it integrates the effect of rTMS into the neural model by considering rTMS as an external input source. Second, we work with the summed output of all neurons (eLFP) which we consider to be comparable to the local field potential (LFP) for two reasons: a) we defined a network of interconnected excitatory and inhibitory neurons, whose synaptic activity lead to local changes of the electric field, and b) we neither included long range connections from or to different networks nor did we look at the action potentials of a small subset of neurons which would be compatible to physiological measures like multi-unit activity (MUA). Third, we fit the model with experimentally validated parameter values
Regarding the choice of our model, we consider a random graph
The network size has no statistical relevance. This scale-freeness of the model has important consequences: a) the results can be transferred on networks of different sizes and b) we can predict local field potentials, cortical potentials and even potentials on the surface of the head. This complies with experimental findings that indicate the independence of scale
The results of the simulation itself aim at identifying how band power depends on rTMS frequency and intensity, and, in doing so, advance the development of rTMS as an efficient therapeutic treatment for mental disorders
The results replicate
We observe an increase in the gamma band power for all stimulus intensities and frequencies above 0.5 Hz. These findings agree with
In the alpha band, the simulation shows that changes strongly depend on rTMS intensity. This observation accounts for contradictory experimental results, regarding the effect of rTMS on the power in the EEG alpha band
As seen in recent research studies and confirmed by these findings, the rTMS intensity does significantly affect the EEG power across all frequency bands. The stimulation intensity influences the network activity by causing a higher number of spikes at the neuronal level and accordingly changes of the eLFP with increasing intensity. It is important to note that the effects are studied on a neuronal network level which means that our model neglects potential side effects of the applied rTMS energy.
This model does not explain all experimentally observed results of rTMS. Our simulation and most
In addition, we assume a uniform perturbation of the network by the rTMS stimulation. It is left open how the stimulation affects remote brain areas not directly affected by the magnetic field. The effects of rTMS on remote, not directly stimulated cortical areas are not included in this model. We believe that first and foremost, it is vital to understand the effects of rTMS on the directly stimulated brain areas and to close the gap between model behaviour and EEG measurement data.
Our model for rTMS experiments offers an opportunity to corroborate and predict experimental results, which can be used to further develop the model. This approach could be extended to allow the development of models that depict pathological activity like depression, schizophrenia or epilepsy and the effects of rTMS on abnormal brain activity, representing an important preliminary step towards the clinical use of rTMS.
Our simulation is based on a random network
The discrete time evolution for each neuron is given by the input function Ii(t) and the output function Ei(t) with a threshold value of Θi = −60 mV for the generation of an action potential (AP), and with constant c denoting the change in postsynaptic potential triggered by the arrival of an AP at the synaptic cleft
Whether the synapses are of the inhibitory or the excitatory type is coded in the randomly generated adjacency matrix aij
Network size | Connection probability p | Average number of neighbours k |
1000 | 0.8799 | 120 |
2000 | 0.9400 | 120 |
4000 | 0.9700 | 120 |
8000 | 0.9850 | 120 |
16000 | 0.9925 | 120 |
For determining the stimulus intensities, we assume that a single TMS pulse triggers the simultaneous depolarization of neurons
The spectral power of the output ρ(t) is calculated using the Fast-Fourier-Transform (FFT) and averaged within the common frequency bands (delta: 0.5–4 Hz, theta: 4–8 Hz, alpha: 8–12 Hz, beta: 12–24 Hz, gamma: 24–48 Hz). The estimates of the band power are statistically analysed using STATISTICA 6.1. We perform a repeated measures ANOVA with band power as dependent variable for the 17 runs (random factor), the repeated measures factor TIME (2 steps: PRE, POST), and 3 fixed factors: TMS-FREQUENCY (6 steps: 0.5 Hz, 1 Hz, 2 Hz, 5 Hz, 10 Hz, 20 Hz), TMS-INTENSITY (4 steps: 4, 6, 8, 10) and FREQUENCY-BAND (5 steps: delta, theta, alpha, beta, gamma).