Network Effects in Neurostimulation (2)

Dataset used to training

The dataset we used is from Paulk, 2022, in which single pulse-induced cortico-cortical evoked potentials (CCEPs) were recorded. The data is from F-TRACT database.

Reasons of training with original Deco model

As mentioned at the end of the previous blog, I derived a modified model.

Correlation is significantly higher in critical region

Analysis of the phase transition line

Calculate the phase transition line

\[ \begin{aligned} \frac{\mathrm{d}S_{i}(t)}{\mathrm{d}t} &= -\frac{S_{i}}{\tau_{S}} + (1-S_{i})\gamma H(x_i) \\ H(x_i) &= \frac{ax_{i}-b}{1-\exp (-d(ax_{i}-b))} \\ x_{i} &= wJ_{N}S_{i} + GJ_{N}\sum_{j}^{} C_{ij}S_{j} \end{aligned} \]

• First we consider \(G=0\)

\[ \tau_S \gamma (1-S_i)H(wJ_N S_i) = S_i \]

At critical point, \[ \tau_S \gamma H(wJ_N S_i)+1 = \tau_S \gamma (1-S_i) wJ_N \frac{\mathrm{d}H}{\mathrm{d}x}(wJ_N S_i) \]

\[ w = w_0 \approx 2.9185 \]

• For \(G>0\), denote \(\lambda\) as the spectral radius of struc_conn_matrix.

If there exists a high stable state, we have the following lemma: Lemma \(\exists k \in \{1,\cdots ,N\}\), such that \[ \sum_{j=1}^{N} \mathbf{C}_{kj}S_{j} \leqslant \lambda S_{k}. \]

proof Since \(C\) is symmetric, we have (https://en.wikipedia.org/wiki/Spectral_radius)

\[ \left\| \mathbf{C}\mathbf{S} \right\|_{2}\leqslant \lambda \left\| \mathbf{S} \right\|_{2}, \]

i.e., \[ \sqrt{\sum_{k=1}^{N} \left( \sum_{j=1}^{N} C_{kj}S_j \right) ^{2}}\leqslant \lambda \sqrt{\sum_{k=1}^{N} S_k^{2}} \]

Thus, \(\exists k \in \{1,\cdots ,N\}\), such that \[ \sum_{j=1}^{N} C_{kj}S_j \leqslant \lambda S_{k} \]

The existence of the high stable state is equivalent to \[ S_i = \tau_S \gamma (1-S_i)H(J_N (w S_i + G\sum_{j=1}^{N} C_{ij}S_j)) \]

Since \(H\) is increasing, by the lemma we know that \(\exists k \in \{1,\cdots ,N\}\), \[ S_k = \tau_S \gamma (1-S_k)H(J_N (w S_k + G\sum_{j=1}^{N} C_{kj}S_j)) \leqslant \tau_S \gamma (1-S_k)H(J_N (w + \lambda G)S_k) \]

This can only happen when \(w + \lambda G \geqslant w_0\).

Information in N2 component implies effective connectivity

Further problems

Refrence

Paulk, A.C., Zelmann, R., Croker, B., et al. (2022). Local and distant cortical responses to single pulse intracranial stimulation in the human brain are differentially modulated by specific stimulation parameters. Brain Stimulation. 15, 491-508