MEG simulations using local propagation of traveling waves calculated using neural field equations
Vitaly M. Verkhlyutov, Evgenii Olegovich Burlakov · Zenodo (CERN European Organization for Nuclear Research) · 2023
Abstract Algorithms are shown here that reproduce the propagation of the electrical activity of the cortical surface of the human brain in the form of a traveling wave, which has (1) a sinusoidal profile (2) a wave profile calculated according to the neural field equations of a traveling wave. A feature of the current algorithms is the gradual increase in the activated cortical tissue, in contrast to our previous solutions, when the instantaneous course of this process was reproduced. The influence of functional inhomogeneities on the spread process of the waves of profile (2) is studied. The influence of neural field microstructure on the process of the spread of traveling waves We study the following two-dimensional homogenized Amari neural field equation \(\partial_tu(t,\mathbf{x},\mathbf{y})=-u(t,\mathbf{x},\mathbf{y})+\int_{\mathbb{R}^2}\int_Y\omega(|\mathbf{x}-\mathbf{x}'|,\mathbf{y}-\mathbf{y}')f(u(t,\mathbf{x}',\mathbf{y}'))d\mathbf{y}'d\mathbf{x}', \) where \(\mathbf{x},\,\mathbf{x}'\in\mathbb{R}^2\) and \(\mathbf{y}\in Y=[0,1)^2\). The weight distribution function \(\omega(\mathbf{x},\mathbf{y})\) is taken to be positive function in \(\mathbf{L}^1(\mathbb{R}^2_+,Y)\), and periodically modulated for the local scale variable \(\mathbf{y}\). We denote by \(\langle\omega\rangle\) the average of connectivity function \(\omega \) over the unit cell, i.e. \(\langle\omega\rangle(\mathbf{x})=\int_Y\omega(\mathbf{x},\mathbf{y})d\mathbf{y}\quad\forall\mathbf{x}\in\mathbb{R}^2. \) For the activation function \(f:\mathbb{R}\rightarrow\mathbb{R}\) we assume that \(0\leq f (u)\leq 1\). The firing-rate function \(f(u)\) for Amari model is chosen to be the Heaviside step function. Thus, we have \( \partial_tu(t,\mathbf{x},\mathbf{y})=-u(t,\mathbf{x},\mathbf{y})+\int_{\mathbb{R}^2}\int_Y\omega(|\mathbf{x}-\mathbf{x}'|,\mathbf{y}-\mathbf{y}')H(u(t,\mathbf{x}',\mathbf{y}'))d\mathbf{y}'d\mathbf{x}'.\label{ama_mod}\) We make the following notations \(\mathcal{W}_l(\mathbf{y})= \left( \begin{matrix}\displaystyle a|U'(b)|\int_0^{2\pi}\omega\left(\sqrt{2a^2-2a^2\cos(\phi)},\mathbf{y}\right)\cos(l\phi)d\phi&\displaystyle b|U'(a)|\int_0^{2\pi}\omega\left(\sqrt{a^2+b^2-2ba\cos(\phi)},\mathbf{y}\right)\cos(l\phi)d\phi\\\\\displaystyle a|U'(b)|\int_0^{2\pi}\omega\left(\sqrt{b^2+a^2-2ba\cos(\phi)},\mathbf{y}\right)\cos(l\phi)d\phi&\displaystyle b|U'(a)|\int_0^{2\pi}\omega\left(\sqrt{2b^2-2b^2\cos(\phi)},\mathbf{y}\right)\cos(l\phi)d\phi \end{matrix} \right),\) \(\Phi_l(\mathbf{y})=\left( \begin{matrix} \varphi_l(a,\mathbf{y})\\ \varphi_l(b,\mathbf{y})\end{matrix} \right)\), \(\mu=(\lambda+1)|U'(a)||U'(b)|\), where \(U'(a) \ \mbox{and} \ U'(b)\) are the slopes of the tangent lines to the radial component of the wave at the intersection points \(a \ \mbox{and} \ b\) with the threshold of neuronal activation, for all \(l\in\mathbb{Z}\). We also make the following notations: \(\hat\Phi_{lmn}=\int_Y\Phi_l(y_1,y_2)e^{-2i\pi my_1}e^{-2i\pi ny_2}dy_1dy_2\) and \(\widehat{\mathcal{W}}_{lmn}=\int_Y\mathcal{W}_l(\mathbf{x},(y_1,y_2))e^{-2i\pi my_1}e^{-2i\pi ny_2}dy_1dy_2.\) We now calculate \(\lambda^\pm_{lmn}\) as \(\lambda^\pm_{lmn}=\frac{\mu^\pm_{lmn}}{|U'(a)||U'(b)|}-1\), where \(\mu^\pm_{lmn}=\frac{\mathrm{tr}(\widehat{\mathcal{W}}_{lmn})\pm\sqrt{(\mathrm{tr}(\widehat{\mathcal{W}}_{lmn}))^2-4\det(\widehat{\mathcal{W}}_{lmn})}}{2}. \) We can now formulate the following result on the dependence of the spread of radial traveling waves on neural field microstructure (given by the second argument of the function \(\omega\)): If \(\lambda^\pm_{lmn}\) are greater or equal to 0, then the neural field with the corresponding parameters supports the spread of radial traveling waves, whereas for the case when \(\lambda^\pm_{lmn}\) is less than 0, there is no traveling waves in the neural field. The proof of this statement as well as a numerical illustration of its application are given in the additional file The influence of neural field microstructure on the process of the spread of traveling waves.pdf. Demonstration instruction Load NeuField.zip protocol to Brainstorm (https://neuroimage.usc.edu/brainstorm/). This database allows one to view a demonstration of sine and neurofield waves. To make your own calculations, use the create_sigm.m for neurofield calculations by setting LoadSig=1 in load_sigm.m, and then run load_sigm.m. To calculate the new sine, one does not need to run create_sigm.m , but one needs to set LoadSig=0. Next, one needs to upload the calculations to Brainstorm. Additional Files function&script - functions and scripts for Matlab. Alr - Connectivity matrix for a surface. CorLR - Triangulated cortical surface. Travelling pulses in neural fields.pdf - preprint on the topic. RSSCDNFE.pdf - preprint on the topic. The influence of neural field microstructure on the process of the spread of traveling waves.pdf - proof of the result on the topic and a numerical illustration. Acknowledgments The reported study was funded by RFBR, project number 20-015-00475.