Macroscopic and microscopic spectral properties of brain networks during local and global synchronization

We introduce a practical and computationally not demanding technique for inferring interactions at various microscopic levels between the units of a network from the measurements and the processing of macroscopic signals. Starting from a network model of Kuramoto phase oscillators, which evolve adaptively according to homophilic and homeostatic adaptive principles, we give evidence that the increase of synchronization within groups of nodes (and the corresponding formation of synchronous clusters) causes also the defragmentation of the wavelet energy spectrum of the macroscopic signal. Our methodology is then applied to getting a glance into the microscopic interactions occurring in a neurophysiological system, namely, in the thalamocortical neural network of an epileptic brain of a rat, where the group electrical activity is registered by means of multichannel EEG. We demonstrate that it is possible to infer the degree of interaction between the interconnected regions of the brain during different types of brain activities and to estimate the regions' participation in the generation of the different levels of consciousness.

Figure 1

(a) Wavelet decomposition of the macroscopic signal, produced by a monolayer network of phase oscillators under the effect of adaptive mechanism. Lower row: momentum distributions of the wavelet energy and the network topologies. Panel (b) refers to $t_1=500$ s (i.e., before adaptation starts to take place in the system), while panel (c) is obtained for $t_2=500$ s (i.e., after the adaptive process has produced a time independent network's topology). Both instants at which the lower panels are obtained, are indicated with vertical arrows in panel (a). ${\lambda }_1=0.5,T=100$ s. See main text for all other specifications.

Figure 2

(a) Schematic illustration of the two-layered network. Top right panels: $W_1(f,t)$ [panel (b)] and $W_2(f,t)$ [panel (c)] (see text for definitions), and the momentum distributions of the wavelet energy $W_1(f,t^{*})$ [blue curve in panel (b)] and $W_2(f,t^{*})$ [red curve in panel (c)], obtained for $t^{*}=1400$ s. (d) The surface of wavelet product Eq. (6). ${\lambda }_1=0.5,{\lambda }_2=0.005,T=100$ s. See main text for all other specifications.

Figure 3

Five-layered network ($M=5$) for ${\lambda }_1=0.5,{\lambda }_2=0.005,T=100$ s. The upper left panel reports the two similarity functions of Eqs. (7) and (8) (see text for definitions). All other panels report the wavelet energy spectra $W\left(f\right)$ (left plots) and the effective frequency distributions $N\left(f\right)$ (right histograms) for the different layers, calculated at $t^{*}=1500$ s.

Figure 4

(a) Schematic illustration of the experimental setup. The panel is a drawing of the rat's brain, with marked regions of cortex and thalamus, which contain the location of the recording electrodes that actually register the group neuronal activity by means of local-field potential (LFP) and the activity of the single cells (unit recordings). (b) The set of registered neurophysiological signals, reflecting the activity of single cells (unit S1 and unit PO) and the group activity in cortex and thalamus (LFP S1 and LFP PO). (c–e) Wavelet decompositions of the macroscopic signals. (c) $W_{\mathrm{PO}}(s,t)$ obtained from signal LFP PO, (d) $W_{\mathrm{S}1}(s,t)$ obtained from signal LFP S1, and (e) $W(s,t)=W_{\mathrm{S}1}(s,t)\times W_{\mathrm{PO}}(s,t)$. The solid circles and the solid squares shows the main spectral component of the signals, taken from the single cell in thalamus (unit PO) and cortex (unit S1), respectively. $s=1/f$ indicates the timescale, and $f$ the linear frequency. The instant of time at which there is the onset of SWD is shown by an arrow and marked by a vertical white dashed line.

Figure 5

Typical fragments of registered EEG signals, reflecting the activity of the neuronal group in cortex and thalamus during the epileptic discharge (SWD, a), during active wakefulness (AW, b), and during deep slow-wave sleep (DSWS, c). The epochs corresponding to the onset (SWDO) and end (SWDE) of SWD, to AW and DSWS are marked by rectangles.

Figure 6

Left panel: the ratio of the wavelet energies [Eq. (9)], calculated for all considered states (columns) and all channels (rows). Right panel: the energy values [Eq. (9)] averaged over the cortical (solid bars) and thalamical (empty bars) channels. The error bars stand for the maximal deviation within the channel set. Once again, SWDO, SWDE, AW, and DSWS stay for the SWD onset, the SWD end, the active wakefulness, and the deep slow-wave sleep, respectively.

Figure 7

(a) The mean value of the strength of interaction between the different parts of the thalamocortical neuronal network. The values [calculated for whole network ($w_{ij}^1$), for the cortex ($w_{ij}^2$), and for the thalamus ($w_{ij}^3$)] are averages of the coefficients $w_{i,j}$ over the considered parts of the brain. $w_{i,j}$ are estimated via Eq. (7), based on the similarity of the wavelet spectra. (b) The schematic illustration of the coefficients $w_{i,j}$, reflecting the degree of interaction between the different parts of the cortex and thalamus, for all considered states of brain network and frequency bands. The values of $w_{i,j}$ are shown by the increase (or decrease) of the line width, which connect the corresponded brain structures. Other stipulations are as described in the caption of Fig. 6.

Figure 8

(a) The mean values of the strength of interaction between the different parts of the thalamocortical neuronal network, calculated for the time interval corresponding to the development of the SWD. The values $w_{ij}^2$ (for the cortex) and $w_{ij}^3$ (for the thalamus) are averages of the coefficients $w_{i,j}$ over the considered parts of the brain. $w_{i,j}$ are estimated via Eq. (7), based on the similarity of the wavelet spectra. The values $u_{ij}^2$ (for the cortex) and $u_{ij}^3$ (for the thalamus) are also averages of the coefficients $u_{i,j}$ over the same parts of the brain, where $u_{i,j}$ are estimated based on the wavelet coherence [25]. The EEG samples, corresponded to the first seconds of the SWD development (b) (vicinity of time moment $t_1$) and (c) $t_2$.