Table 1.
All parameters have been adapted from Jansen et al. [35] except for the sinusoidal input in Eq. (6).
Figure 1.
Chaotic behavior of an isolated, periodically driven cortical column.
(A) Maximal Lyapunov Exponent (MLE) for different driving amplitudes and frequencies. Parameter values are those given in Table 1. (B) Regularity parameter obtained in the same conditions as in panel A. (C) Example of a chaotic signal obtained for a driving frequency f = 8.5 Hz and scaled driving amplitude δ/C = 0.49 Hz. With these parameters MLE = 5.24 and Reg = 0.37. (D) Power spectrum of the signal shown in C. This complex spectrum shows a narrow peak at the driving frequency (see inset).
Figure 2.
Regularity of the coupled system.
(A) Topology of the network of cortical macrocolumns studied below. The network is constructed using the Barabási-Albert algorithm with m0 = 1 initial nodes (see Materials and Methods for details). (B) Regularity parameter averaged over 1 network of 50 coupled cortical columns (see panel (A)), for 20 different realizations of the initial conditions and for varying excitatory and inhibitory coupling intensities. Darker regions indicate less regular dynamics (chaotic), whereas lighter regions indicate more regular dynamics. Annotations in the plot indicate parameter values that will be studied later in Figs. 3, 4A, 4C, 6A and 6B.
Figure 3.
Node-pair correlation and dynamical clustering.
(A) Maximal Cmax(τ) (y axis) and average regularity (color coded) between pairs of cortical columns for scaled α/C = 0.56 and β/C = 0.26 (point 1 in Fig. 2B). Connected (not connected) pairs of nodes are shown as triangles (circles). The dynamical evolution of selected node pairs is shown in panels B-E. In (E) the two nodes are synchronized at zero lag, and one of the time traces has been shifted horizontally for clarity.
Figure 4.
Excitatory/inhibitory segregation of cortical columns.
(A) Example of a scale-free network characterized by all nodes exhibiting a dominant excitatory dynamics, illustrated by empty circles. Here α/C = 0.790 and β/C = 0.037 (Fig. 2B, point 2). (B) Average distribution of the activity ⟨ye(t)−yi(t)⟩, obtained from the analysis of 50 different scale-free architectures using different random seeds for the same α and β parameters as in panel A. The extreme values in the tail of the distribution are not represented. (C) The same scale-free network of panel A but now for the case of a mostly inhibitory intercolumnar coupling, with β/C = 0.190 compared to α/C = 0.075 (Fig. 2B, point 3). Here full circles represent inhibitory nodes, and empty circles denote again excitatory nodes. (D) Average distribution of ⟨ye(t)−yi(t)⟩ corresponding to panel C.
Figure 5.
Excitatory-Inhibitory Segregation index (EIS).
The EIS index quantifies the relative distribution of excitatory- and inhibitory-dominated dynamics. Three different domains exist, labeled by ‘N’ (no segregation), ‘L’ (low segregation) and ‘S’(high segregation).
Figure 6.
Topological organization of the excitation-inhibition segregation.
Average activity of all nodes of the networks as a function of their degree, corresponding to the two cases analyzed in Figs. 4A and 4C, respectively. Color coding denotes the regularity for each node in the networks, and upper panels show the average regularity in the nodes’ dynamics as well as its standard deviation.
Figure 7.
Segregation of two coupled Wilson-Cowan oscillators.
(A) Coupling scheme between the two oscillators. (B) Time traces of the subtracted signal xi−yi for the two segregated oscillators, in an excitatory-dominated coupling scheme (Kexc = 1, Kinh = 0.1). (C) Resulting segregation map for different excitatory and inhibitory coupling strengths.