Figure 1.
(a) The thirteen different motifs of size 3. (b–e) Connectivity matrices, and (f–i) Structural motif counts for each cortical network. Data (from the CoCoMac database [67], [68]) and algorithms are available at the brain connectivity toolbox website [69].
Figure 2.
Synchronization in motifs of Hodgkin-Huxley neurons.
Dynamics of common driving motif (M3) versus common driving motifs with resonant sources (M6, M9 and M3+1) in motifs of excitatory delayed-coupled neurons with delay . First and second columns (panels a, b, e, f, i, j, m, n, q, r) correspond to individual spiking time traces of neurons, whereas the third and forth columns (panels c, d, g, h, k, l, o, p, s, t) correspond to the cross-correlation functions of the corresponding single time series and average over 40 trials respectively. Descending rows show motifs M3, M6, M9 and M3+1, respectively.
Figure 3.
Synchronization dynamics and incoherence in Hodgkin-Huxley neurons.
(a) Incoherence in motif M3 does not depend on time delay. Colors indicate the different nodes. (b–d) Top panels show incoherence for M6, where colors represent different nodes, and bottom panels show crosscorrelations for M6 (blue) and M3 (black). Continuous lines indicate the cross-correlation coefficients at zero time lag, and dashed lines indicate the maximum cross-correlation coefficients for all time lags. Panels b, c and d represent pairs of nodes: 1–2, 1–3, and 2–3 respectively. Phase, anti-phase synchrony, and asynchrony can be found in motif M6 depending on the time delay (see exemplar time traces in supplementary Fig. S2). Results are averaged over 40 trials.
Figure 4.
Synchronization in motifs of populations of Izhikevich neurons.
Panels (a–t) as per Fig. 2, for populations of 500 (400 excitatory and 100 inhibitory) spiking neurons and delay .
Figure 5.
Synchronization in motifs of neural mass models.
Panels (a–t) as per Fig. 2, but for neural mass models with coupling strength c = 0.01, and delay .
Figure 6.
Robustness of the synchronization with respect to mismatch in the delays.
The top schemes (a) illustrate the motifs of neurons considered. Motifs M6′, M9′ and M3+1′ have one connection with delay , and all the other connections have delays of 6 ms. The bottom panels show the zero-lag crosscorrelation between nodes 1 and 3 in motifs of Hodgkin-Huxley neurons (b) and in motifs of neural mass models with c = 0.01 (c) averaged over 40 trials for varying
. Panel (d) shows the same as (c) but across a broader range of
. Plot colors correspond to motifs as per panel a.
Figure 7.
Zero-lag crosscorrelation between neural masses 1 and 3 for different common-driving motifs.
Top: Common-driving motifs, labeled as per Sporns and Kötter (2004) [9], see Fig. c1 a. Bidirectional connections (red stars) indicate active resonance pairs. Top row panels compare common driving (M3) to common driving with resonance pairs (M6, M9 and M3+1) for varying coupling (panels a and b) and varying delay (panel c). Bottom row panels compare common driving (M3) to a ring of mutually coupled nodes (M13), and to common driving plus a bidirectional connection between 1 and 3 (M8) as a function of coupling (panels d and e) and time delay (panel f). Curve colors correspond to the motifs depicted on the top of the figure.
Figure 8.
Propagation of the effect of a resonance pair along a chain.
(a) A resonance pair (nodes N and N-1) arbitrarily distant from a pair of commonly driven neural masses (1 and 3). (b) A seven-node chain configuration with a common-driving motif at the edge. (c) Zero-lag cross-correlation functions between nodes 1 and 3 for different chain sizes as illustrated in panel (a) are shown in solid lines. Thin dashed line represents the chain of panel (a) without the feedback connection from node N-1 to node N. (d) Zero-lag cross-correlation functions between every other node in the chain depicted in panel (b) are shown in solid lines. Thin dashed line represents the chain of panel (b) without the feedback connection from node 6 to node 7.
Figure 9.
Fast transient behavior and onset of synchronization.
(a) Example of time-trace synchronization following random initial conditions (starting at time = 0) and consequent to a brief perturbing current (green bar) at time = 1000 ms in motif M6 with c = 0.01. (b) averaged over 400 trials with c = 0.01 in motifs M3 compared to M6. (c) Exponent
estimated from
averaged over 400 trials on the interval between 1200 and 2400 ms for varying coupling strengths in motifs M3, M6 and M9. Delay
.
Figure 10.
Fine-tuning can enhance synchronization.
(a) Crosscorrelation averaged over 40 trials, (b) dominant oscillatory frequencies of neural masses 1 (green) and 2 (magenta) as a function of the mismatch on the input current over node 2. (c) Dominant oscillatory frequencies of neural masses 1 (green) and 2 (magenta) for varying coupling strength.
Figure 11.
Effect of resonance chains on the synchronization.
(a) Loops of reciprocally connected versus unidirectional connected loops. (b) Zero-lag cross-correlation between neural masses 1 and 3 with neural mass 2 connected to bidirectional or unidirectional chains of varying length. Blue dashed line highlights the effect of the resonance pair, and green (magenta) dashed line highlights the effect of the resonance triplet (quad). Red (yellow) curve represents the cross-correlation averaged over 40 trials for reciprocally (unidirectionally) connected loops. The coupling strength is 0.01, and delay 10 ms.
Figure 12.
Propagation of synchrony to pairs of nodes at higher orders of distance.
Common driving to first (1,1′), second (2,2′) and n-th (n,n′) order for the resonance-induced pair (a), a unidirectional input (b), and simple common driving (c). (d) to (g): Zero-lag crosscorrelation for the different types of common driving from the first to the forth order versus the coupling strength. (h) Zero-lag crosscorrelations between pairs of nodes (n,n′) as a function of the distance from the driver node A. (i) Maximum (non-zero-lag) crosscorrelations as a function of the distance from the driver node A. Red, yellow and black curves represent the crosscorrelation averaged over 40 trials for the system depicted in (a), (b) and (c) respectively.