Self-organized criticality and scale-free properties in emergent functional neural networks

Recent studies on complex systems have shown that the synchronization of oscillators, including neuronal ones, is faster, stronger, and more efficient in small-world networks than in regular or random networks. We show that the functional structures in the brain can be self-organized to both small-world and scale-free networks by synaptic reorganization via spike timing dependent synaptic plasticity instead of conventional Hebbian learning rules. We show that the balance between the excitatory and the inhibitory synaptic inputs is critical in the formation of the functional structure, which is found to lie in a self-organized critical state.

Figure 1

(a) Connection probability of the STDP network $⟨k∕N⟩$ in the range between 0.03 (dark) and 0.6 (light), (b) phase coherence $\Gamma$ between 0.4 (dark) and 1.0 (light), and (c) the ratio of the clustering coefficients between our model and the random network with the same connection probability $C∕C_{\mathrm{rand}}$ between 0.5 (light) and 7.5 (dark) in the parameter space of $G_{\max }$ and $G_{\mathrm{inh}}$ in log-log scale. The results are averaged over 30 different initial conditions.

Figure 2

Log-log plots of degree distributions of the functional STDP network for the optimal synaptic couplings. (a) In-degree probability distribution and (b) out-degree probability distribution for $N=1000$ and $N=10000$. The scaling exponents are ${\gamma }_{\mathrm{in}}\approx 1.7$ and ${\gamma }_{\mathrm{out}}\approx 1.5$ in the intermediate $k$. The results are averaged over 100(40) runs with different initial conditions for $N=1000$ $\left(10000\right)$.

Figure 3

(a) Distribution of the size of the change of the total synaptic coupling strengths and (b) the power spectrum of the fluctuation of the synaptic strengths in log-log scale showing power-law decays with exponents ${\gamma }_s\approx 2.5$ and ${\gamma }_f\approx 2.0$.

Figure 4

Phase coherence $\Gamma$ of the STDP network (solid lines) and the random networks (dashed lines) when a suprathreshold external stimulus is subjected to all the neurons (a) and only half of neurons (b), with a random initial phase of the neurons.