Self-organization of feed-forward structure and entrainment in excitatory neural networks with spike-timing-dependent plasticity

Spike-timing dependent plasticity (STDP) is an organizing principle of biological neural networks. While synchronous firing of neurons is considered to be an important functional block in the brain, how STDP shapes neural networks possibly toward synchrony is not entirely clear. We examine relations between STDP and synchronous firing in spontaneously firing neural populations. Using coupled heterogeneous phase oscillators placed on initial networks, we show numerically that STDP prunes some synapses and promotes formation of a feedforward network. Eventually a pacemaker, which is the neuron with the fastest inherent frequency in our numerical simulations, emerges at the root of the feedforward network. In each oscillatory cycle, a packet of neural activity is propagated from the pacemaker to downstream neurons along layers of the feedforward network. This event occurs above a clear-cut threshold value of the initial synaptic weight. Below the threshold, neurons are self-organized into separate clusters each of which is a feedforward network.

Figure 1

Complete graph (a) without a pacemaker and (b) with a pacemaker.

Figure 2

(Color) Phase diagrams for the complete graph in the [(a) and (b)] $\Delta \omega \text{−}{g}_0$ and [(c) and (d)] $\Delta {\omega }_1\text{−}\Delta {\omega }_2$ spaces. One pacemaker neuron is initially present [(a) and (c)] or absent [(b) and (d)]. We run numerical simulations 20 times for each pair of parameter values. We add the red element of the RGB color scheme by the maximum amount divided by 20 when $g_{12}$ survives in a simulation run. Similarly, the green is added when $g_{23}$ survives, and the blue is added when all the neurons get disconnected. Yellow regions appear when both $g_{12}$ and $g_{23}$ survive, since the combination of red and green is yellow. In this case, it turns out that $g_{13}$ also survives. We verified that no other connectivity, such as survival of $g_{13}$ without survival of $g_{12}$ or $g_{23}$, and survival of $g_{21}$, $g_{31}$, or $g_{32}$, appears except at points near phase transitions and resonance. Near phase transitions, we exclude such exceptional runs from the statistics. In the resonance regions (e.g., $\Delta \omega \approx 2.7$ and $g_0\approx 0.4$), the three neurons may remain connected. In this situation, however, synaptic weights keep oscillating, and any pair of the three neurons is not in frequency synchrony. Therefore, we judge such a run as being completely desynchronized and colored blue (labeled D).

Figure 3

(a) Two-neuron network. (b) Threshold for frequency synchrony for the two-neuron networks corresponding to the $\Delta {\omega }_1=0$ line and the $\Delta {\omega }_2=0$ line in Figs. 2c, 2d.

Figure 4

(Color) (a) Feedforward loop. [(b) and (c)] Phase diagrams for the feedforward loop in two different parameter spaces. See Fig. 2 for the color code.

Figure 5

(Color) (a) Fan-in network. [(b) and (c)] Phase diagrams for the fan-in network in the $\Delta {\omega }_1\text{−}\Delta {\omega }_2$ space, with (c) being an enlargement of (b). We set $g_0=0.2$. [(d), (e), and (f)] Phase diagrams in the $\Delta \omega \text{−}\Delta g_0$ space. We set $g_0=0.2$, (d) ${\omega }_1={\omega }_3+0.8$, (e) ${\omega }_1={\omega }_3+1.0$, and (f) ${\omega }_1={\omega }_3+1.2$.

Figure 6

Rastergrams for (a) initial 5 time units and (b) final 5 time units. (c) is an enlargement of (b). We set $N=100$, $g_0=1.0$, and $\sigma =0$. The neurons are aligned according to the order of the inherent frequency.

Figure 7

Time course of the degree of synchrony when $N=100$, $g_0=1.0$, and $\sigma =0$. The values of $r$ are plotted every $10000$ time units.

Figure 8

Final network structure when $N=100$, $g_0=1.0$, and $\sigma =0$. The network is drawn by Pajek [21].

Figure 9

Segregation into clusters when $N=100$, $g_0=0.5$, and $\sigma =0$. Inherent frequencies $(+)$ and actual frequencies after STDP (○) are shown. We estimate the actual frequencies from the phase shifts with bins of width 10 time units.

Figure 10

Results for $N=100$, $g_0=1.0$, with dynamical noise and heterogeneity in the initial synaptic weights. We set [(a) and (b)] $\sigma =0.081$, [(c) and (d)] $\sigma =0.405$, and [(e) and (f)] $\sigma =0.81$. Rastergrams for 5 time units after transient are shown in (a), (c), and (e). The neurons are aligned according to the order of the inherent frequency. Inherent frequencies $(+)$ and actual frequencies (○) are shown in (b), (d), and (f). Because of the dynamical noise, we estimate the actual frequencies from the phase shifts with bins of width ${10}^5$ time units.

Figure 11

Normalized abundance of different three-neuron network patterns. We set $N=100$, $g_0=5.0$, and $\sigma =0$. Circles and squares correspond to the initial and final stages of the simulation runs, respectively.