We simulate 10 networks, and initial weights of each network are randomly selected from uniform, Gaussian, delta (all weights identical), or exponential distributions. After weight initialization, each such network is examined on 10 different sets of network evolution parameters, such as neuron number, learning rates, neuron firing rates, etc. The network connectivity changes due to the action of the different plasticity mechanisms. As observed in Zheng et al. (2013), the network goes through different phases characterized by the number of excitatory-to-excitatory connections present in the network. Eventually, it enters a stable regime where connectivity stays roughly constant. For such stabilized networks we use the Fanmod software (Wernicke, 2005) and its computation of a p-value to analyze network motifs involving 3 and 4 neurons. Here the p-value of a motif is defined as the number of random networks in which it occurred more often than in the original network, divided by the total number of random networks. Therefore, p-values range from 0 to 1, and the smaller the p-value, the more significant is the abundance of the motif. The frequency of a motif occurring in 100 simulated SORN networks is compared to the mean frequency of the motif occurring in 1000 random networks with identical connection probability. We found the network motifs are organized into two distinct groups with p-value = 0 and p-value = 1. Figure 1 shows the group of motifs always with p-value = 0, all of which reveal a feed-forward structure consistent with a synfire-chain topology.

The abundance of feed-forward network motifs among groups of 3 and 4 neurons during the stable phase of network evolution already suggests that the network may be forming synfire-chain like structures. To investigate this, we studied the evolution of the network’s activity patterns and connectivity during its self-organization. Figure 2 shows an example. In Figure 2A we plot the activity of the first 50 neurons during short 500 time step intervals taken at five different time points of the network’s evolution. Excitatory neurons are sorted in all recorded networks according to their activity correlations in the last recorded network (in the stable phase). Thus neurons that are highly correlated during the stable phase are plotted in neighboring rows. While the network initially exhibits quite irregular activity, it spontaneously forms highly structured activity patterns as it develops (also see Figure S1 in the supplementary material, which shows example cross-correlograms of different pairs of neurons). In the particular case shown here, the network forms two subsets of neurons which alternate in exhibiting phases of high firing rates.

Figure 2B shows the evolution of firing correlations among all excitatory neurons in the network. The network forms 8 distinct pools of neurons, with neurons of each pool exhibiting highly synchronized firing. The excitatory weight matrix shown in Figure 2C reveals that the network develops two independent circular synfire chains, which we will refer to as synfire-rings. The layers of synfire rings are identified automatically by applying a threshold to the neurons’ activity correlations. Due to noise and the interaction of multiple forms of plasticity, a neuron’s activity maintains a certain degree of randomness, which leads to positive but non-uniform correlations in each layer. As a result there are some neurons with relatively weaker correlation in each layer in most cases.

In the given example, the first synfire-ring comprises 3 smaller pools of neurons (total of 43 neurons), the second synfire ring comprises 5 larger pools of neurons (total of 157 neurons). The two synfire-rings correspond to two transiently stable activity patterns. As shown in Figure 2, activities of the first 43 and remaining 7 neurons, which belong to different rings, are roughly complementary. If one synfire ring becomes active, it tends to activate the inhibitory neurons and thereby suppress activity in the other synfire ring. After a while, however, the intrinsic plasticity mechanism will increase the firing thresholds of neurons belonging to the active synfire-ring and decrease the firing thresholds of the inactive synfire-ring. Over time, this destabilizes the active synfire-ring and eventually leads to the suppressed synfire-ring taking over. The strong competition between the synfire rings is due to the widespread inhibition with each inhibitory unit receiving input from all excitatory cells in the network and projecting randomly to one fifth of the excitatory cells (compare Methods).

We next investigate how the sizes of neuronal pools and their connectivity depend on the target firing rates of the neurons in a 200 excitatory neuron network with fixed initial connectivity. The parameter H^IP_i sets the target firing rate for the i-th excitatory neuron. These target firing rates are drawn from a uniform distribution in [μ_IP − σ_HIP, μ_IP + σ_HIP].

We first fix σ_HIP = 0 and study the influence of the target firing rate μ_IP. As the target firing rate of the neurons increases, the variability of the sizes of neuronal pools increases. Figure 3A plots the average maximum and minimum pool sizes as a function of μ_IP. For large μ_IP, the maximum layer size tends to get bigger and the minimum layer size tends to be smaller. In addition, the variability of the maximum and especially the minimum layer sizes tends to be largest for the biggest μ_IP. Figure 3B compares the histograms of pool sizes for different μ_IP. The distribution is very narrow for small μ_IP (green bars corresponding to μ_IP = 0.025) and very broad for large μ_IP (red bars corresponding to μ_IP = 0.125). In all cases, the final distribution of synaptic strength is lognormal-like which means some weights are way stronger than others. This is shown in Figure 3C, which plots this distribution for different μ_IP.

We next fix μ_IP = 0.1 and study the effect of the interval size σ_HIP of the target firing rates. In a similar way, the diversity of pool sizes grows as σ_HIP increases. This holds true for σ_HIP ≤ 0.06 as shown in Figure 4A. However, as σ_HIP increases more and more neurons are close to silent. The minimum target firing rate of some excitatory neurons is as small as ~0.02 when σ_HIP reaches 0.08. These neurons barely fire during the network evolution and barely contribute to structuring the network. Therefore, the effective network size is reduced as σ_HIP is increased. This may explain why the variability in pool sizes shrinks when σ_HIP grows to 0.08. Figure 4B compares the distribution of pool sizes for different values of σ_HIP. The greatest spread of the distribution is obtained for an intermediate value of σ_HIP = 0.06. Figure 4C shows an example of an excitatory weight matrix in the stable regime for σ_HIP = 0.06. The network has developed a single synfire ring with 4 pools of neurons whose sizes range from 44 to 57.

We next study how the number of synfire rings and the number of neuronal pools or layers depends on the overall network size. To this end, we simulate 40 networks with 200–800 excitatory neurons. As a first measure of network structure we define the number of layers present in the network. Figure 5A plots this number as a function of network size (red curve). Not surprisingly, the number of neuronal pools increases as the network gets bigger. As a second index of network structure we measure the fraction of networks of a given size that develop multiple synfire rings. As shown in Figure 5A (blue curve) this fraction increases with network size. For networks of 800 neurons it already reaches a value of 0.4 and the increase with network size seems to be faster than linear for the range of sizes considered. Figure 5B shows a typical example of the excitatory weight matrix in a network with 800 neurons and μ_IP = 0.1, σ_HIP = 0. This network has developed 4 synfire rings of different sizes. Note that the second one from the top is very small. In Figure 5C it is easier to identify it. The sizes of the pools are fairly consistent within a single synfire ring (mean SD is 4.5) but can vary widely across synfire rings (SD is 26.6). The biggest ring in Figure 5B has 12 pools, so if activity runs around in this circle, each neuron is activated only every 12th time step, which is less than intrinsic plasticity wants (compare Methods). As shown in Figure 5C, the biggest ring is roughly active all the time, and unlike the synfire rings in Figure 2, the network could start multiple rings simultaneously. It is worth noting that we achieve the synfire ring structure under a wide range of parameters, excitatory neuron number being one of them. We also run a few simulations with 1000 and 1200 excitatory neurons, which also develop synfire rings (see Figures S2, S3 in supplementary material).

With all forms of plasticity present, the network will develop synfire rings spontaneously and robustly over a large range of parameters as long as the network operates in a healthy regime. The results are fully in line with our previous work since we use same network as Zheng et al. (2013), where we discuss in detail the necessity of the different plasticity mechanisms for the networks behavior. So how do these (circular) feed-forward structures come about?

The formation of synfire-chains can be understood as a process of network self-organization driven largely by the STDP rule. Figure 6 illustrates the process. Consider as an example a strong feed-forward chain from a unit a to a unit b and on to a unit c. According to this structure, there is a high probability that a, b, and c fire in three successive time steps. Standard STDP rules, including the one we are using here, will strengthen the connections in the feed-forward direction and weaken the reverse connections such as the red synapse in Figure 6A. This is because of the nature of the STDP rule, which potentiates “causal” firing patterns (pre before post) and depresses “acausal” firing patterns (post before pre). As shown in Figure 7A, the fraction of bidirectional connections plummets during the first stage of network evolution. Thus, a first relevant mechanism in synfire ring formation is the removal of reciprocal connections.

A second mechanism in synfire ring formation is the establishment of parallel pathways. Consider two units b₁ and b₂ which also happen to be strongly innervated by a (see Figure 6B). Because of this, they will tend to be synchronously active with unit b and their activity will be reliably followed by activation of unit c. Because of this correlation structure (b₁ and b₂ likely being active in the time step before c) the weights from b₁ and b₂ onto c, if present, will have a strong tendency to get potentiated. Thus, STDP will potentiate the “missing” connections from b₁ and b₂ onto c establishing additional parallel pathways connecting a and c. In order for STDP to be able to strengthen these connections, they have to either be present from the beginning or become added by the structural plasticity. With this mechanism operating not just at the level of b but at all levels of the network, synfire-chains will develop (see Figure 6C). Due to the homeostatic activity regulation, at each time step a certain fraction of neurons in the network will tend to be active. This implicitly regulates the range of layer sizes and limits the breadth of the growing chain. Due to the synaptic scaling, every neuron in a layer receives a certain amount of synaptic input. Moreover, since the network has only a finite number of units and each unit tries to maintain a certain average activity level such that activity cannot die out, it is inevitable that such a chain eventually terminates or connects back to itself thereby forming a synfire ring. A ring-like structure has a competitive advantage against a terminating chain during the formative stage of network development, because a synfire ring will reactivate itself while a terminating chain cannot.

STDP alone can not depress existing synapses that are incompatible with the emerging synfire ring structure. For example, connections within one layer of neurons or connections jumping ahead beyond the immediate next layer (compare red synapses in Figure 6B) remain unaltered under perfect synfire chain activity. However, the synaptic scaling mechanism gradually depresses these connections to very small values as the other weights on the synfire route are potentiated. Thus, another relevant mechanism in synfire ring formation is the competition among synaptic weights onto the same target neuron.

The synaptic scaling mechanism we use does not remove any such “spurious” synapses, however. This is achieved by STDP. Due to intrinsic membrane noise of the neurons and fluctuations of intrinsic excitability and inhibitory drive, the network’s activity always maintains a random component—even in its stable phase (compare Figure 2A). As a consequence, neuron a in Figure 6 could fire right after c, which would lead to the removal of a sufficiently depressed spurious connection from a to c. Such events occur only rarely but they suffice to eliminate such spurious connections if they have already been depressed by synaptic scaling. Thus, a final mechanism in synfire ring formation is the removal of spurious connections due to random activity fluctuations. To illustrate this effect, we manually added new connections with rather strong weights of value 0.1 within one layer and between one layer and the layer two steps ahead. These manually added new connections are even stronger than ~70% of the existing connections. Figures 7B,C show the fate of these manually inserted connections that are inconsistent with the dominant synfire-ring structure: within a few thousand time steps their weights decrease to zero as a result of competition among synapses and STDP driven by random activity fluctuations.

The precise outcome of the overall network self-organization depends on the initial conditions (initial network structure and connection weights) and the random activity fluctuations. Feed-forward connections between synfire layers go through strong competition during network evolution as a result of synaptic scaling. Connections that start out strong or are added early have an advantage in this competition and are less prone to removal due to random activity fluctuations. Synapses added in later phases of the network’s evolution are more fragile, which contributes to the stability of already formed synfire rings. Figure 8 shows examples of weight changes of new synapses that have been added through structural plasticity during the network’s stable phase. In Figure 8A we plot synaptic connections that are off any existing synfire ring structure. These connections are removed comparatively quickly. Figure 8B illustrates the fate of newly added synaptic connections that are congruent with an existing synfire ring, i.e., they connect a neuron from one pool to a neuron in the successor pool. Interestingly, even these connections tend to be removed eventually. Due to synaptic scaling, they have to compete with many other connections along the synfire ring, which limits their growth and makes them prone to elimination due to random activity fluctuations. It should be clarified that Figures 7B,C (unlike Figure 7A) and Figure 8 all study the network in its stable phase. That is the synfire chain is already formed, which is analogous to a prewired synfire chain. In all of these cases, the synfire chain is indeed restored after our perturbation. Figures 7B,C, 8B are similar, but they are different in terms of new synapse weight and position definition.

It should be mentioned that the network becomes rather rigid only after synfire chains have been formed. This is important to maintain the stability of synfire chains. At the beginning of network evolution, however, newer synapses are freely competing. During the development from a randomly initialized network to synfire chains, many new synapses are added and stabilized. Besides that, newer synapses could also survive in synfire chains when the network is driven by appropriate strong structured external input (not shown).

As mentioned above, every plasticity mechanism is important for the development of synfire structure. In simulations, we did not observe the formation of synfire rings in networks without synaptic normalization, structural plasticity or STDP of the excitatory connections. Intrinsic plasticity and inhibitory STDP both try to maintain a low average firing rate of excitatory cells, and the formation of synfire structure relies on the presence of both of them. If we switch off one of them, the network suffers from big activity fluctuations from time to time, which usually stop the formation of synfire structure or lead to abnormal network structures exhibiting both extra-large and single-neuron layers (Figure S4 in the supplementary material).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.