### The self-organized growth of synfire patterns

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The connectivity of the neocortex displays striking regularities that account for its functional specialization. Synaptic contacts reorganize during preand post-natal development via epigenetic factors involving interactions between extrinsic and intrinsic sources of neural activity. We propose a neural network that models the development of such structures as a process of spatiotemporal pattern formation. We show the spontaneous and simultaneous emergence of ordered chains of synaptic connectivity and wave-like propagation of neural activity. These simple linear structures were first called “synfire chains” by Moshe Abeles. They consist of synchronous groups, feed-forward connections and waves of activity. In our model, starting from a disordered network with broad diffuse connectivity and low stochastic activity, some synaptic connections become selected and strengthened to the detriment of others. This “focusing of innervation” is accompanied by a gradual increase in sustained correlated firing. In a true selforganized fashion, connections and correlations reinforce each other through heterosynaptic cooperation, while the global stability of the network is maintained through a constraint of heterosynaptic competition. Hebbian plasticity is one of the major principles underlying the development and tuning of the nervous system. Cortical neurons are highly sensitive to nearly synchronous inputs amongst afferent connections, while synaptic contacts are rewarded by successful transmission events. This work describes how a neural network may become spontaneously structured as a result of these two principles, which are given a straightforward mathematical form. We propose a very simple model based on binary neurons, fixed weight increments and uniform graphs, and show that regularities already emerge under these conditions. In a fully-connected network of N {0, 1}-valued excitatory neurons, cell i fires at time t with probability P[xi(t) = 1] = 1 / (1 + exp(−(Vi(t) − θ) / T)). The “temperature” T determines the amount of noise in the system. The membrane potential is given by Vi(t) = ∑j wij(t) xj(t − τij), where wij is the weight of j→i and τij a fixed transmission delay (time is discrete, in steps of roughly 1 ms). Synaptic weight modification occurs in three simple stages. First, all weights are potentiated according to a Hebbian rule: for each j ≠ i such that xj(t − τij) = xi(t) = 1, a small positive increment α is added to wij (Fig. a). Second, competition pulls all converging and diverging weight sums, ∑j wij(t) and ∑i wij(t), towards a fixed target s0. This is done by correcting wij with a term −γ ∂H / ∂wij, where H measures a global quadratic difference between the sums and s0. Finally, weights are clipped, when necessary, to remain in the interval [0, 1]. The network also contains a few “seed” neurons, which send out stronger connections than average. Each seed neuron reliably triggers the same group of cells, thereby creating an initial synchronous pool P0. Through its repeated (yet not necessarily periodic) activation, P0 starts recruiting neurons “downstream” and eventually becomes the root of a developing synfire chain P0→P1→P2→ ... (Fig. b). The growth of the chain is recursive and akin to the growth by accretion of a crystal from an inhomogeneity. Under the dual activityconnectivity dynamics, the aggregation of Pk+1 by Pk is a form of “Darwinian” evolution. In a first phase, noise acts as a diversification mechanism, proposing multiple candidates—neurons that happened to fire after Pk and whose contacts from Pk increased by α (Fig. a). In a second phase, competition driven by s0 acts to select among these: the large pool of candidates shrinks and rounds up a final set of winners Pk+1. This accretion process is not strictly iterative, as successive pools in the chain develop over broadly overlapping periods of time. As soon as a new pool Pk+1 reaches a critical mass, its activity is high enough to trigger the growth of the next pool Pk+2. Thus, the chain extends in length before widening and, in a spatially reorganized view of the network, typically presents a “beveled” head of immature pools at the end of a mature trunk (Fig. b). Fig. c reveals the growing profile of a chain in four snapshots of the network’s total activity following the activation of a seed neuron at different times. At a larger, mesoscopic scale of organization, Fig. d illustrates the concurrent growth of multiple chains and hints at the possibility of wave synchronization and coalescence, via dynamical binding (arrows), in a “self-woven tapestry” of synfire patterns.