The structure and plasticity of synaptic connectivity underlies much of our ability to process and store information. In the past two decades, spike timing dependent plasticity (STDP) has become one of the main foundations of theoretical research on plasticity and learning. Previous works that explored theoretically the change in the synaptic efficacy according to STDP mainly focused on local approximated terms that depend on the firing rate of the pre and the post synaptic neurons and the synaptic efficacies between them (for example: Babadi and Abbott 2013). We recently developed an analytical framework in which the average change in the synaptic efficacy, due to STDP, can be evaluated precisely in recurrent networks of Poisson neurons with arbitrary connectivity. In this framework, the local learning plasticity rule of Babadi and Abbott arises in a systematic manner as the first order term of an expansion in the strength of synaptic efficacies. We show that higher order terms lead to an effective interaction between synapses of different neurons that can significantly alter the dynamics, and affect the global structure of the neural network in steady state. As an example, we examine the plasticity in a recurrent network that includes two subpopulations of inhibitory and excitatory neurons. We consider the spontaneous formation of wide synfire chains, in which distinct groups of neurons share similar connectivity, and sequentially project to each other. It has been hypothesized that this architecture underlies the synchronous neural activity observed in the premotor nucleus of songbirds HVC (Hahnloser et al.2002 , Long et al. 2010). We show that the high order terms in our theory can promote the formation of wide synfire chains without the need to introduce structural constraints or correlated external input.
 B. Babadi, and L.F. Abbott. PLoS computational biology 9(2): e1002906 (2013).
 R.H.R. Hahnloser, A.A, Kozhevnikov, and M.S. Fee. Nature 419: 65-70 (2002).
 M.A. Long, D.Z. Jin, and M.S. Fee. Nature 468: 394-399 (2010).