Gilson et al 2009a-d,2010 are a 5-part paper series in Biological Cybernetics for deducing recurrent network structure from a plasticity rule.
Here, I have my notes on a follow-up review paper:
Gilson, M., Burkitt, A., and van Hemmen, J.L. (2010). STDP in Recurrent Neuronal Networks. Front Comput Neurosci 4.
I might refer to the original papers for details.
Change in weight of $j\to i$ synapse during duration $T$ is:
$\Delta J_{ij}(t)=\eta \left( \sum_{t-T\le t_j^n \le t} w^{in} + \sum_{t-T\le t_i^m \le t} w^{out} + \sum_{t-T\le t_j^n,t_i^m \le t} W(\cdot,t_j^n-t_i^m) \right)$ ...(2)
where $t_j^n$ and $t_i^m$ are pre- and post-synaptic spike times indexed by $n$ and $m$; and the three RHS terms are weight changes due to pre-, post- and joint activity. In particular the third term is due to the STDP window.
If learning is sufficiently slow then spike trains can be averaged over resulting in a learning dynamics:
$\dot{J_{ij}}=f(J_{ij};\nu_j,\nu_i)+g(J_{ij};C_{ij},d_{ij}^{ax}-d_{ij}^b)$ ....(4)
where firing rate:
$\nu_i(t)\equiv \frac{1}{T}\int_{t-T}^t \langle S_i(t') \rangle dt'$ ...(5),
and the spike-time covariance coefficient:
$C_{ij}(t,u)\equiv \frac{1}{T}\int_{t-T}^t \langle S_i(t')S_J(t'+u) \rangle dt' - \nu_i(t)\nu_j(t)$ ...(6)
$d_{ij}^{ax}$ and $d_{ij}^b$ are axonal (pre-synaptic) and back-propagation (post-synaptic) delays respectively. $d_{ij}^{dend}$ the dendritic EPSP forward propagation delay does not enter.
[What about higher order correlations -- don't they enter?]
[Integration of triplet rules, what does that give?]
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