Original framework of GLMCC

In the previous study¹⁴, we developed a method of estimating the connectivity by fitting the generalized linear model to a cross-correlogram, GLMCC. We designed the GLM function as

where t is the time from the spikes of the reference neuron. a(t) represents large-scale fluctuations in the cross-correlogram in a window [-W,W] (W=50 ms). By discretizing the time in units of Δ(=1ms), a(t) is represented as a vector a→=(a1,a2,…,aM) (M=2W/Δ). J12 (J21) represents a possible synaptic connection from the reference (target) neuron to the target (reference) neuron. The temporal profile of the synaptic interaction is modeled as f(t)=exp(-t-dτ) for t>d and f(t)=0 otherwise, where τ is the typical timescale of synaptic impact and d is the transmission delay. Here we have chosen τ=4 ms, and let the synaptic delay d be selected from 1, 2, 3, and 4 ms for each pair.

The parameters θ→={J12,J21,a→} are determined with the maximum a posteriori (MAP) estimate, that is, by maximizing the posterior distribution or its logarithm:

where {ti} are the relative spike times. The log-likelihood is obtained as

where npre is the number of spikes of presynaptic neuron (j). Here we have provided the prior distribution of a→ that penalizes a large gradient of a(t) and uniform prior for {J12,J21}

where the hyperparameter γ representing the degree of flatness of a(t) was chosen as γ=2×10-4 [ms-1].