Neural network inference from its time series: FDR-based inference on Izhikevich simulation using cross-correlation as metric

Simulation of Izhikevich neurons (10 and 100 neurons):

In the above simulation, all neurons are independent (no synaptic connections). Here is a distribution of

$C_{ij}^\mathrm{max}$ for uncorrelated neurons (i.e. the "distribution of the test statistic under the null hypothesis"):

Now here is a set of neurons with some synaptic connections between them. (Not easy at all to discern the underlying graphical structure.)

We consider the distribution of

$p$ -values for synaptic pairs in the Izhikevich model as a function of the "synaptic strength" constant

$\alpha$ :

The above shows that a synaptic strength of

$\alpha \approx 10$ is necessary to discern the connectivity structure in the Izhikevich model. Note that

$\alpha \approx 10$ does not yield a totally trivial (i.e. deterministic firing) relationship between two synaptically connected neurons. Here is a trace of two neurons with connectivity of

$\alpha = 10$ that shows that the relationship is not necessarily deterministic:

For that matter,

$\alpha = 20$ doesn't yield deterministic relations either, but there are more (almost) coincident spikes between the two:

(Note: An independent characterization for the "interpretation" of

$\alpha$ could be an empirically derived probability that the post-synaptic neuron fires within some interval of receiving the pre-synaptic spike.)

Next, let's try some inference on a model of

$N=10$ neurons with

$M=10$ synapses. (My eventual target is to run simulations on

$N=100$ or even

$N=10^3$ neurons.) I visualize the inference result as an adjacency matrix, where a filled green rectangle represents a correctly inferred edge, unfilled green rectangle represents an uninferred edge (false negative) and a filled red rectangle represents a falsely inferred edge (false positive). All inference instances are for

$\alpha=10$ .