Emergence of Scale-free Graphs in Dynamical Spiking Neural Networks

Abstract

Emergence of Scale-free Graphs in Dynamical SpikingThe model consists of a number (about 1000-3000) of neuronal groups, connected randomly (weights chosen from Gaussian distribution N(0,1)) by the group leaders -neurons chosen to interconnect every group with others. The group's synchronization depends on the input received from the group leader and, on the other hand, the activity of the leader resembles the activity of the group. figure 2 showing neurons 1000 to 1500 within 2000ms-3000ms timeframe. Please note the similarities of these plots (actually it is a self-similarity). The number and length of straight horizontal lines (each symbolizing a bursting activity) in both plots is approximately the same. Fig. 6. The dependency between spiking activity and vertex degree before (left) and after thresholding the graph (right). In either case we observe a clearly monotone dependency which supports our preferential attachment hypothesis. kernel. This procedure blurs the spike train significantly, but lets one receive non zero product of two of such trains even if corresponding spikes are shifted. The product is later integrated to obtain synchronization strength, a measure we introduced to describe similarity between spike trains. REFERENCES DYNAMICSThe simulation was carried out on two levels, on both of them synchronously:• Initialization phase -each group was simulated synchronously over one time step (1ms). The initial input to every group was 0 plus some slight Gaussian noise (applied to every neuron independently).• After this phase, weighted summation of group leader output activities is performed and given as input activity to group leaders in the next step.• Each group was simulated synchronously over one time step, with the group leader activity and a slight Gaussian noise as an input for every neuron.• Steps 2 and 3 were repeated until the end of simulation (in this case up to 12000 steps).As an output, the simulation produced a significant number of spike trains (3000 neurons, each over more than 10000 time steps) that had to be compared with respect to a measure of synchronization computed in the following manner:• Each spike train was blurred by a convolution with kernel exp − x 10 2 , see figure 2.• The transformed spike train of every two neurons was then multiplied and integrated. The integral (real number) was interpreted as a measure of synchronization. The blur was necessary, to assure similarity between two spike trains that were in fact roughly similar, but corresponding spikes were shifted by several of time steps in either direction. It is worth noting that this measure strongly supports bursting -two units giving continuous spike response in the same time gain much similarity in the sense above. Note that this measure is significant only if spikes actually occur, two empty spike trains are similar in some sense, but in terms of a proposed measure their similarity is zero. INTRODUCTIONThis research is based on previous results presented in [3], which dealt with a discreete model, that ...

Spectra of winner-take-all stochastic neural networks

“…This observation allowed us to show in Piekniewski & Schreiber (2008) that asymptotically the charge-flow networks are scale-free with exponent 2, see ibidem as well as Piersa, Piekniewski & Schreiber (2010), in agreement with the empirical findings as quoted above. We have also argued there that even though the spin glass model we propose may be regarded quite specific, its large scale behaviour and in particular its winner-take-all approximation is presumably universal for a large class of networks where each formal neuron represents a computational unit exhibiting some non-trivial internal structure and memory, for instance a group of biological or artificial neurons (see Piekniewski, 2007) whose internal state requires more complicated labeling than just {−1, +1} as in the original Sherrington-Kirkpatrick model, whence the N-valued labels in our model.…”