Linear stability analysis of retrieval state in associative memory neural networks of spiking neurons

We study associative memory neural networks of the Hodgkin-Huxley type of spiking neurons in which multiple periodic spatiotemporal patterns of spike timing are memorized as limit-cycle-type attractors. In encoding the spatiotemporal patterns, we assume the spike-timing-dependent synaptic plasticity with the asymmetric time window. Analysis for periodic solution of retrieval state reveals that if the area of the negative part of the time window is equivalent to the positive part, then crosstalk among encoded patterns vanishes. Phase transition due to the loss of the stability of periodic solution is observed when we assume fast α function for direct interaction among neurons. In order to evaluate the critical point of this phase transition, we employ Floquet theory in which the stability problem of the infinite number of spiking neurons interacting with α function is reduced to the eigenvalue problem with the finite size of matrix. Numerical integration of the single-body dynamics yields the explicit value of the matrix, which enables us to determine the critical point of the phase transition with a high degree of precision