Abstract
We present an analytical framework that allows the quantitative study of statistical dynamic properties of networks with adaptive nodes that have memory and is used to examine the emergence of oscillations in networks with response failures. The frequency of the oscillations was quantitatively found to increase with the excitability of the nodes and with the average degree of the network and to decrease with delays between nodes. For networks of networks, diverse cluster oscillation modes were found as a function of the topology. Analytical results are in agreement with large-scale simulations and open the horizon for understanding network dynamics composed of finite memory nodes as well as their different phases of activity.
Highlights
Networks and networks of networks are important concepts in many fields of physics[1,2,3]
We develop an analytical framework, which allows the examination of the statistical properties of such networks and their dynamics, based on stochastic equations in the mean-field limit
The dynamics are in continues time, if all the delays between connected neurons are equal, the situation can be simplified to an iterative map
Summary
Where [1 − R(i − 1)]k is the probability that no presynaptic neuron evoked a spike at the previous time step, (1 − fext∙d) is the probability that a neuron was not stimulated since the last step by synaptic noise and the last term stands for random fluctuations in finite networks which scale with N−0.5 (Gaussian random variable with a zero mean and a variance equals to < pstk(i) > ·(1 −< pstk(i) > )/(Ck·N), where < · > averages over the noise term). Both analytical and simulation results clearly indicate that the oscillation frequency increases with fc and with the average incoming connectivity and decreases with the delay, d (Fig. 2a–c). It represents a non-local mechanism for zero-lag synchronization, since the orange and the a
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