Architectural and functional connectivity in scale-free integrate-and-fire networks

Abstract

Using integrate-and-fire networks, we study the relationship between the architectural connectivity of a network and its functional connectivity as characterized by the network's dynamical properties. We show that dynamics on a complex network can be controlled by the topology of the network, in particular, scale-free functional connectivity can arise from scale-free architectural connectivity, in which the architectural degree correlation plays a crucial role.

Topological effects on dynamics in complex pulse-coupled networks of integrate-and-fire type

“…The main results of this paper address the question of how the pulse rate of the dynamical units depends on the underlying connectivity statistics in three example complex IF networks of increasingly complex topology, which include an uncorrelated network [11] and two scale-free networks grown through preferential attachment [41][42][43]. While these networks are sufficiently idealized to allow for explicit solution, they also progressively incorporate features conjectured to be present in realistic neuronal networks, including scale-free distribution of incoming node degrees and clustering [32][33][34]36,[44][45][46][47].…”

The role of fluctuations in coarse-grained descriptions of neuronal networks

“…These two requirements leave us with two possible branches of the γ versus λ dependence as expressed by Equation (3.26), as shown in Figure 3.1. Numerical simulations of the corresponding I&F network (3.1) indicate that the lower branch is stable [113,114]. Note that the scaling factor φ(N ) in the coupling constant λ can be of O(1), and the network will still maintain finite firing rates in the large-size limit, N → ∞.…”

“…and 24) respectively [66,114]. Note that the distribution (3.23) behaves as a power law when the neuronal incoming degree k is large.…”

“…In the network we discuss in this section, the neurons have their incoming degrees distributed according to a scalefree law, i.e., the distribution has power-law tails, while each neuron only sends spikes to one postsynaptic neuron [113,114]. The network grows in stages, beginning with one synaptic connection between a pair of neurons.…”

“…The statistical models, exemplified by the kinetic theory [1, 4, 8, 24, 31, 32, 34, 36, 48, 50, 55, 62, 87-91, 95, 143] or the field-theoretic approach [19-21, 26-28, 93, 94], take synaptic-input fluctuations into account, and therefore also provide a description for the network processing of ference between the theory of complex neuronal networks and the standard complexnetwork theory is in that, unlike most social and technological networks, neuronal networks must be described by directed graphs, and thus require an additional step in their description. This step consists of deriving the statistics of the directed neuronto-neuron connections from their undirected counterparts [113,114]. For three example networks, we incorporate these statistics into their mean-field description in the high-conductance limit, in order to find the corresponding firing-rate statistics.…”