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Spontaneous emergence of neuronal activity avalanches characterized by power-law distributions is known to occur in different types of nervous tissues suggesting that nervous systems may operate at a critical regime. Here, we explore the possible relation of this dynamical state with the underlying topology in a small-size network of interconnected Morris-Lecar neurons. Studying numerically different topological configurations, we find that, very close to the efficient small-world situation, the system self-organizes near to a critical branching process with observable distributions in the proximity of a power law with exponents similar to those reported in the experimental literature. Therefore, we conclude that the observed scaling is intimately related only with the small-world topology.

The spontaneous emergence of neuronal activity avalanches showing power-law distributions has been found in superficial layers of cortex, both in vitro [

Consider a set of coupled Morris-Lecar neurons forming a ring through weighted synaptic interactions. We use the version of the Morris-Lecar (ML) model [^{++} channels and voltage, gated delayed rectifier K^{+} channels, are present. The calcium current plays the role of Na^{+} current in the original HH system. Thus, the membrane potential equation, under the action of an input

The functions

In these equations,

The system described by (

Time evolution of the membrane potential obtained with the Morris-Lecar model.

The synaptic interaction is described as

The network is built under the assumption that each neuron is connected to the remaining

Temporal evolution starts from a set of initial conditions selected randomly from a uniform distribution on

Raster plot showing the network’s temporal evolution during an avalanche for the case

For each shortcuts’ proportion, we calculated the avalanche size—measured as the number of

Temporal evolution of the avalanche size plotted versus the avalanche number for

Normalized probability distribution of the activity avalanches sizes (a) and branching rate (b). From top to bottom, networks with ^{2}), ^{2}), ^{2}), ^{2}),

Lifetime distribution behaves in a quite remarkable similar fashion as shown in Figure

Normalized probability distribution of the activity avalanches lifetimes. From top to bottom, networks with

Next, we characterize the topology of the weighted network for different values of

(a) Global and local efficiencies and cost. (b) Behavior of normalized mean

Further insights can be obtained studying the system branching parameter,

It seems surprising for us that this small network may produce such differences for the distribution of sizes and lifetimes. It is revealing that, for the case identified as an efficient SW, the branching parameter approaches the value expected for the critical state. Indeed it is remarkable that, our approach not being a realistic one, results close to those experimentally obtained can be mimicked. This is particularly true considering the inclusion of the nonrealistic

Our work with this tiny network (when compared with a real nervous system) suggests that there is no need to define an

Juan Luis Cabrera gratefully thanks R. Kuske for her hospitality during a sabbatical leave in the Mathematics Department of the University of British Columbia where this work was initially written. The authors thanks H. de Nobrega for his contribution to the software’s initial versions. Juan Luis Cabrera acknowledges the research support from IVIC-141 grant; Johans Hoenicka acknowledges the support from IVIC’s undergraduate fellowships. Correspondence and requests for materials should be addressed to Juan Luis Cabrera.