  We study a simple model for a neuron function in a collective brain system.
The neural network is composed of uncorrelated random scale-free network for
eliminating the degree correlation of dynamical processes. The interaction of
neurons is supposed to be isotropic and idealized. This neuron dynamics is
similar to biological evolution in extremal dynamics with isotropic locally
interaction but has different time scale. The evolution of neuron spike takes
place according to punctuated patterns similar to the avalanche dynamics. We
find that the evolutionary dynamics of this neuron function exhibit
self-organized criticality which shows power-law behavior of the avalanche
sizes. For a given network, the avalanche dynamic behavior is not changed with
different degree exponents of networks, $\gamma \geq 2.4$ and refractory period
correspondent to the memory effect, $T_r$. In addition, the avalanche size
distributions exhibit the power-law behavior in a single scaling region in
contrast to other networks. However, the return time distributions displaying
spatiotemporal complexity have three characteristic time scaling regimes.
