There is a numerical error, which has led to a large magnification of some of the synchronization effects on equilibrium conditions in Fig. 3. In numerical programming, we set the spike-timing window function to be F A expÿ= for 0 rather than for > 0; this was expected not to give an appreciable difference since in the ideal sense the case 0 has measure zero and the corresponding probability should vanish. In practice, however, such events of perfect synchronization turn out to happen frequently in simulations due to the discrete evolution time step (t 0:02 ms). We have thus corrected this to obtain the accurate results shown in Fig. 1 below. It is observed that both mechanisms suggested are still effective in inducing unimodal synaptic strength through synchronous neural activity, although they do not achieve sharply equalized synaptic strengths under the employed simulation conditions: The data plotted by the dashed line, related to the first mechanism, show that C V first decreases with the network size N but increases again for large N. There appears to be little difference for the data plotted by the dotted line, related to the second mechanism, because synchronization is not so sharp in neural firing. For the data represented by the solid line, the synaptic currents are observed to fall off too quickly to fire neurons repeatedly, so C V remains close to unity. Instead, the validity of the second mechanism is shown by the data represented by the solid gray line, obtained under the new condition of an additional external direct current injected into the neurons. This additional current, which may be associated with the background noise in the real neural system, lowers the firing threshold of the neurons and facilitates repeated firing even with the rapid decrease of synaptic current. We expect that under optimized conditions substantially more equalized synaptic efficacy should be achieved. It would thus be of interest to probe how to maximize the effects of both mechanisms, which is left for further study. FIG. 1. Coefficient deviation C V versus network size N. The time delay and the characteristic time are given by d ; a 0; 2 (in milliseconds) for the data plotted by the solid line, 6; 2 by the dashed line, 2; 8 by the dotted line, and 4; 200 by the solid gray line.In the latter case, external direct current I dc 6:1 A=cm 2 is also applied.
A trendy method to understand the brain is to make a map representing the structural network of the brain, also known as the connectome, on the scale of a brain region. Indeed analysis based on graph theory provides quantitative insights into general topological principles of brain network organization. In particular, it is disclosed that typical brain networks share the topological properties, such as small-world and scale-free, with many other complex networks encountered in nature. Such topological properties are regarded as characteristics of the optimal neural connectivity to implement efficient computation and communication; brains with disease or abnormality show distinguishable deviations in the graph theoretical analysis. Considering that conventional models in graph theory are, however, not adequate for direct application to the neural system, we also discuss a model for explaining how the neural connectivity is organized.
It is commonly believed that spike timings of a postsynaptic neuron tend to follow those of the presynaptic neuron. Such orthodromic firing may, however, cause a conflict with the functional integrity of complex neuronal networks due to asymmetric temporal Hebbian plasticity. We argue that reversed spike timing in a synapse is a typical phenomenon in the cortex, which has a stabilizing effect on the neuronal network structure. We further demonstrate how the firing causality in a synapse is perturbed by synchronous neural activity and how the equilibrium property of spike-timing dependent plasticity is determined principally by the degree of synchronization. Remarkably, even noise-induced activity and synchrony of neurons can result in equalization of synaptic efficacy.
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