Effect of Structure on Function in Model Nerve Nets

A theoretical analysis has been made on the effect of the pattern of interneuronal connectivity in model nerve nets on the activity of these nets. Two types of nets have been investigated: one in which the likelihood of a connection between a given neuron and any other element in the net is given by a Poisson probability distribution, and a second type in which the pattern of interconnection follows a Gaussian distribution. An analytical treatment is presented of the equations for noiseless nets in these two conditions. The principal result is that nets with Poisson connectivity law are activated by extraneous firing of a single neuron and continue in spontaneous activity indefinitely. On the other hand, similar nets in which the connections are, however, distributed according to a normal connectivity law, exhibit a definite threshold and produce spontaneous activity only subsequent to extraneous activation of a substantial fraction of the population. Moreover, spontaneous activity in Gaussian nets, but not in Poisson nets, becomes extinguished if the number of active neurons falls below the critical threshold. Some neuroanatomical implications are discussed which suggest that the pyramidal system of the cerebral cortex and other neuronal systems histologically characterized by large numbers of synapses per neuron may incorporate a Gaussian connectivity law, whereas a Poisson law may be characteristic of these cortical layers and nuclei primarily containing granule cells