A dynamical behaviour of active regions in randomly connected neural networks

The field of the randomly connected neural network is approximately formulated by Griffith's equation, regarding the network as being continuous. An integral representation of Griffith's equation is derived. If a relative refractory period can be ignored, it is $\text{X(x,t)}\text{=1}\text{∫}\text{o}\text{∞}\text{ds}\text{∫}\text{?vs}\text{vs}\text{dn}\text{kv}\text{2t}\text{e}{}^{\mathrm{?avs}}\text{X(x?n, t?s) ? θ}$ where X(x, t) corresponds to the firing rate and θ means the threshold of the neural firing, τ the absolute refractory period and v the velocity for the spike potential travelling down the axon. The above equation is formally analogous to Caianiello's equation, but the former describes the more macroscopic behaviour of the neural network than the latter. With the aid of computer simulation, appropriate solutions are successfully obtained.In regions where X = 1, neurones are firing at a high constant rate of $\text{1}\text{τ}$ (active regions). In regions where X = 0, there is no firing of neurones (resting regions). In the neural net for which $\text{0 <}\text{a}{}^2\text{τθ}\text{k}\text{< 1}$, the net is generally a mixture of the active regions and of the resting regions. In the case that a large active region is in contact with a large resting region, the propagation velocity of boundary between the two regions tends to the velocity u given by $\text{u = (1 ?}\text{2a}{}^2\text{τθ}\text{k}\text{)v}$. This expression of velocity u was deduced from the fact that there exists a solution of the type X(x, t) = 1 (ut ? x) for equation (A). In the case of $\text{0 <}\text{a}{}^2\text{τθ}\text{k}\text{< 0 · 5}$, the active region grows and in the case of $\text{0 · 5 <}\text{a}{}^2\text{τθ}\text{k}\text{< 1}$, the resting region grows. A fatigue effect is introduced, for which it is hard for neurones to maintain firing states. In this case an active region of definite width L propagates with constant velocity u′. The dependence of L and u′ on characteristics of neural network and on the fatigue effect is investigated.