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In this paper, applying the theory of semigroups of operators to evolution family and Banach fixed point theorem, we prove the existence and uniqueness of an (a) almost automorphic (weighted pseudo almost automorphic) mild solution of the semilinear evolution equation x (t) = A(t)x(t) + f (t, x(t)) in Banach space under conditions.

The electric activity of neuron and collective behaviors of neurons can be modulated by autapse, which can be described by self-feedback current in close loop with time delay being considered. Distribution of electric autapses in a local area can introduce heterogeneity in the network and thus traveling wave emits from this area. In this paper, diversity in time delay of electric autapse is considered and collision between emitting waves from different local areas driven by electric autapses under different time delays is observed. In the numerical studies, neurons in the square area with 15×15 (and/or 20×20) nodes are connected electric autapses with different time delays and target-like waves are induced and converted into, spiral waves after continuous collision between wave fronts. It is found that a group of spiral waves can emerge in the network, or coexist with target waves under appropriate coupling intensity due to time delay diversity in autapse and these waves can regulate the collective behaviors of neurons as continuous pacemakers.

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