Let Z and R denote the sets of all integers and real numbers, respectively. Throughout the paper, the norm ‖ u ‖ = max ( i,j ) | u$_{ij}$ | , where u = { u$_{ij}$ } = ( u$_{11}$, . . . , u$_{1}$$_{n}$, . . . , u$_{m}$$_{1}$ . . . , u$_{mn}$ ) ∈ R m × $^{n}$, will be used. Suppose that θ = { θ$_{p}$ } and ζ = { ζ$_{p}$ } , p ∈ Z , are sequences of real numbers such that the first one is strictly ordered, | θ$_{p}$ | → ∞ as | p | → ∞ , and the second one satisfies θ$_{p}$ ≤ ζ$_{p}$ ≤ θ$_{p}$$_{+1}$ for all p ∈ Z . The sequence ζ is not necessarily strictly ordered. We say that a function is of γ − type, and denote it by γ ( t ) , if γ ( t ) = ζ$_{p}$ for θ$_{p}$ ≤ t < θ$_{p}$$_{+1}$, p ∈ Z . One can affirm, for example, that 2 [ t + 1 2 ] is a γ − type function with θ$_{p}$ = 2 p − $_{1}$, ζ$_{p}$ = 2 p. Fix a non-negative number τ ∈ R and let C 0 be the set of all continuous functions mapping the interval [ − τ, 0] into R , with the uniform norm ‖ φ ‖ 0 = max t ∈ [ − τ, 0] | φ ( t ) | . Moreover, we denote by C the set consisting of continuous functions mapping the interval [ − τ, 0] into R m × $^{n}$, with the uniform norm ‖ φ ‖ 0 = max t ∈ [ − τ, 0] ‖ φ ( t ) ‖ . In the present study, we propose to investigate retarded SICN N s with functional response on piece- wise constant argument of the following form, where f : C 0 × C 0 → R is a continuous functional. In network (2.3), the terms x$_{klt}$ and x$_{klγ}$$_{(}$$_{t}$$_{)}$ must be understood in the way used for functional differential equations [68]-[70]. That is, x$_{klt}$ ( s ) = x$_{kl}$ ( t + s ) and x$_{klγ}$$_{(}$$_{t}$$_{)}$ ( s ) = x$_{kl}$ ( γ ( t ) + s ) for s ∈ [ − τ, 0] . Let us clarify that the argument function γ ( t ) is of the alternate type. Fix an integer p and consider the function on the interval [ θ$_{p}$, θ$_{p}$$_{+1}$ ) . Then, the function γ ( t ) is equal to ζ$_{p}$. If the argument t satisfies θ$_{p}$ ≤ t < ζ$_{p}$, then γ ( t ) > t and it is of advanced type. Similarly, if ζ$_{p}$ < t < θ$_{p}$$_{+1}$, then γ ( t ) < t and, hence, it is of delayed type. Consequently, it is worth noting that the SICN N (2.3) is with alternate constancy of argument. It is known that γ ( t ) is the most general among piecewise constant argument functions [22]. Our model is much more general than the equations investigated in [71]-[74], where the delay is constant τ = 1 and it is equal to the step of the greatest integer function [ t ] . Differential equations with functional response on the piecewise constant argument were first introduced in the paper [75]. In the present study, we apply the theory to the analysis of neural networks. All previous authors were at most busy with terms of the form x ( γ ( t )) . Thus, one can say that retarded functional differential equations with piecewise constant argument in the most general form is investigated in this paper. Since the model (2.3) is a new one, we have to investigate not only the existence of almost periodic solutions and their stability, but also common problems of the existence and uniqueness of solutions, their continuation to infinity and boundedness. One can easily see that system (1.2) is a particular case of (2.3). Additionally, results of the present 5 dx$_{ij}$ dt = − a$_{ij}$x$_{ij}$ − ∑ C$_{kl}$ ∈ N$_{r}$ ( i,j ) C kl ij f ( x$_{klt}$, x$_{klγ}$$_{(}$$_{t}$$_{)}$ ) x$_{ij}$ + L$_{ij}$ ( t ) , (2.3) 2 Preliminaries