conditions ( C 1) and ( C 2) that for each i = 1 , 2 , . . . , m, j = 1 , 2 , . . . , n, there exist positive numbers K$_{ij}$ such that | u$_{ij}$ ( s, τ ) | ≤ K$_{ij}$e − λ$_{ij}$ ( s − τ $^{)}$, s ≥ τ. Suppose that M$_{f}$c < 1 , where c = max ( i,j ) ( K$_{ij}$ λ$_{ij}$ + pδK$_{ij}$ 1 − e − λ$_{ij}$ψ ( ω ) ) ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij and δ = max 1 ≤ k ≤ $_{p}$δ$_{k}$ . In the remaining parts of the paper, we will denote and M$_{F}$ = max η ∈ Λ ‖ F ( η ) ‖ , H$_{0}$ = M$_{F}$ 1 − M$_{f}$c max ( i,j ) ( K$_{ij}$ λ$_{ij}$ + pδK$_{ij}$ 1 − e − λ$_{ij}$ψ ( ω ) ) and d = ( M$_{f}$ + H$_{0}$L$_{f}$ ) max ( i,j ) K$_{ij}$ ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij . The following conditions are also needed. In the next section, we will deal with the existence, uniqueness and attractiveness property of the bounded solutions of SICNN (1.2). According to the results of [77, 78], a bounded on R function y ( s ) = { y$_{ij}$ ( s ) } , i = 1 , 2 , . . . , m, j = 1 , 2 , . . . , n, is a solution of the impulsive system (2.7) if and only if the equation is valid. The existence of bounded on R solutions of (2.7) is considered in the next assertion. Lemma 3.1 Suppose that the conditions ( C 1) − ( C 5) are fulfilled. For each solution ζ = { ζ$_{k}$ }$_{k}$$_{∈}$$_{Z}$ of (1.3) , there exists a unique bounded on R solution φ$_{ζ}$ ( s ) = { φ ij ζ ( s ) } , i = 1 , 2 , . . . , m, j = 1 , 2 , . . . , n of (2.7) such that sup s ∈ R ‖ φ$_{ζ}$ ( s ) ‖ ≤ H$_{0}$. Proof. Fix a solution ζ = { ζ$_{k}$ }$_{k}$$_{∈}$$_{Z}$ of (1.3) and take into account the set D$_{0}$ ⊂ P C ( R ) consisting of functions of the form z ( s ) = { z$_{ij}$ ( s ) } which have discontinuities at the points s$_{k}$, k ∈ Z such that ‖ z ‖$_{∞}$ ≤ H$_{0}$, where ‖ z ‖$_{∞}$ = sup s ∈ R ‖ z ( s ) ‖ . The set D$_{0}$ is complete [77]. Define an operator Π on D$_{0}$ as 7 (Π z ( s ))$_{ij}$ = − ∫ s −∞ u$_{ij}$ ( s, τ ) [ ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij f ( z$_{hl}$ ( τ )) z$_{ij}$ ( τ ) − L$_{ij}$ ( ψ − $^{1}$( τ ) , ζ ) ] dτ − ∑ −∞ <s$_{k}$<s u$_{ij}$ ( s, s$_{k}$ +) [ ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij f ( z$_{hl}$ ( s$_{k}$ )) z$_{ij}$ ( s$_{k}$ ) − ζ ij k ] δ$_{k}$, y$_{ij}$ ( s ) = − ∫ s −∞ u$_{ij}$ ( s, τ ) [ ∑ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij f ( y$_{hl}$ ( τ )) y$_{ij}$ ( τ ) − L$_{ij}$ ( ψ − $^{1}$( τ ) , ζ ) ] dτ − ∑ −∞ <s$_{k}$<s u$_{ij}$ ( s, s$_{k}$ +) [ C$_{hl}$ ∈ N$_{r}$ ( i,j ) C hl ij f ( y$_{hl}$ ( s$_{k}$ )) y$_{ij}$ ( s$_{k}$ ) − ζ ij k ] δ$_{k}$ (C5) ( M$_{f}$ + H$_{0}$L$_{f}$ ) c < 1; (C6) − λ + d + p ψ ( ω ) ln(1 + δd ) < 0 . 3 Bounded solutions