This manuscript introduces the square-mean doubly weighted pseudo almost automorphy and also square-mean doubly weighted pseudo almost automorphy in the sense of Stepanov $\left(S_l^2\right)$ over time scales. We derive results for a general stochastic dynamic system on time scales which can model a stochastic cellular neural network with time shifting delays on time scales. The coefficients are considered to be doubly weighted Stepanov-like pseudo almost automorphic functions in square-mean sense which is more general than weighted pseudo almost automorphic functions. We present several new and key results such as composition theorem for such functions on time scale. These results play a crucial role in order to study qualitative properties of nonlinear differential equations. Furthermore, we study the existence of a unique solution of stochastic delay cellular neural network on time scales. These results improve and extend the previous works in this direction. At the end, a numerical example is given to illustrate the analytical findings.

该手稿介绍了Stepanov的平方均值双重加权伪几乎自同构以及平方均值双重加权伪几乎自同构 $（{\mathrm{小号}}_{升}^2）$随着时间的推移。我们在时标上推导了一个通用随机动力学系统的结果，该系统可以在时标上具有时移延迟的情况下对随机细胞神经网络进行建模。在平方均值的意义上，系数被认为是双重加权的类似于Stepanov的伪几乎自纯函数，比加权的伪几乎自纯函数更普遍。我们提出了一些新的关键结果，例如时间尺度上此类函数的合成定理。这些结果对于研究非线性微分方程的定性性质起着至关重要的作用。此外，我们研究了时标上随机延迟细胞神经网络的唯一解决方案的存在。这些结果改进并扩展了先前在该方向上的工作。在最后，