A cell must sense and respond to its environment. One aspect of this is regulation of gene activity by interactions of for example, transcription factors. When interaction types can be well enough approximated as either of inhibition or activation, you can model them with boolean networks and when levels matter, some have found moderate success with recurrent neural networks under a restriction on what NN nodes represent, to ease interpretability.
That is not the same thing as saying your genome unrolls neural networks. What it is actually saying is that the complexity of the best performing neural network indicates the richness of the underlying computation, and the predictive accuracy of the model captures the functional equivalence of the respective computations (in the cell and in the neural network). The learned model is the instance and neural networks are the class, it is a mistake to place emphasis on neural network, they are merely a way of packaging the computational model of interest.
A cursory review here: https://www.sciencedirect.com/science/article/pii/S001048251...
If you have time for videos: https://www.youtube.com/watch?v=vA9D727AGHI , https://www.youtube.com/watch?v=ZX-GrO2qXQM
More relevant to this article is the study of cancer within the framework of evolutionary game theory, as done here: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2768082/
There is a precise correspondence between that formulation and one done in terms of online sequential prediction or the so called experts algorithm. All of these are models and still quite limited at that but biology is complex and the future lies in the development and leveraging of these rich connections.