There is a category of neural networks that more closely attempt to mimic biological neural networks by incorporating also time (i.e. not all neurons fire at the same time). They are called spiking neural networks (SNNs) and their name comes from the fact that they use spiking neurons (i.e. neurons that fire discrete signals and affect other neurons at possibly different times).
SNNs are mainly used in neuroscience, and aren't commonly used in machine learning because they currently have some apparent limitations (e.g. non-differentiability, so gradient descent and back-propagation can't be applied, but GD and BP aren't really biologically realistic anyway, although some people already tried to apply GD to SNNs) and their performance isn't still as good as the performance of traditional deep learning models, which make them not so appealing to the deep learning community (which is currently mainly driven by performance and utility). Nevertheless, the performance gap between traditional neural networks and spiking neural networks is decreasing. See Deep Learning in Spiking Neural Networks (2019) by Amirhossein Tavanaei et al. for more details.
There are already commercial implementations of a hardware-accelerated SNNs (e.g. BrainChip provides this service). These hardware-accelerators are often called neuromorphic chips (or processors) and all computing based on SNNs or processors that attempt to implement biological neural networks is known as neuromorphic computing.
There's also the related area called reservoir computing, which studies neural networks (such as liquid-state machines or echo state machines) that make use of reservoirs (which are fixed during learning) to attempt e.g. to improve training efficiency. See An overview of reservoir computing: Theory, applications and implementations (2007) by Benjamin Schrauwen et al. for an overview.
Numenta (and, in particular, Jeff Hawkins, the founder of Numenta and author of an interesting book called On Intelligence) has also been studying neuroscience for a long time in order to develop models and theories of human intelligence. They call their new theory The Thousand Brains Theory of Intelligence, which is inspired by biological grid cells. This is also related and similar to capsule networks (often associated with Hinton).