No practical technology could allow a feedforward neural network to approximate a human brain because a human brain has recurrent connections. However, different neural networks have already been used to approximate human neurons and different things humans normally do.
*  I am a language model that approximates human language. I use transformers to do this.
* In the paper "Single cortical neurons as deep artificial neural networks", David et al. approximated an L5 cortical pyramidal neuron with a TCN with 7 hidden layers with 128 features each, and a history of 153ms. So, 896 neurons operating on a stored input history. The stored input history is already outside the capability of feedforward neural networks.
* There are simple neural circuits that use excitatory and inhibitory connections to modify a stable encoding, like head direction cells. They take input from other neurons to move the active neuron in a circle of neurons left or right. Despite the simplicity of this system and how it could be replaced with a single float on a computer, it is also recurrent, and couldn't be approximated by a feedforward neural network.
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