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On the Wikipedia page we can read the basic structure of an artificial neuron (a model of biological neurons) which consist:

  • Dendrites - acts as the input vector,
  • Soma - acts as the summation function,
  • Axon - gets its signal from the summation behavior which occurs inside the soma.

I've checked deep learning Wikipedia page, but I couldn't find any references to dendrites, soma or axons.

Which type of artificial neural network implements or can mimic such a model most closely?

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Only a small portion of the habituation, sensitization, and classical conditioning behavior of neurons has been primitively simulated in ANN systems. Simulation of actin cytoskeletal machinery1 and other agents of neural plasticity, central to learning new domains, is in its beginnings2. As of this writing, the complexity of neuron activation dwarfs the models being used in working commercial ANN systems, but the research continues along multiple fronts.

  • The neuroscience of learning3,
  • Parallel hardware approaches that better support ANN simulation accuracy4, 5, and
  • Dynamic frameworks6

This list and the examples referenced in the superscripts, with links below, represent a tiny sample of the information available and the work in progress.

References

[1] Molecular Cell Biology. 4th edition.; Lodish H, Berk A, Zipursky SL, et al.; New York: W. H. Freeman; 2000.; Section 18.1 The Actin Cytoskeleton

[2] NEURON Software; Yale U

[3] Molecular Cell Biology. 4th edition.; Lodish H, Berk A, Zipursky SL, et al.; New York: W. H. Freeman; 2000.; Section 21.7 Learning and Memory

[4] Artificial Neural Networks on Massively Parallel Computer Hardware; Udo Seiffert; University of Magdeburg, Germany

[5] NeuroGrid; Stanford U

[6] Explanation of Dynamic Computational Graph frameworks

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ANNs approximate biological neuronal networks. The approximation began with extreme simplicity in the early perceptron design. Spiking networks are examples of more accurate approximations. More accurate still, are complex simulations of neuron behavior that therefore necessitate significant computing resources.

If you are interested in a mathematical overview on analysis of biological neuron models I can recommend Dynamical Systems in Neuroscience by Eugene Izhikevich.

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Most artificial neurons model biological neurons but in a very simplistic way. Nowadays, the main aim is to achieve better performance at prediction tasks. However, there is a body of literature in neuroscience that looks at computational models of neurons. Neurons are complicated cells and our understanding of neurons is still not complete.

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A computational model that attempts to closely mimic the human neural networks is Numenta's hierarchical temporal memory (which has not yet received much attention from the machine learning community). In their models, they explicitly model and implement dendrites and other biological concepts.

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