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In artificial neural networks, activation functions are used for neurons, i.e. the sigmoid activation:

Sigmoid function

Which can be implemented in code as (in Python):

def sigmoid(x):
    return 1 / (1 + math.exp(-x))

How can we implement a biological activation function, such as the Hodgkin-Huxley model, whose mathematical form is:

enter image description here

Where:

  • Cm: Capacitance
  • Vm: Membrane potential

As mentioned on the Wikipedia page,

The typical Hodgkin–Huxley model treats each component of an excitable cell as an electrical element (as shown in the figure). The lipid bilayer is represented as a capacitance (Cm). Voltage-gated ion channels are represented by electrical conductances (gn, where n is the specific ion channel) that depend on both voltage and time. Leak channels are represented by linear conductances (gL). The electrochemical gradients driving the flow of ions are represented by voltage sources (En) whose voltages are determined by the ratio of the intra- and extracellular concentrations of the ionic species of interest. Finally, ion pumps are represented by current sources (Ip).[clarification needed] The membrane potential is denoted by Vm.

EDIT: In addition to implementing a biological activation function to neurons in an artificial neural network, does this network simulate (in a simplified way) networks of biological neurons?

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    $\begingroup$ You can't just plug the HH model directly into a typical machine learning-style neural net (ANN), as they work differently. The HH model is a set of differential equations that operate in continuous time, whereas ANNs operate instantaneously (in the feedforward case) or in discrete steps (in the recurrent case). That said, there's a huge literature on simulating biological neural nets at many levels of abstraction (including HH). $\endgroup$ – user20160 Feb 23 '18 at 7:15
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This is basically done by so-called Spiking Neural Networks. They are based on Leaky Integrate and Fire models which are essentially Hodgson-Huxley with an extra resistive current term.

You can find a reference here:

Training Deep Spiking Neural Networks using Backpropagation -- Lee, et. al.

and here:

https://www.frontiersin.org/articles/10.3389/fnins.2016.00508/full

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