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Neurons are a special type of biological cell, serving as the organic computing units in a neuronal system. In previous studies, researchers built a computational model mirroring neurons in the biological system. In order to build such a model, it was essential to emulate the biological mechanism of the neuron.

According to biological research, a neuron consists of three key parts: the cell body, dendrites, and axons, as shown in figure 3.2. The cell body offers the energy supply for neuronal activities, where metabolism and other biochemical processes are carried out, dendrites are the ports to receive information from other neurons, and axons are the gateways that transmit the excitatory information to other neurons. In addition, the synapse is the structure in which one neuron interacts with another neuron for communication.

Figure 3.2. Key elements of a biological neuron.

According to neurobiological research results, the information processing and transmission mechanism of the biological neuron has the following characteristics:

1 Information integration: A neuron can integrate different neurotransmitters transmitted by other neurons into an overall response.

2 Potential difference: Defined as the difference between the electrical potentials inside and outside the cell membrane, and the differences in neuronal state.

3 Transmission threshold: The membrane potential changes constantly when a neuron receives information. If the fixed potential value is exceeded, an action potential is transmitted along an axon as an electrical pulse. Since the threshold is used, the neuron is a nonlinear system.

In reference to the characteristics of a biological neuron and its functions, an artificial neuron is an approximation to a biological neuron, which is a simple mathematical model illustrated in figure 3.3.

Figure 3.3. Mathematical model of an artificial neuron.

Mimicking a biological neuron, the mathematical model of an artificial neuron can be represented as follows: the input vector x represents the signals (from the dendrites), the weight vector w corresponds to the strength of the pathway (dendrite and synapse), and the summation node ∑ and the activation function φ(·) represent the integration and activation of the input signals (the cell body) and the thresholder output y (along the axon), respectively. Such an artificial neuron model can be formulated as follows:

v=∑i=1mxiwi+by=φ(v),(3.1)

where w_i represents the weight for the input signal component x_i, b is a bias, v is the inner product of the input vector and the weight vector, and y represents the output of the neuron after a nonlinear activation. Please note that the neuron based on the inner product is not the only type of neuron in artificial neural systems. For example, there are also quadratic neurons dedicated to extracting nonlinear features directly (Fan et al 2017a, 2017b). Discussion of these is outside the scope of this chapter.

Given the mathematical model of the neuron, the aforementioned single-hidden layer neural network can be formulated into equation (3.2). The structure of the single-hidden layer neural network is shown in figure 3.4. The corresponding mathematical formulas are as follows:

h(1)=φ(1)∑i=1mxiwi(1)+b(1)y=φ(2)∑j=1nhj(1)wj(2)+b(2),(3.2)

where x∈Rm is an input vector, h(1)∈Rn is an output vector of the hidden layer, w(1)∈Rm×n and b(1)∈Rn are the weight matrix and bias for the input to the hidden layer, and φ(1) and φ(2) are the corresponding activation functions, respectively. A multi-hidden-layer neural network can be obtained when the number of hidden layers exceeds 1, which is an extension of the single-hidden-layer neural network.

Figure 3.4. Architecture of the neural network with a single-hidden layer.