We explained in the last class, why do we need a neural network? After answering the questions of machine learning and deep learning, what kind of problems are good at solving , in this lesson we will start to explain the components of neural networks, neurons, and how do neurons work? How to compose a neural network, and the mathematical expression of neurons .
First of all, let's take a look at what a neuron is. Let's use the neuron of the organism as an analogy. For example, how do the neurons of the human brain work and structure successfully? First we see the following picture.
As shown in the simple schematic diagram, it is composed of dendrites, axons, and a nucleus connecting the dendrites and axons. The dendrites are responsible for receiving signals from other neurons. The nerve cell nucleus collects the amount of signals transmitted by the dendrites. If the amount of signal exceeds a threshold, the nucleus transmits the signals to other neurons through the axons. The specific working process is shown in the figure below:
After the nerve cell receives the signal from the dendrites, it starts to turn red and judges whether it should output a signal. If the sum of the input signal is less than the threshold , it will not output any signal, otherwise, it will output a fixed signal. Why there is a threshold requirement is actually very easy to understand. The reason why we laugh when watching comedies is because the signal transmitted by the nerve brain cells exceeds our laugh point , so we laughed, and the laugh point is the threshold . The output signal is laugh and non-smile, which can also be expressed in this way. Smile is represented by 1 and non-smile is represented by 0.
The entire complete process can be demonstrated by the following animation:
The working structure of sorting neurons is as follows:
The sum of the signals from multiple other neurons becomes the input of the neuron.
If the sum of this signal exceeds the inherent threshold of the neuron, it will fire.
The output signal of the neuron can be represented by the digital signal 0 and 1. Even if there are multiple output terminals, the value is the same.
Based on the above three points, we try to express the working structure of neurons in a mathematical way.
We can use mathematical expressions to express the input signal. Since the input signal is the output signal from the adjacent neuron, according to the input signal at point 3, it can also be expressed with two kinds of information, with or without information. Therefore, it is expressed by the variable X. When the signal is input As follows.
Next, we use a mathematical formula to express the output signal. According to the third point, the output signal can be used to indicate whether there is ignition or not. The output signal is represented by two kinds of information, so the variable y represents the output signal as shown below.
From point 1 and point 2, it can be seen that whether a neuron fires or not is determined based on the sum of the input signals from other neurons , but the summation method should not be a simple summation. For example, we are watching a comedy sketch. The reason why Zhong laughed is because we saw the actor's exaggerated performance, not just heard the lines. In the process of watching a comedy performance, it is obvious that the amount of signal sent by the optic nerve cell is significantly higher than the signal brought by hearing. Therefore, the sum of the input signal and the brain are summed by changing the weight . So how does this change weight to sum up mathematically express it? Look carefully at the picture below.
We get the formula 1 in the figure, W1X1+ W2X2 + W3X3 .
According to the second point we summarized, the neuron does not ignite when the sum of the signals exceeds the threshold and the ignition does not exceed the threshold. Therefore, using Formula 1, the ignition condition can be expressed by the following Formula 2.
Next, we will graph the ignition condition formula 2 with the sum of the input signals of the neuron as the output signal of the neuron on the horizontal axis and y as the vertical axis to graphically express the formula 2 as shown in the figure below. When the sum of the input signals is less than θ, y takes the value 0, otherwise y takes the value 1.
If we want to use a function to represent this graph, we need to use the following unit step function . Its graphic is shown in the figure below.
Use the unit step function. Formula 2 can be expressed by Formula 3.
The following table can prove that Formula 2 and Formula 3 are equivalent.
Finally, leave a homework for everyone.
Assignment: If you are a product manager, and you put forward the requirements for the realization of neurons for algorithm engineers, how would you mention it? If you are an algorithm engineer, what kind of requirements do you need to be able to give a mathematical representation of neurons?
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