Question

Assign weights and threshold values in the following neural network so that the output neuron fires only when \(x_{1}\) and \(x_{3}\) have the value 1 and \(x_{2}\), has the value 0 . Remember that weights can be negative.

Short answer

Set weights: \(w_1 = 1\), \(w_2 = -2\), \(w_3 = 1\) and threshold: \(T = 1\).

Step-by-step solution

Understanding the Problem

We need to design a neural network with weights and thresholds so that it functions as a logical AND gate for inputs \(x_1 = 1\) and \(x_3 = 1\), and \(x_2 = 0\). The output neuron must fire (output = 1) only under these conditions.

Choosing Weights

Allocate weights for each input. Since the output should be activated when \(x_1 = 1\) and \(x_3 = 1\), assign these inputs positive weights: \(w_1 = 1\) for \(x_1\), and \(w_3 = 1\) for \(x_3\). Assign a negative weight \(w_2 = -2\) to \(x_2\) to ensure that its input value being 1 will more than counteract the positive influence of either of the other inputs.

Setting the Threshold

To ensure that the neuron fires only when all conditions (\(x_1 = 1\), \(x_3 = 1\), and \(x_2 = 0\)) are satisfied, set the threshold \(T\) to 1. This ensures that the net input \(f(x) = w_1 \, x_1 + w_2 \, x_2 + w_3 \, x_3\) must meet or exceed 1 to activate the neuron.

Checking the Solution

Verify the conditions:1. \(x_1 = 1, x_2 = 0, x_3 = 1\): The net input is \(1\times1 + (-2)\times0 + 1\times1 = 2\), which is greater than the threshold, so the neuron fires.2. Any other combination results in a net input less than 1 (either 0 or negative), so the neuron does not fire. For example, \(x_1 = 1, x_2 = 1, x_3 = 1\) gives \(1\times1 - 2\times1 + 1\times1 = 0\).

Key concepts

Logical AND Gate

A logical AND gate is a fundamental concept in digital circuits and neural network design. It outputs a true, or 1, only when all its inputs are true. In simpler terms, it's like a rule saying, 'only if all required conditions are met, I'll give the go-ahead.'
In the neural network designed for this exercise, the AND gate is used to ensure the neuron fires when specific conditions are satisfied: both inputs \( x_1 \) and \( x_3 \) must be 1, while \( x_2 \) must be 0. This setup allows the neuron to behave just like an AND gate, giving an output of 1 only when the inputs meet these criteria.
Understanding how an AND gate works is pivotal when creating a neural network model because it helps in setting the rules for the neuron firing conditions.

Weights and Thresholds

Weights and thresholds play a crucial role in determining how a neural network processes inputs and generates the desired outputs.
Weights are the values assigned to each input of the neuron to adjust the importance of each input in the final decision. In our exercise, for instance, the weights were chosen as 1 for \( x_1 \) and \( x_3 \), and -2 for \( x_2 \). These weights ensure that \( x_1 \) and \( x_3 \) positively influence the neuron's firing. Meanwhile, the negative weight on \( x_2 \) counteracts the positive effect when \( x_2 \) is 1, preventing the neuron from firing unless all conditions are specifically satisfied.
A threshold is the critical value that the neuron's weighted sum must surpass for the neuron to fire. In this exercise, the chosen threshold was set to 1, meaning that only when the weighted sum meets or exceeds 1 does the neuron produce an output of 1. Configuring the right weights and threshold is essential for the neuron to function as an intended logical AND gate.

Neuron Firing Conditions

Neuron firing conditions determine when a neuron in a neural network activates and sends a signal forward. Think of it as the neuron's rulebook -- it checks if all the rules are followed before triggering an action.
In this specific scenario, we set the firing conditions such that the neuron only fires if \( x_1 = 1 \), \( x_3 = 1 \), and \( x_2 = 0 \). These conditions are established through the clever selection of weights and a threshold. As previously discussed, weights of 1 for \( x_1 \) and \( x_3 \), a weight of -2 for \( x_2 \), and a threshold of 1 ensure that any deviation from the desired input combination results in the neuron not firing.
This firing condition ensures that only the specific combination of inputs will produce a sufficient total sum to surpass the threshold, illustrating how firing conditions can shape a neuron's response in a neural network.