**Introduction**

In artificial neural networks, an activation function is a mathematical function applied to the weighted sum of a neuron's inputs plus a bias term. This function determines the neuron's output, introducing essential non-linearity into the network. Without activation functions, neural networks would be limited to linear transformations, severely restricting their ability to model complex real-world relationships.

**Purpose**

**Non-linearity:**Activation functions help neural networks learn complex patterns in data that linear models cannot capture. This gives neural networks superior flexibility and the power to represent intricate functions.**Gradient-based learning:**Most activation functions are differentiable (have derivatives), facilitating the backpropagation algorithm, the cornerstone of neural network training. The derivatives provide information about how to adjust weights and biases to reduce errors.**Decision-making:**Activation functions can model threshold-like behaviors, making the neuron "fire" (activate) only when the input signal surpasses a certain strength. This emulates the decision-making process seen in biological neurons.

**Types of Activation Functions**

**Commonly Used:****Sigmoid:**Maps inputs to a value between 0 and 1, creating a smooth S-shaped curve. Traditionally popular, but can suffer from vanishing gradients in deep networks.**Tanh (Hyperbolic Tangent):**Similar to the sigmoid but has a range of -1 to 1. Also susceptible to vanishing gradients.**ReLU (Rectified Linear Unit):**Very popular due to its simplicity and ability to mitigate vanishing gradients. It outputs 0 for negative inputs and the input itself for positive inputs.**Leaky ReLU:**A variation of ReLU that introduces a small non-zero slope for negative inputs, addressing the "dying ReLU" problem.

**Other Notable Functions:****Softmax:**Often used in the output layer for multi-class classification, producing a probability distribution over output classes.**Swish:**A relatively new function that demonstrates improvement over ReLU in deeper networks.**Linear:**Sometimes used in output layers where linear relationships are desired. Provides no non-linearity.

**Choosing an Activation Function**

The choice of activation function depends on various factors:

**Network Architecture:**ReLU variations generally perform well in hidden layers, while softmax is common in output layers for classification tasks.**Problem Type:**For regression problems, linear output activations might suffice. Classification problems often necessitate sigmoid or softmax.**Sparsity:**ReLU promotes sparsity (many zero outputs), which can improve computational efficiency.**Vanishing Gradients:**Leaky ReLU, Swish, and similar functions can help mitigate vanishing gradients in very deep networks.