A feedforward neural network is an artificial neural network in which information flows in a single direction – inputs are multiplied by weights to obtain outputs (inputs-to-output). It contrasts with a recurrent neural network, in which loops allow information from later processing stages to feed back to earlier stages. Feedforward multiplication is essential for backpropagation, because feedback, where the outputs feed back to the very same inputs and modify them, forms an infinite loop which is not possible to differentiate through backpropagation. This nomenclature appears to be a point of confusion between some computer scientists and scientists in other fields studying brain networks.

The two historically common activation functions are both sigmoids, and are described by

$${\displaystyle y(v_{i})=\tanh(v_{i})~~{\text{and}}~~y(v_{i})=(1+e^{-v_{i}})^{-1}.}$$

The first is a hyperbolic tangent that ranges from -1 to 1, while the other is the logistic function, which is similar in shape but ranges from 0 to 1. Here ${\displaystyle y_{i}}$ is the output of the ${\displaystyle i}$-th node (neuron) and ${\displaystyle v_{i}}$ is the weighted sum of the input connections. Alternative activation functions have been proposed, including the rectifier and softplus functions. More specialized activation functions include radial basis functions (used in radial basis networks, another class of supervised neural network models).

In recent developments of deep learning, the rectified linear unit (ReLU) is more frequently used as one of the possible ways to overcome the numerical problems related to the sigmoids.

Learning occurs by changing connection weights after each piece of data is processed, based on the amount of error in the output compared to the expected result. This is an example of supervised learning, and is carried out through backpropagation.

We can represent the degree of error in an output node ${\displaystyle j}$ in the ${\displaystyle n}$-th data point (training example) by ${\displaystyle e_{j}(n)=d_{j}(n)-y_{j}(n)}$, where ${\displaystyle d_{j}(n)}$ is the desired target value for ${\displaystyle n}$-th data point at node ${\displaystyle j}$, and ${\displaystyle y_{j}(n)}$ is the value produced at node ${\displaystyle j}$ when the ${\displaystyle n}$-th data point is given as an input.

The node weights can then be adjusted based on corrections that minimize the error in the entire output for the ${\displaystyle n}$-th data point, given by