When is the last time you used a sigmoid function?

The sigmoid logistic function was introduced in a series of three papers by Pierre François Verhulst between 1838 and 1847, who devised it as a model of population growth by adjusting the exponential growth model, under the guidance of Adolphe Quetelet (Figure 1).

Figure 1 Verhulst's logistic curve, contrasted with a logarithmic curve

For a 1963 example of how a sigmoid function and curve can be used, see Reference 1. An electro-optical instrument measures the capacity of red blood cell membranes as their internal pressure increases due to the diffusion of water into the cells via a gradual deceasing of salt concentration of the fluid surrounding the cell. This test yields an osmotic fragility curve. A direct sigmoid curve or a derivative curve can then be used to fit the data and then recorded.

In today’s modern world of artificial intelligence (AI), the sigmoid function is used in artificial neural networks (Reference 6) to determine the relationships between biological and artificial neural networks.

See how the sigmoid function can also be used in machine learning (ML) in a data center. In Reference 7, section 2.2.2 on Forward Propagation, the activation function mimics the biological neuron firing within a network by mapping the nodal input values to an output in the range (0,1). It is given by the sigmoidal logistic function.

Steve Taranovich is a senior technical editor at EDN with 45 years of experience in the electronics industry.


  1. An Instrument for Automatically Recording the Osmotic Fragility Curve of Red Cells and or its Derivative, D. Danon, E. H. Freit, Y. F. Freit, and Y. Lipkin, IEEE Transactions on Bio-medical Electronics, 1963
  2. The Origins of Logistic Regression, J.S. Cramer, Tinbergen Institute Discussion Paper, Nov. 2002
  3. Continuous Output—The Sigmoid Function, Dr. Mark Humphrys, School of Computing, Dublin City University
  4. Sigmoid Function, Science Direct
  5. MathWorks Sigmoid Function
  6. Sigmoid Functions and their usage in Artificial Neural Networks
  7. Machine Learning Applications for Data Center Optimization, Jim Gao, Google
  8. Sigmoidal Problems