Logistic Population Growth and Neural Activation

Introduction:

The population of most of the species follow an exponential growth in an open and infinitely expandable environment. However, in practice this assumption won't hold good. If we consider an isolated environment with limited resource that would better represent the typical environmental condition in nature. Till the point these resources are in excess to the total consumption of the specie population there would be an exponential growth as if the system had infinite resources but when the total consumption of resources by the specie population reaches near the total resources available there is a competition for survival which begins within the ecosystem. 

A typical ecosystem will contain multiple species competing for the resources and therefore this exponential growth is curtailed by the food availability, competition for mating, disease, predation and other such factors. More the competition for the resources, lesser the growth and eventually it slows down to zero as population reaches its carrying capacity (K) for that particular environment. The result of such a growth is an 'S' shaped population growth curve. This phenomenon is known as Logistic Growth and the curve representing this is known as Logistic Curve. Below is the differential equation representing logistic curve-

Solving this equation gives us -

The same model can be used to explain the effect of advertisement and word of mouth influence in technology diffusion, spread of disease or saturation of market sales. 

Interesting insights:

The solution to the differential equation representing logistic growth is very similar to Sigmoid function used extensively in mathematics and machine learning. In neural networks sigmoid function is used to emulate the neural signal activation inside the brain. The neurons use both mechanical and ionic mechanism on their dendrites (input) to produce electric signals which are propagated on axons (output). These electric activations are similar to sigmoid function applied on the resultant of dendrite signals. This gives a small inkling that our neurons might also follow logistic growth model in signal propagation.

Hough Transform

Introduction

The Hough Transform (named after Paul Hough) is a feature extraction technique which is widely used in Digital Image Processing and related fields. 

Where to use

In automated analysis of digital images, a subproblem often arises of detecting simple shapes, such as straight lines, circles or ellipses. In many cases an edge detector can be used as a pre-processing stage to obtain image points or image pixels that are on the desired curve in the image space. Due to imperfections in either the image data or the edge detector, however, there may be missing points or pixels on the desired curves as well as spatial deviations between the ideal line/circle/ellipse and the noisy edge points as they are obtained from the edge detector. For these reasons, it is often non-trivial to group the extracted edge features to an appropriate set of lines, circles or ellipses. The purpose of the Hough transform is to address this problem by making it possible to perform groupings of edge points into object candidates by performing an explicit voting procedure over a set of parameterized image objects.  

How it works

Hough Transform uses a voting procedure. This voting procedure is carried out in a parameter space, from which object candidates are obtained as local maxima in a so-called accumulator space that is explicitly constructed by the algorithm for computing the Hough transform.