# Exploring the mathematical principles underlying signal statistics in sensory transduction

Vemula, Sai 1; Norwich, Kenneth 1,3; Wong, Willy 2,1

1. Institute of Biomaterials and Biomedical Engineering, University of Toronto; 2. Edward S. Rogers Sr. Department of Electrical & Computer Engineering, University of Toronto; 3. Department of Physiology, University of Toronto

**Introduction**

The study of perception involves the understanding of how information from the environment is conveyed to our senses, and how this information is processed by our sensory systems. The pathophysiology underlying sensory impairments (e.g. central auditory processing disorder or retinitis pigmentosa) can be viewed as deficits in information processing by our sensory organs. Developing mathematical models of sensation is critical for understanding and treating these impairments, as they can help uncover fundamental principles of the physiological systems. The entropic model of sensation, which incorporates the properties of information theory, aims to characterize sensory processing generally. The entropic model of sensation posits that perception is a result of measuring the entropy (i.e. the uncertainty) of a stimulus being presented to us. In other words, we gain information about our environment when we discern a specific quantity or type of stimulus from many possibilities.

In this model, sensory receptors can be regarded as drawing samples of stimuli from a larger population. For example, cilia on olfactory receptor neurons must sample odorant molecules in the nasal cavity, which is the first step of sensory transduction. Defining the relationship between the variance and mean of a stimulus presented to our senses is integral to the model. Is there a general theory that explains the relationship between the variance and mean of sensory stimuli?* *If this relationship is general, then it should be observed in all the senses.

**Approach**

Probing the mean and variance relationship involves considering the mechanism of transduction between the physical signal and sensory epithelium. This will likely involve modelling complex systems. One such example includes using Markov processes as a model of population dynamics. Such a model can describe the interaction between (say) odorant molecules together with binding sites on the cilia on the olfactory epithelium. A specific prediction of this model is that there exists a functional relationship between the mean population of binding molecules and their variance. This is surprising mathematically since for the normal distribution, the first and second moments are not generally connected. However, in real world systems, this makes good sense: larger quantities are generally associated with larger errors or fluctuations. In such a model, this will lead to a Poisson-gamma distribution which gives rise to the types of mean-variance relationship found in olfaction as well as other sensory modalities.

**Impact**

Developing a general theory of stimulus fluctuations will further strengthen our understanding of the principles which govern sensory transduction. Such principles will invariably help with furthering basic physiological science, as well as the diagnosis of disease and the development of sensory aids for impairment.