Exploring and updating the mathematical model behind theolfactory sense
Radu Cimpean1
1.Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence,
University of Groningen, The Netherlands

Summary. Communication is a key aspect when it comes to the existence and functioning ofsystems. For the biological systems, neurons fulfil this role via chemical and electrical modes. Thereis an exchange of chemical substances that either excites or inhibits the relevant components till aspike, also called action potential, is reached and thus the neuron has relayed the information. As asystem of interest, the research will focus further on the olfactory system.

The basic function of the olfactory system is represented by the identification of odors as singularor mixed chemical compounds. Given the amount of research conducted on different species, the firstquestion to be asked is the following: Is there common ground regarding the olfactory sense across different
classes of animals? As an answer, several articles offer some details on the matter [1], [6], [4]. Accordingto [1], after reviewing carefully the existing literature, came to the conclusion that ”notwithstanding
mechanistic differences between species, the general principles of olfactory organization are shared bymany animals”. This means that the organization of the olfactory pathway displays striking similarities
and it spans across a broad phylogenetical array of animals (from insects to vertebrates and even humans).
Similar conclusions have been reached by [6], [4]. and by taking into account these findings, we can infercertain commonalities when discussing the olfactory system across species. This allows us to make useof literature sources that used different species (e.g. fruit flies, zebrafish) to build a general view on theolfaction principles. For a better understanding, the case of human olfaction will be further exemplified.
The chemical molecules are entering the nasal cavity and reach the olfactory epithelium, where thebinding to the Olfactory Receptor Neurons (ORN) occurs, triggering a two pathway response. The
information is further relayed to the olfactory bulb (OB). The exchange occurs in ball shaped components
called glomeruli, where the closely related ORN synapse with other neurons, called mitral cells (MC),
one for each glomerulus. The MC have a major role in conveying the odor information, through a process
called whitening. Since there can be a lot of redundancy in the information gathered in the glomeruli
from the ORN, the OB performs the above mentioned process with the purpose of decorrelating theactivity patterns, aspect achieved via the interaction between the MC, which are excitatory by natureand granule cells (GC)/interneurons (IN), which are inhibitory. As a result, a preprocessed, decorelated
pattern of signal activity is obtained, which is further fed to the higher areas of the brain for analysis
and identification.

Figure 1: Whitening process in the olfactory bulb (left) and connectivity matrices(right) (adapted from[7].)Given the above brief biological description of the olfactory process, the interest of this research liesin the mathematical modeling. The base model is provided by [5] in the form of equations of motion:Ẋ = −H0 GY (Y ) − αx X + I
Ẏ = W0 GX (X) − αY Y + Ic ,where
• X-signal from the MC; Y -signal from the GC;• GX , GY cell output as continuous functions, proportional to the cell firing frequency;• H0 - n × m matrix describing the synaptic connection MC → GC; W0 - m × n matrix describing
the synaptic connection GC → MC;• I - input, result of the sniffing process; Ic - input control, coming from higher areas of the brain;• αX , αY - parameters.
Moreover, [5] further indicates that the OB system may behave like a harmonic oscillator with damping, thus treating it as a group of coupled non-linear oscillators.
By reducing the biological complexity of the OB to the damped harmonic oscillator approach, severalissues arise. One such issue is that the model does not account for the whitening process. This aspect canbe confirmed by observing the diagrams in [7] and [2]. The outcomes indicated there do not fully match
the proposed oscillator model. Another issue concerns the parameters (αX , αY ) for the OB system. For
instance, in [5], they are taken as fixed values related to the firing rate of the neurons.
Given these issues, several research directions are being explored. As a starting point, it would be ofinterest to account for the whitening process by providing a function based approach to the parameters(αX , αY ). This would entail modeling them as characteristic functions, and further considering obtaining
a well-posedness result, or simply put, getting a unique solution whose value will only modify slightly if wemake a small change in the initial conditions. Another key aspect is the optimization of the connectivity
matrices H0 and W0 , again towards the goal of expressing the whitening process more accurately.
Therefore, it would be of interest to make changes to the above given model to incorporate theaforementioned biological findings that could be further used in analysing more accurately the behaviorof the olfactory system. This will provide valuable insight into applications for an artificial nose, that
could be used in detecting chemical indicators of certain diseases.

References[1] B. Ache and J. Young, ”Olfaction: Diverse species, conserved principles”. Neuron, 48:417–430, 2005.
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[4] B. Kaupp, ”Olfactory signalling in vertebrates and insects: differences and commonalities”. Nature
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[7] A. Wanner and R.W. Friedrich, ”Whitening of odor representations by the wiring diagram of theolfactory bulb”. Nat Neurosci.,23(3):433-442, 2020.