Stochastic Models and Methods for the Study of Olfaction

The general aim of mathematical neuroscience is to provide a quantitative basis for describing what nervous systems do and to investigate the general principles by which they operate. One of the areas of interest concerns the relationship between sensory stimuli, e.g. input via sight, smell, taste, sound or touch, and the resulting neural response, which usually take the form of a spike train. A spike train is a sequence of recorded times at which a neuron fires a so-called action potential, an electrical impulse which plays a central role in cell-to-cell communication within the brain, or between the brain and other body systems such as the motor system. Spike trains are considered to be the primary mode of information transmission in the nervous system. In this project, we aim to understand more about a particular neuronal system, the insect sense of smell (olfaction). Many insect species, e.g. moths, are known to depend critically on the olfaction sensory system. Indeed, the male has typically only several hours/days to reach an appropriate female and to spread his genetic information, and he is guided by a faint smell, i.e. a pheromone trail released by a female. The pheromone doses are extremely low, yet the insect olfactory system is known to exhibit probably the greatest sensitivity in all the animal kingdom. We pursue two main goals: a) developing a mathematical description of a biophysical model for olfactory neurons, from the pheromone-receptor interaction to the membrane voltage and spikes; b) providing a description of coding schemes and analysing the efficiency of neural communication. For goal a), we intend to describe the pheromone-receptor interaction via chemical reaction network theory; the resulting receptor signal then changes the membrane voltage of a neuron, modelled as a certain stochastic process. The spike times are then obtained as hitting times of the membrane voltage to a time-varying threshold, yielding a so-called point process. Using the modelled spike trains, we investigate different coding schemes and properties of these codes, e.g. the response latency (time from the stimulus onset to the first spike evoked by the stimulus) or the decoding precision dynamics under changing stimulus conditions. Further, we would like to understand how well the stimulus value can be determined from the noisy neuronal response and how well we can discriminate between spontaneous (not related to the smelling of pheromones) and evoked (due to the smelling of the pheromones) spikes. To be able to do thes e analyses, we need to develop new mathematical techniques, such as filtering and parameter estimation techniques, which are simulation-based and require efficient and property preserving numerical schemes for the newly developed models (stochastic differe ntial equation systems) to be successful.

The topic of the project concerns modelling olfactory neurons of moths and the neuronal response to pheromones received by moths. lt involves Part a) olfactory neuron modelling: the transformation of the external signal, i.e. the pheromone concentration in the air, to the internal signal, i.e. the concentration of activated receptors (Step 1); the description of the changes in the membrane voltage determined by the internal signal (Step 2); the firing mechanism describing how the spikes are generated (Step 3); Part b) a search for alternative coding schemes and their efficiency for neural communication. In framework of the present research the following main results were obtained. 1) Studying computational models of the olfactory system, using the Stochastic Simulation Algorithm (SSA), i.e. the continuous-time Markov process exactly describing the chemical reactions involved, the Chemical Langevin Equation (CLE), derived as an approximation of the SSA, as well as a new model using a special Ornstein-Uhlenbeck process. The latter, in contrast to both other models, also takes extrinsic noise into account, but is not exact like the SSA. 2) Developing new numerical methods for the CLE, in particular, we have derived splitting methods for CLEs, which provide a new approach to treat the square-root term in the diffusion coefficient. We have also proved mean-square convergence for these nonlinear SDE systems.