# Exploring and statistically learning an excitable stochastic-dynamical model

*Sebastiano Grazzi, Frank van der Meulen, Marcin Mider, Moritz Schauer*

## 1. Introduction

We are a group of statisticians located in the Netherlands with interest in models describing phenomena evolving over time. Statistical inference for these models cannot be done for example by simple linear regression, as this does not consider the temporal inter-dependencies of the process. Our research has lead to a statistical procedure for the FitzHugh-Nagumo model, which is a prototype of such a model.

The procedure takes as input an observed trajectory (a curve showing the change of the modelled quantity over time) and returns likely intervals of the parameters which govern the dynamics of the model. With the collaboration of Nextjournal we intend to create a living Nextjournal document as essential companion for this estimation procedure. We will combine our tool for statistical analysis of data from such models together with an intuitive interface aimed at visualising and exploring the behaviour of the models best describing the data.

## 2. Statement of the problem

The FitzHugh-Nagumo model is a versatile yet simple model for the *evolution* of excitable or oscillatory physical processes. Recurring natural events such as West Antarctic ice shelf collapses, the firing of neurons or rapid population expansions of defoliating insect populations typically occur neither in a completely periodic nor completely irregular fashion. The cause is the interplay between the *noise* coming from outside factors interacting with the system in a random or non-predictable way and the *rhythm* created by the delayed interaction between excitatory and inhibitory components of the system. This interplay creates a manifold of possible recurrence patterns. The FitzHugh-Nagumo model can express such variability by introducing a single hidden or latent component and a single source of noise.

*Figure 1: A trajectory of the FitzHugh-Nagumo model: the evolution of the quantity of interest X and the introduced latent variable Y on the X-Y plane (left) and the evolution of the quantity of interest X over time (right). A small perturbation of the system can result in a large, non-linear excursion of X, the so called spike.*

Because of its flexibility, this model has been widely used in many scientific fields. There is practical importance to match the free parameters to the real phenomena at hand. The task of statisticians is indeed to fit real data and infer the parameters of the model. This complex problem is the subject of active and promising research of the team members of this project [1]. The methodologies appearing in the literature are difficult to implement and difficult to tune. An easy to use procedure has not appeared yet. Providing automatic solutions for the inference problem is not of much use if the results are not easily understood. This is why it is important to allow scientists to visually explore the estimated models and make the estimation results tangible.

## 3. Objectives

Together with the specialists of Nextjournal we can package the statistical method, provide an interface to visualise and explore the interaction between the parameters and the dynamical patterns created and finally interactively reconnect the inferential results with the natural phenomenon. This can be achieved in tight collaboration between our team and Nextjournal. The members of our team have already implemented many statistical tools in Julia and intend to continue to work in this language. A primitive toy implementation of a visualisation of the model has been created by the team member Moritz Schauer [link] from the FieldPlay applet available on GitHub by Andrei Kashcha [2].

*Figure 2: The effect on the evolution of the particles when changing one parameter of the model. Each particle represents a possible value of the quantity of interest and the latent variable as point in the 2-dimensional plane. The particles move according to the dynamics of the model through this plane. Changing the parameters affects the dynamics of the particles, making spikes or excursions less likely. Though visually pleasing, this prototype does not produce very satisfactory results, a prominent limitation is that the applet in which the visualisation is generated only gives access to very crude forms of "randomness".*

We develop the new interface from this prototype, connecting the dynamics on the plane with the dynamics of the quantity of interest over time as in Figure 1. The Bayesian statistical procedure we are proposing explores the range of likely parameters given the data. The user can observe the interaction between the particle trajectories and these parameters. This will help the scientist to understand and even act on the results reported by the statistical procedure.

*Figure 3: A mock interface for the visualisation of the inferential results together with the dynamics of the model: box-plots representing likely intervals of the parameters given the observed data (left), the dynamics of the particles in the X-Y plane (centre), the evolution of the system over time (right).*

## 4. Plan of action

The work will be divided into two tasks which can be worked on concurrently by two groups of team members. One group will be working on the statistical procedure of taking data as input and returning estimation results. The second group will be working in tight cooperation with the specialists of Nextjournal in order to create the interface for visualising the model and the statistical results. The creation of the corresponding Nextjournal components is initially disentangled from the statistical methodology so that the two groups can work simultaneously. Only in a second stage, the two groups will converge in order to make the statistical procedure available online in Nextjournal and embed it together with the graphical interface created for the visualisation of the model. The whole project requires close collaboration between the team members and Nextjournal team and we think it will be a learning experience for both sides. We intend therefore to dedicate part of the scholarship to refund travels to Berlin that will facilitate our work together with Nextjournal. In two months we will provide the interface, the statistical method and the full documentation, supported by a framework of tests.

## 5. Management plan

## 6. Conclusions

The key idea of the project is to create attractive Nextjournal components that will give means to produce and reproduce explorable research to a wide range of scientists of diverse background; by means of making our own work accessible. Due to the generality of the statistical procedure and the graphical interface, the program can be expanded and translated to other interesting models and applications. This project is embedded in a larger research project on statistical methods for stochastic dynamical systems that team members have been working on over the past 5 to 10 years. The project will make Nextjournal a natural host for scientific research using these types of models, including our own ongoing research on topics such as past and current climatic warming events. Finally, a short report about the results obtained within this project is suitable for a traditional venue such as a journal for statistical computing.

## 7. References

- [1] F.H. van der Meulen and M. Schauer (2017): Bayesian estimation of incompletely observed diffusions,
*Stochastics*90(5), 641-662. - [2] Andrei Kashcha (2017-2018): A vector field explorer (applet available at https://github.com/anvaka/fieldplay).