Interest Rates and Brain Waves as Coupled Oscillators

Introduction

The Kuramoto Model is a nonlinear differential equation that governs the behavior of n oscillators. The model can induce phase couplings, where the oscillators align with each other. This property has been helpful in modeling real-world phenomena that involve trends. This model is also particularly helpful in modeling cyclical data.

Variable Definitions

The Basic Kuramoto Model

The derivative with respect to time tells us how oscillator i’s position on the unit circle will change from the current time step to the next. This model assumes uniform infleuence and fully-connected coupling (all oscillators are coupled with all other oscillators).

Modified Kuramoto Model

For both biological and financial applications, the Kuramoto model becomes more insightful when we add non-uniform connections and phase delays. In the context of biomedical engineering, non uniform connections can be thought of as certain parts of the brain more strongly influencing some parts of the brain versus others. In the financial context, non-uniform connections can represent the varying correlations between different financial assets. Phase delay can be thought of in both cases as varying information travel times between different oscillators in the network.

The modified Kuramoto model with phase shifts and non-uniform couplings takes the form

Equations for Modeling Data

the phase change of oscillator i minus the average phase change of all oscillators at that time step.

We implemented both the Bandpass filter and the Hilbert transform via the Scipy library. The exact equations can be found in the Scipy Signal Documentation.

To model the K matrix for the brain data, we utilize the Phase Locking Value Matrix (PLV), an

equation often used in neurological fields. The PLV is defined as

Datasets

We obtained rates data from the Daily Treasury Rate Archives provided by the US Department of the Treasury. Our dataset contains the 3 month, 6 month, 1 year, 2 year, 3 year, 5 year, 7 year, 10 year, 20 year, and 30 year tenors.

We downloaded EEG data from the EEG Brainwave Dataset on Kaggle. The data was collected across four subjects, each of whom were measured in three different stages of cognition (concentrating, neutral and relaxed) with four electrodes.

Data Preparation: Interest Rates

Data Preparation: EEG

Simulation

Visualization

Results

Figure 1. Simulated yield curve evolution

Figure 2. Simulated brainwave development