A mathematical model for the two-learners problem

Jan Saputra Müller, Carmen Vidaurre, Martijn Schreuder, Frank C. Meinecke, Paul Von Bünau, Klaus Robert Müller

Research output: Contribution to journalArticlepeer-review

43 Citations (Scopus)


Objective. We present the first generic theoretical formulation of the co-adaptive learning problem and give a simple example of two interacting linear learning systems, a human and a machine. Approach. After the description of the training protocol of the two learning systems, we define a simple linear model where the two learning agents are coupled by a joint loss function. The simplicity of the model allows us to find learning rules for both human and machine that permit computing theoretical simulations. Main results. As seen in simulations, an astonishingly rich structure is found for this eco-system of learners. While the co-adaptive learners are shown to easily stall or get out of sync for some parameter settings, we can find a broad sweet spot of parameters where the learning system can converge quickly. It is defined by mid-range learning rates on the side of the learning machine, quite independent of the human in the loop. Despite its simplistic assumptions the theoretical study could be confirmed by a real-world experimental study where human and machine co-adapt to perform cursor control under distortion. Also in this practical setting the mid-range learning rates yield the best performance and behavioral ratings. Significance. The results presented in this mathematical study allow the computation of simple theoretical simulations and performance of real experimental paradigms. Additionally, they are nicely in line with previous results in the BCI literature.

Original languageEnglish
Article number036005
JournalJournal of Neural Engineering
Issue number3
Publication statusPublished - 2017 Mar 21

Bibliographical note

Funding Information:
This work was supported in part by the Deutsche Forschungsgesellschaft (DFG SPP 1527, MU 987/14-1), the Federal Ministry for Education and Research (BMBF) as well as by the Brain Korea 21 Plus Program funded by the Korean government and by the MINECO Grant RyC-2014-15671. We thank our reviewers for their helpful suggestions.

Publisher Copyright:
© 2017 IOP Publishing Ltd.


  • brain-computer interfacing
  • co-adaptation
  • linear models
  • mathematical models
  • theoretical models

ASJC Scopus subject areas

  • Biomedical Engineering
  • Cellular and Molecular Neuroscience


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