Abstract: Let $C$ be a class of dynamical systems satisfying prescribed constraints, $R$ an equivalence relation on dynamical systems, and $f$ a target dynamical system. The triple $(C,R,f)$ defines a constrained realization problem: does there exist $g\in C$ such that $gRf$? Such problems arise in mathematical neuroscience, where $C$ is determined by biological data-access constraints. These constraints are either macroscopic, reflecting that the brain is an isolated system, or microscopic, reflecting that synapses process information locally. We discuss two such instances in this talk. 

 

The two instances enter the problem from opposite directions. The first is motivated by reservoir-based approximation of dynamical systems. Here we start from a desired equivalence relation $R$, namely topological conjugacy between a target dynamics and an internal recurrent dynamics, and construct $g\in C$ realizing $R$ under the macroscopic constraint. The second is motivated by biologically plausible synaptic learning rules. Here we start from the microscopic constraint and propose a candidate $R$, namely equivalence at the level of expected dynamics, together with a $g\in C$ that realizes a chosen target $f$ under this $R$. 

 

While these two instances illustrate the framework, they do not yet constitute a study of the interplay among $C$, $R$, and $f$. We close with future directions toward such a study: given $C$ inherited from the brain, what $R$ is distinguishable, and what $f$ can be realized?

 

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Organizers:

Chueh-Hsin Chang (CCU), Jia-Yuan Dai (NTHU), Bo-Chih Huang (CCU), Chih-Chiang Huang (CCU), Yuya Tokuta (Kyoto U.), Chang-Hong Wu (NYCU)