Pithya

A Parallel Tool for Parameter Synthesis of Piecewise Multi-Affine Dynamical Systems

Nikola Beneš, Luboš Brim, Martin Demko, Samuel Pastva, David Šafránek

What is it that you do, anyway?

Two dimensional repressilator

Primitive biological switch: depending on the parameters,

either X or Y will dominate.

\frac{dX}{dt} = \mathbf{k} \cdot \frac{5^5}{5^5+Y^5} - 0.1 \cdot X
dXdt=k⋅5555+Y5−0.1⋅X\frac{dX}{dt} = \mathbf{k} \cdot \frac{5^5}{5^5+Y^5} - 0.1 \cdot X
\frac{dY}{dt} = \frac{5^5}{5^5+X^5} - \mathbf{p} \cdot Y
dYdt=5555+X5−p⋅Y\frac{dY}{dt} = \frac{5^5}{5^5+X^5} - \mathbf{p} \cdot Y

Slides won't cut it? Try Pithya at pithya.ics.muni.cz

Autonomous ODE System

Piecewise Multi-affine ODE System

Autonomous ODE System

Abstract State Space

 

Property

HUCTL_P
HUCTLPHUCTL_P

Piecewise Multi-affine ODE System

Abstract State Space

 

Property

HUCTL_P
HUCTLPHUCTL_P

Coloured Model Checking

\begin{aligned} stable & =\ \downarrow x : \mathbf{AG} \mathbf{EF} x \\ bistable & =\ stable \land \exists y \in stable : \neg \textbf{EF} y \end{aligned}
stable= ↓x:AGEFxbistable= stable∧∃y∈stable:¬EFy\begin{aligned} stable & =\ \downarrow x : \mathbf{AG} \mathbf{EF} x \\ bistable & =\ stable \land \exists y \in stable : \neg \textbf{EF} y \end{aligned}

Slides won't cut it? Try Pithya at pithya.ics.muni.cz

What's next?

Slides won't cut it? Try Pithya at pithya.ics.muni.cz

  • δ-decision based abstractions
    • ​most current techniques are local and/or limited to specific types of equations
  • Noise and/or environment input

Discrete Bifurcation Analysis

  • Bifurcation points
    • boundary points between parameter regions with equivalent behaviour
  • Long term behaviour