Michael Margaliot

Eyal Weiss, Michael Margaliot

Many complex systems in biology, physics, and engineering include a large number of state-variables, and measuring the full state of the system is often impossible. Typically, a set of sensors is used to measure part of the state-variables. A system is called observable if these measurements allow to reconstruct the entire state of the system. When the system is not observable, an important and practical problem is how to add a \emph{minimal} number of sensors so that the system becomes observable. This minimal observability problem is practically useful and theoretically interesting, as it pinpoints the most informative nodes in the system. We consider the minimal observability problem for an important special class of Boolean networks, called conjunctive Boolean networks (CBNs). Using a graph-theoretic approach, we provide a necessary and sufficient condition for observability of a CBN with $n$ state-variables, and an efficient~$O(n^2)$-time algorithm for solving the minimal observability problem. We demonstrate the usefulness of these results by studying the properties of a class of random CBNs.

Eyal Weiss, Michael Margaliot

The dynamics of linear positive systems map the positive orthant to itself. In other words, it maps a set of vectors with zero sign variations to itself. This raises the following question: what linear systems map the set of vectors with $k$ sign variations to itself? We address this question using tools from the theory of cooperative dynamical systems and the theory of totally positive matrices. This yields a generalization of positive linear systems called $k$-positive linear systems, that reduces to positive systems for $k=1$. We describe applications of this new type of systems to the analysis of nonlinear dynamical systems. In particular, we show that such systems admit certain explicit invariant sets, and for the case $k=2$ establish the Poincaré-Bendixson property for any bounded trajectory.

Eyal Weiss, Michael Margaliot, Guy Even

Given a conjunctive Boolean network (CBN) with $n$ state-variables, we consider the problem of finding a minimal set of state-variables to directly affect with an input so that the resulting conjunctive Boolean control network (CBCN) is controllable. We give a necessary and sufficient condition for controllability of a CBCN; an $O(n^2)$-time algorithm for testing controllability; and prove that nonetheless the minimal controllability problem for CBNs is NP-hard.

Eyal Weiss, Michael Margaliot

Smillie (1984) proved an interesting result on the stability of nonlinear, time-invariant, strongly cooperative, and tridiagonal dynamical systems. This result has found many applications in models from various fields including biology, ecology, and chemistry. Smith (1991) has extended Smillie's result and proved entrainment in the case where the vector field is time-varying and periodic. We use the theory of linear totally nonnegative differential systems developed by Schwarz (1970) to give a generalization of these two results. This is based on weakening the requirement for strong cooperativity to cooperativity, and adding an additional observability-type condition.

Michael Margaliot, Samuel Coogan

We consider contractive systems whose trajectories evolve on a compact and convex state-space. It is well-known that if the time-varying vector field of the system is periodic then the system admits a unique globally asymptotically stable periodic solution. Obtaining explicit information on this periodic solution and its dependence on various parameters is important both theoretically and in numerous applications. We develop an approach for approximating such a periodic trajectory using the periodic trajectory of a simpler system (e.g. an LTI system). Our approximation includes an error bound that is based on the input-to-state stability property of contractive systems. We show that in some cases this error bound can be computed explicitly. We also use the bound to derive a new theoretical result, namely, that a contractive system with an additive periodic input behaves like a low pass filter. We demonstrate our results using several examples from systems biology.

Ron Teichner, Ron Meir, Michael Margaliot

Given a time-series of noisy measured outputs of a dynamical system z[k], k=1...N, the Identifying Regulation with Adversarial Surrogates (IRAS) algorithm aims to find a non-trivial first integral of the system, namely, a scalar function g() such that g(z[i]) = g(z[j]), for all i,j. IRAS has been suggested recently and was used successfully in several learning tasks in models from biology and physics. Here, we give the first rigorous analysis of this algorithm in a specific setting. We assume that the observations admit a linear first integral and that they are contaminated by Gaussian noise. We show that in this case the IRAS iterations are closely related to the self-consistent-field (SCF) iterations for solving a generalized Rayleigh quotient minimization problem. Using this approach, we derive several sufficient conditions guaranteeing local convergence of IRAS to the correct first integral.