News & Events

Malaria is one of the most important global health challenges, with almost 1 million dead a year, mostly children in Africa. Mathematical models can help determine the likely impact of interventions such as bed nets or drug distribution. The way a single infection is modeled can strongly influence the result, whether it is an exponential distribution for a duration of constant infectiousness, samples from empirical data, or a bottom-up mechanistic model. Within each of these categories, there are a variety of approaches with different levels of realism. We review other approaches and present a new mechanistic model, the results of which are compared to malariatherapy data.

The Cockroach is a quick and nimble runner, navigating a variety of environments with an adeptness that eludes any robot. These specific characteristics have inspired biologists, mathematicians, engineers, and roboticists to investigate rapidly running cockroaches. At these speeds, previous studies have shown that cockroach locomotion is driven by a feed-forward architecture in which a neuronal system rhythmically activates muscles and legs, ballistically driving the body over rough terrain. In contrast, slipping or mis-stepping at slow speeds may require reflexes, much the same as in the human body, that help the insect recover from perturbations. The importance of reflexes, via proprioceptive sensors in the legs, and how they can modulate movement is the primary concern of this research. Integrating feedback in neuro-mechanical locomotion models, we can investigate the importance of different types of feedback (e.g. position of, velocity of, or load on a leg). Understanding how feedback is utilized in these systems will provide insight into the remarkable ability of insects and animals to adapt and modulate their running behavior to interact effectively with their environments.

The reliability and predictability of neuronal network dynamics is a central question in neuroscience. We present a numerical analysis of the dynamics of all-to-all pulsed-coupled Hodgkin-Huxley (HH) neuronal networks. Since this is a non-smooth dynamical system, we propose a pseudo-Lyapunov exponent (PLE) that captures the long-time predictability of HH neuronal networks. The PLE can capture very well the dynamical regimes of the network. Furthermore, we present an efficient library-based numerical method for simulating HH neuronal networks. Our pre-computed high resolution data library can allow us to avoid resolving the spikes in detail and to use large numerical time steps for evolving the HH neuron equations. By using the library-based method, we can evolve the HH networks using time steps one order of magnitude larger than the typical time steps used for resolving the trajectories without the library, while achieving comparable resolution in statistical quantifications of the network activity. Moreover, our large time steps using the library method can overcome the stability requirement of standard ODE methods for the original dynamics.

We develop a neuromechanical model for running insects that includes a simplified hexapedal leg geometry with agonist-antagonist muscle pairs actuating each leg joint. Restricting to dynamics in the horizontal plane and neglecting leg masses, we reduce the model to three degrees of freedom describing translational and yawing motions. The muscles are driven by stylized action potentials characteristic of fast motoneurons, and modeled using activation via calcium release and nonlinear length and shortening velocity dependence. Parameter values are based on measurements from depressor muscles and observations of kinematics and dynamics of the cockroach Blaberus discoidalis; in particular, motoneuronal inputs are chosen to approximately achieve joint torques that are consistent with measured ground reaction forces. We show that the model has stable double-tripod gaits over the animal's speed range, that its dynamics at preferred speeds matches those observed, and that it maintains stable gaits, with low frequency yaw deviations, when subject to random perturbations in foot touchdown timing and action potential input timing. We explain this in terms of the low-dimensional dynamics.

Solving the Boltzmann - BGK equation, we investigate a flow generated by an infinite plate oscillating with frequency ω. Geometrical simplicity of the problem allows a solution in the entire range of dimensionless frequency variation 0 ≤ωτ≤∞, where τ is a properly defined relaxation time. A transition from viscoelastic behavior of Newtonian fluid (ωτ→0 ) to purely elastic dynamics in the limit ωτ→∞ is discovered. The relation of the derived solutions to nanofluidics is demonstrated on a solvable example of a "plane oscillator." The results from the derived formulae are favorably compared with experimental data on various nanoresonators operating in a wide range of both frequency and pressure variation. The universal relation for the dissipation rate in oscillating flows valid in both Newtonian and Non-Newtonian regimes is derived and compared with experimental data covering huge ranges of frequency (10^3 ≤ω≤10^9 Hz) and linear dimension (10^{-2}≤L≤10^{-6} m) variation.

Synchronization is the science of order in time and studies the ways rhythms become spontaneously organized. It is ubiquitous in nature and in engineering applications: groups of fireflies, neurons or pacemaker cells synchronize spontaneously; fish move in formations to escape predators and improve foraging; robots can coordinate to accomplish tasks more efficiently.

In this talk I will present a mathematical formalism, based on input/output operators and graph theory, to study the emergent behavior of networks of interconnected systems. Following a system theory approach, in the first step, the system is decomposed into smaller isolated subsystems by ignoring interconnections. In the second step, the information from the isolated systems is combined with the information about the interconnections to draw conclusions on the behavior of the overall system.

First I will employ the proposed methodology to relate, for a class of networked systems, the cohesiveness of the network to the connectivity of the underlying communication graph. Cohesiveness is characterized by providing a finite L2 gain condition (depending on the graph connectivity) for the interconnected system. Applications range from coordination problems, where there are conflicting objectives, to the study of aggregation phenomena, where perturbations of the nominal systems must be taken into account. Both scenarios arise in networks of biological and engineered coordinating systems. Then I will characterize synchronization for a class of nonlinear coupled systems where each system is described by input-output interconnections of heterogeneous subsystems. These interconnection structures are common in the modeling of biochemical networks. Specific examples include synchronization of genetic oscillators (regulating the circadian clocks of living organisms) like the Goodwin oscillator and the repressilator.