Putting the Cycle Back into Business Cycle Analysis

Are business cycles mainly a response to persistent exogenous shocks, or do they instead reflect a strong endogenous mechanism which produces recurrent boom-bust phenomena? In this paper we present evidence in favor of the second interpretation and we highlight the set of key elements that influence our answer. The elements that tend to favor this type of interpretation of business cycles are (i) slightly extending the frequency window one associates with business cycle phenomena, (ii) allowing for strategic complementarities across agents that arise due to financial frictions, and (iii) allowing for a locally unstable steady state in estimation. (JEL E22, E24, E23, E44)

Dynamics of Multiple-Choice Decision Making

Neuroscientists have carried out comprehensive experiments to reveal the neural mechanisms underlying the perceptual decision making that pervades daily life. These experiments have illuminated salient features of decision making, including probabilistic choice behavior, the ramping activity of decision-related neurons, and the dependence of decision time and accuracy on the difficulty of the task. Spiking network models have reproduced these features, and a two-dimensional mean field model has demonstrated that the saddle node structure underlies two-alternative decision making. Here, we reduced a spiking network model to an analytically tractable, partial integro-differential system and characterized not only multiple-choice decision behaviors but also the time course of neural activities underlying decisions, providing a mechanistic explanation for the observations noted in the experiments. First, we observed that a two-bump unstable steady state of the system is responsible for two-choice decision making, similar to the saddle node structure in the two-dimensional mean field model. However, for four-choice decision making, three types of unstable steady states collectively predominate the time course of the evolution from the initial state to the stable states. Second, the time constant of the unstable steady state can explain the fact that four-choice decision making requires a longer time than two-choice decision making. However, the quicker decision, given a stronger motion strength, cannot be explained by the time constant of the unstable steady state. Rather, the decision time can be attributed to the projection coefficient of the difference between the initial state and the unstable steady state on the eigenvector corresponding to the largest positive eigenvalue.

Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators

We present an estimator-based control design procedure for flow control, using reduced-order models of the governing equations linearized about a possibly unstable steady state. The reduced-order models are obtained using an approximate balanced truncation method that retains the most controllable and observable modes of the system. The original method is valid only for stable linear systems, and in this paper, we present an extension to unstable linear systems. The dynamics on the unstable subspace are represented by projecting the original equations onto the global unstable eigenmodes, assumed to be small in number. A snapshot-based algorithm is developed, using approximate balanced truncation, for obtaining a reduced-order model of the dynamics on the stable subspace.The proposed algorithm is used to study feedback control of two-dimensional flow over a flat plate at a low Reynolds number and at large angles of attack, where the natural flow is vortex shedding, though there also exists an unstable steady state. For control design, we derive reduced-order models valid in the neighbourhood of this unstable steady state. The actuation is modelled as a localized body force near the trailing edge of the flat plate, and the sensors are two velocity measurements in the near wake of the plate. A reduced-order Kalman filter is developed based on these models and is shown to accurately reconstruct the flow field from the sensor measurements, and the resulting estimator-based control is shown to stabilize the unstable steady state. For small perturbations of the steady state, the model accurately predicts the response of the full simulation. Furthermore, the resulting controller is even able to suppress the stable periodic vortex shedding, where the nonlinear effects are strong, thus implying a large domain of attraction of the stabilized steady state.

A Positive Real Eigenvalue Condition for the Determination of Unstable Steady States in Chemical Reaction Networks

The interesting dynamical behaviours exhibiting in chemical reaction systems, such as multiple steady states and undamped oscillations, often result from unstable steady states. A positive real eigenvalue condition is proposed which gives a necessary and sufficient condition for the determination of an unstable steady state having a positive real eigenvalue in general isothermal reaction networks. Formulas are developed to construct an unstable steady state and a set of positive rate constants. The applications are illustrated by three examples. Two give rise to oscillations and one admits multiple steady states.

A-Type K+ Current Can Act as a Trigger for Bursting in the Absence of a Slow Variable

Models of bursting in single cells typically include two subsystems with different timescales. Variations in one or more slow variables switch the system between a silent and a spiking state. We have developed a model for bursting in the pituitary lactotroph that does not include any slow variable. The model incorporates fast, noninactivating calcium and potassium currents (the spike-generating mechanism), as well as the fast, inactivating A-type potassium current (IA). IA is active only briefly at the beginning of a burst, but this brief impulse of IA acts as a burst trigger, injecting the spike trajectory close to an unstable steady state. The spiraling of the trajectory away from the steady state produces a period of low-amplitude spiking typical of lactotrophs. Increasing the conductance of A-type potassium current brings the trajectory closer to the unstable steady state, increasing burst duration. However, this also increases interburst interval, and for larger conductance values, all activity stops. To our knowledge, this is the first example of a physiologically based, single-compartmental model of bursting with no slow subsystem.