Local/global analysis of the stationary solutions of some neural field equations

Neural or cortical fields are continuous assemblies of mesoscopic models, also called neural masses, of neural populations that are fundamental in the modeling of macroscopic parts of the brain. Neural fields are described by nonlinear integro-differential equations. The solutions of these equations represent the state of activity of these populations when submitted to inputs from neighbouring brain areas. Understanding the properties of these solutions is essential in advancing our understanding of the brain. In this paper we study the dependency of the stationary solutions of the neural fields equations with respect to the stiffness of the nonlinearity and the contrast of the external inputs. This is done by using degree theory and bifurcation theory in the context of functional, in particular infinite dimensional, spaces. The joint use of these two theories allows us to make new detailed predictions about the global and local behaviours of the solutions. We also provide a generic finite dimensional approximation of these equations which allows us to study in great details two models. The first model is a neural mass model of a cortical hypercolumn of orientation sensitive neurons, the ring model. The second model is a general neural field model where the spatial connectivity isdescribed by heterogeneous Gaussian-like functions.Comment: 38 pages, 9 figure

A constrained optimization problem in quantum statistical physics

In this paper, we consider the problem of minimizing quantum free energies under the constraint that the density of particles is fixed at each point of Rd, for any d $\ge$ 1. We are more particularly interested in the characterization of the minimizer, which is a self-adjoint nonnegative trace class operator, and will show that it is solution to a nonlinear self-consistent problem. This question of deriving quantum statistical equilibria is at the heart of the quantum hydrody-namical models introduced by Degond and Ringhofer. An original feature of the problem is the local nature of constraint, i.e. it depends on position, while more classical models consider the total number of particles in the system to be fixed. This raises difficulties in the derivation of the Euler-Lagrange equations and in the characterization of the minimizer, which are tackled in part by a careful parametrization of the feasible set

Illusions in the Ring Model of visual orientation selectivity

The Ring Model of orientation tuning is a dynamical model of a hypercolumn of visual area V1 in the human neocortex that has been designed to account for the experimentally observed orientation tuning curves by local, i.e., cortico-cortical computations. The tuning curves are stationary, i.e. time independent, solutions of this dynamical model. One important assumption underlying the Ring Model is that the LGN input to V1 is weakly tuned to the retinal orientation and that it is the local computations in V1 that sharpen this tuning. Because the equations that describe the Ring Model have built-in equivariance properties in the synaptic weight distribution with respect to a particular group acting on the retinal orientation of the stimulus, the model in effect encodes an infinite number of tuning curves that are arbitrarily translated with respect to each other. By using the Orbit Space Reduction technique we rewrite the model equations in canonical form as functions of polynomials that are invariant with respect to the action of this group. This allows us to combine equivariant bifurcation theory with an efficient numerical continuation method in order to compute the tuning curves predicted by the Ring Model. Surprisingly some of these tuning curves are not tuned to the stimulus. We interpret them as neural illusions and show numerically how they can be induced by simple dynamical stimuli. These neural illusions are important biological predictions of the model. If they could be observed experimentally this would be a strong point in favour of the Ring Model. We also show how our theoretical analysis allows to very simply specify the ranges of the model parameters by comparing the model predictions with published experimental observations.Comment: 33 pages, 12 figure

Tax Reform in Two-Sector General Equilibrium

We use a two-sector open economy model with an imperfectly competitive non traded sector to investigate the dynamic and steady-state effects of three tax reforms : [i] two revenue-neutral tax reforms shifting the tax burden from labor to consumption taxes and [ii] one labor tax reform keeping the marginal tax wedge constant. Regardless of its form, a tax restructuring crowds-in consumption and investment and raises employment. While tax multipliers for overall output are always positive, their size depends on the type of the tax reform and the financing scheme. Interestingly, the trade balance plays a key role in determining the relative size of sectoral tax multipliers : whereas the long-term tax multiplier is always slightly higher in the traded sector than in the non traded sector, this result is reversed in the short-term. Finally, time horizon matters in determining the relationships between both overall and sectoral tax multipliers and labor responsiveness.non traded goods ; investment ; employment ; tax multiplier

Sectoral Effects of Tax Reforms in an Open Economy

We use a neoclassical open economy model with traded and non traded goods to investigate the sectoral effects of three tax reforms: i) two revenue-neutral shifting the tax burden from labor to consumption taxes and ii) one labor tax restructuring keeping the marginal tax wedge constant. Regardless of its type, a tax reform crowds-in both consumption and investment and raises employment. Whereas tax reforms have a small impact on GDP, they exert substantial effects on sectoral outputs which move in opposite direction in the short-run. The sensitivity analysis reveals that raising the elasticity of labor supply or reducing the tradable content in consumption expenditure amplifies the heterogeneity in sectoral output responses. Finally, allowing for the markup to depend on the number of competitors, we find that a substantial share of sectoral output variations can be attributed to the change in the markup triggered by firm entry.Non Traded Goods; Employment; Current Account; Tax Reform.

Permanent vs Temporary Fiscal Expansion in a Two-Sector Small Open Economy Model

This contribution shows that the duration of a fisscal shock together with sectoral capital intensity matter in determining the dynamic and steady-state effects in an intertemporal-optimizing two-sector small open economy model. First, unlike a permanent shock, net foreign asset position always worsens in the long-run after a transitory fiscal expansion. Second, steady-state changes in physical capital depend on sectoral capital-labor ratios but their signs may be reversed compared to the corresponding permanent public policy. Third, investment and the current account may now adjust non monotonically. Fourth, a temporary fiscal shock always crowds-out (crowds-in) investment in the long-run whenever the non traded (traded) sector is more capital intensive.current account; government spending; nontraded goods; temporary shocks

Fiscal shocks in a two sector open economy

We use a two-sector neoclassical open economy model with traded and non-traded goods to investigate both the aggregate and the sectoral effects of temporary fiscal shocks. One central finding is that both sectoral capital intensities and labor supply elasticity matter in determining the response of key economic variables. In particular, the model can produce a drop in investment and in the current account, in line with empirical evidence, only if the traded sector is more capital intensive than the non-traded sector, and labor is supplied elastically. Irrespective of sectoral capital intensities, a fiscal shock raises the relative size of the non-traded sector substantially in the short-run. Additionally, allowing for the markup to depend on the number of competitors, the two-sector model can produce the real exchange rate depreciation found in the data. Finally, markup variations triggered by firm entry modify substantially the response of the real wage and the sectoral composition of GDP in the short-run.Non-traded Goods; Fiscal Shocks; Investment; Current Account.

Sectoral Effects of Tax Reforms in an Open Economy

We use a neoclassical open economy model with traded and non traded goods to investigate the sectoral effects of three tax reforms: i) two revenue-neutral shifting the tax burden from labor to consumption taxes and ii) one labor tax restructuring keeping the marginal tax wedge constant. Regardless of its type, a tax reform crowds-in both consumption and investment and raises employment. Whereas tax reforms have a small impact on GDP, they exert substantial effects on sectoral outputs which move in opposite direction in the short-run. The sensitivity analysis reveals that raising the elasticity of labor supply or reducing the tradable content in consumption expenditure amplifies the heterogeneity in sectoral output responses. Finally, allowing for the markup to depend on the number of competitors, we find that a substantial share of sectoral output variations can be attributed to the change in the markup triggered by firm entry.Non Traded Goods; Employment; Current Account; Tax Reform

Unanticipated vs. Anticipated Tax Reforms in a Two-Sector Open Economy.

We use a two-sector neoclassical open economy model with traded and non-traded goods to investigate the effects of unanticipated and anticipated tax reforms. First, an unanticipated tax reform produces an expansion of GDP, labor, and investment, while an anticipated tax reform has opposite effects before the implementation of the labor tax cut. Quantitatively, if the traded sector is more capital intensive, GDP increases by 1.6 percentage points or declines by 2.8 percentage points after three years, depending on whether the tax cut is unanticipated or anticipated. Second, we find that GDP change masks a wide dispersion in sectoral output responses. Importantly, in all scenarios, a tax reform substantially raises the relative size of the non-traded sector while traded output always drops. Allowing for the markup to depend on the number of competitors, we find that a significant share of GDP change can be attributed to the competition channel while the dispersion of sectoral output responses is amplified. Finally, the workers only benefit from the labor tax cut if the tax change is unanticipated and the traded sector is more capital intensive.Non Traded Goods; Investment; Tax Reform; Anticipation effects.

Fiscal Shocks in a Two-Sector Open Economy

We use a two-sector neoclassical open economy model with traded and non-traded goods to investigate both the aggregate and the sectoral effects of temporary fiscal shocks. One central finding is that both sectoral capital intensities and labor supply elasticity matter in determining the response of key economic variables. In particular, the model can produce a drop in investment and in the current account, in line with empirical evidence, only if the traded sector is more capital intensive than the non-traded sector, and labor is supplied elastically. Irrespective of sectoral capital intensities, a fiscal shock raises the relative size of the non-traded sector substantially in the short-run. Additionally, allowing for the markup to depend on the number of competitors, the two-sector model can produce the real exchange rate depreciation found in the data. Finally, markup variations triggered by firm entry modify substantially the response of the real wage and the sectoral composition of GDP in the short-run.Non-traded Goods; Fiscal Shocks; Investment; Current Account