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We investigate a reaction-diffusion-chemotaxis system that describes the immune response during an inflammatory attack. The model is a modification of the system proposed in Penner, Ermentrout, and Swigon [SIAM J. Appl. Dyn. Syst., 11 (2012), pp. 629-660]. We introduce a logistic term in the immune cell dynamics to reproduce the macrophages' activation, allowing us to describe the disease evolution from the early stages to the acute phase. We focus on the appearance of pattern solutions and their stability. We discover steady-state (Turing) and wave instabilities and classify the bifurcations deriving the corresponding amplitude equations. We study stationary radially symmetric solutions and show that they reproduce various inflammatory aggregates observed in the clinical practice. Moreover, the model supports oscillating-in-time spatial patterns, thus giving a theoretical explanation of the periodic appearance of inflammatory eruptions typical of recurrent erythema multiforme. A detailed numerical bifurcation analysis indicates that the inclusion of the logistic growth term is crucial for the occurrence of a sequence of bifurcations leading to spatio-temporal chaos. In the parameter space, there are large regions where the model system displays critical behavior.

We construct square and target patterns solutions of the FitzHugh–Nagumo reaction–diffusion system on planar bounded domains. We study the existence and stability of stationary square and super-square patterns by performing a close to equilibrium asymptotic weakly nonlinear expansion: the emergence of these patterns is shown to occur when the bifurcation takes place through a multiplicity-two eigenvalue without resonance. The system is also shown to support the formation of axisymmetric target patterns whose amplitude equation is derived close to the bifurcation threshold. We present several numerical simulations validating the theoretical results.

In this paper we study radially symmetric solutions for our recently proposed reaction–diffusion–chemotaxis model of Multiple Sclerosis. Through a weakly nonlinear expansion we classify the bifurcation at the onset and derive the amplitude equations ruling the formation of concentric demyelinating patterns which reproduce the concentric layers observed in Balò sclerosis and in the early phase of Multiple Sclerosis. We present numerical simulations which illustrate and fit the analytical results.

Background: A type II odontoid fracture, if unstable, can cause spinal cord damage. In this case, it is essential to choose the correct treatment—but the issues of what the correct treatment is and which of the different surgical options is best are quite controversial. In this paper we present strategies for treatment of type II odontoid fracture. Materials and Methods: Thirty consecutive cases of type II odontoid fracture were treated at the Division of Neurosurgery at Villa Sofia Hospital in Palermo (23 cases) and at the Neurosurgical Clinic, University Hospital of Palermo (seven cases), from January 2011 to August 2016. Four patients were treated with external immobilization. Twenty-six patients underwent a surgical procedure. Results: There was no mortality related to the surgical procedure. One patient had a pre- and postoperative neurological deficit, and remained tetraparetic. Follow-up radiological studies in the surgically treated group showed bone union in 21 patients and stable fibrous union in one. Conclusion: In our and other authors’ experience, when the direction of the fracture line is down and forward, external immobilization can be sufficient for healing. Anterior odontoid screw fixation can be considered the treatment of choice for unstable odontoid fractures (with a horizontal, down and back, or comminuted fracture line) without dislocation or with dislocation less than 7 mm. When the odontoid fracture is associated with a Jefferson fracture or dislocation greater than 7 mm, stabilization of C1–C2 may be necessary. In this case, placement of screws in the dens and in the joints through a single approach represents the most valid technique. In the case of an inveterate fracture of the dens with severe C1–C2 dislocation, the surgical operation that offers the best prospects is posterior stabilization, utilizing the Guo technique.

In this paper we investigate the complex dynamics originated by a cross-diffusion-induced subharmonic destabilization of the fundamental subcritical Turing mode in a predator-prey reaction-diffusion system. The model we consider consists of a two-species Lotka-Volterra system with linear diffusion and a nonlinear cross-diffusion term in the predator equation. The taxis term in the search strategy of the predator is responsible for the onset of complex dynamics. In fact, our model does not exhibit any Hopf or wave instability, and on the basis of the linear analysis one should only expect stationary patterns; nevertheless, the presence of the nonlinear cross-diffusion term is able to induce a secondary instability: due to a subharmonic spatial resonance, the stationary primary branch bifurcates to an out-of-phase oscillating solution. Noticeably, the strong resonance between the harmonic and the subharmonic is able to generate the oscillating pattern albeit the subharmonic is below criticality. We show that, as the control parameter is varied, the oscillating solution (subT mode) can undergo a sequence of secondary instabilities, generating a transition toward chaotic dynamics. Finally, we investigate the emergence of subT-mode solutions on two-dimensional domains: when the fundamental mode describes a square pattern, subharmonic resonance originates oscillating square patterns. In the case of subcritical Turing hexagon solutions, the internal interactions with a subharmonic mode are able to generate the so-called "twinkling-eyes" pattern.