Structural stability of invasion graphs for Lotka-Volterra systems.

In this paper, we study in detail the structure of the global attractor for the Lotka-Volterra system with a Volterra-Lyapunov stable structural matrix. We consider the invasion graph as recently introduced in Hofbauer and Schreiber (J Math Biol 85:54, 2022) and prove that its edges represent all the heteroclinic connections between the equilibria of the system. We also study the stability of this structure with respect to the perturbation of the problem parameters. This allows us to introduce a definition of structural stability in ecology in coherence with the classical mathematical concept where there exists a detailed geometrical structure, robust under perturbation, that governs the transient and asymptotic dynamics.

Informational Structures and Informational Fields as a Prototype for the Description of Postulates of the Integrated Information Theory.

Informational Structures (IS) and Informational Fields (IF) have been recently introduced to deal with a continuous dynamical systems-based approach to Integrated Information Theory (IIT). IS and IF contain all the geometrical and topological constraints in the phase space. This allows one to characterize all the past and future dynamical scenarios for a system in any particular state. In this paper, we develop further steps in this direction, describing a proper continuous framework for an abstract formulation, which could serve as a prototype of the IIT postulates.

Sensitivity of Optimal Solutions to Control Problems for Second Order Evolution Subdifferential Inclusions.

In this paper the sensitivity of optimal solutions to control problems described by second order evolution subdifferential inclusions under perturbations of state relations and of cost functionals is investigated. First we establish a new existence result for a class of such inclusions. Then, based on the theory of sequential [Formula: see text]-convergence we recall the abstract scheme concerning convergence of minimal values and minimizers. The abstract scheme works provided we can establish two properties: the Kuratowski convergence of solution sets for the state relations and some complementary [Formula: see text]-convergence of the cost functionals. Then these two properties are implemented in the considered case.

Numerical simulation for a neurotransmitter transport model in the axon terminal of a presynaptic neuron.

Neurotransmitters in the terminal bouton of a presynaptic neuron are stored in vesicles, which diffuse in the cytoplasm and, after a stimulation signal is received, fuse with the membrane and release its contents into the synaptic cleft. It is commonly assumed that vesicles belong to three pools whose content is gradually exploited during the stimulation. This article presents a model that relies on the assumption that the release ability is associated with the vesicle location in the bouton. As a modeling tool, partial differential equations are chosen as they allow one to express the continuous dependence of the unknown vesicle concentration on both the time and space variables. The model represents the synthesis, concentration-gradient-driven diffusion, and accumulation of vesicles as well as the release of neuroactive substances into the cleft. An initial and boundary value problem is numerically solved using the finite element method (FEM) and the simulation results are presented and discussed. Simulations were run for various assumptions concerning the parameters that govern the synthesis and diffusion processes. The obtained results are shown to be consistent with those obtained for a compartment model based on ordinary differential equations. Such studies can be helpful in gaining a deeper understanding of synaptic processes including physiological pathologies. Furthermore, such mathematical models can be useful for estimating the biological parameters that are included in a model and are hard or impossible to measure directly.

Compartment model of neuropeptide synaptic transport with impulse control.

In this paper a mathematical description of a presynaptic episode of slow synaptic neuropeptide transport is proposed. Two interrelated mathematical models, one based on a system of reaction diffusion partial differential equations and another one, a compartment type, based on a system of ordinary differential equations (ODE) are formulated. Processes of inflow, calcium triggered activation, diffusion and release of neuropeptide from large dense core vesicles (LDCV) as well as inflow and diffusion of ionic calcium are represented. The models assume the space constraints on the motion of inactive LDCVs and free diffusion of activated ones and ions of calcium. Numerical simulations for the ODE model are presented as well. Additionally, an electronic circuit, reflecting the functional properties of the mathematically modelled presynaptic slow transport processes, is introduced.

Model of neurotransmitter fast transport in axon terminal of presynaptic neuron.

In this paper a methodology of mathematical description of the synthesis, storage and release of the neurotransmitter during the fast synaptic transport is presented. The proposed model is based on the initial and boundary value problem for a parabolic nonlinear partial differential equation (PDE). Presented approach enables to express space and time dependences in the process: rate of vesicular replenishment, gradients of vesicular concentration and, through the boundary conditions, the location of docking and release sites. The model should be a good starting point for future numerical simulations since it is based on thoroughly studied parabolic equation. In the article classical and variational formulation of the problem is presented and the unique solution is shown to exist. The model is referred to the model based on ordinary differential equations (ODEs), created by Aristizabal and Glavinovic (AG model). It is shown that, under some assumptions, AG model is a special case of the introduced one.