Darya V. Verveyko

Turing-like structures in a functional model of cortical spreading depression

Cortical spreading depression (CSD) along with migraine waves and spreading depolarization events with stroke or injures are the front-line examples of extreme physiological behaviors of the brain cortex which manifest themselves via the onset and spreading of localized areas of neuronal hyperactivity followed by their depression. While much is known about the physiological pathways involved, the dynamical mechanisms of the formation and evolution of complex spatiotemporal patterns during CSD are still poorly understood, in spite of the number of modeling studies that have been already performed. Recently we have proposed a relatively simple mathematical model of cortical spreading depression which counts the effects of neurovascular coupling and cerebral blood flow redistribution during CSD. In the present study, we address the main dynamical consequences of newly included pathways, namely, the changes in the formation and propagation speed of the CSD front and the pattern formation features in two dimensions. Our most notable finding is that the combination of vascular-mediated spatial coupling with local regulatory mechanisms results in the formation of stationary Turing-like patterns during a CSD event.

Mathematical model of chaotic oscillations and oscillatory entrainment in glycolysis originated from periodic substrate supply.

We study the influence of periodic influx on a character of glycolytic oscillations within the forced Selkov system. We demonstrate that such a simple system demonstrates a rich variety of dynamical regimes (domains of entrainment of different order (Arnold tongues), quasiperiodic oscillations, and chaos), which can be qualitatively collated with the known experimental data. We determine detailed dynamical regimes exploring the map of Lyapunov characteristic exponents obtained in numerical simulations of the Selkov system with periodic influx. In addition, a special study of the chaotic regime and the scenario of its origin in this system was evaluated and discussed.

Cluster structure of condensed media

This paper presents the results of a theoretical study of cluster formation in condensed media for simple and multi-atomic liquids, as an example. A two-parameter function is proposed by the authors for the density of the probability distribution of clusters, which corresponds to a disordered condensed media, depending on the number of constituent particles. Using this function, the universal mean, most probable, and rms numbers of particles in clusters of noble gases, liquid metals, and some organic liquids were calculated. As well, the cluster component of total entropy for the substances under study was computed.

Mechanisms for the target patterns formation in a stochastic bistable excitable medium

We study the features of formation and evolution of spatiotemporal chaotic regime generated by autonomous pacemakers in excitable deterministic and stochastic bistable active media using the example of the FitzHugh – Nagumo biological neuron model under discrete medium conditions. The following possible mechanisms for the formation of autonomous pacemakers have been studied: 1) a temporal external force applied to a small region of the medium, 2) geometry of the solution region (the medium contains regions with Dirichlet or Neumann boundaries). In our work we explore the conditions for the emergence of pacemakers inducing target patterns in a stochastic bistable excitable system and propose the algorithm for their analysis.

Raindrops of synaptic noise on dual excitability landscape: an approach to astrocyte network modelling

The most abundant non-neuronal cells in the brain, astrocytes, populate all parts of the central nervous system (CNS). Astrocytic calcium activity ranging from subcellular sparkles to intercellular waves is believed to be the key to a plethora of regulatory pathways in the central nervous system from synaptic plasticity to blood flow regulation. Modeling of the calcium wave initiation and transmission and their spatiotemporal dynamics is therefore an important step stone in understanding the crucial cogs of cognition. Astrocytes are active sensors of ongoing neuronal and synaptic activity, and neurotransmitters diffusing from the synaptic cleft make a strong impact on the astrocytic activity. Here we propose a model describing the patterns of calcium wave formation at a single cell level and discuss the interplay between astrocyte shape the calcium waves dynamics driven by local stochastic surges of glutamate simulating synaptic activity.

Sodium–Calcium Exchanger Can Account for Regenerative Ca2+ Entry in Thin Astrocyte Processes

Calcium transients in thin astrocytic processes can be important in synaptic plasticity, but their mechanism is not completely understood. Clearance of synaptic glutamate leads to increase in astrocytic sodium. This can electrochemically favor the reverse mode of the Na/Ca-exchanger (NCX) and allow calcium into the cell, accounting for activity-dependent calcium transients in perisynaptic astrocytic processes. However, cytosolic sodium and calcium are also allosteric regulators of the NCX, thus adding kinetic constraints on the NCX-mediated fluxes and providing for complexity of the system dynamics. Our modeling indicates that the calcium-dependent activation and also calcium-dependent escape from the sodium-mediated inactive state of the NCX in astrocytes can form a positive feedback loop and lead to regenerative calcium influx. This can result in sodium-dependent amplification of calcium transients from nearby locations or other membrane mechanisms. Prolonged conditions of elevated sodium, for example in ischemia, can also lead to bistability in cytosolic calcium levels, where a delayed transition to the high-calcium state can be triggered by a short calcium transient. These theoretical predictions call for a dedicated experimental estimation of the kinetic parameters of the astrocytic Na/Ca-exchanger.

Spatio-temporal cerebral blood flow perfusion patterns in cortical spreading depression

Cortical spreading depression (CSD) is an example of one of the most common abnormalities in biophysical brain functioning. Despite the fact that there are many mathematical models describing the cortical spreading depression (CSD), most of them do not take into consideration the role of redistribution of cerebral blood flow (CBF), that results in the formation of spatio-temporal patterns. The paper presents a mathematical model, which successfully explains the CBD role in the CSD process. Numerical study of this model has revealed the formation of stationary dissipative structures, visually analogous to Turing structures. However, the mechanism of their formation is not diffusion. We show these structures occur due to another type of spatial coupling, that is related to tissue perfusion rate. The proposed model predicts that at similar state of neurons the distribution of blood flow and oxygenation may by different. Currently, this effect is not taken into account when the Blood oxygen-level dependent (BOLD) contrast imaging used in functional magnetic resonance imaging (fMRI). Thus, the diagnosis on the BOLD signal can be ambiguous. We believe that our results can be used in the future for a more correct interpretation of the data obtained with fMRI, NIRS and other similar methods for research of the brain activity.

Computational model of cerebral blood flow redistribution during cortical spreading depression

In recent decades modelling studies on cortical spreading depression (CSD) and migraine waves successfully contributed to formation of modern view on these fundamental phenomena of brain physiology. However, due to the extreme complexity of object under study (brain cortex) and the diversity of involved physiological pathways, the development of new mathematical models of CSD is still a very relevant and challenging research problem. In our study we follow the functional modelling approach aimed to map the action of known physiological pathways to the specific nonlinear mechanisms that govern formation and evolution of CSD wave patterns. Specifically, we address the role of cerebral blood flow (CBF) redistribution that is caused by excessive neuronal activity by means of neurovascular coupling and mediates a spatial pattern of oxygen and glucose delivery. This in turn changes the local metabolic status of neural tissue. To build the model we simplify the web of known cell-to-cell interactions within a neurovascular unit by selecting the most relevant ones, such as local neuron-induced elevation of extracellular potassium concentration and biphasic response of arteriole radius. We propose the lumped description of distance-dependent hemodynamic coupling that fits the most recent experimental findings.

NON-TURING MECHANISM OF SELF-SUSTAINED STRUCTURE FORMATION

We study the mechanism of experimentally observed phase waves and clusters in yeast extracts (cells with destroyed membranes) placed into the unstirred medium (gel). As a mathematical model, the di...

Mathematical Model for Temperature Regulation of Self-Sustained Glycolytic Oscillations in a Closed Reactor

We base on the Selkov system [1] to construct the model for temperaturecontrol of glycolytic reaction in a closed spatial reactor. To establish acorrespondence with the experiment [2] we add the slow catalytic term xwhich describes the small value of additional substrate influx and productoutflow and introduce a temperature-dependent coefficient satisfying theArrhenius law... The considered model explains the key experimentally observed phenomena[2]: 1) decaying of the average concentrations of reagents duringthe reaction, 2) Arrhenius-type temperature dependence for frequency of oscillations,3) change of the form of oscillations with the temperature growth,4) modulations of oscillations induced by a periodic temperature variation.The addition of the diffusion terms to the system (1) allows to reproducethe emerging of glycolytic travelling waves observed in a closed reactor inthe presence of a temperature gradient [2]. Comparison of the dynamics of travelling waves in the numerical solution with the experimental data [2]permits to propose a new method to estimate the diffusion coefficients of reagents in the case of a chemical reaction occurring in a dense media