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The present work is devoted to investigation of operator inclusions in partially ordered spaces and application of the obtained results to differential inclusions. We consider the inclusion Υ(x,x)∋y with respect to the unknown x∈X,where Υ:X×X⇒Y is a set-valued mapping, X and Y are partially ordered spaces. It is assumed that the mapping Υ is order covering with respect to the first argument and antitone with respect to the second argument. We prove that for any x0 ∈ X, if the set G(x0) contains an element y0 such that y ≼ y0, then there exists a solution to the inclusion under consideration, which satisfies the inequality x ≼ x0. This statement is applied to investigation of a Cauchi problem for the differential inclusion f(t,x,ẋ,ẋ)∋0 with a bound for the derivative of the unknown function ẋ(t)∈B(t) (here f:[a,b]×Rn×Rn×Rn ⇒Rm,B:[a,b]⇒Rn). We obtain conditions of solvability in the space of absolutely continuous functions, conditions of existence of a solution with the least derivative, and derive the solutions estimates. The latter results are applied to the analysis of the dynamic Walrasian-Evans-Samuelson model of economic processes, which can be reduced to a system of implicit differential inclusions. We establish the existence of the equilibrium and obtain estimates of the equilibrium prices., Advances in Systems Science and Applications, Выпуск 1 2022, Pages 176-191

In the space of nonempty closed subsets of a given metric space, we define a vector-valued metric which differs from the known vector generalizations of the Hausdorff metric. The metric proposed appears in the analysis of multi-valued maps with possibly unbounded images. It allows to get more information about distances between elements of the corresponding sets and turns out to be convenient in fixed point theorems for multi-valued maps.

We formulate and prove the theorem on well-posedness of abstract Volterra equations in metric spaces. We consider nonlinear nonlocal integral Volterra equation involving generalizing equations typically used in mathematical neuroscience. We investigate solutions that tend to zero at any moment with unbounded growth of the spatial variable. In the literature such solutions are called «spatially localized solutions» or «bumps». They correspond to normal brain functioning. For the main equation, we consider an impulsive control problem, where the control parameters are moments, when the solution discontinue, and corresponding jumps’ values. Such control systems model electrical stimulation used in the presence of various diseases of central nervous system. We define suitable complete metric (not linear) space. Using the aforementioned theorem, we obtain conditions for existence and uniqueness of solution to this equation and continuous dependence of this solution on the control.

We propose method of studying implicit singular differential equations based on the results of the covering mapping theory. The article consists of three sections. In the first section we give the necessary designations and definitions and formulate the theorem on Lipcshitz perturbations of covering mappings. In the second section we introduce special spaces of measurable functions making it possible to study singular equations by methods of functional analysis and formulate the results about the Nemytskii operator in those spaces. In the last section we provide conditions for the resolubility of the Cauchy problem for implicit singular differential equations.

One of the major challenges in neurobiology remains understanding the relationship between complex neural network dynamics underlying spatially structured activity states and the corresponding neural circuitry for which the electromagnetic field is macroscopically measurable via electroencephalogram (EEG) or local field potentials. Such macroscopic electrical activity in the neocortex is naturally studied in the framework of cortical networks. However, since the number of neurons and synapses in even a small piece of cortex is immense, a suitable modeling approach is to take a continuum limit of the neural networks and, thus, consider so-called neural field models of the brain cortex. This modeling framework involves integro-differential equations or Volterra integral equations and goes back to the seminal papers by Wilson, Cowan and Amari in the 1970’s. In recent years, such neural fields have been used to model a wide range of neurobiological phenomena, including orientation tuning in primary visual cortex, short term working memory, control of head direction, motion perception, geometric visual hallucinations, EEG rhythms, and wave propagation in cortical slices and in vivo. The aforementioned framework, however, does not take into account the heterogeneity in the cortical structure. Recent works in neuroscience have drawn attention to homogenized neural field models where the brain heterogeneity is captured by a special parameter. Such models are obtained from heterogeneous neural field models by means of homogenization: the two scale convergence method developed by Nguetseng. These investigations have been restricted to a one-dimensional case though. We take a step in the direction of considering a more realistic two-dimensional variation of the homogenized neural field model. We use pinning function technique and spectral properties of Hilbert–Schmidt integral operators to establish existence and stability of localized stationary activity states. Various approximations and numerical approaches, which are frequently used in the mathematical neuroscience, need to be justified rigorously. Using the fixed point theorems and convergence techniques in functional spaces, we investigate the well-posedness aspects of the homogeneous and homogenized neural field models, thus justifying implementation of numerical schemes. We also justify the approximations of continuous neural fields by network models, thus, proving the validity of various disctetization methods. Using compactness in functional spaces and topological degree theory, we justify the approximation of smooth activation functions by the Heaviside unit step function in the case of localized stationary solutions for the n-dimensional homogenized neural field model. The latter result is of particular importance in the aforementioned homogenization procedure. The present thesis illustrates that methods of functional analysis employed in mathematical neuroscience may be very beneficial. En av hovedutfordringene i nevrobiologi består i å forstå sammenhengen mellom den komplekse nettverks-dynamikken som ligger under de romlige aktivitetstilstandene i hjernebarken og makroskopiske målinger av den korresponderende elektriske kretsverk aktiviteten ved hjelp av elektroencefalogram (EEG) og lokale felt potensialer. Slik makroskopisk elektrisk aktivitet in neocortex beskrives gjerne ved hjelp av fyringsrate modeller. Men, siden antallet nevroner og synapser i selv en svært liten del av hjernebarken er enormt stort, så er det naturlig å ta kontinuumsgrensen av disse fyringsrate modellene. Dette betyr at en studerer fyringsaktiviteten i hjernebarken ved hjelp av såkalte nevrofelt modeller. Slike rammeverk for modellering baseres på integro-differensial likninger eller Volterra integral likninger. Disse rammeverkene går tilbake til banebrytende arbeider av Wilson, Cowan og Amari på 1970–tallet. I de senere årene har en brukt nevrofeltmodeller til å beskrive ett vidt sett av nevrobiologiske fenomener, som f. eks. inkludering av orientering tuning i den primære visuelle hjernebarken, korttidshukommelse, kontroll av hode retning, persepsjon, visuelle hallusinasjoner, EEG rytmer og bølgeforplantning i snitt av hjernebarken og i levende vev. En svakhet med mange nevrofelt-modeller er at de ikke tar hensyn til heterogeniteten som er til stede i den kortikale strukturen. I noen nylig publiserte arbeider i nevrovitenskap tar en hensyn til heterogeniteten ved hjelp av en spesiell parameter. Slike modeller er utledet fra heterogene nevrofelt modeller ved hjelp av en homogeniseringsmetode basert på to-skala konvergens metoden til Nguetseng. Disse studiene er imidlertid begrenset til en rom dimensjon. I denne avhandlingen ser vi på en realistisk to-dimensjonal situasjon for en en-populasjon homogenisert nevrofelt modell. Vi bruker pinning funksjonsteknikken til å avgjøre eksistens av romlig lokaliserte tilstander og spektral egenskapene til Hilbert–Schmidt integral operatorer til å bestemme stabiliteten til disse tilstandene. Det er viktigå rettferdiggjøre de ulike approksimasjonene og de numeriske skjemaene som brukes i matematisk nevrovitenskap rigorøst. Ved å bruke fikspunkt teoremer of konvergensteknikker i funksjonsrom, studerer vi velformulerthet av homogene og homogeniserte nevrofelt modeller. Vi rettferdiggjør også implementering av numeriske skjemaer. Vi begrunner også approksimasjonen av kontinuerlige nevrofelt modeller ved hjelp av diskrete nettverks modeller, hvilket innebærer at vi rettferdiggjør ulike diskretiseringsmetoder. Ved å bruke kompakthet resultat for funksjonsrom og gradteori, rettferdiggjør vi approksimasjonen av glatte fyringsrate funksjoner med Heaviside-step funksjon når vi studerer lokaliserte stasjonære løsninger av den n dimensjonale homogeniserte nevrofelt modellen. Det sistnevnte resultatet er spesielt viktig i den tidligere nevnte homogeniseringsprosedyren. Den foreliggende avhandlingen viser at det er svært fordelaktig å bruke funksjonalanalytiske metoder i matematisk nevrovitenskap.