Spatially Localized Synchronous Oscillations in Synaptically Coupled Neuronal Networks: Conductance-based Models and Discrete Maps

We study how spatiotemporal chaos in dynamical systems can be controlled by stochastically returning them to their initial conditions. Focusing on discrete nonlinear maps, we analyze how key measures of chaos-the Lyapunov exponent and butterfly velocity, which quantify sensitivity to initial perturbations and the ballistic spread of information, respectively-are reduced by stochastic resetting. We identify a critical resetting rate that induces a dynamical phase transition, characterized by the simultaneous vanishing of the Lyapunov exponent and butterfly velocity, effectively arresting the spread of information. These theoretical predictions are validated and illustrated with numerical simulations of the celebrated logistic map and its lattice extension. Beyond discrete maps, our findings are applicable to virtually any chaotic extended classical many-body system.

A new class of bifurcations is defined in discrete dynamical systems, and methods for their diagnostics and the analysis of their properties are presented. The TQ-bifurcations considered are implemented in discrete mappings and are related to the qualitative rearrangement of the shape of trajectories in an extended space of states. Within the demonstration of the main capabilities of the toolkit, an analysis is carried out of a logistic mapping in a domain to the right of the period-doubling limit point. Five critical values of the parameter are found for which the geometric structure of the trajectories of the mapping experiences a qualitative rearrangement. In addition, an analysis is carried out of the so-called “trace map,” which arises in the problems of quantum-mechanical description of various properties of discrete crystalline and quasicrystalline lattices.

Discrete dynamical systems - generalities on dynamical systems, examples of discrete dynamical systems, problems one-dimensional dynamical systems - preliminaries on one-dimensional systems, nonlinear one-dimensional systems, global behaviour and conjugacy, parameter space analysis, problems functions and matrices - matrices, matrices as linear operators, derivative and first order approximation, problems discrete linear dynamical systems - linear and affine processes, stability and instability of the origin, the origin as a saddle point, Eigenvalues with modulus I, problems nonlinear dynamical systems - global stability of equilibrium points, sinks, repellers and saddles, bounded invariant sets, bifurcation, problems chaotic behaviour - attractors, chaotic dynamical systems, dimension, Lyapunov exponents, problems analysis of four dynamical systems - a blood cell population model, predator-prey models, the Lorenz model, neural networks, problems.

Consider a discrete dynamical system given by a square matrix $M \in \mathbb{Q}^{d \times d}$ and a starting point $s \in \mathbb{Q}^d$. The orbit of such a system is the infinite trajectory $\langle s, Ms, M^2s, \ldots\rangle$. Given a collection $T_1, T_2, \ldots, T_m \subseteq \mathbb{R}^d$ of semialgebraic sets, we can associate with each $T_i$ an atomic proposition $P_i$ which evaluates to true at time $n$ if, and only if, $M^ns \in T_i$. This gives rise to the LTL Model-Checking Problem for discrete linear dynamical systems: given such a system $(M,s)$ and an LTL formula over such atomic propositions, determine whether the orbit satisfies the formula. The main contribution of the present paper is to show that the LTL Model-Checking Problem for discrete linear dynamical systems is decidable in dimension 3 or less.

The traditional models of distributed computing systems employ the mathematical formalisms of discrete event dynamical systems along with Petri Nets. However, such models are complex to analyze and computationally expensive. Interestingly, the evolving distributed computing systems closely resemble the discrete dynamical systems. This paper constructs and analyzes the model of distributed computing systems by applying mathematical formalisms of the discrete dynamical systems. The proposed model embeds the partial ordering of states under happened-before relation in the set of globally observable states. The stability of the proposed model is analyzed using the first order linear approximation.

This chapter will present periodic motions in the Duffing oscillator investigated through the mapping structures of discrete implicit maps. The discrete implicit maps will be obtained from the differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions in such a nonlinear oscillator will be predicted analytically through nonlinear algebraic equations of implicit maps, and the corresponding stability and bifurcation analysis of periodic motion in the bifurcation trees will be carried out. The bifurcation trees of periodic motions will also be presented through the harmonic amplitudes of the discrete Fourier series. Finally, from the analytical prediction, numerical simulation results of periodic motions will be presented for the verification of the analytical prediction. The harmonic amplitude spectrums will also be presented, and the corresponding analytical expressions of periodic motions can be obtained approximately.KeywordsDuffing OscillatorPeriodic MotionBifurcating TreeMain Harmonic AmplitudesDiscrete Implicit MapsThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This textbook provides a broad introduction to continuous and discrete dynamical systems. With its hands-on approach, the text leads the reader from basic theory to recently published research material in nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing. Dynamical Systems with Applications Using Python takes advantage of Python’s extensive visualization, simulation, and algorithmic tools to study those topics in nonlinear dynamical systems through numerical algorithms and generated diagrams. After a tutorial introduction to Python, the first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations. The second part of the book deals with discrete dynamical systems and progresses to the study of both continuous and discrete systems in contexts like chaos control and synchronization, neural networks, and binary oscillator computing. These later sections are useful reference material for undergraduate student projects. The book is rounded off with example coursework to challenge students’ programming abilities and Python-based exam questions. This book will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a range of disciplines, such as biology, chemistry, computing, economics, and physics. Since it provides a survey of dynamical systems, a familiarity with linear algebra, real and complex analysis, calculus, and ordinary differential equations is necessary, and knowledge of a programming language like C or Java is beneficial but not essential.

External excitatory stimuli can terminate bursting in neural network models

The dynamics of a differential algebraic equation takes place on a lower dimensional manifold in phase space. Applying a numerical integration scheme, it is natural to ask if and how this geometric property is preserved by the discrete dynamical system. In the index-1 case answers to this question are obtained from the singularly perturbed case treated by Nipp and Stoffer, Numer. Math. 70 (1995), 245–257, for Runge-Kutta methods and in K. Nipp and D. Stoffer, Numer. Math. 74 (1996), 305–323, for linear multistep methods. As main result of this paper it is shown that also for Runge-Kutta methods and linear multistep methods applied to a index-2 problem of Hessenberg form there is a (attractive) invariant manifold for the discrete dynamical system and this manifold is close to the manifold of the differential algebraic equation.

Basal ganglia circuits in Parkinson’s disease constitute a prominent example of neural system of coupled inhibitory and excitatory neurons, which exhibit synchronous activity [1]. The phase-locking of neural activity in this system, as analyzed in experimental data intraoperatively recorded in Parkinsonian patients, exhibits intermittent temporal dynamics [2]. The mechanisms of this intermittent synchronous dynamics make the subject of this study. The conductance-based network models of subthalamo-pallidal circuitry of basal ganglia have being shown to reproduce the intermittent temporal patterns of phase-locking observed in experiments with high fidelity [3]. We start with this network model of synaptically coupled excitatory subthalamic and inhibitory pallidal bursting neurons. We further reduce it to the model of two mutually coupled inhibitory neurons with self-inhibition. This network is presented at Figure ​Figure1.1. While real subthalamic neurons are excitatory, the resulting two-neurons network is adequate simplification of the original circuitry under assumption of high fidelity of responses of inhibitory pallidal cells. While this reduction may be associated with the loss of some other biologically realistic features, it produces a relatively generic circuit, therefore the studied mechanism may be quite general too. Figure 1 The network architecture. Each connection is inhibitory synapse. We used geometric dynamical systems and singular perturbation methods to reduce the full model to a simpler set of equations. Mathematical analysis was completed using three slow variables with two different time scales. Intermittently synchronous oscillations are generated by overlapped spiking which crucially depend on the geometry of slow phase plane and the interplay between slow variables as well as the strength of synapses. Two slow variables are responsible for the generation of out-of-phase stable solution and the other slower variable for irregular and intermittent activity pattern. The results of analysis can be traced to particular values of biophysical parameters (synaptic strength and parameters of calcium dynamics), which are known to be impacted in Parkinson’s disease.

systems. In this paper, we firstly study the Noether symmetry and Noether exact invariants of discrete nonconservative dynamical systems. Secondly, we study perturbation to symmetry and adiabatic invariants of discrete dynamical systems. Finally, we give an example to illustrate the application of these results.

The stability and bifurcation properties of one-dimensional discrete dynamical systems with positivity, which are derived from continuous ones by tropical discretization, are studied. The discretized time interval is introduced as a bifurcation parameter in the discrete dynamical systems, and the emergence condition of an additional bifurcation, flip bifurcation, is identified. The correspondence between the discrete dynamical systems with positivity and the ultradiscrete ones derived from them is discussed. It is found that the derived ultradiscrete max-plus dynamical systems can retain the bifurcations of the original continuous ones via tropical discretization and ultradiscretization.

A class of kick‐excited self‐adaptive dynamical systems is formed and proposed. The class is characterized by nonlinear (inhomogeneous) external periodic excitation (as regards to the coordinates of excited systems) and is remarkable for its objective regularities: the phenomenon of “discrete” (“quantized”) oscillation excitation and strong self‐adaptive stability. The main features of these systems are studied both numerically and analytically on the basis of a general model: a pendulum under inhomogeneous action of a periodic force which is referred to as a kicked pendulum. Multiple bifurcation diagram for the attractor set of the system under consideration is obtained and analyzed. The complex dynamics, evolution and the fractal boundaries of the multiple attractor basins in state space corresponding to energy and phase variables are obtained, traced and discussed. A two‐dimensional discrete map is derived for this case. A general treatment of the class of kick‐excited self‐adaptive dynamical systems is made by putting it in correspondence to a general class of dissipative twist maps and showing that the latter is an immanent tool for general description of its behavior.

The set of trajectories for discrete dynamical systems (DDS) in the space is investigated. These sets are the solutions for difference equations in a metric space (space of nonempty convex compacts with the Hausdorff metric). On the basis of the comparison principle the general theorems on stability in terms of two measures were established. Applying the Minkowskij theory of mixed volumes, for some classes of nonlinear DDS in space the finite-dimensional comparison systems were constructed. The stability in terms of two measures and Lyapunov stability of fixed points for DDS in space were studied. The examples of studies of certain dynamical systems were given to illustrate the effectiveness of obtained results.

1. The recruitment of evoked fast inhibitory postsynaptic currents (IPSCs) and excitatory postsynaptic currents (EPSCs) was examined using whole cell voltage-clamp recordings from layer V pyramidal neurons in slices of rat somatosensory cortex. Synaptic currents were evoked with graded electrical stimulation to assess the relative activation of IPSCs and EPSCs. Fast GABAA ergic IPSCs were selectively recorded by holding cells at potentials equal to EPSC reversal (approximately 0 mV). EPSCs were likewise isolated by holding cells at IPSC reversal potential (about -75 mV). 2. As stimulus intensities were increased, the magnitude of the postsynaptic currents also increased. Over the range of stimuli applied (2-10 V), EPSCs did not exhibit an upper limit. However, fast gamma-aminobutyric acid-A (GABAA-mediated IPSCs reached a maximum at intensities approximately 2 times threshold. 3. The limit on fast inhibition was unresponsive to alterations in N-methyl-D-aspartate (NMDA)-mediated excitation. Exposure to nominally magnesium-free solutions or to the NMDA antagonist 3-[(RS)-2-carboxypiperazin-4-yl]-propyl-1-phosphonic acid did not affect the fast IPSC maximum. Shifts in the input-output curves for submaximal activation of IPSCs were seen, which were attributed to polysynaptic excitation. 4. Blockade of alpha-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid/kainate (non-NMDA) receptors with 6-cyano-7-nitro-quinoxaline-2,3-dione (CNQX) completely abolished synaptically driven, fast GABAA-mediated inhibition. These findings suggested that neocortical inhibitory cells could be driven exclusively through non-NMDA transmission. 5. By comparison, in hippocampal CA1 pyramidal neurons maximal fast inhibition was sensitive to both NMDA and non-NMDA receptor blockade. 6. The results in neocortex were corroborated by direct intracellular recordings from layer V-VI interneurons. Non-NMDA receptor blockade with CNQX prevented synaptic activation of action potentials in these cells, even during cotreatment with magnesium-free solution. 7. Together, these results suggest that recruitment of GABA(A) ergic IPSCs in neocortex is ultimately driven via glutamatergic afferents arriving at non-NMDA receptors on interneurons. Properties limiting fast inhibition would favor the propagation of enhanced excitatory activity through the neuronal network.