Analyzing synchronized clusters in neuron networks

Improved Mathematical Models of Parkinson's Disease with Hopf Bifurcation and Huntington's Disease with Chaos

Using delay differential equations to study mathematical models of Parkinson's disease and Huntington's disease is important to show how important it is for synchronization between basal ganglia loops to work together. We used the delay circuit RLC (resistor, inductor, capacitor) model to show how the direct pathway and the indirect pathway in the basal ganglia excite and inhibit the motor cortex, respectively. A term has been added to the mathematical model without time delay in the case of the hyperdirect pathway. It is proposed to add a non-linear term to adjust the synchronization. We studied Hopf bifurcation conditions for the proposed models. The desynchronization of response times between the direct pathway and the indirect pathway leads to different symptoms of Parkinson's disease. Tremor appears when the response time in the indirect pathway increases at rest. The simulation confirmed that tremor occurs and the motor cortex is in an inhibited state. The direct pathway can increase the time delay in the dopaminergic pathway, which significantly increases the activity of the motor cortex. The hyperdirect pathway regulates the activity of the motor cortex. The simulation showed bradykinesia occurs when we switch from one movement to another that is less exciting for the motor cortex. A decrease of GABA in the striatum or delayed excitation of the substantia nigra from the subthalamus may be a major cause of Parkinson's disease. An increase in the response time delay in one of the pathways results in the chaotic movement characteristic of Huntington's disease.

Endowing networks with desired symmetries and modular behavior

Symmetries in a network regulate its organization into functional clustered states. Given a generic ensemble of nodes and a desirable cluster (or group of clusters), we exploit the direct connection between the elements of the eigenvector centrality and the graph symmetries to generate a network equipped with the desired cluster(s), with such a synthetical structure being furthermore perfectly reflected in the modular organization of the network's functioning. Our results solve a relevant problem of designing a desired set of clusters and are of generic application in all cases where a desired parallel functioning needs to be blueprinted.

Transitional cluster dynamics in a model for delay-coupled chemical oscillators

Cluster synchronization is a fundamental phenomenon in systems of coupled oscillators. Here, we investigate clustering patterns that emerge in a unidirectional ring of four delay-coupled electrochemical oscillators. A voltage parameter in the experimental setup controls the onset of oscillations via a Hopf bifurcation. For a smaller voltage, the oscillators exhibit simple, so-called primary, clustering patterns, where all phase differences between each set of coupled oscillators are identical. However, upon increasing the voltage, secondary states, where phase differences differ, are detected, in addition to the primary states. Previous work on this system saw the development of a mathematical model that explained how the existence, stability, and common frequency of the experimentally observed cluster states could be accurately controlled by the delay time of the coupling. In this study, we revisit the mathematical model of the electrochemical oscillators in order to address open questions by means of bifurcation analysis. Our analysis reveals how the stable cluster states, corresponding to experimental observations, lose their stability via an assortment of bifurcation types. The analysis further reveals complex interconnectedness between branches of different cluster types. We find that each secondary state provides a continuous transition between certain primary states. These connections are explained by studying the phase space and parameter symmetries of the respective states. Furthermore, we show that it is only for a larger value of the voltage parameter that the branches of secondary states develop intervals of stability. For a smaller voltage, all the branches of secondary states are completely unstable and are, therefore, hidden to experimentalists.

Computational modeling predicts regulation of central pattern generator oscillations by size and density of the underlying heterogenous network

Central pattern generators are characterized by a heterogeneous cellular composition, with different cell types playing distinct roles in the production and transmission of rhythmic signals. However, little is known about the functional implications of individual variation in the relative distributions of cells and their connectivity patterns. Here, we addressed this question through a combination of morphological data analysis and computational modeling, using the pacemaker nucleus of the weakly electric fish Apteronotus leptorhynchus as case study. A neural network comprised of 60-110 interconnected pacemaker cells and 15-30 relay cells conveying its output to electromotoneurons in the spinal cord, this nucleus continuously generates neural signals at frequencies of up to 1 kHz with high temporal precision. We systematically explored the impact of network size and density on oscillation frequencies and their variation within and across cells. To accurately determine effect sizes, we minimized the likelihood of complex dynamics using a simplified setup precluding differential delays. To identify natural constraints, parameter ranges were extended beyond experimentally recorded numbers of cells and connections. Simulations revealed that pacemaker cells have higher frequencies and lower within-population variability than relay cells. Within-cell precision and between-cells frequency synchronization increased with the number of pacemaker cells and of connections of either type, and decreased with relay cell count in both populations. Network-level frequency-synchronized oscillations occurred in roughly half of simulations, with maximized likelihood and firing precision within biologically observed parameter ranges. These findings suggest the structure of the biological pacemaker nucleus is optimized for generating synchronized sustained oscillations.

Automatic differentiation of human induced pluripotent stem cells toward synchronous neural networks on an arrayed monolayer of nanofiber membrane

Automatic differentiation of human-induced pluripotent stem cells (hiPSCs) facilitates the generation of cortical neural networks and studies of brain functions. Here, we present a method of directed differentiation of hiPSCs with a substrate made of a honeycomb microframe and a monolayer of crosslinked gelatin nanofibers in the form of an array of nanofiber membranes. Neural precursor cells (NPCs) were firstly derived from hiPSCs and then placed on the nanofiber membranes for automatically controlled neural differentiation over a long period. Due to the strong modulation of the substrate stiffness and permeability, most cells were found in the center area of the honeycomb compartments, giving rise to regular and inter-connected cortical neural clusters. More importantly, the neural activities of the clusters were synchronized proving the reliability of the method. Our results showed that the self-organization, as well as the neural activities of differentiating neural cells, were more efficient in the nanofiber membrane compared to the types of the substrate such as glass and nanofiber-covered glass. In addition to the inherent advantages such as manpower saving and fewer risks of contamination and human error, automatic differentiation avoided undesired shaking which might have critical effects on the formation of synchronous neural clusters. STATEMENT OF SIGNIFICANCE: : Synchronization of cortical neural activities is essential for information processing and human cognition. By automated differentiation of human induced pluripotent stem cells on arrayed monolayer of nanofiber membrane, synchronous neural clusters could be formed. Such an approach would allow creating a variety of neural networks with regular and interconnected clusters for systematic studies of human cortical functions.

First-order route to antiphase clustering in adaptive simplicial complexes

This Letter investigates the transition to synchronization of oscillator ensembles encoded by simplicial complexes in which pairwise and higher-order coupling weights alter with time through a rate-based adaptive mechanism inspired by the Hebbian learning rule. These simultaneously evolving disparate adaptive coupling weights lead to a phenomenon in that the in-phase synchronization is completely obliterated; instead, the antiphase synchronization is originated. In addition, the onsets of antiphase synchronization and desynchronization are manageable through both dyadic and triadic learning rates. The theoretical validation of these numerical assessments is delineated thoroughly by employing Ott-Antonsen dimensionality reduction. The framework and results of the Letter would help understand the underlying synchronization behavior of a range of real-world systems, such as the brain functions and social systems where interactions evolve with time.

Phase synchronization and measure of criticality in a network of neural mass models

Synchronization has an important role in neural networks dynamics that is mostly accompanied by cognitive activities such as memory, learning, and perception. These activities arise from collective neural behaviors and are not totally understood yet. This paper aims to investigate a cortical model from this perspective. Historically, epilepsy has been regarded as a functional brain disorder associated with excessive synchronization of large neural populations. Epilepsy is believed to arise as a result of complex interactions between neural networks characterized by dynamic synchronization. In this paper, we investigated a network of neural populations in a way the dynamics of each node corresponded to the Jansen-Rit neural mass model. First, we study a one-column Jansen-Rit neural mass model for four different input levels. Then, we considered a Watts-Strogatz network of Jansen-Rit oscillators. We observed an epileptic activity in the weak input level. The network is considered to change various parameters. The detailed results including the mean time series, phase spaces, and power spectrum revealed a wide range of different behaviors such as epilepsy, healthy, and a transition between synchrony and asynchrony states. In some points of coupling coefficients, there is an abrupt change in the order parameters. Since the critical state is a dynamic candidate for healthy brains, we considered some measures of criticality and investigated them at these points. According to our study, some markers of criticality can occur at these points, while others may not. This occurrence is a result of the nature of the specific order parameter selected to observe these markers. In fact, The definition of a proper order parameter is key and must be defined properly. Our view is that the critical points exhibit clear characteristics and invariance of scale, instead of some types of markers. As a result, these phase transition points are not critical as they show no evidence of scaling invariance.

One-way dependent clusters and stability of cluster synchronization in directed networks

Cluster synchronization in networks of coupled oscillators is the subject of broad interest from the scientific community, with applications ranging from neural to social and animal networks and technological systems. Most of these networks are directed, with flows of information or energy that propagate unidirectionally from given nodes to other nodes. Nevertheless, most of the work on cluster synchronization has focused on undirected networks. Here we characterize cluster synchronization in general directed networks. Our first observation is that, in directed networks, a cluster A of nodes might be one-way dependent on another cluster B: in this case, A may remain synchronized provided that B is stable, but the opposite does not hold. The main contribution of this paper is a method to transform the cluster stability problem in an irreducible form. In this way, we decompose the original problem into subproblems of the lowest dimension, which allows us to immediately detect inter-dependencies among clusters. We apply our analysis to two examples of interest, a human network of violin players executing a musical piece for which directed interactions may be either activated or deactivated by the musicians, and a multilayer neural network with directed layer-to-layer connections.

Mechanisms of cluster formation in networks with directed links differ from those in undirected networks. Lodi et al. propose a method to compute interdependencies among clusters of nodes in directed networks. They show that clusters can be one-way dependent, as found in social and neural networks.

Delays induced cluster synchronization in chaotic networks

We study networks of coupled oscillators and analyze the role of coupling delays in determining the emergence of cluster synchronization. Given a network topology and a particular arrangement of the coupling delays over the network connections, different patterns of cluster synchronization may emerge. We focus on a simple ring network of six bidirectionally coupled identical oscillators, for which with two different values of the delays, a total of eight cluster synchronization patterns may emerge, depending on the assignment of the delays to the ring connections. We analyze stability of each of the patterns and find that for large enough coupling strength and specific values of the delays, they can all be stabilized. We construct an experimental ring of six bidirectionally coupled Colpitts oscillators, with delayed connections obtained by coupling the oscillators via RF cables of appropriate length. We find that experimental observations of cluster synchronization are in essential agreement with theoretical predictions. We also verify our theory in a fully connected network of fifty nodes for which connections are randomly assigned to be either undelayed or delayed with a given probability.