Key Bifurcations of Bursting Polyrhythms in 3-Cell Central Pattern Generators

We identify and describe the key qualitative rhythmic states in various 3-cell network motifs of a multifunctional central pattern generator (CPG). Such CPGs are neural microcircuits of cells whose synergetic interactions produce multiple states with distinct phase-locked patterns of bursting activity. To study biologically plausible CPG models, we develop a suite of computational tools that reduce the problem of stability and existence of rhythmic patterns in networks to the bifurcation analysis of fixed points and invariant curves of a Poincare´ return maps for phase lags between cells. We explore different functional possibilities for motifs involving symmetry breaking and heterogeneity. This is achieved by varying coupling properties of the synapses between the cells and studying the qualitative changes in the structure of the corresponding return maps. Our findings provide a systematic basis for understanding plausible biophysical mechanisms for the regulation of rhythmic patterns generated by various CPGs in the context of motor control such as gait-switching in locomotion. Our analysis does not require knowledge of the equations modeling the system and provides a powerful qualitative approach to studying detailed models of rhythmic behavior. Thus, our approach is applicable to a wide range of biological phenomena beyond motor control

The Role of Plasma Membrane ATPase Pumps in the Regulation of Rhythmic Activity in Electrically Excitable Cells

Membrane bound ion pumps have long been studied in a housekeeping role, and it is well known that they play a major part in creating the ionic gradients which determine the electrical excitability in a cell. Recent work has begun to highlight other, more direct roles for ion pumps in rhythm generation and information processing. As many pumps obtain energy for active ion transport from adenosine triphosphate (ATP) hydrolysis, they can exchange ions in an electrically asymmetric manner, generating an outward current, which along with ion channel currents, drives the membrane potential of the cell. Membrane potential is a major determining characteristic for how information is transferred between neurons, and so in persistently active excitable cells, pumps can provide a considerable contribution to neuron dynamics. Specialized networks of neurons and non-neural cells which drive rhythmic behaviors such as breathing and locomotion, must robustly produce useful patterns for the animal under dynamic behavioral goals in a highly variable environment. Here we will focus on two well-studied classes of ATPase pumps (the plasma membrane calcium ATPase pump (PMCA) and the sodium-potassium ATPase pump (Na+/K+ pump)) and investigate the role of these pumps in two rhythm generating biological subsystems with a combination of modeling and experimental approaches. In a model of a leech heartbeat central pattern generator, we demonstrate how the neuromodulator myomodulin can regulate the temporal properties of rhythm generation through effects on the hyperpolarization-activated current and the Na+/K+ pump current, and discuss the benefits of modulators which target multiple currents. With this model, we also show how synaptic inhibition can support a functional pattern when pump current is downregulated. Then, in a model of interstitial cells of Cajal (ICC) in the muscular syncytium of the intestinal walls, we demonstrate that due to the importance of complex intracellular calcium oscillations in the generation of ICC rhythms, the PMCA pump can play a major role in regulating the temporal properties of rhythm generation. We discuss rhythm generation mechanisms in both systems and predict parameter domains of multistability which correspond to both functional and pathological states of rhythm generation

Multistable Decision Switches for Flexible Control of Epigenetic Differentiation

It is now recognized that molecular circuits with positive feedback can induce two different gene expression states (bistability) under the very same cellular conditions. Whether, and how, cells make use of the coexistence of a larger number of stable states (multistability) is however largely unknown. Here, we first examine how autoregulation, a common attribute of genetic master regulators, facilitates multistability in two-component circuits. A systematic exploration of these modules' parameter space reveals two classes of molecular switches, involving transitions in bistable (progression switches) or multistable (decision switches) regimes. We demonstrate the potential of decision switches for multifaceted stimulus processing, including strength, duration, and flexible discrimination. These tasks enhance response specificity, help to store short-term memories of recent signaling events, stabilize transient gene expression, and enable stochastic fate commitment. The relevance of these circuits is further supported by biological data, because we find them in numerous developmental scenarios. Indeed, many of the presented information-processing features of decision switches could ultimately demonstrate a more flexible control of epigenetic differentiation

Complex Dynamics in Dedicated / Multifunctional Neural Networks and Chaotic Nonlinear Systems

We study complex behaviors arising in neuroscience and other nonlinear systems by combining dynamical systems analysis with modern computational approaches including GPU parallelization and unsupervised machine learning. To gain insights into the behaviors of brain networks and complex central pattern generators (CPGs), it is important to understand the dynamical principles regulating individual neurons as well as the basic structural and functional building blocks of neural networks. In the first section, we discuss how symbolic methods can help us analyze neural dynamics such as bursting, tonic spiking and chaotic mixed-mode oscillations in various models of individual neurons, the bifurcations that underlie transitions between activity types, as well as emergent network phenomena through synergistic interactions seen in realistic neural circuits, such as network bursting from non-intrinsic bursters. The second section is focused on the origin and coexistence of multistable rhythms in oscillatory neural networks of inhibitory coupled cells. We discuss how network connectivity and intrinsic properties of the cells affect the dynamics, and how even simple circuits can exhibit a variety of mono/multi-stable rhythms including pacemakers, half-center oscillators, multiple traveling-waves, fully synchronous states, as well as various chimeras. Our analyses can help generate verifiable hypotheses for neurophysiological experiments on central pattern generators. In the last section, we demonstrate the inter-disciplinary nature of this research through the applications of these techniques to identify the universal principles governing both simple and complex dynamics, and chaotic structure in diverse nonlinear systems. Using a classical example from nonlinear laser optics, we elaborate on the multiplicity and self-similarity of key organizing structures in 2D parameter space such as homoclinic and heteroclinic bifurcation curves, Bykov T-point spirals, and inclination flips. This is followed by detailed computational reconstructions of the spatial organization and 3D embedding of bifurcation surfaces, parametric saddles, and isolated closed curves (isolas). The generality of our modeling approaches could lead to novel methodologies and nonlinear science applications in biological, medical and engineering systems

Multi-Stability and Pattern-Selection in Oscillatory Networks with Fast Inhibition and Electrical Synapses

A model or hybrid network consisting of oscillatory cells interconnected by inhibitory and electrical synapses may express different stable activity patterns without any change of network topology or parameters, and switching between the patterns can be induced by specific transient signals. However, little is known of properties of such signals. In the present study, we employ numerical simulations of neural networks of different size composed of relaxation oscillators, to investigate switching between in-phase (IP) and anti-phase (AP) activity patterns. We show that the time windows of susceptibility to switching between the patterns are similar in 2-, 4- and 6-cell fully-connected networks. Moreover, in a network (N = 4, 6) expressing a given AP pattern, a stimulus with a given profile consisting of depolarizing and hyperpolarizing signals sent to different subpopulations of cells can evoke switching to another AP pattern. Interestingly, the resulting pattern encodes the profile of the switching stimulus. These results can be extended to different network architectures. Indeed, relaxation oscillators are not only models of cellular pacemakers, bursting or spiking, but are also analogous to firing-rate models of neural activity. We show that rules of switching similar to those found for relaxation oscillators apply to oscillating circuits of excitatory cells interconnected by electrical synapses and cross-inhibition. Our results suggest that incoming information, arriving in a proper time window, may be stored in an oscillatory network in the form of a specific spatio-temporal activity pattern which is expressed until new pertinent information arrives

Stability Analysis of Phase-Locked Bursting in Inhibitory Neuron Networks

Networks of neurons, which form central pattern generators (CPGs), are important for controlling animal behaviors. Of special interest are configurations or CPG motifs composed of reciprocally inhibited neurons, such as half-center oscillators (HCOs). Bursting rhythms of HCOs are shown to include stable synchrony or in-phase bursting. This in-phase bursting can co-exist with anti-phase bursting, commonly expected as the single stable state in HCOs that are connected with fast non-delayed synapses. The finding contrasts with the classical view that reciprocal inhibition has to be slow or time-delayed to synchronize such bursting neurons. Phase-locked rhythms are analyzed via Lyapunov exponents estimated with variational equations, and through the convergence rates estimated with Poincar\\u27e return maps. A new mechanism underlying multistability is proposed that is based on the spike interactions, which confer a dual property on the fast non-delayed reciprocal inhibition; this reveals the role of spikes in generating multiple co-existing phase-locked rhythms. In particular, it demonstrates that the number and temporal characteristics of spikes determine the number and stability of the multiple phase-locked states in weakly coupled HCOs. The generality of the multistability phenomenon is demonstrated by analyzing diverse models of bursting networks with various inhibitory synapses; the individual cell models include the reduced leech heart interneuron, the Sherman model for pancreatic beta cells, the Purkinje neuron model and Fitzhugh-Rinzel phenomenological model. Finally, hypothetical and experiment-based CPGs composed of HCOs are investigated. This study is relevant for various applications that use CPGs such as robotics, prosthetics, and artificial intelligence

Neural dynamics of perceptual competition

This research aims to understand the neural dynamics and mechanisms underlying perceptual bistability. In perceptual rivalry, ambiguous sensory information leads to dynamic changes in the perceptual interpretation of fixed stimuli. This phenomenon occurs when participants receive sensory stimuli that support two or more distinct interpretations; this results in spontaneous alternations between possible perceptual interpretations. Perceptual rivalry has been widely studied across different sensory modalities including vision, audition, and to a limited extent, in the tactile domain. Common features of perceptual rivalry across various ambiguous visual and auditory paradigms characterise the randomness of switching times and their dependence on input strength manipulations (Levelt's propositions). Binocular rivalry occurs when the two eyes are presented with incompatible stimuli and perception alternates between these two stimuli. This phenomenon has been investigated in two types of experiments: 1) Traditional experiments where the stimulus is fixed, 2) Eye-swap experiments in which stimulus periodically swaps between eyes many times per second~\citep{logothetis1996rivalling}. In spite of the rapid swapping between eyes, perception can be stable for many seconds with specific stimulus parameter configurations. Wilson introduced a two-stage, hierarchical model to explain both types of experiments~\citep{wilson2003computationala}. Wilson's model and other rivalry models have been only studied with bifurcation analysis for fixed inputs and different dynamical behaviours that can occur with periodic forcing have yet to be explored. Here I report 1) a more complete description of the complex dynamics in the unforced Wilson model, 2) a bifurcation analysis with periodic forcing. Previously, bifurcation analysis of the Wilson model with fixed inputs has revealed three main types of dynamical behaviours: Winner-take-all (WTA), Rivalry oscillations (RIV), Simultaneous activity (SIM). The results presented here reveal richer dynamics including mixed-mode oscillations (MMOs) and period-doubling cascade which corresponds to low amplitude WTA (LAWTA) oscillations. On the other hand, studying rivalry models with numerical continuation shows that periodic forcing with high frequency (e.g. 18 Hz, known as flicker) modulates the three main types of behaviours that occur with fixed inputs by the forcing frequency (WTA-Mod, RIV-Mod, SIM-Mod). However, the dynamical behaviour will be different with low frequency periodic forcing (around 1.5Hz, so-called swap), and in addition to WTA-Mod and SIM-Mod, cycle skipping and multi-cycle skipping behaviour exist, which can also lead to chaotic dynamics. This research provides a framework for either assessing binocular rivalry models for consistency checks against empirical results, or for better understanding neural dynamics and the mechanisms necessary to implement a minimal binocular rivalry model.\\ At present, it remains unclear whether the general characteristics of perceptual rivalry are preserved with tactile stimuli. I introduce a simple tactile stimulus capable of generating perceptual rivalry and explore whether general features of perceptual rivalry from other modalities extend to the tactile domain. In these experiments, vibrotactile stimuli consisted of anti-phase sequences of high and low intensity high-frequency pulses, each followed by a silent interval, delivered to the right and left index fingers. Participants perceived the stimulus as either one simultaneous pattern of vibration on both hands (SIM), or patterns of vibration that jumped from one hand to the other hand, giving a sensation of apparent movement (AM). For long stimulus presentations, perception switches back and forth between these two percepts. Furthermore, my results confirm that Levelt's proposition II extends to tactile bistability, and that the stochastic characteristics of irregular perceptual alternations agree with non-tactile modalities. An analysis of correlations between subsequent perceptual phases reveals a significant positive correlation at lag 1 (as found in visual bistability), and a negative correlation for lag 2 (in contrast with visual bistability). In this study, a mathematical model of tactile rivalry is developed that focuses on accurately reproducing the dynamics of the perceptual alternations. The model of tactile rivalry presented here consists of two processing stages; first stage for producing perceptual alternations; and a second stage for encoding the percept types (SIM and AM). Putative neural populations of the first stage could be located early in the somatosensory pathway at brainstem nuclei, and the neural populations of the second stage could be located within area 3b of the primary somatosensory cortex, based on excitatory and lagged inhibitory components of their receptive fields. The powerful combination of bifurcation analysis along with optimisation tools have been used to tune certain features of the model, resulting in a good qualitative and quantitative match to my experimental data. As well as capturing the dynamical characteristics specific to the perceptual interpretations in tactile rivalry, the model presented here is able to produce the general characteristics of perceptual rivalry including Levelt's proposition, short-tailed skewness of reversal time distributions and a scaling property of this distribution's first three moments.Engineering and Physical Sciences Research Council (EPSRC

Gene Networks of Fully Connected Triads with Complete Auto-Activation Enable Multistability and Stepwise Stochastic Transitions

abstract: Fully-connected triads (FCTs), such as the Oct4-Sox2-Nanog triad, have been implicated as recurring transcriptional motifs embedded within the regulatory networks that specify and maintain cellular states. To explore the possible connections between FCT topologies and cell fate determinations, we employed computational network screening to search all possible FCT topologies for multistability, a dynamic property that allows the rise of alternate regulatory states from the same transcriptional network. The search yielded a hierarchy of FCTs with various potentials for multistability, including several topologies capable of reaching eight distinct stable states. Our analyses suggested that complete auto-activation is an effective indicator for multistability, and, when gene expression noise was incorporated into the model, the networks were able to transit multiple states spontaneously. Different levels of stochasticity were found to either induce or disrupt random state transitioning with some transitions requiring layovers at one or more intermediate states. Using this framework we simulated a simplified model of induced pluripotency by including constitutive overexpression terms. The corresponding FCT showed random state transitioning from a terminal state to the pluripotent state, with the temporal distribution of this transition matching published experimental data. This work establishes a potential theoretical framework for understanding cell fate determinations by connecting conserved regulatory modules with network dynamics. Our results could also be employed experimentally, using established developmental transcription factors as seeds, to locate cell lineage specification networks by using auto-activation as a cipher.The article is published at http://journals.plos.org/plosone/article?id=10.1371/journal.pone.010287

Requirements for efficient cell-type proportioning: regulatory timescales, stochasticity and lateral inhibition

The proper functioning of multicellular organisms requires the robust establishment of precise proportions between distinct cell-types. This developmental differentiation process typically involves intracellular regulatory and stochastic mechanisms to generate cell-fate diversity as well as intercellular mechanisms to coordinate cell-fate decisions at tissue level. We thus surmise that key insights about the developmental regulation of cell-type proportion can be captured by the modeling study of clustering dynamics in population of inhibitory-coupled noisy bistable systems. This general class of dynamical system is shown to exhibit a very stable two-cluster state, but also frustrated relaxation, collective oscillations or steady-state hopping which prevents from timely and reliably reaching a robust and well-proportioned clustered state. To circumvent these obstacles or to avoid fine-tuning, we highlight a general strategy based on dual-time positive feedback loops, such as mediated through transcriptional versus epigenetic mechanisms, which improves proportion regulation by coordinating early and flexible lineage priming with late and firm commitment. This result sheds new light on the respective and cooperative roles of multiple regulatory feedback, stochasticity and lateral inhibition in developmental dynamics

Noise-activated barrier crossing in multi-attractor spiking networks

Noise-activated transitions between coexisting attractors are investigated in a chaotic spiking network. At low noise level, attractor hopping consists of discrete bifurcation events that conserve the memory of initial conditions. When the escape probability becomes comparable to the intra-basin hopping probability, the lifetime of attractors is given by a detailed balance where the less coherent attractors act as a sink for the more coherent ones. In this regime, the escape probability follows an activation law allowing us to assign pseudo-activation energies to limit cycle attractors. These pseudo-energies introduce a useful metric for evaluating the resilience of biological rhythms to perturbations