A group-theoretic approach to rings of coupled biological oscillators

In this paper, a general approach for studying rings of coupled biological oscillators is presented. This approach, which is group-theoretic in nature, is based on the finding that symmetric ring networks of coupled non-linear oscillators possess generic patterns of phaselocked oscillations. The associated analysis is independent of the mathematical details of the oscillators' intrinsic dynamics and the nature of the coupling between them. The present approach thus provides a framework for distinguishing universal dynamic behaviour from that which depends upon further structure. In this study, the typical oscillation patterns for the general case of a symmetric ring of n coupled non-linear oscillators and the specific cases of three- and five-membered rings are considered. Transitions between different patterns of activity are modelled as symmetry-breaking bifurcations. The effects of one-way coupling in a ring network and the differences between discrete and continuous systems are discussed. The theoretical predictions for symmetric ring networks are compared with physiological observations and numerical simulations. This comparison is limited to two examples: neuronal networks and mammalian intestinal activity. The implications of the present approach for the development of physiologically meaningful oscillator models are discussed.     

Hexapodal gaits and coupled nonlinear oscillator models

The general, model-independent features of different networks of six symmetrically coupled nonlinear oscillators are investigated. These networks are considered as possible models for locomotor central pattern generators (CPGs) in insects. Numerical experiments with a specific oscillator network model are briefly described. It is shown that some generic phase-locked oscillation-patterns for various systems of six symmetrically coupled nonlinear oscillators correspond to the common forward-walking gaits adopted by insects. It is also demonstrated that transitions observed in insect gaits can be modelled as standard symmetry-breaking bifurcations occurring in such systems. The present analysis, which leads to a natural classification of hexapodal gaits by symmetry and to natural sequences of gait bifurcations, relates observed gaits to the overall organizational structure of the underlying CPG. The implications of the present results for the development of simplified control systems for hexapodal walking robots are discussed.     

Pulse coupled oscillators and the phase resetting curve

Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.     

Hard-wired central pattern generators for quadrupedal locomotion

Animal locomotion is generated and controlled, in part, by a central pattern generator (CPG), which is an intraspinal network of neurons capable of producing rhythmic output. In the present work, it is demonstrated that a hard-wired CPG model, made up of four coupled nonlinear oscillators, can produce multiple phase-locked oscillation patterns that correspond to three common quadrupedal gaits — the walk, trot, and bound. Transitions between the different gaits are generated by varying the network's driving signal and/or by altering internal oscillator parameters. The above in numero results are obtained without changing the relative strengths or the polarities of the system's synaptic interconnections, i.e., the network maintains an invariant coupling architecture. It is also shown that the ability of the hard-wired CPG network to produce and switch between multiple gait patterns is a model-independent phenomenon, i.e., it does not depend upon the detailed dynamics of the component oscillators and/or the nature of the inter-oscillator coupling. Three different neuronal oscillator models — the Stein neuronal model, the Van der Pol oscillator, and the FitzHugh-Nagumo model -and two different coupling schemes are incorporated into the network without impeding its ability to produce the three quadrupedal gaits and the aforementioned gait transitions.     

The logical analysis of continuous, non-linear biochemical control networks

We propose a mapping to study the qualitative properties of continuous biochemical control networks which are invariant to the parameters used to describe the networks but depend only on the logical structure of the networks. For the networks, we are able to place a lower limit on the number of steady states and strong restrictions on the phase relations between components on cycles and transients. The logical structure and the dynamical behavior for a number of simple systems of biological interest, the feedback (predator-prey) oscillator, the bistable switch, the phase dependent switch, are discussed. We discuss the possibility that these techniques may be extended to study the dynamics of large many component systems.     

Robustness of Oscillatory Behavior in Correlated Networks

Understanding network robustness against failures of network units is useful for preventing large-scale breakdowns and damages in real-world networked systems. The tolerance of networked systems whose functions are maintained by collective dynamical behavior of the network units has recently been analyzed in the framework called dynamical robustness of complex networks. The effect of network structure on the dynamical robustness has been examined with various types of network topology, but the role of network assortativity, or degree–degree correlations, is still unclear. Here we study the dynamical robustness of correlated (assortative and disassortative) networks consisting of diffusively coupled oscillators. Numerical analyses for the correlated networks with Poisson and power-law degree distributions show that network assortativity enhances the dynamical robustness of the oscillator networks but the impact of network disassortativity depends on the detailed network connectivity. Furthermore, we theoretically analyze the dynamical robustness of correlated bimodal networks with two-peak degree distributions and show the positive impact of the network assortativity.     

Heartbeat control in the medicinal leech: A model system for understanding the origin,coordination, and modulation of rhythmic motor patterns

We have analyzed in detail the neuronal network that generates heartbeat in the leech. Reciprocally inhibitory pairs of heart interneurons form oscillators that pace the heartbeat rhythm. Other heart interneurons coordinate these oscillators. These coordinating interneurons, along with the oscillator interneurons, form an eight-cell timing oscillator network for heartbeat. Still other interneurons, along with the oscillator interneurons, inhibit heart motor neurons, sculpting their activity into rhythmic bursts. Critical switch interneurons interface between the oscillator interneurons and the other premotor interneurons to produce two alternating coordination states of the motor neurons. The periods of the oscillator interneurons are modulated by endogenous RFamide neuropeptides. We have explored the ionic currents and graded and spike-mediated synaptic transmission that promote oscillation in the oscillator interneurons and have incorporated these data into a conductance-based computer model. This model has been of considerable predictive value and has led to new insights into how reciprocally inhibitory neurons produce oscillation. We are now in a strong position to expand this model upward, to encompass the entire heartbeat network, horizontally, to elucidate the mechanisms of FMRFamide modulation, and downward, to incorporate cellular morphology. By studying the mechanisms of motor pattern formation in the leech, using modeling studies in conjunction with parallel physiological experiments, we can contribute to a deeper understanding of how rhythmic motor acts are generated, coordinated, modulated, and reconfigured at the level of networks, cells, ionic currents, and synapses. © 1995 John Wiley & Sons, Inc.     

Local network parameters can affect inter-network phase lags in central pattern generators

Weakly coupled phase oscillators and strongly coupled relaxation oscillators have different mechanisms for creating stable phase lags. Many oscillations in central pattern generators combine features of each type of coupling: local networks composed of strongly coupled relaxation oscillators are weakly coupled to similar local networks. This paper analyzes the phase lags produced by this combination of mechanisms and shows how the parameters of a local network, such as the decay time of inhibition, can affect the phase lags between the local networks. The analysis is motivated by the crayfish central pattern generator used for swimming, and uses techniques from geometrical singular perturbation theory.     

Self-consistent proteomic field theory of stochastic gene switches

We present a self-consistent field approximation approach to the problem of the genetic switch composed of two mutually repressing/activating genes. The protein and DNA state dynamics are treated stochastically and on an equal footing. In this approach the mean influence of the proteomic cloud created by one gene on the action of another is self-consistently computed. Within this approximation a broad range of stochastic genetic switches may be solved exactly in terms of finding the probability distribution and its moments. A much larger class of problems, such as genetic networks and cascades, also remain exactly solvable with this approximation. We discuss, in depth, certain specific types of basic switches used by biological systems and compare their behavior to the expectation for a deterministic switch.     

Coupling of a slow and a fast oscillator can generate bursting

A general mechanism underlying bursting is proposed and described. It consists of two coupled nonlinear oscillators with different frequencies, where the slower oscillator alternatively switches the faster one on and off. This mechanism is shown to work in an extended Bonhoefer-van der Pol oscillator as well as in a modified version of the Hodgkin-Huxley equations.     

Modeling genetic switches with positive feedback loops

In this paper, we develop a new methodology to design synthetic genetic switch networks with multiple genes and time delays, by using monotone dynamical systems. We show that the networks with only positive feedback loops have no stable oscillation but stable equilibria whose stability is independent of the time delays. In other words, such systems have ideal properties for switch networks and can be designed without consideration of time delays, because the systems can be reduced from functional spaces to Euclidian spaces. Therefore, we can ensure that the designed switches function correctly even with uncertain delays. We first prove the basic properties of the genetic networks composed of only positive feedback loops, and then propose a procedure to design the switches, which drastically simplifies analysis of the switches and makes theoretical analysis and design tractable even for large-scaled systems. Finally, to demonstrate our theoretical results, we show biologically plausible examples by designing a synthetic genetic switch with experimentally well investigated lacI, tetR, and cI genes for numerical simulation.     

Neural rate equations for bursting dynamics derived from conductance-based equations

A method of obtaining rate equations from conductance-based equations is developed and applied to fast-spiking and bursting neocortical neurons. It involves splitting systems of conductance-based equations into fast and slow subsystems, and averaging the effects of fast terms that drive the slowly varying quantities by showing that their average is closely proportional to the firing rate. The dependence of the firing rate on the injected current is then approximated in the analysis. The resulting behavior of the slow variables is then substituted back into the fast equations, with the further approximation of replacing the fast voltages in these terms by effective values. For bursting neurons the method yields two coupled limit-cycle oscillators: a self-exciting oscillator for the slow variables that commences limit-cycle oscillations at a critical current and modulates a fast spike-generating oscillator, thereby leading to slowly modulated bursts with a group of spikes in each burst. The dynamics of these coupled oscillators are then verified against those of the conductance-based equations. Finally, it is shown how to place the results in a form suitable for use in mean-field equations for neural population dynamics.     

Modeling the dual pacemaker system of the tau mutant hamster

Circadian pacemakers in many animals are compound. In rodents, a two-oscillator model of the pacemaker composed of an evening (E) and a morning (M) oscillator has been proposed based on the phenomenon of "splitting" and bimodal activity peaks. The authors describe computer simulations of the pacemaker in tau mutant hamsters viewed as a system of mutually coupled E and M oscillators. These mutant animals exhibit normal type 1 PRCs when released into DD but make a transition to a type 0 PRC when held for many weeks in DD. The two-oscillator model describes particularly well some recent behavioral experiments on these hamsters. The authors sought to determine the relationships between oscillator amplitude, period, PRC, and activity duration through computer simulations. Two complementary approaches proved useful for analyzing weakly coupled oscillator systems. The authors adopted a "distinct oscillators" view when considering the component E and M oscillators and a "system" view when considering the system as a whole. For strongly coupled systems, only the system view is appropriate. The simulations lead the authors to two primary conjectures: (1) the total amplitude of the pacemaker system in tau mutant hamsters is less than in the wild-type animals, and (2) the coupling between the unit E and M oscillators is weakened during continuous exposure of hamsters to DD. As coupling strength decreases, activity duration (alpha) increases due to a greater phase difference between E and M. At the same time, the total amplitude of the system decreases, causing an increase in observable PRC amplitudes. Reduced coupling also increases the relative autonomy of the unit oscillators. The relatively autonomous phase shifts of E and M oscillators can account for both immediate compression and expansion of activity bands in tau mutant and wild-type hamsters subjected to light pulses.     

Coupling oscillations and switches in genetic networks

Switches (bistability) and oscillations (limit cycle) are omnipresent in biological networks. Synthetic genetic networks producing bistability and oscillations have been designed and constructed experimentally. However, in real biological systems, regulatory circuits are usually interconnected and the dynamics of those complex networks is often richer than the dynamics of simple modules. Here we couple the genetic Toggle switch and the Repressilator, two prototypic systems exhibiting bistability and oscillations, respectively. We study two types of coupling. In the first type, the bistable switch is under the control of the oscillator. Numerical simulation of this system allows us to determine the conditions under which a periodic switch between the two stable steady states of the Toggle switch occurs. In addition we show how birhythmicity characterized by the coexistence of two stable small-amplitude limit cycles, can easily be obtained in the system. In the second type of coupling, the oscillator is placed under the control of the Toggleswitch. Numerical simulation of this system shows that this construction could for example be exploited to generate a permanent transition from a stable steady state to self-sustained oscillations (and vice versa) after a transient external perturbation. Those results thus describe qualitative dynamical behaviors that can be generated through the coupling of two simple network modules. These results differ from the dynamical properties resulting from interlocked feedback loops systems in which a given variable is involved at the same time in both positive and negative feedbacks. Finally the models described here may be of interest in synthetic biology, as they give hints on how the coupling should be designed to get the required properties.     

Synchronization of coupled nonidentical genetic oscillators

The study of the collective dynamics of synchronization among genetic oscillators is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. Genetic oscillators are biochemical networks, which can generally be modelled as nonlinear dynamic systems. We show in this paper that many genetic oscillators can be transformed into Lur'e form by exploiting the special structure of biological systems. By using a control theory approach, we provide a theoretical method for analysing the synchronization of coupled nonidentical genetic oscillators. Sufficient conditions for the synchronization as well as the estimation of the bound of the synchronization error are also obtained. To demonstrate the effectiveness of our theoretical results, a population of genetic oscillators based on the Goodwin model are adopted as numerical examples.     

Frequency and burst duration in oscillating neurons and two-cell networks

We study the relationship of injected current to oscillator period in single neurons and two-cell model networks formed by reciprocal inhibitory synapses. Using a Morris-Lecar-like model, we identify two qualitative types of oscillatory behavior for single model neurons. The classical oscillator behavior is defined as type A. Here the burst duration is relatively constant and the frequency increases with depolarization. For oscillator type B, the frequency first increases and then decreases when depolarized, due to the variable burst duration. Our simulations show that relatively modest changes in the maximal inward and outward conductances can move the oscillator from one type to another. Cultured stomatogastric ganglion neurons exhibit both A and B type behaviors and can switch between the two types with pharmacological manipulation. Our simulations indicate that the stability of a two-cell network with injected current can be extended with inhibitory coupling. In addition, two-cell networks formed from type A or type B oscillators behave differently from each other at lower synaptic strengths.     

"Rapid" rhythmic discharges of sympathetic nerves: sources, mechanisms of generation, and physiological relevance

Like virtually all other physiological control systems, the sympathetic nervous system controlling cardiovascular function is characterized by the presence of rhythmic activity. These include slow rhythms with frequencies at or below that of the respiration and rapid rhythms with frequencies at or above that of the heart beat. The rapid rhythms are the subject of this review. The specific questions entertained are as follows: (1) Are the rapid cardiac-related and 10-Hz rhythms inherent to central sympathetic networks, or are they imposed on sympathetic nerve discharge (SND) by extrinsic periodic inputs? (2) Does basal SND arise from an anatomically circumscribed "vasomotor center" composed of pacemaker neurons in the rostral ventrolateral medulla or from an anatomically distributed network oscillator composed of different types of brainstem neurons, none of which necessarily have intrinsic pacemaker properties? (3) Are the rapid rhythms generated by single circuits or by systems of coupled oscillators, each with a separate target? (4) Are the rapid rhythms in SND simply by-products of the sympathetic generating mechanisms, or do they subserve selective and special functions, such as the formulation of differential patterns of spinal sympathetic outflow that support particular behaviors? The controversial aspects of these issues and the state-of-the-art analytical methods used to study them are stressed in this review.     

Transient dynamics of genetic regulatory networks

We present an approximation scheme for deriving reaction rate equations of genetic regulatory networks. This scheme predicts the timescales of transient dynamics of such networks more accurately than does standard quasi-steady state analysis by introducing prefactors to the ODEs that govern the dynamics of the protein concentrations. These prefactors render the ODE systems slower than their quasi-steady state approximation counterparts. We introduce the method by examining a positive feedback gene regulatory network, and show how the transient dynamics of this network are more accurately modeled when the prefactor is included. Next, we examine the repressilator, a genetic oscillator, and show that the period, amplitude, and bifurcation diagram defining the onset of the oscillations are better estimated by the prefactor method. Finally, we examine the consequences of the method to the dynamics of reduced models of the phage lambda switch, and show that the switching times between the two states is slowed by the presence of the prefactor that arises from protein multimerization and DNA binding.