Nonlinear behavior of sinusoidally forced pyloric pacemaker neurons

U(1) dynamics in neuronal activities

Neurons convert external stimuli into action potentials, or spikes, and encode the contained information into the biological nervous system. Despite the complexity of neurons and the synaptic interactions in between, rate models are often adapted to describe neural encoding with modest success. However, it is not clear whether the firing rate, the reciprocal of the time interval between spikes, is sufficient to capture the essential features for the neuronal dynamics. Going beyond the usual relaxation dynamics in Ginzburg-Landau theory for statistical systems, we propose that neural activities can be captured by the U(1) dynamics, integrating the action potential and the "phase" of the neuron together. The gain function of the Hodgkin-Huxley neuron and the corresponding dynamical phase transitions can be described within the U(1) neuron framework. In addition, the phase dependence of the synaptic interactions is illustrated and the mapping to the Kinouchi-Copelli neuron is established. It suggests that the U(1) neuron is the minimal model for single-neuron activities and serves as the building block of the neuronal network for information processing.

Mode locking in periodically forced gradient frequency neural networks

We study mode locking in a canonical model of gradient frequency neural networks under periodic forcing. The canonical model is a generic mathematical model for a network of nonlinear oscillators tuned to a range of distinct frequencies. It is mathematically more tractable than biological neuron models and allows close analysis of mode-locking behaviors. Here we analyze individual modes of synchronization for a periodically forced canonical model and present a complete set of driven behaviors for all parameter regimes available in the model. Using a closed-form approximation, we show that the Arnold tongue (i.e., locking region) for k:m synchronization gets narrower as k and m increase. We find that numerical simulations of the canonical model closely follow the analysis of individual modes when forcing is weak, but they deviate at high forcing amplitudes for which oscillator dynamics are simultaneously influenced by multiple modes of synchronization.

Visualization of currents in neural models with similar behavior and different conductance densities

Conductance-based models of neural activity produce large amounts of data that can be hard to visualize and interpret. We introduce visualization methods to display the dynamics of the ionic currents and to display the models’ response to perturbations. To visualize the currents’ dynamics, we compute the percent contribution of each current and display them over time using stacked-area plots. The waveform of the membrane potential and the contribution of each current change as the models are perturbed. To represent these changes over a range of the perturbation control parameter, we compute and display the distributions of these waveforms. We illustrate these procedures in six examples of bursting model neurons with similar activity but that differ as much as threefold in their conductance densities. These visualization methods provide heuristic insight into why individual neurons or networks with similar behavior can respond widely differently to perturbations.

The nervous system contains networks of neurons that generate electrical signals to communicate with each other and the rest of the body. Such electrical signals are due to the flow of ions into or out of the neurons via proteins known as ion channels. Neurons have many different kinds of ion channels that only allow specific ions to pass. Therefore, for a neuron to produce an electrical signal, the activities of several different ion channels need to be coordinated so that they all open and close at certain times.

Researchers have previously used data collected from various experiments to develop detailed models of electrical signals in neurons. These models incorporate information about how and when the ion channels may open and close, and can produce numerical simulations of the different ionic currents. However, it can be difficult to display the currents and observe how they change when several different ion channels are involved.

Alonso and Marder used simple mathematical concepts to develop new methods to display ionic currents in computational models of neurons. These tools use color to capture changes in ionic currents and provide insights into how the opening and closing of ion channels shape electrical signals.

The methods developed by Alonso and Marder could be adapted to display the behavior of biochemical reactions or other topics in biology and may, therefore, be useful to analyze data generated by computational models of many different types of cells. Additionally, these methods may potentially be used as educational tools to illustrate the coordinated opening and closing of ion channels in neurons and other fundamental principles of neuroscience that are otherwise hard to demonstrate.

Nonlinear resonances and multi-stability in simple neural circuits

This article describes a numerical procedure designed to tune the parameters of periodically driven dynamical systems to a state in which they exhibit rich dynamical behavior. This is achieved by maximizing the diversity of subharmonic solutions available to the system within a range of the parameters that define the driving. The procedure is applied to a problem of interest in computational neuroscience: a circuit composed of two interacting populations of neurons under external periodic forcing. Depending on the parameters that define the circuit, such as the weights of the connections between the populations, the response of the circuit to the driving can be strikingly rich and diverse. The procedure is employed to find circuits that, when driven by external input, exhibit multiple stable patterns of periodic activity organized in complex tuning diagrams and signatures of low dimensional chaos.

Noise effects on robust synchronization of a small pacemaker neuronal ensemble via nonlinear controller: electronic circuit design

In this paper, we report on the synchronization of a pacemaker neuronal ensemble constituted of an AB neuron electrically coupled to two PD neurons. By the virtue of this electrical coupling, they can fire synchronous bursts of action potential. An external master neuron is used to induce to the whole system the desired dynamics, via a nonlinear controller. Such controller is obtained by a combination of sliding mode and feedback control. The proposed controller is able to offset uncertainties in the synchronized systems. We show how noise affects the synchronization of the pacemaker neuronal ensemble, and briefly discuss its potential benefits in our synchronization scheme. An extended Hindmarsh-Rose neuronal model is used to represent a single cell dynamic of the network. Numerical simulations and Pspice implementation of the synchronization scheme are presented. We found that, the proposed controller reduces the stochastic resonance of the network when its gain increases.

Synergistic plasticity of intrinsic conductance and electrical coupling restores synchrony in an intact motor network

Motor neurons of the crustacean cardiac ganglion generate virtually identical, synchronized output despite the fact that each neuron uses distinct conductance magnitudes. As a result of this variability, manipulations that target ionic conductances have distinct effects on neurons within the same ganglion, disrupting synchronized motor neuron output that is necessary for proper cardiac function. We hypothesized that robustness in network output is accomplished via plasticity that counters such destabilizing influences. By blocking high-threshold K⁺ conductances in motor neurons within the ongoing cardiac network, we discovered that compensation both resynchronized the network and helped restore excitability. Using model findings to guide experimentation, we determined that compensatory increases of both G_A and electrical coupling restored function in the network. This is one of the first direct demonstrations of the physiological regulation of coupling conductance in a compensatory context, and of synergistic plasticity across cell- and network-level mechanisms in the restoration of output.

DOI:http://dx.doi.org/10.7554/eLife.16879.001

Neurons can communicate with each other by releasing chemicals called neurotransmitters, or by forming direct connections with each other known as gap junctions. These direct connections allow electrical impulses to flow from one neuron to another via pores in the membranes between the cells. Unlike communication via neurotransmitters, gap junctions are usually thought to be hard-wired and unchanging over the life of the animal.

Lane et al. recorded electrical activity in a network of neurons that generates rhythmic heart contractions in the Jonah crab. Neurons in this network usually all fire an electrical impulse at the same time, which is crucial to make sure that the whole heart contracts at the same time. The experiments show that drugs that block potassium channel pores in the membrane cause the neurons to fire too much and at different times to each other.

However, the network of neurons soon adapted to the changes caused by the drugs and returned to working as normal. Mimicking these changes in a computer model of the neuron network, together with experimental data, showed that changes to the gap junctions play a major role in restoring normal activity to the network.

The next step following on from this research is to understand how a network of neurons ‘senses’ that it is not working normally and changes its electrical activity.

DOI:http://dx.doi.org/10.7554/eLife.16879.002

Single electrode dynamic clamp with StdpC

Dynamic clamp is a powerful approach for electrophysiological investigations allowing researchers to introduce artificial electrical components into target neurons to simulate ionic conductances, chemical or electrotonic inputs or connections to other cells. Due to the rapidly changing and potentially large current injections during dynamic clamp, problematic voltage artifacts appear on the electrode used to inject dynamic clamp currents into a target neuron. Dynamic clamp experiments, therefore, typically use two separate electrodes in the same cell, one for recording membrane potential and one for injecting currents. The requirement for two independent electrodes has been a limiting factor for the use of dynamic clamp in applications where dual recordings of this kind are difficult or impossible to achieve. The recent development of an active electrode compensation (AEC) method has overcome some of these prior limitations, permitting artifact-free dynamic clamp experimentation with a single electrode. Here we describe an AEC method for the free dynamic clamp software StdpC. The AEC component of StdpC is the first such system implemented for the use of non-expert users and comes with a set of semi-automated configuration and calibration procedures that facilitate its use. We briefly introduce the AEC method and its implementation in StdpC and then validate it with an electronic model cell and in two different biological preparations.

Chaotic phase synchronization in bursting-neuron models driven by a weak periodic force

We investigate the entrainment of a neuron model exhibiting a chaotic spiking-bursting behavior in response to a weak periodic force. This model exhibits two types of oscillations with different characteristic time scales, namely, long and short time scales. Several types of phase synchronization are observed, such as 1:1 phase locking between a single spike and one period of the force and 1:l phase locking between the period of slow oscillation underlying bursts and l periods of the force. Moreover, spiking-bursting oscillations with chaotic firing patterns can be synchronized with the periodic force. Such a type of phase synchronization is detected from the position of a set of points on a unit circle, which is determined by the phase of the periodic force at each spiking time. We show that this detection method is effective for a system with multiple time scales. Owing to the existence of both the short and the long time scales, two characteristic phenomena are found around the transition point to chaotic phase synchronization. One phenomenon shows that the average time interval between successive phase slips exhibits a power-law scaling against the driving force strength and that the scaling exponent has an unsmooth dependence on the changes in the driving force strength. The other phenomenon shows that Kuramoto's order parameter before the transition exhibits stepwise behavior as a function of the driving force strength, contrary to the smooth transition in a model with a single time scale.

Interaction between telencephalic signals and respiratory dynamics in songbirds

The mechanisms by which telencephalic areas affect motor activities are largely unknown. They could either take over motor control from downstream motor circuits or interact with the intrinsic dynamics of these circuits. Both models have been proposed for telencephalic control of respiration during learned vocal behavior in birds. The interactive model postulates that simple signals from the telencephalic song control areas are sufficient to drive the nonlinear respiratory network into producing complex temporal sequences. We tested this basic assumption by electrically stimulating telencephalic song control areas and analyzing the resulting respiratory patterns in zebra finches and in canaries. We found strong evidence for interaction between the rhythm of stimulation and the intrinsic respiratory rhythm, including naturally emerging subharmonic behavior and integration of lateralized telencephalic input. The evidence for clear interaction in our experimental paradigm suggests that telencephalic vocal control also uses a similar mechanism. Furthermore, species differences in the response of the respiratory system to stimulation show parallels to differences in the respiratory patterns of song, suggesting that the interactive production of respiratory rhythms is manifested in species-specific specialization of the involved circuitry.

Dynamic clamp with StdpC software

Dynamic clamp is a powerful method that allows the introduction of artificial electrical components into target cells to simulate ionic conductances and synaptic inputs. This method is based on a fast cycle of measuring the membrane potential of a cell, calculating the current of a desired simulated component using an appropriate model and injecting this current into the cell. Here we present a dynamic clamp protocol using free, fully integrated, open-source software (StdpC, for spike timing-dependent plasticity clamp). Use of this protocol does not require specialist hardware, costly commercial software, experience in real-time operating systems or a strong programming background. The software enables the configuration and operation of a wide range of complex and fully automated dynamic clamp experiments through an intuitive and powerful interface with a minimal initial lead time of a few hours. After initial configuration, experimental results can be generated within minutes of establishing cell recording.

Encoding of spatio-temporal input characteristics by a CA1 pyramidal neuron model

The in vivo activity of CA1 pyramidal neurons alternates between regular spiking and bursting, but how these changes affect information processing remains unclear. Using a detailed CA1 pyramidal neuron model, we investigate how timing and spatial arrangement variations in synaptic inputs to the distal and proximal dendritic layers influence the information content of model responses. We find that the temporal delay between activation of the two layers acts as a switch between excitability modes: short delays induce bursting while long delays decrease firing. For long delays, the average firing frequency of the model response discriminates spatially clustered from diffused inputs to the distal dendritic tree. For short delays, the onset latency and inter-spike-interval succession of model responses can accurately classify input signals as temporally close or distant and spatially clustered or diffused across different stimulation protocols. These findings suggest that a CA1 pyramidal neuron may be capable of encoding and transmitting presynaptic spatiotemporal information about the activity of the entorhinal cortex-hippocampal network to higher brain regions via the selective use of either a temporal or a rate code.

Pyramidal neurons in the hippocampus are crucially involved in learning and memory functions, but the ways in which they contribute to the processing of sensory inputs and their internal representation remain mostly unclear. The principal neurons of the CA1 region of the hippocampus are surrounded by at least 21 different types of interneurons. This feature, together with the fact that CA1 pyramidal dendrites associate two major glutamatergic inputs arriving from the entorhinal cortex, makes it laborious to track the ‘how’ and ‘what’ of synaptic integration. The present study tries to shed light on the ‘what’, that is, the information content of the CA1 discharge pattern. Using a detailed biophysical CA1 neuron model, we show that the output of the model neuron contains spatial and temporal features of the incoming synaptic input. This information lies in the temporal pattern of the inter-spike intervals produced during the bursting activity which is induced by the temporal coincidence of the two activated synaptic streams. Our findings suggest that CA1 pyramidal neurons may be capable of capturing features of the ongoing network activity via the use of a temporal code for information transfer.

Robust microcircuit synchronization by inhibitory connections

Microcircuits in different brain areas share similar architectural and biophysical properties with compact motor networks known as central pattern generators (CPGs). Consequently, CPGs have been suggested as valuable biological models for understanding of microcircuit dynamics and particularly, their synchronization. We use a well known compact motor network, the lobster pyloric CPG to study principles of intercircuit synchronization. We couple separate pyloric circuits obtained from two animals via artificial synapses and observe how their synchronization depends on the topology and kinetic parameters of the computer-generated synapses. Stable in-phase synchronization appears when electrically coupling the pacemaker groups of the two networks, but reciprocal inhibitory connections produce more robust and regular cooperative activity. Contralateral inhibitory connections offer effective synchronization and flexible setting of the burst phases of the interacting networks. We also show that a conductance-based mathematical model of the coupled circuits correctly reproduces the observed dynamics illustrating the generality of the phenomena.

Neuronal synchrony: peculiarity and generality

Synchronization in neuronal systems is a new and intriguing application of dynamical systems theory. Why are neuronal systems different as a subject for synchronization? (1) Neurons in themselves are multidimensional nonlinear systems that are able to exhibit a wide variety of different activity patterns. Their "dynamical repertoire" includes regular or chaotic spiking, regular or chaotic bursting, multistability, and complex transient regimes. (2) Usually, neuronal oscillations are the result of the cooperative activity of many synaptically connected neurons (a neuronal circuit). Thus, it is necessary to consider synchronization between different neuronal circuits as well. (3) The synapses that implement the coupling between neurons are also dynamical elements and their intrinsic dynamics influences the process of synchronization or entrainment significantly. In this review we will focus on four new problems: (i) the synchronization in minimal neuronal networks with plastic synapses (synchronization with activity dependent coupling), (ii) synchronization of bursts that are generated by a group of nonsymmetrically coupled inhibitory neurons (heteroclinic synchronization), (iii) the coordination of activities of two coupled neuronal networks (partial synchronization of small composite structures), and (iv) coarse grained synchronization in larger systems (synchronization on a mesoscopic scale).

Probing the dynamics of identified neurons with a data-driven modeling approach

In controlling animal behavior the nervous system has to perform within the operational limits set by the requirements of each specific behavior. The implications for the corresponding range of suitable network, single neuron, and ion channel properties have remained elusive. In this article we approach the question of how well-constrained properties of neuronal systems may be on the neuronal level. We used large data sets of the activity of isolated invertebrate identified cells and built an accurate conductance-based model for this cell type using customized automated parameter estimation techniques. By direct inspection of the data we found that the variability of the neurons is larger when they are isolated from the circuit than when in the intact system. Furthermore, the responses of the neurons to perturbations appear to be more consistent than their autonomous behavior under stationary conditions. In the developed model, the constraints on different parameters that enforce appropriate model dynamics vary widely from some very tightly controlled parameters to others that are almost arbitrary. The model also allows predictions for the effect of blocking selected ionic currents and to prove that the origin of irregular dynamics in the neuron model is proper chaoticity and that this chaoticity is typical in an appropriate sense. Our results indicate that data driven models are useful tools for the in-depth analysis of neuronal dynamics. The better consistency of responses to perturbations, in the real neurons as well as in the model, suggests a paradigm shift away from measuring autonomous dynamics alone towards protocols of controlled perturbations. Our predictions for the impact of channel blockers on the neuronal dynamics and the proof of chaoticity underscore the wide scope of our approach.

StdpC: a modern dynamic clamp

With the advancement of computer technology many novel uses of dynamic clamp have become possible. We have added new features to our dynamic clamp software StdpC ("Spike timing-dependent plasticity Clamp") allowing such new applications while conserving the ease of use and installation of the popular earlier Dynclamp 2/4 package. Here, we introduce the new features of a waveform generator, freely programmable Hodgkin-Huxley conductances, learning synapses, graphic data displays, and a powerful scripting mechanism and discuss examples of experiments using these features. In the first example we built and 'voltage clamped' a conductance based model cell from a passive resistor-capacitor (RC) circuit using the dynamic clamp software to generate the voltage-dependent currents. In the second example we coupled our new spike generator through a burst detection/burst generation mechanism in a phase-dependent way to a neuron in a central pattern generator and dissected the subtle interaction between neurons, which seems to implement an information transfer through intraburst spike patterns. In the third example, making use of the new plasticity mechanism for simulated synapses, we analyzed the effect of spike timing-dependent plasticity (STDP) on synchronization revealing considerable enhancement of the entrainment of a post-synaptic neuron by a periodic spike train. These examples illustrate that with modern dynamic clamp software like StdpC, the dynamic clamp has developed beyond the mere introduction of artificial synapses or ionic conductances into neurons to a universal research tool, which might well become a standard instrument of modern electrophysiology.

A Network of Electronic Neural Oscillators Reproduces the Dynamics of the Periodically Forced Pyloric Pacemaker Group

Low-dimensional oscillators are a valuable model for the neuronal activity of isolated neurons. When coupled, the self-sustained oscillations of individual free oscillators are replaced by a collective network dynamics. Here, dynamical features of such a network, consisting of three electronic implementations of the Hindmarsh-Rose mathematical model of bursting neurons, are compared to those of a biological neural motor system, specifically the pyloric CPG of the crustacean stomatogastric nervous system. We demonstrate that the network of electronic neurons exhibits realistic synchronized bursting behavior comparable to the biological system. Dynamical properties were analyzed by injecting sinusoidal currents into one of the oscillators. The temporal bursting structure of the electronic neurons in response to periodic stimulation is shown to bear a remarkable resemblance to that observed in the corresponding biological network. These findings provide strong evidence that coupled nonlinear oscillators realistically reproduce the network dynamics experimentally observed in assemblies of several neurons.

Analysis and simulation of gain control and precision in crayfish visual interneurons

Impulse trains in sustaining and dimming fibers of crayfish optic lobe (in situ) were elicited with sinusoidal extrinsic current and sine-wave illumination. Extrinsic currents and currents derived from postsynaptic potentials (PSPs) were used to compute the time course of the spike train with an adaptive integrate-and-fire model. The neurons exhibit variations in gain and spike timing precision related to the frequency of stimulation. These phenomena are influenced by spike-frequency adaptation and nonlinearities in the PSP. Dimming fibers exhibit relatively strong spike-frequency adaptation and an associated increase in gain with the frequency of sinusoidal extrinsic current and sine-wave illumination. The dimming fiber IPSP promotes spike train rectification, and rectification contributes to spike timing precision. Sustaining fibers exhibit weaker spike-frequency adaptation and the gain of the current-elicited response is less sensitive to stimulus frequency. The sustaining fiber excitatory PSP, however, exhibits a strong frequency-dependent nonlinearity that influences the frequency response. Spike timing precision is a function of stimulus frequency in all cells and it is enhanced by rectification of the discharge and/or resonance. In rectified responses the jitter in spike times is closely related to the variance in the times the membrane potential reaches spike threshold. These gain and spike timing results are well approximated by the simulated responses. Because the nonlinearity of the sustaining fiber PSP entails a high rate of depolarization, the PSP can increase the precision of spike timing by 10- to 100-fold compared with the response to pure sine-wave stimuli. This enhanced precision has implications for crayfish oculomotor reflexes that are driven by sustaining fibers and highly sensitive to impulse timing during transient excitation.

Motor-unit coherence during isometric contractions is greater in a hand muscle of older adults

The purpose of this study was to quantify the strength of motor-unit coherence from the first dorsal interosseus muscle in young and old adults using data obtained in a previous study, where no differences in motor-unit synchronization between the two groups were observed. The strength of motor-unit coherence was quantified from 47 motor-unit pairs in 11 young adults (age 24.1 +/- 4.1 yrs) and from 48 motor-unit pairs in 14 old adults (age 70.4 +/- 5.9 yrs). The strength of motor-unit coherence was greater in old adults, particularly at low frequencies of 5-9 Hz (85% greater in old adults at 5 Hz). In addition, the older adults expressed an extra oscillation at approximately 12-13 Hz that was not present in the young subjects. These data demonstrate that common oscillatory inputs to motor neurons (motor-unit coherence) are enhanced in older adults despite no age-related difference in the strength of shared inputs (synchronization). Furthermore, the data emphasize that measures of motor-unit synchronization and coherence highlight different features of the same common input, and a coherence analysis may be a more sensitive tool to characterize shared input to motor neurons.

Amplitude and frequency dependence of spike timing: implications for dynamic regulation

The spike-time reliability of motoneurons in the Aplysia buccal motor ganglion was studied as a function of the frequency content and the relative amplitude of the fluctuations in the neuronal input, calculated as the coefficient of variation (CV). Measurements of spike-time reliability to sinusoidal and aperiodic inputs, as well as simulations of a noisy leaky integrate-and-fire neuron stimulated by spike trains drawn from a periodically modulated process, demonstrate that there are three qualitatively different CV-dependent mechanisms that determine reliability: noise-dominated (CV < 0.05 for Aplysia motoneurons) where spike timing is unreliable regardless of frequency content; resonance-dominated (CV approximately 0.05-0.25) where reliability is reduced by removal of input frequencies equal to motoneuron firing rate; and amplitude-dominated (CV >0.35) where reliability depends on input frequencies greater than motoneuron firing rate. In the resonance-dominated regime, changes in the activity of the presynaptic inhibitory interneuron B4/5 alter motoneuron spike-time reliability. The increases or decreases in reliability occur coincident with small changes in motoneuron spiking rate due to changes in interneuron activity. Injection of a hyperpolarizing current into the motoneuron reproduces the interneuron-induced changes in reliability. The rate-dependent changes in reliability can be understood from the phase-locking properties of regularly spiking motoneurons to periodic inputs. Our observations demonstrate that the ability of a neuron to support a spike-time code can be actively controlled by varying the properties of the neuron and its input.

Synaptic modulation of the interspike interval signatures of bursting pyloric neurons

The pyloric network of the lobster stomatogastric nervous system is one of the best described assemblies of oscillatory neurons producing bursts of action potentials. While the temporal patterns of bursts have been investigated in detail, those of spikes have received less attention. Here we analyze the intraburst firing patterns of pyloric neurons and the synaptic interactions shaping their dynamics in millisecond time scales not performed before. We find that different pyloric neurons express characteristic, cell-specific firing patterns in their bursts. Nonlinear analysis of the interspike intervals (ISIs) reveals distinctive temporal structures ('interspike interval signatures'), which are found to depend on the synaptic connectivity of the network. We compare ISI patterns of the pyloric dilator (PD), lateral pyloric (LP), and ventricular dilator (VD) neurons in 1) normal conditions, 2) after blocking glutamatergic synaptic connections, and 3) in various functional configurations of the three neurons. Manipulation of the synaptic connectivity results in characteristic changes in the ISI signatures of the postsynaptic neurons. The intraburst firing pattern of the PD neuron is regularized by the inhibitory synaptic connection from the LP neuron as revealed in current-clamp experiments and also as reconstructed with a dynamic clamp. On the other hand, mutual inhibition between the LP and VD neurons tend to produce more irregular bursts with increased spike jitter. The results show that synaptic interactions fine-tune the output of pyloric neurons. The present data also suggest a way of processing of synaptic information: bursting neurons are capable of encoding incoming signals by altering the fine structure of their intraburst spike patterns.

Inhibitory synchronization of bursting in biological neurons: dependence on synaptic time constant

Using the dynamic clamp technique, we investigated the effects of varying the time constant of mutual synaptic inhibition on the synchronization of bursting biological neurons. For this purpose, we constructed artificial half-center circuits by inserting simulated reciprocal inhibitory synapses between identified neurons of the pyloric circuit in the lobster stomatogastric ganglion. With natural synaptic interactions blocked (but modulatory inputs retained), these neurons generated independent, repetitive bursts of spikes with cycle period durations of approximately 1 s. After coupling the neurons with simulated reciprocal inhibition, we selectively varied the time constant governing the rate of synaptic activation and deactivation. At time constants <or=100 ms, bursting was coordinated in an alternating (anti-phase) rhythm. At longer time constants (>400 ms), bursts became phase-locked in a fully overlapping pattern with little or no phase lag and a shorter period. During the in-phase bursting, the higher-frequency spiking activity was not synchronized. If the circuit lacked a robust periodic burster, increasing the time constant evoked a sharp transition from out-of-phase oscillations to in-phase oscillations with associated intermittent phase-jumping. When a coupled periodic burster neuron was present (on one side of the half-center circuit), the transition was more gradual. We conclude that the magnitude and stability of phase differences between mutually inhibitory neurons varies with the ratio of burst cycle period duration to synaptic time constant and that cellular bursting (whether periodic or irregular) can adopt in-phase coordination when inhibitory synaptic currents are sufficiently slow.