Robert J. Butera

Respiratory rhythm generation in neonatal and adult mammals: the hybrid pacemaker–network model

We review a new unified model of respiratory rhythm generation - the hybrid pacemaker-network model. This model represents a comprehensive synthesis of cellular and network mechanisms that can theoretically account for rhythm generation in different functional states, from the most reduced states in the neonatal nervous system in vitro to the intact adult system in vivo. The model incorporates a critical neuronal kernel consisting of a network of excitatory neurons with state-dependent, oscillatory bursting or pacemaker properties. This kernel, located in the pre-Bötzinger complex of the ventrolateral medulla, provides a rudimentary pacemaker network mechanism for generating an inspiratory rhythm, revealed predominately in functionally reduced states in vitro. In vivo the kernel is embedded in a larger network that interacts with the kernel via inhibitory synaptic connections that provide the dynamic control required for the evolution of the complete pattern of inspiratory and expiratory network activity. The resulting hybrid of cellular pacemaker and network properties functionally endows the system with multiple mechanisms of rhythm generation. New biophysically realistic mathematical models of the hybrid pacemaker-network have been developed that illustrate these concepts and provide a computational framework for investigating interactions of cellular and network processes that must be analyzed to understand rhythm generation.

Phase response characteristics of model neurons determine which patterns are expressed in a ring circuit model of gait generation

Abstract. In order to assess the relative contributions to pattern-generation of the intrinsic properties of individual neurons and of their connectivity, we examined a ring circuit composed of four complex physiologically based oscillators. This circuit produced patterns that correspond to several quadrupedal gaits, including the walk, the bound, and the gallop. An analysis using the phase response curve (PRC) of an uncoupled oscillator accurately predicted all modes exhibited by this circuit and their phasic relationships – with the caveat that in certain parameter ranges, bistability in the individual oscillators added nongait patterns that were not amenable to PRC analysis, but further enriched the pattern-generating repertoire of the circuit. The key insights in the PRC analysis were that in a gait pattern, since all oscillators are entrained at the same frequency, the phase advance or delay caused by the action of each oscillator on its postsynaptic oscillator must be the same, and the sum of the normalized phase differences around the ring must equal to an integer. As suggested by several previous studies, our analysis showed that the capacity to exhibit a large number of patterns is inherent in the ring circuit configuration. In addition, our analysis revealed that the shape of the PRC for the individual oscillators determines which of the theoretically possible modes can be generated using these oscillators as circuit elements. PRCs that have a complex shape enable a circuit to produce a wider variety of patterns, and since complex neurons tend to have complex PRCs, enriching the repertoire of patterns exhibited by a circuit may be the function of some intrinsic neuronal complexity. Our analysis showed that gait transitions, or more generally, pattern transitions, in a ring circuit do not require rewiring the circuit or any changes in the strength of the connections. Instead, transitions can be achieved by using a control parameter, such as stimulus intensity, to sculpt the PRC so that it has the appropriate shape for the desired pattern(s). A transition can then be achieved simply by changing the value of the control parameter so that the first pattern either ceases to exist or loses stability, while a second pattern either comes into existence or gains stability. Our analysis illustrates the predictive value of PRCs in circuit analysis and can be extended to provide a design method for pattern-generating circuits.

Sequential optimal design of neurophysiology experiments

Adaptively optimizing experiments has the potential to significantly reduce the number of trials needed to build parametric statistical models of neural systems. However, application of adaptive methods to neurophysiology has been limited by severe computational challenges. Since most neurons are high-dimensional systems, optimizing neurophysiology experiments requires computing high-dimensional integrations and optimizations in real time. Here we present a fast algorithm for choosing the most informative stimulus by maximizing the mutual information between the data and the unknown parameters of a generalized linear model (GLM) that we want to fit to the neurons activity. We rely on important log concavity and asymptotic normality properties of the posterior to facilitate the required computations. Our algorithm requires only low-rank matrix manipulations and a two-dimensional search to choose the optimal stimulus. The average running time of these operations scales quadratically with the dimensionality of the GLM, making real-time adaptive experimental design feasible even for high-dimensional stimulus and parameter spaces. For example, we require roughly 10 milliseconds on a desktop computer to optimize a 100-dimensional stimulus. Despite using some approximations to make the algorithm efficient, our algorithm asymptotically decreases the uncertainty about the model parameters at a rate equal to the maximum rate predicted by an asymptotic analysis. Simulation results show that picking stimuli by maximizing the mutual information can speed up convergence to the optimal values of the parameters by an order of magnitude compared to using random (nonadaptive) stimuli. Finally, applying our design procedure to real neurophysiology experiments requires addressing the nonstationarities that we would expect to see in neural responses; our algorithm can efficiently handle both fast adaptation due to spike history effects and slow, nonsystematic drifts in a neurons activity.

Periodicity, Mixed-Mode Oscillations, and Quasiperiodicity in a Rhythm-Generating Neural Network

We studied patterns of oscillatory neural activity in the network that generates respiratory rhythm in mammals. When isolated in vitro, this network spontaneously generates an inspiratory-related motor rhythm, with stable amplitude from cycle to cycle. We show that progressively elevating neuronal excitability in vitro causes periodic modulation of this inspiratory rhythm, evoking (in order): mixed-mode oscillations, quasiperiodicity, and ultimately disorganized aperiodic activity. Thus, the respiratory network oscillator follows a well defined sequence of behavioral states characterized by dynamical systems theory, which includes discrete stages of periodic and quasiperiodic amplitude modulation and progresses (according to theory) to aperiodic chaos-like behavior. We also observed periodic, mixed-mode periodic, and quasiperiodic breathing patterns in neonatal rodents and human infants in vivo, suggesting that breathing patterns generated by the intact nervous system reflect deterministic neural activity patterns in the underlying rhythm-generating network.

A mathematical criterion based on phase response curves for stability in a ring of coupled oscillators

Abstract. Canavier et al. (1997) used phase response curves (PRCs) of individual oscillators to characterize the possible modes of phase-locked entrainment of an N-oscillator ring network. We extend this work by developing a mathematical criterion to determine the local stability of such a mode based on the PRCs. Our method does not assume symmetry; neither the oscillators nor their connections need be identical. To use these techniques for predicting modes and determining their stability, one need only determine the PRC of each oscillator in the ring either experimentally or from a computational model. We show that network stability cannot be determined by simply testing the ability of each oscillator to entrain the next. Stability depends on the number of neurons in the ring, the type of mode, and the slope of each PRC at the point of entrainment of the respective neuron. We also describe simple criteria which are either necessary or sufficient for stability and examine the implications of these results.

Estimating action potential thresholds from neuronal time-series: new metrics and evaluation of methodologies

The estimation of action potential thresholds is a subjective process, which we quantified by surveying experienced electrophysiologists via a software application that allowed them to select action potential thresholds from several presented neuronal time series. Independent of this survey, we derived two nonparametric techniques for automating the detection of an action potential threshold from the time-series of intracellular recordings. Both methods start with a phase-space representation of the action potential (dV/dt versus V). Method I detects the maximum slope in the phase space, while Method 11 detects the maximum second derivative in the phase space. These two methods, as well as five additional methods in the literature, were tested on three data sets representing a variety of action potential shapes, the same three datasets that were used in the electrophysiologist survey. The database of user responses was used to provide an external benchmark against which to statistically evaluate all seven methods. Method 11, as well as the curvature-based Methods VI and VII, provided the best results tracking both absolute and relative changes in threshold versus the other nonparametric methods (peak of second and third time derivatives). The one parametric method evaluated, detection of threshold crossing of the first temporal derivative, performed comparably to these methods, provided that an appropriate threshold was chosen. We conclude that Methods 11, VI, and VII were the best methods evaluated due to their performance across a wide range of action potential shapes and the fact that they are nonparametric. Our user database of responses may be useful to other investigators interested in developing additional methods in that it quantifies what has often been a subjective estimate.

A methodology for achieving high-speed rates for artificial conductance injection in electrically excitable biological cells

We present a novel approach to implementing the dynamic-clamp protocol (Sharp et al., 1993), commonly used in neurophysiology and cardiac electrophysiology experiments. Our approach is based on real-time extensions to the Linux operating system. Conventional PC-based approaches have typically utilized single-cycle computational rates of 10 kHz or slower. In thispaper, we demonstrate reliable cycle-to-cycle rates as fast as 50 kHz. Our system, which we call model reference current injection (MRCI); pronounced merci is also capable of episodic logging of internal state variables and interactive manipulation of model parameters. The limiting factor in achieving high speeds was not processor speed or model complexity, but cycle jitter inherent in the CPU/motherboard performance. We demonstrate these high speeds and flexibility with two examples: 1) adding action-potential ionic currents to a mammalian neuron under whole-cell patch-clamp and 2) altering a cells intrinsic dynamics via MRCI while simultaneously coupling it via artificial synapses to an internal computational model cell. These higher rates greatly extend the applicability of this technique to the study of fast electrophysiological currents such fast a currents and fast excitatory/inhibitory synapses.

Phase sensitivity and entrainment in a modeled bursting neuron

A model of neuron R15 in Aplysia was used to study the mechanisms determining the phase-response curve (PRC) of the cell in response to both extrinsic current pulses and modeled synaptic input and to compare entrainment predictions from PRCs with those from actual simulations. Over the range of stimulus parameters studied, the PRCs of the model exhibited minimal dependence upon stimulus amplitude, and a strong dependence upon stimulus duration. State-space analysis of the effect of transient current pulses provided several important insights into the relationship between the PRC and the underlying dynamics of the model, such as a correlation between the prestimulus concentration of Ca2+ and the poststimulus phase of the oscillation. The system nullclines were also found to provide well-defined limits upon the perturbatory extent of a hyperpolarizing input. These results demonstrated that experimentally applied current pulses are sufficient to determine the shape of the PRC in response to a synaptic input, provided that the duration of the current pulse is of a duration similar to that of the evoked synaptic current. Furthermore, we found that predictions of phase-locked 1:m entrainment from PRCs were valid, even when the duration of the periodically applied pulses were a significant portion of the control limit cycle.

Analysis of the Effects of Modulatory Agents on a Modeled Bursting Neuron: Dynamic Interactions Between Voltage and Calcium Dependent Systems

In a computational model of the bursting neuron R15, we have implemented proposed mechanisms for the modulation of two ionic currents (IR andISI) that play key roles in regulating its spontaneous electrical activity. The model was sufficient to simulate a wide range of endogenous activity in the presence of various concentrations of serotonin (5-HT) or dopamine (DA). The model was also sufficient to simulate the responses of the neuron to extrinsic current pulses and the ways in which those responses were altered by 5-HT or DA. The results suggest that the actions of modulatory agents and second messengers on this neuron, and presumably other neurons, cannot be understood on the basis of their direct effects alone. It is also necessary to take into account the indirect effects of these agents on other unmodulated ion channels. These indirect effects occur through the dynamic interactions of voltage-dependent and calcium-dependent processes.

MRCI: a flexible real-time dynamic clamp system for electrophysiology experiments

We present a real-time simulation system that enables modeled dynamical systems to interact with physical experimental systems, and is specifically aimed towards execution of the dynamic clamp protocol. Model reference current injection (MRCI) operates under Real-Time Linux (RT-Linux or RTL) and provides a simple equation-oriented language for describing dynamical system models. Features include scripting of commands to implement repeatable protocols, the ability to output pre-computed waveforms through any variable or parameter of the model, the means to conduct time measurements and assess the computational performance of the real-time system, and an installation program that installs the software and accompanying device drivers with minimal input from the user. Tested models operate as fast as 30 kHz, with actual maximum rates dependent on model complexity. We present sample models that exhibit the main features of the modeling language. Experiments demonstrate the abilities of the system by creating a hybrid network of real and simulated neurons, and playing a pre-defined synaptic waveform into a synaptic conductance variable. We conclude by introducing a waveform reconstruction technique that is useful for establishing the presence of significant experimental error in implementations of the dynamic clamp protocol.