Bioimpedance modeling to monitor astrocytic response to chronically implanted electrodes.

The widespread adoption of neural prosthetic devices is currently hindered by our inability to reliably record neural signals from chronically implanted electrodes. The extent to which the local tissue response to implanted electrodes influences recording failure is not well understood. To investigate this phenomenon, impedance spectroscopy has shown promise for use as a non-invasive tool to estimate the local tissue response to microelectrodes. Here, we model impedance spectra from chronically implanted rats using the well-established Cole model, and perform a correlation analysis of modeled parameters with histological markers of astroglial scar, including glial fibrillary acid protein (GFAP) and 4',6-diamidino-2- phenylindole (DAPI). Correlations between modeled parameters and GFAP were significant for three parameters studied: Py value, R(o) and |Z|(1 kHz), and in all cases were confined to the first 100 microm from the interface. Py value was the only parameter also correlated with DAPI in the first 100 microm. Our experimental results, along with computer simulations, suggest that astrocytes are a predominant cellular player affecting electrical impedance spectra. The results also suggest that the largest contribution from reactive astrocytes on impedance spectra occurs in the first 100 microm from the interface, where electrodes are most likely to record electrical signals. These results form the basis for future approaches where impedance spectroscopy can be used to evaluate neural implants, evaluate strategies to minimize scar and potentially develop closed-loop prosthetic devices.

Intrinsic bursters increase the robustness of rhythm generation in an excitatory network.

The pre-Botzinger complex (pBC) is a vital subcircuit of the respiratory central pattern generator. Although the existence of neurons with pacemaker-like bursting properties in this network is not questioned, their role in network rhythmogenesis is unresolved. Modeling is ideally suited to address this debate because of the ease with which biophysical parameters of individual cells and network architecture can be manipulated. We modeled the parameter variability of experimental data from pBC bursting pacemaker and nonpacemaker neurons using a modified version of our previously developed pBC neuron and network models. To investigate the role of pacemakers in networkwide rhythmogenesis, we simulated networks of these neurons and varied the fraction of the population made up of pacemakers. For each number of pacemaker neurons, we varied the amount of tonic drive to the network and measured the frequency of synchronous networkwide bursting produced. Both excitatory networks with all-to-all coupling and sparsely connected networks were explored for several levels of synaptic coupling strength. Networks containing only nonpacemakers were able to produce networkwide bursting, but with a low probability of bursting and low input and output ranges. Our results indicate that inclusion of pacemakers in an excitatory network increases robustness of the network by more than tripling the input and output ranges compared with networks containing no pacemakers. The largest increase in dynamic range occurs when the number of pacemakers in the network is greater than 20% of the population. Experimental tests of our model predictions are proposed.

A simple model of dynamic interactions between respiratory centers.

There is a general consensus that neurons responsible for the generation of a respiratory rhythm are located in the rostral ventrolateral medulla. However, there is still controversy regarding the necessary components for respiratory rhythm generation. Both inspiratory neurons in the preBotzinger Complex (preBOtC) and pre-inspiratory neurons located more rostrally to this anatomical structure referred to as the parafacial respiratory group (pFRG) have been proposed to be essential for respiratory rhythmogenesis. To study the dynamical interactions between preBOtC and pFRG neurons, we use a canonical model that describes each neuron population as a phase oscillator. We assume that the oscillators are weakly coupled with pFRG neurons providing stimulation to preBOtC neurons and preBOtC neurons providing inhibitory drive to pFRG neurons. In our mathematical study, we explore plausible mechanisms that may account for the complex interactions between I and pre-I neuron. In particular, we show that reduced excitability of inspiratory in preBOtC may lead to the phenomena known as "quantal slowing".

An implementation of a simple neuron model in field programmable analog arrays.

A minimal neuron model, the Morris-Lecar model, is implemented on field programmable analog arrays (FPAAs). Our approach is to solve the differential equation describing the model in a similar way a computer solves the same problem: numerically integrate the differential equation by making arithmetic operations on voltage mode circuits of the FPAAs. The results demonstrate that biologically relevant dynamics can be observed from the electronic neuron despite limitations on the configurability of the FPAAs. Such models can be run accurately in real-time or many orders of magnitude faster than real-time. FPAAs are feasible candidates for implementation of neuron models using off-the-shelf software-reconfigurable analog circuit elements.

Models of respiratory rhythm generation in the pre-Bötzinger complex. III. Experimental tests of model predictions.

We used the testable predictions of mathematical models proposed by Butera et al. to evaluate cellular, synaptic, and population-level components of the hypothesis that respiratory rhythm in mammals is generated in vitro in the pre-Bötzinger complex (pre-BötC) by a heterogeneous population of pacemaker neurons coupled by fast excitatory synapses. We prepared thin brain stem slices from neonatal rats that capture the pre-BötC and maintain inspiratory-related motor activity in vitro. We recorded pacemaker neurons extracellularly and found: intrinsic bursting behavior that did not depend on Ca(2+) currents and persisted after blocking synaptic transmission; multistate behavior with transitions from quiescence to bursting and tonic spiking states as cellular excitability was increased via extracellular K(+) concentration ([K(+)](o)); a monotonic increase in burst frequency and decrease in burst duration with increasing [K(+)](o); heterogeneity among different cells sampled; and an increase in inspiratory burst duration and decrease in burst frequency by excitatory synaptic coupling in the respiratory network. These data affirm the basis for the network model, which is composed of heterogeneous pacemaker cells having a voltage-dependent burst-generating mechanism dominated by persistent Na(+) current (I(NaP)) and excitatory synaptic coupling that synchronizes cell activity. We investigated population-level activity in the pre-BötC using local "macropatch" recordings and confirmed these model predictions: pre-BötC activity preceded respiratory-related motor output by 100-400 ms, consistent with a heterogeneous pacemaker-cell population generating inspiratory rhythm in the pre-BötC; pre-BötC population burst amplitude decreased monotonically with increasing [K(+)](o) (while frequency increased), which can be attributed to pacemaker cell properties; and burst amplitude fluctuated from cycle to cycle after decreasing bilateral synaptic coupling surgically as predicted from stability analyses of the model. We conclude that the pacemaker cell and network models explain features of inspiratory rhythm generation in vitro.

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 cell's 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.

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.

Models of respiratory rhythm generation in the pre-Bötzinger complex. I. Bursting pacemaker neurons.

A network of oscillatory bursting neurons with excitatory coupling is hypothesized to define the primary kernel for respiratory rhythm generation in the pre-Bötzinger complex (pre-BötC) in mammals. Two minimal models of these neurons are proposed. In model 1, bursting arises via fast activation and slow inactivation of a persistent Na+ current INaP-h. In model 2, bursting arises via a fast-activating persistent Na+ current INaP and slow activation of a K+ current IKS. In both models, action potentials are generated via fast Na+ and K+ currents. The two models have few differences in parameters to facilitate a rigorous comparison of the two different burst-generating mechanisms. Both models are consistent with many of the dynamic features of electrophysiological recordings from pre-BötC oscillatory bursting neurons in vitro, including voltage-dependent activity modes (silence, bursting, and beating), a voltage-dependent burst frequency that can vary from 0.05 to >1 Hz, and a decaying spike frequency during bursting. These results are robust and persist across a wide range of parameter values for both models. However, the dynamics of model 1 are more consistent with experimental data in that the burst duration decreases as the baseline membrane potential is depolarized and the model has a relatively flat membrane potential trajectory during the interburst interval. We propose several experimental tests to demonstrate the validity of either model and to differentiate between the two mechanisms.

Models of respiratory rhythm generation in the pre-Bötzinger complex. II. Populations Of coupled pacemaker neurons.

We have proposed models for the ionic basis of oscillatory bursting of respiratory pacemaker neurons in the pre-Bötzinger complex. In this paper, we investigate the frequency control and synchronization of these model neurons when coupled by excitatory amino-acid-mediated synapses and controlled by convergent synaptic inputs modeled as tonic excitation. Simulations of pairs of identical cells reveal that increasing tonic excitation increases the frequency of synchronous bursting, while increasing the strength of excitatory coupling between the neurons decreases the frequency of synchronous bursting. Low levels of coupling extend the range of values of tonic excitation where synchronous bursting is found. Simulations of a heterogeneous population of 50-500 bursting neurons reveal coupling effects similar to those found experimentally in vitro: coupling increases the mean burst duration and decreases the mean burst frequency. Burst synchronization occurred over a wide range of intrinsic frequencies (0.1-1 Hz) and even in populations where as few as 10% of the cells were intrinsically bursting. Weak coupling, extreme parameter heterogeneity, and low levels of depolarizing input could contribute to the desynchronization of the population and give rise to quasiperiodic states. The introduction of sparse coupling did not affect the burst synchrony, although it did make the interburst intervals more irregular from cycle to cycle. At a population level, both parameter heterogeneity and excitatory coupling synergistically combine to increase the dynamic input range: robust synchronous bursting persisted across a much greater range of parameter space (in terms of mean depolarizing input) than that of a single model cell. This extended dynamic range for the bursting cell population indicates that cellular heterogeneity is functionally advantageous. Our modeled system accounts for the range of intrinsic frequencies and spiking patterns of inspiratory (I) bursting cells found in the pre-Bötzinger complex in neonatal rat brain stem slices in vitro. There is a temporal dispersion in the spiking onset times of neurons in the population, predicted to be due to heterogeneity in intrinsic neuronal properties, with neurons starting to spike before (pre-I), with (I), or after (late-I) the onset of the population burst. Experimental tests for a number of the model's predictions are proposed.

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

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.

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.

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

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.

Dissection and reduction of a modeled bursting neuron.

An 11-variable Hodgkin-Huxley type model of a bursting neuron was investigated using numerical bifurcation analysis and computer simulations. The results were applied to develop a reduced model of the underlying subthreshold oscillations (slow-wave) in membrane potential. Two different low-order models were developed: one 3-variable model, which mimicked the slow-wave of the full model in the absence of action potentials and a second 4-variable model, which included expressions accounting for the perturbational effects of action potentials on the slow-wave. The 4-variable model predicted more accurately the activity mode (bursting, beating, or silence) in response to application of extrinsic stimulus current or modulatory agents. The 4-variable model also possessed a phase-response curve that was very similar to that of the original 11-variable model. The results suggest that low-order models of bursting cells that do not consider the effects of action potentials may erroneously predict modes of activity and transient responses of the full model on which the reductions are based. These results also show that it is possible to develop low-order models that retain many of the characteristics of the activity of the higher-order system.

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 and ISI) 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.