Search

The Hodgkin-Huxley (HH) model is the basis for numerous neural models. There are two negative feedback processes in the HH model that regulate rhythmic spiking. The first is an outward current with an activation variable n that has an opposite influence to the excitatory inward current and therefore provides subtractive negative feedback. The other is the inactivation of an inward current with an inactivation variable h that reduces the amount of positive feedback and therefore provides divisive feedback. Rhythmic spiking can be obtained with either negative feedback process, so we ask what is gained by having two feedback processes. We also ask how the different negative feedback processes contribute to spiking. We show that having two negative feedback processes makes the HH model more robust to changes in applied currents and conductance densities than models that possess only one negative feedback variable. We also show that the contributions made by the subtractive and divisive feedback variables are not static, but depend on time scales and conductance values. In particular, they contribute differently to the dynamics in Type I versus Type II neurons.

This paper illustrates an informatic technique for inferring and quantifying the dynamic role of a single intrinsic current in a mechanism of neural bursting activity. We analyze the patterns of the most dominant currents in a model of half-center oscillation in the leech heartbeat central pattern generator. We find that the patterns of dominance change substantially over a cycle, allowing different local reductions to be applied to the model. The result is a hybrid dynamical systems model, which is a piecewise representation of the mechanism combining multiple vector fields and discrete state changes. The simulation of such a model tests explicit hypotheses about the mechanism and is a novel way to retain both mathematical clarity and scientific detail in answering mechanistic questions about a complex model. Several insights into the central mechanism of "escape-release" in the model are elucidated by this analysis and compared with previous studies. The broader application and extension of this technique is also discussed.

This work presents a neuroinformatic method for deriving mechanistic descriptions of fine-structured neural activity. This is a new development in the computer-assisted analysis of dynamics in conductance-based models, which is illustrated using single compartment models of an action potential. A sequence of abstract, qualitative motifs is inferred from this analysis, forming a template that is independent of the specific equations from which they were abstracted. The template encodes the assumptions behind the model reduction steps used to derive the motifs, and so specifies quantitative information about their domains of validity. The template representation of a mechanism is converted to a hybrid dynamical system, which is simulated as a sequence of low-dimensional reduced models (in this example, phase plane models) with appropriate switching conditions taken from the motifs. We demonstrate the validity of the template on a detailed single neuron model of spiking taken from the literature, and show that the corresponding hybrid system simulation closely mimics the spiking dynamics of the full model.

The large number of variables involved in many biophysical models can conceal potentially simple dynamical mechanisms governing the properties of its solutions and the transitions between them as parameters are varied. To address this issue, we extend a novel model reduction method, based on “scales of dominance,” to multi-compartment models. We use this method to systematically reduce the dimension of a two-compartment conductance-based model of a crustacean pyloric dilator (PD) neuron that exhibits distinct modes of oscillation—tonic spiking, intermediate bursting and strong bursting. A cursory analysis of the scales of dominance in a trajectory of this sixteen-variable model leads to a globally-reduced, nine-variable model. In a finer analysis we divide the trajectory into intervals dominated by a smaller number of variables, resulting in a locally-reduced hybrid model whose dimension varies between two and six in different temporal regimes. Both reduced models exhibit the same modes of oscillation as the sixteen-dimensional model over a comparable parameter range. The reduced models highlight low-dimensional organizing structure in the dynamics of the PD neuron, and the dependence of its oscillations on parameters such as the maximal conductances of calcium currents. Our technique could be used to build hybrid low-dimensional models from any large multi-compartment conductance-based model in order to analyze the interactions between different modes of activity.

Studies of the degrees of freedom and "synergies" in musculoskeletal systems rely critically on algorithms to estimate the "dimension" of kinematic or neural data. Linear algorithms such as principal component analysis (PCA) are the most popular. However, many biological data (or realistic experimental data) may be better represented by nonlinear sets than linear subspaces. We evaluate the performance of PCA and compare it to two nonlinear algorithms [Isomap and our novel pointwise dimension estimation (PD-E)] using synthetic and motion capture data from a robotic arm with known kinematic dimensions, as well as motion capture data from human hands. We find that PCA can lead to more accurate dimension estimates when considering additional properties of the PCA residuals, instead of the dominant method of using a threshold of variance captured. In contrast to the single integer dimension estimates of PCA and Isomap, PD-E provides a distribution and range of estimates of fractal dimension that identify the heterogeneous geometric structure in the experimental data. A strength of the PD-E method is that it associates a distribution of dimensions to the data. Since there is no a priori reason to assume that the sets of interest have a single dimension, these distributions incorporate more information than a single summary statistic. Our preliminary findings suggest that fewer than ten DOFs are involved in some hand motion tasks. Contrary to common opinion regarding fractal dimension methods, PD-E yielded reasonable results with reasonable amounts of data. Given the complex nature of experimental and biological data, we conclude that it is necessary and feasible to complement PCA with methods that take into consideration the nonlinear properties of biological systems for a more robust estimation of their DOFs.

Near an orbit of interest in a dynamical system, it is typical to ask which variables dominate its structure at what times. What are its principal local degrees of freedom? What local bifurcation structure is most appropriate? We introduce a combined numerical and analytical technique that aids the identification of structure in a class of systems of nonlinear ordinary differential equations (ODEs) that are commonly applied in dynamical models of physical processes. This "dominant scale" technique prioritizes consideration of the influence that distinguished "inputs" to an ODE have on its dynamics. On this basis a sequence of reduced models is derived, where each model is valid for a duration that is determined self-consistently as the system’s state variables evolve. The characteristic time scales of all sufficiently dominant variables are also taken into account to further reduce the model. The result is a hybrid dynamical system of reduced differential-algebraic models that are switched at discrete eve...

In hippocampal slice models of epilepsy, two behaviors are seen: short bursts of electrical activity lasting 100 msec and seizure-like electrical activity lasting seconds. The bursts originate from the CA3 region, where there is a high degree of recurrent excitatory connections. Seizures originate from the CA1, where there are fewer recurrent connections. In attempting to explain this behavior, we simulated model networks of excitatory neurons using several types of model neurons. The model neurons were connected in a ring containing predominantly local connections and some long-distance random connections, resulting in a small-world network connectivity pattern. By changing parameters such as the synaptic strengths, number of synapses per neuron, proportion of local versus long-distance connections, we induced “normal,” “seizing,” and “bursting” behaviors. Based on these simulations, we made a simple mathematical description of these networks under well-defined assumptions. This mathematical description explains how specific changes in the topology or synaptic strength in the model cause transitions from normal to seizing and then to bursting. These behaviors appear to be general properties of excitatory networks.

We describe a novel computational approach to reduce detailed models of central pattern generation to an equationless mapping that can be studied geometrically. Changes in model parameters, coupling properties, or external inputs produce qualitative changes in the mapping. These changes uncover possible biophysical mechanisms for control and modulation of rhythmic activity. Our analysis does not require knowledge of the equations that model the system, and so provides a powerful new approach to studying detailed models, applicable to a variety of biological phenomena beyond motor control. We demonstrate our technique on a motif of three reciprocally inhibitory cells that is able to produce multiple patterns of bursting rhythms. In particular, we examine the qualitative geometric structure of two-dimensional maps for phase lag between the cells.

Centrosome amplification (CA) and resultant chromosomal instability have long been associated with tumorigenesis. However, exacerbation of CA and relentless centrosome declustering engender robust spindle multipolarity (SM) during mitosis and may induce cell death. Recently, we demonstrated that a noscapinoid member, reduced bromonoscapine, (S)-3-(R)-9-bromo-5-(4,5-dimethoxy-1,3-dihydroisobenzofuran-1-yl)-4-methoxy-6-methyl-5,6,7,8-tetrahydro-[1,3]dioxolo-[4,5-g]isoquinoline (Red-Br-nos), induces reactive oxygen species (ROS)-mediated autophagy and caspase-independent death in prostate cancer PC-3 cells. Herein, we show that Red-Br-nos induces ROS-dependent DNA damage that resulted in high-grade CA and SM in PC-3 cells. Unlike doxorubicin, which causes double-stranded DNA breaks and chronic G2 arrest accompanied by 'templated' CA, Red-Br-nos-mediated DNA damage elicits de novo CA during a transient S/G2 stall, followed by checkpoint abrogation and mitotic entry to form aberrant mitotic figures with supernumerary spindle poles. Attenuation of multipolar phenotype in the presence of tiron, a ROS inhibitor, indicated that ROS-mediated DNA damage was partly responsible for driving CA and SM. Although a few cells (∼5%) yielded to aberrant cytokinesis following an 'anaphase catastrophe', most mitotically arrested cells (∼70%) succumbed to 'metaphase catastrophe,' which was caspase-independent. This report is the first documentation of rapid de novo centrosome formation in the presence of parent centrosome by a noscapinoid family member, which triggers death-inducing SM via a unique mechanism that distinguishes it from other ROS-inducers, conventional DNA-damaging agents, as well as other microtubule-binding drugs.

We examine multistability of several coexisting bursting patterns in a central pattern generator network composed of three Hodgkin-Huxley type cells coupled reciprocally by inhibitory synapses. We establish that the control of switching between bursting polyrhythms and their bifurcations are determined by the temporal characteristics, such as the duty cycle, of networked interneurons and the coupling strength asymmetry. A computationally effective approach to the reduction of dynamics of the nine-dimensional network to two-dimensional Poincare return mappings for phase lags between the interneurons is presented.

This paper illustrates an informatic technique for inferring and quantifying the dynamic role of a single intrinsic current in a mechanism of neural bursting activity. We analyze the patterns of the most dominant currents in a model of half-center oscillation in the leech heartbeat central pattern generator. We find that the patterns of dominance change substantially over a cycle, allowing different local reductions to be applied to the model. The result is a hybrid dynamical systems model, which is a piecewise representation of the mechanism combining multiple vector fields and discrete state changes. The simulation of such a model tests explicit hypotheses about the mechanism and is a novel way to retain both mathematical clarity and scientific detail in answering mechanistic questions about a complex model. Several insights into the central mechanism of “escape-release” in the model are elucidated by this analysis and compared with previous studies. The broader application and extension of this technique is also discussed.