Experimental observations of the intracellular recorded electrical activity in individual neurons show that the temporal behavior is often chaotic. We discuss both our own observations on a cell from the stomatogastric central pattern generator of lobster and earlier observations in other cells. In this paper we work with models with chaotic neurons, building on models by Hindmarsh and Rose for bursting, spiking activity in neurons. The key feature of these simplified models of neurons is the presence of coupled slow and fast subsystems. We analyze the model neurons using the same tools employed in the analysis of our experimental data. We couple two model neurons both electrotonically and electrochemically in inhibitory and excitatory fashions. In each of these cases, we demonstrate that the model neurons can synchronize in phase and out of phase depending on the strength of the coupling. For normal synaptic coupling, we have a time delay between the action of one neuron and the response of the other. We also analyze how the synchronization depends on this delay. A rich spectrum of synchronized behaviors is possible for electrically coupled neurons and for inhibitory coupling between neurons. In synchronous neurons one typically sees chaotic motion of the coupled neurons. Excitatory coupling produces essentially periodic voltage trajectories, which are also synchronized. We display and discuss these synchronized behaviors using two "distance" measures of the synchronization.

When the classical Hodgkin-Huxley equations are simulated with Na- and K-channel noise and constant applied current, the distribution of interspike intervals is bimodal: one part is an exponential tail, as often assumed, while the other is a narrow gaussian peak centered at a short interspike interval value. The gaussian arises from bursts of spikes in the gamma-frequency range, the tail from the interburst intervals, giving overall an extraordinarily high coefficient of variation--up to 2.5 for 180,000 Na channels when I approximately 7 microA/cm(2). Since neurons with a bimodal ISI distribution are common, it may be a useful model for any neuron with class 2 firing. The underlying mechanism is due to a subcritical Hopf bifurcation, together with a switching region in phase-space where a fixed point is very close to a system limit cycle. This mechanism may be present in many different classes of neurons and may contribute to widely observed highly irregular neural spiking.

1. The gastric mill central pattern generator (CPG) controls the chewing movements of teeth in the gastric mill of the lobster. This CPG has been extensively studied, but the precise mechanism underlying pattern generation is not well understood. The goal of this research was to develop a simplified model that captures the principle, biologically significant features of this CPG. We introduce a simplified neuron model that embodies approximations of well-known membrane currents, and is able to reproduce several global characteristics of gastric mill neurons. A network built with these neurons, using graded synaptic transmission and having the synaptic connections of the biological circuit, is sufficient to explain much of the network's behavior. 2. The cell model is a generalization and extension of the Van der Pol relaxation oscillator equations. It is described by two differential equations, one for current conservation and one for slow current activation. The model has a fast current that may, by adjusting one parameter, have a region of negative resistance in its current-voltage (I-V) curve. It also has a slow current with a single gain parameter that can be regarded as the combination of slow inward and outward currents. 3. For suitable values of the fast current parameter and the slow current parameter, the isolated model neuron exhibits several different behaviors: plateau potentials, postinhibitory rebound, postburst hyperpolarization, and endogenous oscillations. When the slow current is separated into inward and outward fractions with separately adjustable gain parameters, the model neuron can fire tonically, be quiescent, or generate spontaneous voltage oscillations with varying amounts of depolarization or hyperpolarization. 4. The most common form of synaptic interaction in the gastric CPG is reciprocal inhibition. A pair of identical model cells, connected with reciprocal inhibition, oscillates in antiphase if either the isolated cells are endogenous oscillators, or they are quiescent without plateau potentials, or they have plateau potentials but the synaptic strengths are below a critical level. If the isolated cells have widely differing frequencies (or would have if the cells were made to oscillate by adjusting the fast currents), reciprocal inhibition entrains the cells to oscillate with the same frequency but with phases that are advanced or retarded relative to the phases seen when the cells have the same frequency. The frequency of the entrained pair of cells lies between the frequencies of the original cells. The relative phases can also be modified by using very unequal synaptic strengths.(ABSTRACT TRUNCATED AT 400 WORDS)

We explore the effects of stochastic sodium (Na) channel activation on the variability and dynamics of spiking and bursting in a model neuron. The complete model segregates Hodgin-Huxley-type currents into two compartments, and undergoes applied current-dependent bifurcations between regimes of periodic bursting, chaotic bursting, and tonic spiking. Noise is added to simulate variable, finite sizes of the population of Na channels in the fast spiking compartment. During tonic firing, Na channel noise causes variability in interspike intervals (ISIs). The variance, as well as the sensitivity to noise, depend on the model's biophysical complexity. They are smallest in an isolated spiking compartment; increase significantly upon coupling to a passive compartment; and increase again when the second compartment also includes slow-acting currents. In this full model, sufficient noise can convert tonic firing into bursting. During bursting, the actions of Na channel noise are state-dependent. The higher the noise level, the greater the jitter in spike timing within bursts. The noise makes the burst durations of periodic regimes variable, while decreasing burst length duration and variance in a chaotic regime. Na channel noise blurs the sharp transitions of spike time and burst length seen at the bifurcations of the noise-free model. Close to such a bifurcation, the burst behaviors of previously periodic and chaotic regimes become essentially indistinguishable. We discuss biophysical mechanisms, dynamical interpretations and physiological implications. We suggest that noise associated with finite populations of Na channels could evoke very different effects on the intrinsic variability of spiking and bursting discharges, depending on a biological neuron's complexity and applied current-dependent state. We find that simulated channel noise in the model neuron qualitatively replicates the observed variability in burst length and interburst interval in an isolated biological bursting neuron.

Parkinson's disease is a degenerative condition whose severity is assessed by clinical observations of motor behaviors. These are performed by a neurological specialist through subjective ratings of a variety of movements including 10-s bouts of repetitive finger-tapping movements. We present here an algorithmic rating of these movements which may be beneficial for uniformly assessing the progression of the disease. Finger-tapping movements were digitally recorded from Parkinson's patients and controls, obtaining one time series for every 10 s bout. A nonlinear delay differential equation, whose structure was selected using a genetic algorithm, was fitted to each time series and its coefficients were used as a six-dimensional numerical descriptor. The algorithm was applied to time-series from two different groups of Parkinson's patients and controls. The algorithmic scores compared favorably with the unified Parkinson's disease rating scale scores, at least when the latter adequately matched with ratings from the Hoehn and Yahr scale. Moreover, when the two sets of mean scores for all patients are compared, there is a strong (r ¼ 0.785) and significant (p < 0:0015) correlation between them. Parkinson's disease (PD) is a common disease affecting tens of millions of people worldwide. Its cardinal signs are resting tremor, bradykinesia (slowness clumsiness of movement), rigidity, and loss of postural reflexes. The disease evolves slowly and, to adjust medications to the severity of the disease, there is a need for automatic and objective evaluation of movements. Such objective movement assessments would supplement subjective clinical ratings, which are ordinal rather than metric and often show large inter-rater variability. Rather than using a spectral based technique, we rated dynamical features of each individuals' finger-tapping-one of the items from the unified Parkinson's disease rating scale (UPDRS) used for rating the severity of the disease-by using data models based on nonlinear delay differential equations (DDEs). The coefficients of the DDEs are then used to assess the severity of the disease.