Hans A. Braun

Homoclinic bifurcation in a Hodgkin-Huxley model of thermally sensitive neurons.

We study global bifurcations of the chaotic attractor in a modified Hodgkin-Huxley model of thermally sensitive neurons. The control parameter for this model is the temperature. The chaotic behavior is realized over a wide range of temperatures and is visualized using interspike intervals. We observe an abrupt increase of the interspike intervals in a certain temperature region. We identify this as a homoclinic bifurcation of a saddle-focus fixed point which is embedded in the chaotic attractors. The transition is accompanied by intermittency, which obeys a universal scaling law for the average length of trajectory segments exhibiting only short interspike intervals with the distance from the onset of intermittency. We also present experimental results of interspike interval measurements taken from the crayfish caudal photoreceptor, which qualitatively demonstrate the same bifurcation structure. (c) 2000 American Institute of Physics.

Low-Dimensional Dynamics in Sensory Biology 1: Thermally Sensitive Electroreceptors of the Catfish

We report the results of a search for evidence of periodic unstableorbits in the electroreceptors of the catfish. The function of thesereceptor organs is to sense weak external electric fields. Inaddition, they respond to the ambient temperature and to the ioniccomposition of the water. These quantities are encoded by receptorsthat make use of an internal oscillator operating at the level of themembrane potential. If such oscillators have three or more degreesof freedom, and at least one of which also exhibits a nonlinearity,they are potentially capable of chaotic dynamics. By detecting theexistence of stable and unstable periodic orbits, we demonstratebifurcations between noisy stable and chaotic behavior using theambient temperature as a parameter. We suggest that the techniquedeveloped herein be regarded as an additional tool for the analysisof data in sensory biology and thus can be potentially useful instudies of functional responses to external stimuli. We speculatethat the appearance of unstable orbits may be indicative of a stateof heightened sensory awareness by the animal.

Low-Dimensional Dynamics in Sensory Biology 2: Facial Cold Receptors of the Rat

We report the results of a search for evidence of unstable periodic orbits in the sensory afferents of the facial cold receptors of the rat. Cold receptors are unique in that they exhibit a diversity of action potential firing patterns as well as pronounced transients in firing rate following rapid temperature changes. These characteristics are the result of an internal oscillator operating at the level of the membrane potential. If such oscillators have three or more degree of freedom, and at least one of which also exhibits a nonlinearity, they are potentially capable of complex activity. By detecting the existence of unstable periodic orbits, we demonstrate low-dimensional dynamical behavior whose characteristics depend on the temperature range, impulse pattern, and temperature transients.

Neural Synchronization at Tonic-to-Bursting Transitions

We studied the synchronous behavior of two electrically-coupled model neurons as a function of the coupling strength when the individual neurons are tuned to different activity patterns that ranged from tonic firing via chaotic activity to burst discharges. We observe asynchronous and various synchronous states such as out-of-phase, in-phase and almost in-phase chaotic synchronization. The highest variety of synchronous states occurs at the transition from tonic firing to chaos where the highest coupling strength is also needed for in-phase synchronization which is, essentially, facilitated towards the bursting range. This demonstrates that tuning of the neuron’s internal dynamics can have significant impact on the synchronous states especially at the physiologically relevant tonic-to-bursting transitions.

A Mathematical Model of Homeostatic Regulation of Sleep-Wake Cycles by Hypocretin/Orexin

We introduce a physiology-based mathematical model of sleep-wake cycles, suggesting a novel mechanism of homeostatic regulation of sleep. In this model, the homeostatic process is determined by the neuropeptide hypocretin/ orexin, which is a cotransmitter of the lateral hypothalamus. Hypocretin/ orexin neurons are silent during sleep and active during wakefulness. Firing of these neurons is sustained by reciprocal excitatory synaptic connections with local glutamate interneurons. This feedback loop has been simulated with a minimal but physiologically plausible model. It includes 2 simplified Hodgkin-Huxley type neurons that are connected via glutamate synapses, one of which additionally contains hypocretin/orexin as the functionally relevant cotransmitter. During the active state (wakefulness), the synaptic efficacy of hypocretin/orexin declines as a result of the ongoing firing. It recovers during the silent (sleep) state. We demonstrate that these homeostatic changes can account for typical alterations of sleep-wake transitions, for example, introduced by napping, sleep deprivation, or alarm clock. In combination with a circadian input, the model mimics the transitions between silent and firing states in agreement with sleep-wake cycles. These simulation results support the concept of state-dependent alterations of hypocretin/orexin effects as an important homeostatic process in sleep-wake regulation, although additional mechanisms can be involved.

Oscillations, resonances and noise: basis of flexible neuronal pattern generation

Modulation of neuronal impulse pattern is examined by means of a simplified Hodgkin-Huxley type computer model which refers to experimental recordings of cold receptor discharges. This model essentially consists of two potentially oscillating subsystems: a spike generator and a subthreshold oscillator. With addition of noise the model successfully mimics the major types of experimentally recorded impulse patterns and thereby elucidate different resonance behaviors. (1) There is a range of rhythmic spiking or bursting where the spike generator is strongly coupled to the subthreshold oscillator. (2) There is a pacemaker activity of more complex interactions where the spike generator has overtaken part of the control. (3) There is a situation where the two subsystems are decoupled and only resonate with the help of noise.

Stimulus sensitivity and neuromodulatory properties of noisy intrinsic neuronal oscillators

Intrinsic subthreshold oscillations in the membrane potential are a common property of many neurons in the peripheral and central nervous system. When such oscillations are combined with noise, interesting signal encoding and neuromodulatory properties are obtained which allow, for example, sensitivity adjustment or differential encoding of stimuli. Here we demonstrate that a noisy Hodgkin/Huxley-model for subthreshold oscillations, when tuned to maximum sensitivity, can be significantly modulated by even minor physiological changes in the oscillation parameters amplitude or frequency. Given the ubiquity of subthreshold oscillating neurons, it can be assumed that these findings reflect principle encoding properties which are relevant for an understanding of sensitivity and neuromodulation in peripheral and central neurons.

Noise-induced impulse pattern modifications at different dynamical period-one situations in a computer model of temperature encoding.

We used a minimal Hodgkin-Huxley type model of cold receptor discharges to examine how noise interferes with the non-linear dynamics of the ionic mechanisms of neuronal stimulus encoding. The model is based on the assumption that spike-generation depends on subthreshold oscillations. With physiologically plausible temperature scaling, it passes through different impulse patterns which, with addition of noise, are in excellent agreement with real experimental data. The interval distributions of purely deterministic simulations, however, exhibit considerable differences compared to the noisy simulations especially at the bifurcations of deterministically period-one discharges. We, therefore, analyzed the effects of noise in different situations of deterministically regular period-one discharges: (1) at high-temperatures near the transition to subthreshold oscillations and to burst discharges, and (2) at low-temperatures close to and more far away from the bifurcations to chaotic dynamics. The data suggest that addition of noise can considerably extend the dynamical behavior of the system with coexistence of different dynamical situations at deterministically fixed parameter constellations. Apart from well-described coexistence of spike-generating and subthreshold oscillations also mixtures of tonic and bursting patterns can be seen and even transitions to unstable period-one orbits seem to appear. The data indicate that cooperative effects between low- and high-dimensional dynamics have to be considered as qualitatively important factors in neuronal encoding.

Interactions between slow and fast conductances in the Huber/Braun model of cold-receptor discharges

Abstract Transitions between different types of impulse patterns, according to experimentally recorded cold-receptor discharges, can successfully be mimicked with a minimal Hodgkin–Huxley-type simulation, here referred to as the Huber/Braun cold-receptor model. The model consists of two sets of simplified de- and repolarizing ionic conductances responsible for spike generation and slow-wave potentials, respectively. Over a broad temperature range, spike patterns are determined by the periodicity of subthreshold oscillations. At low temperatures, however, the periodicity of the pattern is destroyed and then appears again but with different patterns of different rhythmicity. We demonstrate that these complex transitions originate from the interactions between slow-wave and spike-generating currents.

Effects of noise on different disease states of recurrent affective disorders.

BACKGROUND Nonlinear dynamics are currently proposed to explain the course of recurrent affective disorders. Such a nonlinear disease model predicts complex interactions with stochastic influences, in particular, because both disease dynamics and stochastic influences, such as psychosocial stressors, will vary during the course of the disease. We approach this problem by investigating general effects of noise intensity on different disease states of a nonlinear model for recurrent affective disorders. METHODS A recently developed neurodynamic model is studied numerically. RESULTS Noise can cause unstructured randomness or can maximize periodic order. The frequency of episode occurrence can increase with noise but it can also remain unaffected or even can decrease. The observed effects, thereby, depend critically on both the noise intensity and the internal nonlinear dynamics of the disease model. CONCLUSIONS Our findings indicate that altered stochastic influences can significantly affect the outcome of a dynamic disease. To evaluate the effects of noise, it is essential to know about the underlying dynamics of respective disease states. Therefore, characterization of low-dimensional dynamics might become valuable for disease prediction and control.