The search for chaotic patterns has occupied numerous investigators in neuroscience, as in many other fields of science. Their results and main conclusions are reviewed in the light of the most recent criteria that need to be satisfied since the first descriptions of the surrogate strategy. The methods used in each of these studies have almost invariably combined the analysis of experimental data with simulations using formal models, often based on modified Huxley and Hodgkin equations and/or of the Hindmarsh and Rose models of bursting neurons. Due to technical limitations, the results of these simulations have prevailed over experimental ones in studies on the nonlinear properties of large cortical networks and higher brain functions. Yet, and although a convincing proof of chaos (as defined mathematically) has only been obtained at the level of axons, of single and coupled cells, convergent results can be interpreted as compatible with the notion that signals in the brain are distributed according to chaotic patterns at all levels of its various forms of hierarchy. This chronological account of the main landmarks of nonlinear neurosciences follows an earlier publication [Faure, Korn, C. R. Acad. Sci. Paris, Ser. III 324 (2001) 773–793] that was focused on the basic concepts of nonlinear dynamics and methods of investigations which allow chaotic processes to be distinguished from stochastic ones and on the rationale for envisioning their control using external perturbations. Here we present the data and main arguments that support the existence of chaos at all levels from the simplest to the most complex forms of organization of the nervous system.

The Anterior Burster (AB) neuron of the lobster stomatogastric ganglion displays varied rhythmic behavior when treated with neuromodulators and channel-blocking toxins. We introduce a channelbased model for this neuron and show how bifurcation analysis can be used to investigate the response of this model to changes of its parameters. Two dimensional maps of the parameter space of the model were constructed using computational tools based on the theory of nonlinear dynamical systems. Changes in the intrinsic firing and oscillatory properties of the model AB neuron were correlated with the boundaries of Hopf and saddle-node bifurcations on these maps. Complex rhythmic patterns were observed, with a bounded region of the parameter plane producing bursting behavior of the model neuron. Experiments were performed by treating an isolated AB cell with 4-aminopyridine which selectively reduces g A , the conductance of the transient potassium channel. The model accurately predicts the qualita...

this article is to highlight, through a targeted review of the modeling literature, some of the basic computational roles assigned to neuromodulation and present their possible neural implementation. Due to the diversity and ubiquity of neuromodulatory phenomena, we will not provide a comprehensive review of all neuromodulatory systems in terms of their anatomical loci, detailed biochemical pathways, and individual physiological effects. Nor will we attempt to define it; rather, we will review neuromodulation according to the computational framework provided by a chosen set of modeling studies. Our intent is not to be exhaustive. Many models not mentioned here have discussed how specific neuromodulations can be implemented and how they affect particular aspects of the neural system they consider. We include here a selection of studies that have dealt explicitly with neuromodulation and will help readers understand a specific computational role of neuromodulation.

Various physiological systems display bursting electrical activity (BEA). There exist numerous three-variable models to describe this behavior. However, higher-dimensional models with two slow processes have recently been used to explain qualitative features of the BEA of some

The gastropyloric receptor (GPR) cells are a set of cholinergic/serotonergic mechanosensory neurons that modulate the activity of neural networks in the crab stomatogastric ganglion (STG). Stimulation of these cells evokes a variety of slow modulatory responses in different STG neurons that are mimicked by exogenously applied serotonin (5-HT); these responses include tonic inhibition, tonic excitation and induction of rhythmic bursting. We used pharmacological agonists and antagonists to show that these three classes of modulatory response in the STG neurons are mediated by distinct 5-HT receptor subtypes. GPR stimulation or application of 5-HT or 2-me-5HT (a vertebrate 5-HT3 agonist) inhibited the pyloric constrictor (PY) neurons; these actions were selectively antagonized by gramine. GPR stimulation or application of 5-HT induced rhythmic bursting in the electrically coupled anterior burster (AB) and pyloric dilator (PD) neurons; these effects were antagonized by the 5-HT1c/2 antagonist cinanserin and by atropine at concentrations that do not block muscarinic cholinergic receptors in the crab STG. The 5-HT agonists 5-CT (5-HT1) and a-me-5HT (5-HT2) also induced AB/PD bursting, which was blocked by cinanserin, but not by atropine. GPR stimulation or application of 5-HT and 5-CT evoked tonic excitation of the lateral pyloric (LP) neuron. These effects were blocked by cinanserin. Several other 5-HT agonists and nearly all the vertebrate 5-HT antagonists we tested had little or no effect on the crab pyloric 5-HT receptors. These results provide further evidence that the modulatory sensory GPR neuron uses serotonin to evoke multiple modulatory responses via multiple 5-HT receptors. However, the 5-HT receptors in the crab STG neurons are not pharmacologically similar to vertebrate 5-HT receptors.

Synopsis. The stomatogastric nervous system of decapod crustaceans is an ideal system for the study of the processes underlying the generation of rhythmic movements by the nervous system. In this chapter we review recent work that uses mathematical analyses and computer simulations to understand: 1) the role of individual currents in controlling the activity of neurons, and 2) the effects of electrical coupling on the activity of neuronal oscillators. The aim of this review is to highlight, for the physiologist, what these studies have taught us about the organization and function of single cell and multicellular neuronal oscillators.

The dynamics of individual neurons are crucial for producing functional activity in neuronal networks. An open question is how temporal characteristics can be controlled in bursting activity and in transient neuronal responses to synaptic input. Bifurcation theory provides a framework to discover generic mechanisms addressing this question. We present a family of mechanisms organized around a global codimension-2 bifurcation. The cornerstone bifurcation is located at the intersection of the border between bursting and spiking and the border between bursting and silence. These borders correspond to the blue sky catastrophe bifurcation and the saddle-node bifurcation on an invariant circle (SNIC) curves, respectively. The cornerstone bifurcation satisfies the conditions for both the blue sky catastrophe and SNIC. The burst duration and interburst interval increase as the inverse of the square root of the difference between the corresponding bifurcation parameter and its bifurcation value. For a given set of burst duration and interburst interval, one can find the parameter values supporting these temporal characteristics. The cornerstone bifurcation also determines the responses of silent and spiking neurons. In a silent neuron with parameters close to the SNIC, a pulse of current triggers a single burst. In a spiking neuron with parameters close to the blue sky catastrophe, a pulse of current temporarily silences the neuron. These responses are stereotypical: the durations of the transient intervals–the duration of the burst and the duration of latency to spiking–are governed by the inverse-square-root laws. The mechanisms described here could be used to coordinate

We’re the same … but different: addressing academic divides

Abstract. Many models of bursting electrical activity (BEA) in pancreatic -cells have been proposed. BEA is characterized by a periodic oscillation of the membrane potential consisting of a silent phase during which the membrane potential is varying slowly and an active phase during which the membrane potential is undergoing rapid oscillations. An important experimental observation of BEA is a correlation between the rate of insulin release from -cells and the plateau fraction as a function of glucose concentration. The plateau fraction is the ratio of the duration of the active phase to the period of BEA. In [SIAM J. Appl. Math., 52 (1992), pp. 1627{1650], Pernarowski, Miura, and Kevorkian develop analytical techniques to determine the leading-order plateau fraction for one of the models, namely, the Sherman{Rinzel{Keizer (SRK) model [Biophys. J., 54 (1988), pp. 411{425]. Applicability of these techniques depends critically on the fact that the fast subsystem of the SRK model is an integrable system to leading order. In this paper, we extend the techniques of Pernarowski, Miura, and Kevorkian to a class of models of BEA, namely, those rst-generation models consisting of three rst-order ordinary dierential equations. We show that the fast subsystem of these models can be reformulated as an integrable system to leading order. The relative ease with which this reformulation can be done depends on a biological property of the models, namely, the value of the integer exponent of the activation variable in the description of the voltage-gated K+ current.

To my parents, and my brother iii